shapes: Statistical Shape Analysis

Documented in abind add as.3d backfit banner1 banner4 bgpa bookstein2d bookstein.shpv bookstein.shpv.complex BoxM cbevec cbevectors centroid.size close1 cnt3 complextoreal defh defplotsize2 defplotsize3 del dif.old dis distcov distEuclidean distLogEuclidean distPowerEuclidean distRiemannianLe distRiemPennec estCholesky estEuclid estLogEuclid estLogRiem2 estPowerEuclid estRiemLe fcel fcnt fgpa fgpa.rot fgpa.singleiteration fJ fopa fort.ROTATEANDREFLECT fort.ROTATION fos.REFLECT frechet ftrsq full.procdist genpower Goodall Goodall2D Goodalltest graf groupstack Hessian2 Hotelling I2mat iglogl ild_centroid.size ild_centroid.size.complex ild_centroid.size.mD ild_defh ild_distCholesky ild_distProcrustesFull ild_Enorm ild_estShape ild_estSS ild_kendall.shpv ild_preshape ild_preshape.mat ild_preshape.mD ild_preshapetoicon ild_tanfigure ild_tanfigurefull isodens isologdens isomle isotropy.test James linegrid loglikeiso loglikeiso2 loneFone mahpreshapedist makearray MDSshape MGM movie msh multiply_by_helmert multiply_by_helmert_explicitly multiply_by_helmert_implicitly multiply_by_helmert_implicitly_3d multiply_by_transpose_of_helmert multiply_by_transpose_of_helmert_explicitly multiply_by_transpose_of_helmert_implicitly objfun objfun4 objfuniso oneFone partial.procdist partialwarpgrids partialwarps pcscore2sphere2 ped pedreg permutationtest plot2rwscores plot3darcs plot3Ddata plot3Ddata.static plot3Dmean plot3Dpca plotpairscores plotPDM plotPDM2 plotPDM3 plotPDMbook plotPDMnoaxis plotprinwarp plotproc plotrelwarp plotshapes pns pns4pc pointsPDMnoaxis3 preshape2shape prinwscoregrids procdist procdistreflect procGPA procOPA procrustes2d procWGPA procWGPA1 project projectPNS read.array read.in realtocomplex reassqpr relwarps resampletest rgpa riemdist riemdist.complex riemdist.mD rigidbody rootmat rotateaxes rotatexyz sgpa sh shaperw shapes3d shapes.cva sigmacov sim1 sphere1.f ssriemdist st tangentcoords.partial.inv tanpreshape testmeanshapes testmeanshapes.old testshapes transformations uji2_centroid.size uji3_centroid.size uji_centroid.size uji_centroid.size.complex uji_centroid.size.mD uji_defh uji_distCholesky uji_distProcrustesFull uji_distProcrustesSizeShape uji_Enorm uji_estShape uji_estSS uji_kendall.shpv uji_preshape uji_preshape.mat uji_preshape.mD uji_tanfigure uji_tanfigurefull V vec1 Vinv Vmat

#-----------------------------------------------------------------------
#
# Statistical shape analysis routines
# written by Ian Dryden in R  (see http://cran.r-project.org)
# (c) Ian Dryden
#      version 1.2.8
#                    2003-2025
#
# Includes contributions by many other authors, including
#   Mohammad Faghihi, Kwang-Rae Kim, Alfred Kume,
#   Gregorio Quintana-Orti, Amelia Simo.
#
###########################################################################



tangentcoords.partial.inv = function(v, p, R)
{
  return(matrix(sqrt(1 - sum(v^2)) * c(p) + v, nrow = nrow(p)) %*% t(R))
}

preshape2shape = function(z)
{
  k = nrow(z) + 1
  H = defh(k - 1)
  return(t(H) %*% z)
}


plot3darcs<-function(x,pcno=1,c=1,nn=100,boundary.data=TRUE,view.theta=0,view.phi=0,type="pnss"){
  # points along principal arcs

   pns.out <- x
    k <- pns.out$GPAout$k
    m <- pns.out$GPAout$m
    n.pc <- dim(pns.out$resmat)[1]
    
    rad1<-sqrt(5/k)/50
    rad2<-sqrt(1/k)/50
    
    npts = 100
    arc1 = t(get.prinarc(resmat = pns.out$resmat, PNS = pns.out$PNS, 
        arc = 1, n = npts, boundary.data = boundary.data))
    arc2 = t(get.prinarc(resmat = pns.out$resmat, PNS = pns.out$PNS, 
        arc = 2, n = npts, boundary.data = boundary.data))
    arc3 = t(get.prinarc(resmat = pns.out$resmat, PNS = pns.out$PNS, 
        arc = 3, n = npts, boundary.data = boundary.data))
    PNSmean = pns.out$PNS$mean
    GPAout = pns.out$GPAout
    {
   #     cat("stdev of PNS1 score:", round(sd(pns.out$resmat[1, 
   #         ]), 4), "\n")
   #    cat("stdev of PNS2 score:", round(sd(pns.out$resmat[2, 
   #        ]), 4), "\n")
   #    cat("stdev of PNS3 score:", round(sd(pns.out$resmat[3, 
   #        ]), 4), "\n")
    }
    rng = c * sd(pns.out$resmat[1, ])
    val = c(seq(-rng, 0, length = nn + 1)[-(nn + 1)], 0, seq(0, 
        rng, length = nn + 1)[-1])
    lu.arc1 = t(get.prinarc.value(PNS = pns.out$PNS, arc = 1, 
        res = val))
    rng = c * sd(pns.out$resmat[2, ])
    val = c(seq(-rng, 0, length = nn + 1)[-(nn + 1)], 0, seq(0, 
        rng, length = nn + 1)[-1])
    lu.arc2 = t(get.prinarc.value(PNS = pns.out$PNS, arc = 2, 
        res = val))
    rng = c * sd(pns.out$resmat[3, ])
    val = c(seq(-rng, 0, length = nn + 1)[-(nn + 1)], 0, seq(0, 
        rng, length = nn + 1)[-1])
    lu.arc3 = t(get.prinarc.value(PNS = pns.out$PNS, arc = 3, 
        res = val))
    scores.arc1 = sphere2pcscore(x = arc1)
    scores.arc2 = sphere2pcscore(x = arc2)
    scores.arc3 = sphere2pcscore(x = arc3)
    scores.PNSmean = sphere2pcscore(x = t(PNSmean))
    scores.lu.arc1 = sphere2pcscore(x = lu.arc1)
    scores.lu.arc2 = sphere2pcscore(x = lu.arc2)
    scores.lu.arc3 = sphere2pcscore(x = lu.arc3)
    U1 = matrix(0, npts, nrow(GPAout$pcar))
    U2 = matrix(0, npts, nrow(GPAout$pcar))
    U3 = matrix(0, npts, nrow(GPAout$pcar))
    for (i in 1:npts) {
        for (j in 1:n.pc) {
            U1[i, ] = U1[i, ] + scores.arc1[i, j] * GPAout$pcar[, 
                j]
            U2[i, ] = U2[i, ] + scores.arc2[i, j] * GPAout$pcar[, 
                j]
            U3[i, ] = U3[i, ] + scores.arc3[i, j] * GPAout$pcar[, 
                j]
        }
    }
    U.mean = matrix(0, 1, nrow(GPAout$pcar))
    for (j in 1:n.pc) {
        U.mean = U.mean + scores.PNSmean[j] * GPAout$pcar[, j]
    }
    tan.lu.arc1 = matrix(0, nrow(lu.arc1), nrow(GPAout$pcar))
    tan.lu.arc2 = matrix(0, nrow(lu.arc2), nrow(GPAout$pcar))
    tan.lu.arc3 = matrix(0, nrow(lu.arc3), nrow(GPAout$pcar))
    for (i in 1:nrow(lu.arc1)) {
        for (j in 1:n.pc) {
            tan.lu.arc1[i, ] = tan.lu.arc1[i, ] + scores.lu.arc1[i, 
                j] * GPAout$pcar[, j]
            tan.lu.arc2[i, ] = tan.lu.arc2[i, ] + scores.lu.arc2[i, 
                j] * GPAout$pcar[, j]
            tan.lu.arc3[i, ] = tan.lu.arc3[i, ] + scores.lu.arc3[i, 
                j] * GPAout$pcar[, j]
        }
    }
    shapes.arc1 = array(NA, c(k, m, npts))
    shapes.arc2 = array(NA, c(k, m, npts))
    shapes.arc3 = array(NA, c(k, m, npts))
    H = defh(k - 1)
    for (i in 1:npts) {
      # to convert from in expo map to partial tangent coords and then to icon configuration 
rho<-Enorm(U1[i,])
        shapes.arc1[, , i] = preshape2shape(tangentcoords.partial.inv(v = U1[i, 
            ]*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))
        rho<-Enorm(U2[i,])
        shapes.arc2[, , i] = preshape2shape(tangentcoords.partial.inv(v = U2[i, 
            ]*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))
        rho<-Enorm(U3[i,])
        shapes.arc3[, , i] = preshape2shape(tangentcoords.partial.inv(v = U3[i, 
            ]*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))
    }
    rho<-Enorm(U.mean)
    shapes.PNSmean = preshape2shape(tangentcoords.partial.inv(v = U.mean*sin(rho)/rho, 
        p = H %*% GPAout$mshape, R = diag(m)))
    shapes.lu.arc1 = array(NA, c(k, m, nrow(lu.arc1)))
    shapes.lu.arc2 = array(NA, c(k, m, nrow(lu.arc2)))
    shapes.lu.arc3 = array(NA, c(k, m, nrow(lu.arc3)))
    for (i in 1:nrow(lu.arc1)) {
      rho<-Enorm(tan.lu.arc1[i,])
        shapes.lu.arc1[, , i] = preshape2shape(tangentcoords.partial.inv(v = tan.lu.arc1[i, 
            ]*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))
        rho<-Enorm(tan.lu.arc2[i,])
        shapes.lu.arc2[, , i] = preshape2shape(tangentcoords.partial.inv(v = tan.lu.arc2[i, 
            ]*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))
        rho<-Enorm(tan.lu.arc3[i,])
        shapes.lu.arc3[, , i] = preshape2shape(tangentcoords.partial.inv(v = tan.lu.arc3[i, 
            ]*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))
    }
    h <- defh(k - 1)
    zero <- matrix(0, k - 1, k)
    H <- cbind(h, zero, zero)
    H1 <- cbind(zero, h, zero)
    H2 <- cbind(zero, zero, h)
    H <- rbind(H, H1, H2)
    if (dim(GPAout$pcar)[1] == (3 * (k - 1))) {
        pcarot <- (t(H) %*% GPAout$pcar)
        GPAout$pcar <- pcarot
    }
    if (pcno == 1) {
        shapes.lu.arc <- shapes.lu.arc1
    }
    if (pcno == 2) {
        shapes.lu.arc <- shapes.lu.arc2
    }
    if (pcno == 3) {
        shapes.lu.arc <- shapes.lu.arc3
    }
    if (type == "pca") {
        open3d()
        par3d(windowRect = c(20, 30, 800, 800))
        view3d(view.theta, view.phi)
        plot3d(GPAout$mshape, type = "s", col = rainbow(k), radius = rad1, 
            add = TRUE)
        lines3d(GPAout$mshape, col = rainbow(k), lwd = 5)
        pcu <- GPAout$mshape + c * GPAout$pcasd[pcno] * cbind(GPAout$pcar[1:k, 
            pcno], GPAout$pcar[(k + 1):(2 * k), pcno], GPAout$pcar[(2 * 
            k + 1):(3 * k), pcno])
        pcl <- GPAout$mshape - c * GPAout$pcasd[pcno] * cbind(GPAout$pcar[1:k, 
            pcno], GPAout$pcar[(k + 1):(2 * k), pcno], GPAout$pcar[(2 * 
            k + 1):(3 * k), pcno])
        spheres3d(pcu, radius = rad2, color = "black")
        spheres3d(pcl, radius = rad2, color = "grey")
        for (j in 1:k) {
            lines3d(rbind(pcl[j, ], pcu[j, ]), col = rainbow(k)[j])
            if (j > 1) {
                lines3d(rbind(pcu[j - 1, ], pcu[j, ]), col = "black")
                lines3d(rbind(pcl[j - 1, ], pcl[j, ]), col = "grey")
            }
        }
    }
    if (type == "pnss") {
        open3d()
        par3d(windowRect = c(20, 30, 800, 800))
        view3d(view.theta, view.phi)
        plot3d(shapes.PNSmean, type = "s", col = rainbow(k), 
            radius = rad1, add = TRUE)
        lines3d(shapes.PNSmean, lwd = 5, col = rainbow(k))
        for (i in 1:k) {
            lines3d(t(shapes.lu.arc[i, , ]), col = rainbow(k)[i], 
                lwd = 1, lty = 2)
            spheres3d(head(t(shapes.lu.arc[i, , ]), 1), radius = rad2, 
                color = "black")
            if (i > 1) {
                lines3d((shapes.lu.arc[(i - 1):i, , 1]), col = "black")
            }
            spheres3d(tail(t(shapes.lu.arc[i, , ]), 1), radius = rad2, 
                color = "grey")
            if (i > 1) {
                lines3d((shapes.lu.arc[(i - 1):i, , 201]), col = "grey")
            }
        }
    }
    out <- list(PNSmean = 0, lu.arc = 0)
    out$PNSmean <- shapes.PNSmean
    out$lu.arc <- shapes.lu.arc
    out
}


########
pnss3d<-
function (x, sphere.type = "seq.test", mean.type="Frechet", alpha = 0.1, R = 100, 
          nlast.small.sphere = 1, n.pc = "Full", output=TRUE) 
{
  k = dim(x)[1]
m = dim(x)[2]
n = dim(x)[3]
  if (n.pc =="Full" ) {
    n.pc=m*k-m*(m-1)/2-m
  }
  if (m==2){
    tem1 <- array( 0, c(k,3,n) )
    tem1[,1:2,]<-x
    x<-tem1
    m<-3
  }

#if (n < ((k - 1) * m)) {
#  print("Note: n < (k - 1) * m.")
#  jj<- round( (k-1)*m/n + 0.5)
#  print("Adding extra copies of the data")
#  tem<- array(0,c(k,m,jj*n))
#  tem[,,1:n]<-x
#  for (i in 2:jj){
#    for (j in 1:n){
#    tem[,,(i-1)*n+ j ]<-x[,,j] + 0*matrix( rnorm(k*m), k,m)
#    }
#  }
#  x<-tem
#}
k = dim(x)[1]
m = dim(x)[2]
n = dim(x)[3]

  
  out = pc2sphere2(x = x, n.pc = n.pc, output=output)
  spheredata = t(out$spheredata)
  GPAout = out$GPAout
  pns.out = pns(x = spheredata, sphere.type = sphere.type, mean.type=mean.type, 
                alpha = alpha, R = R, nlast.small.sphere = nlast.small.sphere, output=output)
  pns.out$percent = pns.out$percent * sum(GPAout$percent[1:n.pc])/100
if (output){
  print("Radii of spheres")
  print(pns.out$PNS$radii)
  print("PNS percent explained")
  cat(c(round(pns.out$percent,2),"\n"))
  print("PCA percent explained")
  cat(c(round(GPAout$percent,2),"\n"))
}
  pns.out$GPAout = GPAout
  pns.out$spheredata = spheredata
  return(pns.out)
}

pc2sphere2<-function (x, n.pc, output=TRUE) 
{
  k = dim(x)[1]
  m = dim(x)[2]
  n = dim(x)[3]
  

  GPAout = procGPA(x = x, scale = TRUE, reflect = FALSE, tol1=1e-8,tangentcoords = "partial", 
                   distances = TRUE)
if (output){
  cat("First ", n.pc, " principal components explain ", round(sum(GPAout$percent[1:n.pc]),2), 
      "% of total variance. \n", sep = "")
}
  H = defh(k - 1)
  X.hat = H %*% GPAout$mshape
  S = array(NA, c(k - 1, m, n))
  for (i in 1:n) {
    S[, , i] = H %*% GPAout$rotated[, , i]
  }
  T.c = GPAout$tan #- apply(GPAout$tan, 1, mean)
  out = pcscore2sphere2(n.pc = n.pc, X.hat = X.hat, S = S, Tan = T.c, 
                       V = GPAout$pcar)
  return(list(spheredata = out, GPAout = GPAout))
}



backfit <- function( scores, x , type="pnss", size=1){
npc <- length(scores)

if (type=="pnss"){
  PNS.object<-x
  PNS<-PNS.object$PNS
GPAout<-PNS.object$GPAout
z1 <- PNSe2s(matrix(scores,npc,1),PNS)
pcscores<-c(sphere2pcscore(x=t(z1)))
#note the PC scores are from the inverse exponential map tangent coordinates 
mu <- GPAout$mshape
k<-dim(mu)[1]
m<-dim(mu)[2]
H = defh(k - 1)
U<- GPAout$pcar[,1]*0
  for (j in 1:npc) {
        U = U + pcscores[j] * GPAout$pcar[, j]
  }
# to convert from in expo map to partial tangent coords and then to icon configuration 
rho<-Enorm(U)
xout<-preshape2shape(tangentcoords.partial.inv(v = U*sin(rho)/rho, p = H %*% GPAout$mshape, R = diag(m)))*size
}

if (type=="pca"){
GPAout<- x
pcscores<-scores #assume partial tangent coordinates
mu <- GPAout$mshape
k<-dim(mu)[1]
m<-dim(mu)[2]
H = defh(k - 1)
U<- GPAout$pcar[,1]*0
  for (j in 1:npc) {
        U = U + pcscores[j] * GPAout$pcar[, j]
  }
xout<-preshape2shape(tangentcoords.partial.inv(v = U, p = H %*% GPAout$mshape, R = diag(m)))*size
}

xout
}


projectPNS <- function( x , PNS){
#obtain the PNS scores for new spherical data with respect to a PNS object
  PNSobj <- PNS
  x<-as.matrix(x)
  k <- dim(x)[1]
  n <- dim(x)[2]
  d <- k-1
  scorescheck <- matrix(0,n,d)
currentSphere <- x
for (i in 1:(d-1)){
center <- PNSobj$PNS$orthaxis[[i]]
r <- PNSobj$PNS$dist[i]
res = ( acos(t(center) %*% currentSphere) - r )
scorescheck[,d+1-i]<-t(res)*PNSobj$PNS$radii[i]  #rescale by actual radius of (sub)sphere where fit is carried out
#####
cur.proj = project.subsphere(x = currentSphere,
                  center = center, r = r)
  NestedSphere = rotMat(center) %*% currentSphere
  currentSphere = NestedSphere[1:(k - i), ]/repmat(matrix(sqrt(1 -
                  NestedSphere[nrow(NestedSphere), ]^2), nrow = 1),
                  k - i, 1)
##############
  }

    S1toRadian = atan2(currentSphere[2, ], currentSphere[1, ])
#    meantheta = geodmeanS1(S1toRadian)$geodmean
    meantheta <- PNSobj$PNS$orthaxis[[d]]
    scorescheck[,1] = (mod(S1toRadian - meantheta + pi, 2 * pi) -
        pi     )*    PNSobj$PNS$radii[d]  #rescale by actual radius of fitted circle
    scorescheck
}



pcscore2sphere3 <- 
function (n.pc, X.hat, Xs, Tan, V) 
{
    d = nrow(Tan)
    n = ncol(Tan)
    W = matrix(NA, d, n)
    for (i in 1:n) {
        W[, i] =  acos( sum(Xs[i,]*X.hat) ) * Tan[, i]/sqrt(sum(Tan[, 
            i]^2))
    }
    lambda = matrix(NA, n, d)
    for (i in 1:n) {
        for (j in 1:n.pc) {
            lambda[i, j] = sum(W[, i] * V[, j])
        }
    }
    U = matrix(0, n, d)
    for (i in 1:n) {
        for (j in 1:n.pc) {
            U[i, ] = U[i, ] + lambda[i, j] * V[, j]
        }
    }
    S.star = matrix(NA, n, n.pc + 1)
    for (i in 1:n) {
        U.norm = sqrt(sum(U[i, ]^2))
        S.star[i, ] = c(cos(U.norm), sin(U.norm)/U.norm * lambda[i, 
            1:n.pc])
    }
    return(S.star)
}


fastpns <- function (x, n.pc = "Full", sphere.type = "seq.test", mean.type="Frechet", alpha = 0.1, 
    R = 100, nlast.small.sphere = 1, output = TRUE, pointcolor = 2) 
{
    n <- dim(x)[2]
  pdim <- dim(x)[1]
    if (n.pc == "Full") {
        n.pc = min(c( pdim-1 , n - 1))
    }
    Xs <- t(x)
    for (i in 1:n) {
        Xs[i, ] <- Xs[i, ]/Enorm(Xs[i, ])
    }
    muhat <- apply(Xs, 2, mean)
    muhat <- muhat/Enorm(muhat)
    TT <- Xs
    for (i in 1:n) {
        TT[i, ] <- Xs[i, ] - sum(Xs[i, ] * muhat) * muhat
    }
    pca <- prcomp(TT)
    pcapercent <- sum(pca$sdev[1:n.pc]^2/sum(pca$sdev^2))
    cat(c("Initial PNS subsphere dimension", n.pc + 1, "\n"))
    cat(c("Percentage of variability in PNS sequence", round(pcapercent * 
        100, 2), "\n"))
    TT <- t(TT)
    ans <- pcscore2sphere3(n.pc, muhat, Xs, TT, pca$rotation)
    Xssubsphere <- t(ans)
    out <- pns( (Xssubsphere), sphere.type = sphere.type, mean.type=mean.type, alpha = alpha, 
        R = R, nlast.small.sphere = nlast.small.sphere, output = output, 
        pointcolor = pointcolor)
    out$percent <- out$percent * pcapercent
    cat(c("Percent explained by 1st three PNS scores out of total variability:", 
        "\n", round(out$percent[1:3], 2), "\n"))
    out$spheredata <- (Xssubsphere)
    out$pca <- pca
    out
}




#==================================================================================
# PNS  The Principal Nested Spheres code (PNS) for spheres and shapes has
# been written by Kwang-Rae Kim, and builds closely on the original matlab
# code for PNS by Sungkyu Jung
#==================================================================================

#==================================================================================
pns = function(x,
               sphere.type = "seq.test", mean.type="Frechet", 
               alpha = 0.1,
               R = 100,
               nlast.small.sphere = 1, output=TRUE , pointcolor=2)
{
  n = ncol(x)
  k = nrow(x)
  PNS = list()
  
  if (abs(sum(apply(x ^ 2, 2, sum)) - n) > 1e-8)
  {
    stop("Error: Each column of x should be a unit vector, ||x[ , i]|| = 1.")
  }
  svd.x = svd(x, nu = nrow(x))
  uu = svd.x$u
  maxd = which(svd.x$d < 1e-15)[1]
  if (is.na(maxd) | k > n)
  {
    maxd = min(k, n) + 1
  }
  nullspdim = k - maxd + 1
  d = k - 1
if (output){
  cat("Message from pns() : dataset is on ", d, "-sphere. \n", sep = "")
}
  if (nullspdim > 0)
  {
if (output){
    cat(" .. found null space of dimension ",
        nullspdim,
        ", to be trivially reduced. \n",
        sep = "")
}
  }

if (d==2){
PNS$spherePNS<-t(x)
}

  resmat = matrix(NA, d, n)
  orthaxis = list()
  orthaxis[[d - 1]] = NA
  dist = rep(NA, d - 1)
  pvalues = matrix(NA, d - 1, 2)
  ratio = rep(NA, d - 1)
  currentSphere = x
  if (nullspdim > 0)
  {
    for (i in 1:nullspdim)
    {
      oaxis = uu[, ncol(uu) - i + 1]
      r = pi / 2
      pvalues[i,] = c(NaN, NaN)
      res = acos(t(oaxis) %*% currentSphere) - r
      orthaxis[[i]] = oaxis
      dist[i] = r
      resmat[i,] = res
      NestedSphere = rotMat(oaxis) %*% currentSphere
      currentSphere = NestedSphere[1:(k - i),] /
        repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                             2), nrow = 1), k - i, 1)
      uu = rotMat(oaxis) %*% uu
      uu = uu[1:(k - i),] / repmat(matrix(sqrt(1 - uu[nrow(uu),] ^ 2), nrow = 1), k - i, 1)
if (output){
      cat(d - i + 1,
          "-sphere to ",
          d - i,
          "-sphere, by ",
          "NULL space \n",
          sep = "")
}
    }
  }
  if (sphere.type == "seq.test")
  {
if (output){
    cat(" .. sequential tests with significance level ",
        alpha,
        "\n",
        sep = "")
}
    isIsotropic = FALSE
    for (i in (nullspdim + 1):(d - 1))
    {
      if (!isIsotropic)
      {
        sp = getSubSphere(x = currentSphere, geodesic = "small")
        center.s = sp$center
        r.s = sp$r
        resSMALL = acos(t(center.s) %*% currentSphere) - r.s
        sp = getSubSphere(x = currentSphere, geodesic = "great")
        center.g = sp$center
        r.g = sp$r
        resGREAT = acos(t(center.g) %*% currentSphere) - r.g
        pval1 = LRTpval(resGREAT, resSMALL, n)
        pvalues[i, 1] = pval1
        if (pval1 > alpha)
        {
          center = center.g
          r = r.g
          pvalues[i, 2] = NA
if (output){
          cat(
            d - i + 1,
            "-sphere to ",
            d - i,
            "-sphere, by GREAT sphere, p(LRT) = ",
            pval1,
            "\n",
            sep = ""
          )
}
        } else {
          pval2 = vMFtest(currentSphere, R)
          pvalues[i, 2] = pval2
          if (pval2 > alpha)
          {
            center = center.g
            r = r.g
if (output){
            cat(
              d - i + 1,
              "-sphere to ",
              d - i,
              "-sphere, by GREAT sphere, p(LRT) = ",
              pval1,
              ", p(vMF) = ",
              pval2,
              "\n",
              sep = ""
            )
}
            isIsotropic = TRUE
          } else {
            center = center.s
            r = r.s
if (output){
            cat(
              d - i + 1,
              "-sphere to ",
              d - i,
              "-sphere, by SMALL sphere, p(LRT) = ",
              pval1,
              ", p(vMF) = ",
              pval2,
              "\n",
              sep = ""
            )
}
          }
        }
      } else if (isIsotropic) {
        sp = getSubSphere(x = currentSphere, geodesic = "great")
        center = sp$center
        r = sp$r
if (output){
        cat(
          d - i + 1,
          "-sphere to ",
          d - i,
          "-sphere, by GREAT sphere, restricted by testing vMF distn",
          "\n",
          sep = ""
        )
}
        pvalues[i, 1] = NA
        pvalues[i, 2] = NA
      }
      res = acos(t(center) %*% currentSphere) - r
      orthaxis[[i]] = center
      dist[i] = r
      resmat[i,] = res
      cur.proj = project.subsphere(x = currentSphere,
                                   center = center,
                                   r = r)
      NestedSphere = rotMat(center) %*% currentSphere
      currentSphere = NestedSphere[1:(k - i),] /
        repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                             2), nrow = 1), k - i, 1)
      ###########    
      if (nrow(currentSphere) == 3)
      {
        PNS$spherePNS = t(currentSphere)
      }
      
      if (nrow(currentSphere) == 2)
      {
        PNS$circlePNS = t(cur.proj)
      }
      #############################
    }
  } else if (sphere.type == "BIC") {
if (output){
    cat(" .. with BIC \n")
}
    for (i in (nullspdim + 1):(d - 1))
    {
      sp = getSubSphere(x = currentSphere, geodesic = "small")
      center.s = sp$center
      r.s = sp$r
      resSMALL = acos(t(center.s) %*% currentSphere) - r.s
      sp = getSubSphere(x = currentSphere, geodesic = "great")
      center.g = sp$center
      r.g = sp$r
      resGREAT = acos(t(center.g) %*% currentSphere) - r.g
      BICsmall = n * log(mean(resSMALL ^ 2)) + (d - i + 1 + 1) * log(n)
      BICgreat = n * log(mean(resGREAT ^ 2)) + (d - i + 1) * log(n)
if (output){
      cat("BICsm: ", BICsmall, ", BICgr: ", BICgreat, "\n", sep = "")
}
      if (BICsmall > BICgreat)
      {
        center = center.g
        r = r.g
if (output){
        cat(d - i + 1,
            "-sphere to ",
            d - i,
            "-sphere, by ",
            "GREAT sphere, BIC \n",
            sep = "")
}
      } else {
        center = center.s
        r = r.s
if (output){
        cat(d - i + 1,
            "-sphere to ",
            d - i,
            "-sphere, by ",
            "SMALL sphere, BIC \n",
            sep = "")
}
      }
      res = acos(t(center) %*% currentSphere) - r
      orthaxis[[i]] = center
      dist[i] = r
      resmat[i,] = res
      
      cur.proj = project.subsphere(x = currentSphere,
                                   center = center,
                                   r = r)
      
      NestedSphere = rotMat(center) %*% currentSphere
      currentSphere = NestedSphere[1:(k - i),] /
        repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                             2), nrow = 1), k - i, 1)
      
      ########### 
      if (nrow(currentSphere) == 3)
      {
        PNS$spherePNS = t(currentSphere)
      }
      
      if (nrow(currentSphere) == 2)
      {
        PNS$circlePNS = t(cur.proj)
      }
      #############################
      
    }
  } else if (sphere.type == "small" | sphere.type == "great") {
    pvalues = NaN
    for (i in (nullspdim + 1):(d - 1))
    {
      sp = getSubSphere(x = currentSphere, geodesic = sphere.type)
      center = sp$center
      r = sp$r
      res = acos(t(center) %*% currentSphere) - r
      orthaxis[[i]] = center
      dist[i] = r
      resmat[i,] = res
      
      cur.proj = project.subsphere(x = currentSphere,
                                   center = center,
                                   r = r)
      
      NestedSphere = rotMat(center) %*% currentSphere
      currentSphere = NestedSphere[1:(k - i),] /
        repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                             2), nrow = 1), k - i, 1)
      
      ###########  
      if (nrow(currentSphere) == 3)
      {
        PNS$spherePNS = t(currentSphere)
      }
      
      if (nrow(currentSphere) == 2)
      {
        PNS$circlePNS = t(cur.proj)
      }
      #############################
      
    }
  } else if (sphere.type == "bi.sphere") {
    if (nlast.small.sphere < 0)
    {
      cat("!!! Error from pns(): \n")
      cat("!!! nlast.small.sphere should be >= 0. \n")
      return(NULL)
    }
    mx = (d - 1) - nullspdim
    if (nlast.small.sphere > mx)
    {
      cat("!!! Error from pns(): \n")
      cat("!!! nlast.small.sphere should be <= ",
          mx,
          " for this data. \n",
          sep = "")
      return(NULL)
    }
    pvalues = NaN
    if (nlast.small.sphere != mx)
    {
      for (i in (nullspdim + 1):(d - 1 - nlast.small.sphere))
      {
        sp = getSubSphere(x = currentSphere, geodesic = "great")
        center = sp$center
        r = sp$r
        res = acos(t(center) %*% currentSphere) - r
        orthaxis[[i]] = center
        dist[i] = r
        resmat[i,] = res
        
        cur.proj = project.subsphere(x = currentSphere,
                                     center = center,
                                     r = r)
        
        NestedSphere = rotMat(center) %*% currentSphere
        currentSphere = NestedSphere[1:(k - i),] /
          repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                               2), nrow = 1), k - i, 1)
        
        ###########    
        if (nrow(currentSphere) == 3)
        {
          PNS$spherePNS = t(currentSphere)
        }
        
        if (nrow(currentSphere) == 2)
        {
          PNS$circlePNS = t(cur.proj)
        }
        #############################
        
      }
    }
    if (nlast.small.sphere != 0)
    {
      for (i in (d - nlast.small.sphere):(d - 1))
      {
        sp = getSubSphere(x = currentSphere, geodesic = "small")
        center = sp$center
        r = sp$r
        res = acos(t(center) %*% currentSphere) - r
        orthaxis[[i]] = center
        dist[i] = r
        resmat[i,] = res
        
        cur.proj = project.subsphere(x = currentSphere,
                                     center = center,
                                     r = r)
        
        NestedSphere = rotMat(center) %*% currentSphere
        currentSphere = NestedSphere[1:(k - i),] /
          repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                               2), nrow = 1), k - i, 1)
        
        ###########    
        if (nrow(currentSphere) == 3)
        {
          PNS$spherePNS = t(currentSphere)
        }
        
        if (nrow(currentSphere) == 2)
        {
          PNS$circlePNS = t(cur.proj)
        }
        #############################
        
        
      }
    }
  } else {
    print("!!! Error from pns():")
    print("!!! sphere.type must be 'seq.test', 'small', 'great', 'BIC', or 'bi.sphere'")
    print("!!!   Terminating execution     ")
    return(NULL)
  }
  S1toRadian = atan2(currentSphere[2,], currentSphere[1,])
  meantheta = geodmeanS1(S1toRadian,mean.type=mean.type)$geodmean
  orthaxis[[d]] = meantheta
  resmat[d,] = mod(S1toRadian - meantheta + pi, 2 * pi) - pi
  
if (output){
  par(
    mfrow = c(1, 1),
    mar = c(4, 4, 1, 1),
    mgp = c(2.5, 1, 0),
    cex = 0.8
  )
  plot(
    currentSphere[1,],
    currentSphere[2,],
    xlab = "",
    ylab = "",
    xlim = c(-1, 1),
    ylim = c(-1, 1),
    asp = 1
  )
  abline(h = 0, v = 0)
  points(
    cos(meantheta),
    sin(meantheta),
    pch = 1,
    cex = 3,
    col = "black",
    lwd = 5
  )
  abline(
    a = 0,
    b = sin(meantheta) / cos(meantheta),
    lty = 3
  )
  l = mod(S1toRadian - meantheta + pi, 2 * pi) - pi
  points(
    cos(S1toRadian[which.max(l)]),
    sin(S1toRadian[which.max(l)]),
    pch = 4,
    cex = 3,
    col = "blue"
  )
  points(
    cos(S1toRadian[which.min(l)]),
    sin(S1toRadian[which.min(l)]),
    pch = 4,
    cex = 3,
    col = "red"
  )
  legend(
    "topright",
    legend = c("Geodesic mean", "Max (+)ve from mean", "Min (-)ve from mean"),
    col = c("black", "blue", "red"),
    pch = c(1, 4, 4)
  )
  {
    cat("\n")
    cat(
      "length of BLUE from geodesic mean : ",
      max(l),
      " (",
      round(max(l) * 180 / pi),
      " degree)",
      "\n",
      sep = ""
    )
    cat(
      "length of RED from geodesic mean : ",
      min(l),
      " (",
      round(min(l) * 180 / pi),
      " degree)",
      "\n",
      sep = ""
    )
    cat("\n")
  }
}
  radii = 1
  for (i in 1:(d - 1))
  {
    radii = c(radii, prod(sin(dist[1:i])))
  }
  resmat = flipud0(repmat(matrix(radii, ncol = 1), 1, n) * resmat)
  
if (d>1){
if (output){
  ### plot points on the 3D sphere (pointcolor), with 2D projection (white)
  rgl.sphgrid1()
  sphere1.f(col="white",alpha=0.6)
  sphrad <- 0.015
  spheres3d(-PNS$circlePNS[,2],PNS$circlePNS[,1],PNS$circlePNS[,3],radius=sphrad,col="White")  
  spheres3d(-PNS$spherePNS[,2],PNS$spherePNS[,1],PNS$spherePNS[,3],radius=sphrad,col=pointcolor)
}
yy <- orthaxis[[d-1]]
xx <- c(-yy[2], yy[1] , yy[3])
c1<-Enorm( c(xx[1],xx[2],xx[3])- c(-PNS$circlePNS[1,2],PNS$circlePNS[1,1],PNS$circlePNS[1,3]))
costheta<- 1 - c1^2/2
angle<-(1:201)/(200)*2*pi
centre<- xx*costheta
A<- xx-centre
B<- diag(3)-A%*%t(A)/Enorm(A)**2
bv<-eigen(B)$vectors
b1<-bv[,1]
b2<-bv[,2]
cc<- sin(acos(costheta))* ( cos(angle)%*%t(b1) + sin(angle)%*%t(b2) ) + rep(1,times=201)%*%t(centre)
if (output){
lines3d(cc,col=3,lwd=2)
}

######   
if (output){ 

         lines3d(cc,col=3,lwd=2)

# Note this provides a plot of the PNS mean in gold (updated calculation)


if (mean.type=="Fisher"){
  sum<-0
  for (i in 1:n){
  sum=sum+  ( acos( cc%*%c(-PNS$circlePNS[i,2],PNS$circlePNS[i,1],PNS$circlePNS[i,3])) )**2
}   #different PNS mean
mean0angle<-which.min(sum[1:200])/200*2*pi
meanpt<- sin(acos(costheta))* ( cos(mean0angle)%*%t(b1) + sin(mean0angle)%*%t(b2) ) +t(centre)
spheres3d( meanpt, radius=sphrad * 1.5,col=7, alpha=0.8)
}

if (mean.type=="Frechet"){ 
          ddout<-rep(0,times=n)
          sum2<-rep(0,times=200)
R <- sin(acos(costheta))
          for (jj in 1:200){
                    for (i in 1:n){
          ddout[i]<- ( mod( acos( sum( (cc[jj,]-centre)*(c(-PNS$circlePNS[i,2],PNS$circlePNS[i,1],PNS$circlePNS[i,3])-centre) )/R^2),2*pi)  )**2
                      }
          sum2[jj]<-sum(ddout)
          }
            mean0angle <- which.min(sum2[1:200])/200 * 2 * pi
            meanpt <- sin(acos(costheta)) * (cos(mean0angle) %*% 
                t(b1) + sin(mean0angle) %*% t(b2)) + t(centre)
             spheres3d(meanpt, radius = sphrad * 1.5, col = 7, 
                alpha = 0.8)
}          


}
###########


}
  PNS$scores = t(resmat) 
  PNS$radii = radii
  PNS$pnscircle <- cbind( cbind( cc[,2],-cc[,1]) , cc[,3])
  PNS$orthaxis = orthaxis
  PNS$dist = dist
  PNS$pvalues = pvalues
  PNS$ratio = ratio
  PNS$basisu = NULL
  PNS$mean = c(PNSe2s(matrix(0, d, 1), PNS))




  
  if (sphere.type == "seq.test")
  {
    PNS$sphere.type = "seq.test"
  } else if (sphere.type == "small") {
    PNS$sphere.type = "small"
  } else if (sphere.type == "great") {
    PNS$sphere.type = "great"
  } else if (sphere.type == "BIC") {
    PNS$sphere.type = "BIC"
  } else if (sphere.type == "bi.sphere") {
    PNS$sphere.type = "bi.sphere"
  }
  varPNS = apply(abs(resmat) ^ 2, 1, sum) / n
  total = sum(varPNS)
  propPNS = varPNS / total * 100
  return(list(
    resmat = resmat,
    PNS = PNS,
    percent = propPNS
  ))
}




#high-res sphere plot
#from stackoverflow answer (Mike Wise)
sphere1.f <- function(x0 = 0, y0 = 0, z0 = 0, r = 1, n = 101, ...){
  f <- function(s,t){ 
    cbind(   r * cos(t)*cos(s) + x0,
             r *        sin(s) + y0,
             r * sin(t)*cos(s) + z0)
  }
  persp3d(f, slim = c(-pi/2,pi/2), tlim = c(0, 2*pi), n = n, add = T, ...)
}

#adapted from the package "sphereplot" to remove text and axes (Aaron Robotham)
rgl.sphgrid1 <- 
  function (radius = 1, col.long = "red", col.lat = "blue", deggap = 15, 
            longtype = "H", add = FALSE, radaxis = TRUE, radlab = "Radius") 
  {
    if (add == F) {
      open3d()
    }
    for (lat in seq(-90, 90, by = deggap)) {
      if (lat == 0) {
        col.grid = "grey50"
      }
      else {
        col.grid = "grey"
      }
      plot3d(sph2car1(long = seq(0, 360, len = 100), lat = lat, 
                     radius = radius, deg = T), col = col.grid, add = T, 
             type = "l")
    }
    for (long in seq(0, 360 - deggap, by = deggap)) {
      if (long == 0) {
        col.grid = "grey50"
      }
      else {
        col.grid = "grey"
      }
      plot3d(sph2car1(long = long, lat = seq(-90, 90, len = 100), 
                     radius = radius, deg = T), col = col.grid, add = T, 
             type = "l")
    }
    if (longtype == "H") {
      scale = 15
    }
    if (longtype == "D") {
      scale = 1
    }
    # rgl.sphtext(long = 0, lat = seq(-90, 90, by = deggap), radius = radius, 
    #              text = seq(-90, 90, by = deggap), deg = TRUE, col = col.lat)
    # rgl.sphtext(long = seq(0, 360 - deggap, by = deggap), lat = 0, 
    #             radius = radius, text = seq(0, 360 - deggap, by = deggap)/scale, 
    #              deg = TRUE, col = col.long)
    if (radaxis) {
      radpretty = pretty(c(0, radius))
      radpretty = radpretty[radpretty <= radius]
      #    lines3d(c(0, 0), c(0, max(radpretty)), c(0, 0), col = "grey50")
      for (i in 1:length(radpretty)) {
        #      lines3d(c(0, 0), c(radpretty[i], radpretty[i]), c(0, 
        #                                                        0, radius/50), col = "grey50")
        #     text3d(0, radpretty[i], radius/15, radpretty[i], 
        #            col = "darkgreen")
      }
      #  text3d(0, radius/2, -radius/25, radlab)
    }
  }
sph2car1<-function (long, lat, radius = 1, deg = TRUE) 
{
    if (is.matrix(long) || is.data.frame(long)) {
        if (ncol(long) == 1) {
            long = long[, 1]
        }
        else if (ncol(long) == 2) {
            lat = long[, 2]
            long = long[, 1]
        }
        else if (ncol(long) == 3) {
            radius = long[, 3]
            lat = long[, 2]
            long = long[, 1]
        }
    }
    if (missing(long) | missing(lat)) {
        stop("Missing full spherical 3D input data.")
    }
    if (deg) {
        long = long * pi/180
        lat = lat * pi/180
    }
    return = cbind(x = radius * cos(long) * cos(lat), y = radius * 
        sin(long) * cos(lat), z = radius * sin(lat))
}




pns_biplot<-function(pns, varnames=rownames(q)){
  pns1<-pns
  nd <- dim(pns$resmat)[1]+1
palette(rainbow(nd))
res1 <- cbind( c( (20:(-20))/10*sd( pns1$resmat[1,])) , matrix(0,41,nd-2) )
if (nd>3){
res2 <- cbind( cbind( matrix(0,41,1) , c( (20:(-20))/10*sd( pns1$resmat[2,])) ) , matrix(0,41,nd-3) )
}
else
{
  res2 <- cbind( cbind( matrix(0,41,1) , c( (20:(-20))/10*sd( pns1$resmat[2,])) )  )
}
aa1 <- PNSe2s( t(res1) , pns1$PNS ) -pns1$PNS$mean
aa2 <- PNSe2s( t(res2) , pns1$PNS ) -pns1$PNS$mean
plot(aa1[1,],aa2[1,],xlim=c( min(aa1),max(aa1)) , type="n", col=2,  ylim=c(min(aa2),max(aa2)) ,xlab="PNS1", ylab="PNS2")
for (i in 1:(nd)){
  lines(aa1[i,],aa2[i,],col=i)
arrows( aa1[i,2],aa2[i,2],aa1[i,1],aa2[i,1],col=i)
text( aa1[i,1],aa2[i,1], varnames[i],col=i,cex=1)
}
title("PNS biplot")
palette("default")
}




#==================================================================================
pns4pc = function(x,
                  sphere.type = "seq.test",
                  alpha = 0.1,
                  R = 100,
                  nlast.small.sphere = 1,
                  n.pc = 2)
{
   if (n.pc < 2)
   {
      stop("Error: n.pc should be >= 2.")
   }
   out = pc2sphere2(x = x, n.pc = n.pc)
   spheredata = t(out$spheredata)
   GPAout = out$GPAout
   pns.out = pns(
      x = spheredata,
      sphere.type = sphere.type,
      alpha = alpha,
      R = R,
      nlast.small.sphere = nlast.small.sphere
   )
   pns.out$percent = pns.out$percent * sum(GPAout$percent[1:n.pc]) / 100
   pns.out$GPAout = GPAout
   pns.out$spheredata = spheredata
   return(pns.out)
}

pns.pc = function(x,
                  sphere.type = "seq.test",
                  alpha = 0.1,
                  R = 100,
                  nlast.small.sphere = 0,
                  n.pc = 0)
{
   k = dim(x)[1]
   m = dim(x)[2]
   n = dim(x)[3]
   if (n.pc == 0)
   {
      GPAout = procGPA(
         x = x,
         scale = TRUE,
         reflect = FALSE,
         tangentcoords = "partial",
         distances = FALSE
      )
      spheredata = matrix(NA, k * m, n)
      for (i in 1:n)
      {
         spheredata[, i] = c(GPAout$rotated[, , i])
      }
      pns.out = pns(
         x = spheredata,
         sphere.type = sphere.type,
         alpha = alpha,
         R = R,
         nlast.small.sphere = nlast.small.sphere
      )
      resmat = pns.out$resmat
      PNS = pns.out$PNS
      npts = 200
      prinarc1 = get.prinarc(
         resmat,
         PNS,
         arc = 1,
         n = npts,
         boundary.data = FALSE
      )
      prinarc2 = get.prinarc(
         resmat,
         PNS,
         arc = 2,
         n = npts,
         boundary.data = FALSE
      )
      prinarc1.ar = array(NA, c(k, m, npts))
      prinarc2.ar = array(NA, c(k, m, npts))
      for (i in 1:npts)
      {
         prinarc1.ar[, , i] = matrix(prinarc1[, i], nrow = k)
         prinarc2.ar[, , i] = matrix(prinarc2[, i], nrow = k)
      }
      scores.prinarc1 = shape.pcscores.partial(PCAout = GPAout, x = prinarc1.ar)
      scores.prinarc2 = shape.pcscores.partial(PCAout = GPAout, x = prinarc2.ar)
      out = pns.out
      out$GPAout = GPAout
      out$scores.prinarc1 = scores.prinarc1
      out$scores.prinarc2 = scores.prinarc2
   } else {
      pns.out = pns4pc(
         x = x,
         sphere.type = sphere.type,
         alpha = alpha,
         R = R,
         nlast.small.sphere = nlast.small.sphere,
         n.pc = n.pc
      )
      GPAout = pns.out$GPAout
      resmat = pns.out$resmat
      PNS = pns.out$PNS
      npts = 200
      prinarc1 = get.prinarc(
         resmat,
         PNS,
         arc = 1,
         n = npts,
         boundary.data = FALSE
      )
      prinarc2 = get.prinarc(
         resmat,
         PNS,
         arc = 2,
         n = npts,
         boundary.data = FALSE
      )
      scores.prinarc1 = matrix(NA, npts, n.pc)
      scores.prinarc2 = matrix(NA, npts, n.pc)
      for (g in 1:npts)
      {
         size1 = acos(prinarc1[1, g])
         size2 = acos(prinarc2[1, g])
         scores.prinarc1[g,] = prinarc1[2:(n.pc + 1), g] / (sin(size1) / size1)
         scores.prinarc2[g,] = prinarc2[2:(n.pc + 1), g] / (sin(size2) / size2)
      }
      out = pns.out
      out$scores.prinarc1 = scores.prinarc1
      out$scores.prinarc2 = scores.prinarc2
   }
   return(out)
}

#==================================================================================
rotMat = function(b, a = NULL, alpha = NULL)
{
   if (is.matrix(b))
   {
      if (min(dim(b)) == 1)
      {
         b = c(b)
      } else {
         stop("Error: b should be a unit vector.")
      }
   }
   d = length(b)
   b = b / norm(b, type = "2")
   
   if (is.null(a) & is.null(alpha))
   {
      a = c(rep(0, d - 1), 1)
      alpha = acos(sum(a * b))
   } else if (!is.null(a) & is.null(alpha)) {
      alpha = acos(sum(a * b))
   } else if (is.null(a) & !is.null(alpha)) {
      a = c(rep(0, d - 1), 1)
   }
   if (abs(sum(a * b) - 1) < 1e-15)
   {
      rot = diag(d)
      return(rot)
   }
   if (abs(sum(a * b) + 1) < 1e-15)
   {
      rot = -diag(d)
      return(rot)
   }
   c = b - a * sum(a * b)
   c = c / norm(c, type = "2")
   A = a %*% t(c) - c %*% t(a)
   rot = diag(d) + sin(alpha) * A + (cos(alpha) - 1) * (a %*% t(a) + c %*% t(c))
   return(rot)
}

#==================================================================================
ExpNPd = function(x)
{
   if (is.vector(x))
   {
      x = as.matrix(x)
   }
   d = nrow(x)
   nv = sqrt(apply(x ^ 2, 2, sum))
   Exppx = rbind(matrix(rep(sin(nv) / nv, d), nrow = d, byrow = T) * x, cos(nv))
   Exppx[, nv < 1e-16] = repmat(matrix(c(rep(0, d), 1)), 1, sum(nv < 1e-16))
   return(Exppx)
}

#==================================================================================
LogNPd = function(x)
{
   n = ncol(x)
   d = nrow(x)
   scale = acos(x[d,]) / sqrt(1 - x[d,] ^ 2)
   scale[is.nan(scale)] = 1
   Logpx = repmat(t(scale), d - 1, 1) * x[-d,]
   return(Logpx)
}

#==================================================================================
objfn = function(center, r, x)
{
   return(mean((acos(t(
      center
   ) %*% x) - r) ^ 2))
}

#==================================================================================
getSubSphere = function(x, geodesic = "small")
{
   svd.x = svd(x)
   initialCenter = svd.x$u[, ncol(svd.x$u)]
   c0 = initialCenter
   TOL = 1e-10
   cnt = 0
   err = 1
   n = ncol(x)
   d = nrow(x)
   Gnow = 1e+10
   while (err > TOL)
   {
      c0 = c0 / norm(c0, type = "2")
      rot = rotMat(c0)
      TpX = LogNPd(rot %*% x)
      fit = sphereFit(
         x = TpX,
         initialCenter = rep(0, d - 1),
         geodesic = geodesic
      )
      newCenterTp = fit$center
      r = fit$r
      if (r > pi)
      {
         r = pi / 2
         svd.TpX = svd(TpX)
         newCenterTp = svd.TpX$u[, ncol(svd.TpX$u)] * pi / 2
      }
      newCenter = ExpNPd(newCenterTp)
      center = solve(rot, newCenter)
      Gnext = objfn(center, r, x)
      err = abs(Gnow - Gnext)
      Gnow = Gnext
      c0 = center
      cnt = cnt + 1
      if (cnt > 30)
      {
         break
      }
   }
   i1save = list()
   i1save$Gnow = Gnow
   i1save$center = center
   i1save$r = r
   U = princomp(t(x))$loadings[,]
   initialCenter = U[, ncol(U)]
   c0 = initialCenter
   TOL = 1e-10
   cnt = 0
   err = 1
   n = ncol(x)
   d = nrow(x)
   Gnow = 1e+10
   
   while (err > TOL)
   {
      c0 = c0 / norm(c0, type = "2")
      rot = rotMat(c0)
      TpX = LogNPd(rot %*% x)
      fit = sphereFit(
         x = TpX,
         initialCenter = rep(0, d - 1),
         geodesic = geodesic
      )
      newCenterTp = fit$center
      r = fit$r
      if (r > pi)
      {
         r = pi / 2
         svd.TpX = svd(TpX)
         newCenterTp = svd.TpX$u[, ncol(svd.TpX$u)] * pi / 2
      }
      newCenter = ExpNPd(newCenterTp)
      center = solve(rot, newCenter)
      Gnext = objfn(center, r, x)
      err = abs(Gnow - Gnext)
      Gnow = Gnext
      c0 = center
      cnt = cnt + 1
      if (cnt > 30)
      {
         break
      }
   }
   if (i1save$Gnow == min(Gnow, i1save$Gnow))
   {
      center = i1save$center
      r = i1save$r
   }
   if (r > pi / 2)
   {
      center = -center
      r = pi - r
   }
   return(list(center = c(center), r = r))
}

#==================================================================================
LRTpval = function(resGREAT, resSMALL, n)
{
   chi2 = max(n * log(sum(resGREAT ^ 2) / sum(resSMALL ^ 2)), 0)
   return(pchisq(
      q = chi2,
      df = 1,
      lower.tail = FALSE
   ))
}

#==================================================================================
vMFtest = function(x, R = 100)
{
   d = nrow(x)
   n = ncol(x)
   sumx = apply(x, 1, sum)
   rbar = norm(sumx, "2") / n
   muMLE = sumx / norm(sumx, "2")
   kappaMLE = (rbar * d - rbar ^ 3) / (1 - rbar ^ 2)
   sp = getSubSphere(x = x, geodesic = "small")
   center.s = sp$center
   r.s = sp$r
   radialdistances = acos(t(center.s) %*% x)
   xi_sample = mean(radialdistances) / sd(radialdistances)
   xi_vec = rep(0, R)
   for (r in 1:R)
   {
      rdata = randvonMisesFisherm(d, n, kappaMLE)
      sp = getSubSphere(x = rdata, geodesic = "small")
      center.s = sp$center
      r.s = sp$r
      radialdistances = acos(t(center.s) %*% rdata)
      xi_vec[r] = mean(radialdistances) / sd(radialdistances)
   }
   pvalue = mean(xi_vec > xi_sample)
   return(pvalue)
}

#==================================================================================
geodmeanS1 = function(theta,mean.type="Frechet")
{
   n = length(theta)
  if (mean.type=="Frechet"){ 
#kk candidates angles
   kk <-1000
   meancandi = mod(mean(theta) + 2 * pi * (0:(kk - 1)) / kk, 2 * pi)
   theta = mod(theta, 2 * pi)
   geodvar = rep(0, kk)
   for (i in 1:kk)
   {
      v = meancandi[i]
      dist2 = apply(cbind((theta - v) ^ 2, (theta - v + 2 * pi) ^ 2, (v - theta + 2 * pi) ^
                             2), 1, min)
      geodvar[i] = sum(dist2)
   }
   m = min(geodvar)
   ind = which.min(geodvar)
   geodmean = mod(meancandi[ind], 2 * pi)
   geodvar = geodvar[ind] / n
}
   if (mean.type=="Fisher"){
     mm <- atan2( mean(sin(theta)), mean(cos(theta)) )
     geodmean <- mod(mm, 2*pi)
     geodvar <- 1 - sqrt( mean(sin(theta))**2 + mean(cos(theta))**2 )
   }
   return(list(geodmean = geodmean, geodvar = geodvar))
}

#==================================================================================
PNSe2s = function(resmat, PNS)
{
   dm = nrow(resmat)
   n = ncol(resmat)
   NSOrthaxis = rev(PNS$orthaxis[1:(dm - 1)])
   NSradius = flipud0(matrix(PNS$dist, ncol = 1))
   geodmean = PNS$orthaxis[[dm]]
   res = resmat / repmat(flipud0(matrix(PNS$radii, ncol = 1)), 1, n)
   T = t(rotMat(NSOrthaxis[[1]])) %*%
      rbind(repmat(sin(NSradius[1] + matrix(res[2,], nrow = 1)), 2, 1) *
               rbind(cos(geodmean + res[1,]), sin(geodmean + res[1,])),
            cos(NSradius[1] + res[2,]))
   if (dm > 2)
   {
      for (i in 1:(dm - 2))
      {
         T = t(rotMat(NSOrthaxis[[i + 1]])) %*%
            rbind(repmat(sin(NSradius[i + 1] + matrix(
               res[i + 2,], nrow = 1
            )), 2 + i, 1) * T,
            cos(NSradius[i + 1] + res[i + 2,]))
      }
   }
   if (!is.null(PNS$basisu))
   {
      T = PNS$basisu %*% T
   }
   return(T)
}

#==================================================================================
PNSs2e = function(spheredata, PNS)
{
   if (nrow(spheredata) != length(PNS$mean))
   {
      cat(" Error from PNSs2e() \n")
      cat(" Dimensions of the sphere and PNS decomposition do not match")
      return(NULL)
   }
   if (!is.null(PNS$basisu))
   {
      spheredata = t(PNS$basisu) %*% spheredata
   }
   kk = nrow(spheredata)
   n = ncol(spheredata)
   Res = matrix(0, kk - 1, n)
   currentSphere = spheredata
   for (i in 1:(kk - 2))
   {
      v = PNS$orthaxis[[i]]
      r = PNS$dist[i]
      res = acos(t(v) %*% currentSphere) - r
      Res[i,] = res
      NestedSphere = rotMat(v) %*% currentSphere
      currentSphere = as.matrix(NestedSphere[1:(kk - i),]) /
         repmat(matrix(sqrt(1 - NestedSphere[nrow(NestedSphere),] ^
                               2), nrow = 1), kk - i, 1)
   }
   S1toRadian = atan2(currentSphere[2,], currentSphere[1,])
   devS1 = mod(S1toRadian - rev(PNS$orthaxis)[[1]] + pi, 2 * pi) - pi
   Res[kk - 1,] = devS1
   EuclidData = flipud0(repmat(PNS$radii, 1, n) * Res)
   return(EuclidData)
}

#==================================================================================
randvonMisesFisherm = function(m, n, kappa, mu = NULL)
{
   if (is.null(mu))
   {
      muflag = FALSE
   } else {
      muflag = TRUE
   }
   if (m < 2)
   {
      print("Message from randvonMisesFisherm(): dimension m must be > 2")
      print("Message from randvonMisesFisherm(): Set m to be 2")
      m = 2
   }
   if (kappa < 0)
   {
      print("Message from randvonMisesFisherm(): kappa must be >= 0")
      print("Message from randvonMisesFisherm(): Set kappa to be 0")
      kappa = 0
   }
   b = (-2 * kappa + sqrt(4 * kappa ^ 2 + (m - 1) ^ 2)) / (m - 1)
   x0 = (1 - b) / (1 + b)
   c = kappa * x0 + (m - 1) * log(1 - x0 ^ 2)
   nnow = n
   w = c()
   while (TRUE)
   {
      ntrial = max(round(nnow * 1.2), nnow + 10)
      Z = rbeta(n = ntrial,
                shape1 = (m - 1) / 2,
                shape2 = (m - 1) / 2)
      U = runif(ntrial)
      W = (1 - (1 + b) * Z) / (1 - (1 - b) * Z)
      
      indicator = kappa * W + (m - 1) * log(1 - x0 * W) - c >= log(U)
      if (sum(indicator) >= nnow)
      {
         w1 = W[indicator]
         w = c(w, w1[1:nnow])
         break
      } else {
         w = c(w, W[indicator])
         nnow = nnow - sum(indicator)
      }
   }
   V = UNIFORMdirections(m - 1, n)
   X = rbind(repmat(sqrt(1 - matrix(w, nrow = 1) ^ 2), m - 1, 1) * V, matrix(w, nrow = 1))
   if (muflag)
   {
      mu = mu / norm(mu, "2")
      X = t(rotMat(mu)) %*% X
   }
   return(X)
}

#==================================================================================
UNIFORMdirections = function(m, n)
{
   V = matrix(0, m, n)
   nr = matrix(rnorm(m * n), nrow = m)
   for (i in 1:n)
   {
      while (TRUE)
      {
         ni = sum(nr[, i] ^ 2)
         if (ni < 1e-10)
         {
            nr[, i] = rnorm(m)
         } else {
            V[, i] = nr[, i] / sqrt(ni)
            break
         }
      }
   }
   return(V)
}

#==================================================================================
trans.subsphere = function(x, center)
{
   return(repmat(1 / sqrt(1 - (t(
      center
   ) %*% x) ^ 2), length(center) - 1, 1) *
      (rotMat(center)[-length(center),] %*% x))
}

#==================================================================================
get.prinarc.value = function(PNS, arc, res)
{
   d = length(PNS$orthaxis)
   n = length(res)
   prinarc = matrix(NA, d + 1, n)
   for (g in 1:n)
   {
      newres = matrix(0, d, 1)
      newres[arc] = res[g]
      T = PNSe2s(newres, PNS)
      prinarc[, g] = T
   }
   return(prinarc)
}

#==================================================================================
get.prinarc = function(resmat, PNS, arc, n, boundary.data = FALSE)
{
   d = nrow(resmat)
   if (boundary.data)
   {
      mn = min(resmat[arc,])
      mx = max(resmat[arc,])
   } else {
      mn = -pi * tail(PNS$radii, arc)[1]
      mx = pi * tail(PNS$radii, arc)[1]
   }
   prinarcgrid = seq(mn, mx, length = n)
   prinarc = matrix(NA, d + 1, n)
   for (g in 1:n)
   {
      newres = matrix(0, d, 1)
      newres[arc] = prinarcgrid[g]
      T = PNSe2s(newres, PNS)
      prinarc[, g] = T
   }
   return(prinarc)
}

#==================================================================================
get.prinarc.subsphere = function(resmat,
                                 PNS,
                                 arc,
                                 n,
                                 subsphere = arc,
                                 boundary.data = FALSE)
{
   if (subsphere < arc)
   {
      stop("Error: subsphere >= arc.")
   }
   if (subsphere < 1)
   {
      stop("Error: subsphere >= 1.")
   }
   prinarc = get.prinarc(
      resmat = resmat,
      PNS = PNS,
      arc = arc,
      n = n,
      boundary.data = boundary.data
   )
   d = nrow(resmat)
   prinarc.sub = prinarc
   if (subsphere < d)
   {
      for (i in 1:(d - subsphere))
      {
         prinarc.sub = trans.subsphere(x = prinarc.sub, center = PNS$orthaxis[[i]])
      }
   }
   return(prinarc.sub)
}

#==================================================================================
get.data.subsphere = function(resmat, PNS, x, subsphere)
{
   if (subsphere < 1)
   {
      stop("Error: subsphere >= 1.")
   }
   d = nrow(resmat)
   x.sub = x
   if (subsphere < d)
   {
      for (i in 1:(d - subsphere))
      {
         x.sub = trans.subsphere(x = x.sub, center = PNS$orthaxis[[i]])
      }
   }
   return(x.sub)
}

#==================================================================================
mod = function(x, y)
{
   return(x %% y)
}

#==================================================================================
repmat = function(x, m, n)
{
   return(kronecker(matrix(1, m, n), x))
}

#==================================================================================
flipud0 = function(x)
{
   return(apply(x, 2, rev))
}

#==================================================================================
sphere.obj = function(center, x, is.greatCircle)
{
   di = sqrt(apply((x - repmat(
      matrix(center, ncol = 1), 1, ncol(x)
   )) ^ 2, 2, sum))
   if (is.greatCircle)
   {
      r = pi / 2
   } else {
      r = mean(di)
   }
   sum((di - r) ^ 2)
}

#==================================================================================
sphere.res = function(center, x, is.greatCircle)
{
   center = c(center)
   xmc = x - center
   di = sqrt(apply(xmc ^ 2, 2, sum))
   if (is.greatCircle)
   {
      r = pi / 2
   } else {
      r = mean(di)
   }
   (di - r)
}

#==================================================================================
sphere.jac = function(center, x, is.greatCircle)
{
   center = c(center)
   xmc = x - center
   di = sqrt(apply(xmc ^ 2, 2, sum))
   di.vj = -xmc / repmat(matrix(di, nrow = 1), length(center), 1)
   if (is.greatCircle)
   {
      c(t(di.vj))
   } else {
      r.vj = apply(di.vj, 1, mean)
      c(t(di.vj - repmat(matrix(r.vj, ncol = 1), 1, ncol(x))))
   }
}

#==================================================================================
sphereFit = function(x,
                     initialCenter = NULL,
                     geodesic = "small")
{
   if (is.null(initialCenter))
   {
      initialCenter = apply(x, 1, mean)
   }
   op = nls.lm(
      par = initialCenter,
      fn = sphere.res,
      jac = sphere.jac,
      x = x,
      is.greatCircle = ifelse(geodesic == "great", TRUE, FALSE),
      control = nls.lm.control(maxiter = 1000)
   )
   center = coef(op)
   di = sqrt(apply((x - repmat(
      matrix(center, ncol = 1), 1, ncol(x)
   )) ^ 2, 2, sum))
   if (geodesic == "great")
   {
      r = pi / 2
   } else {
      r = mean(di)
   }
   
   list(center = center, r = r)
}

#==================================================================================
tr = function(x)
{
   return(sum(diag(x)))
}

#==================================================================================
Enormalize = function(x)
{
   return(x / Enorm(x))
}

#==================================================================================
sphere2pcscore = function(x)
{
   n = nrow(x)
   p = ncol(x)
   scores = matrix(NA, n, p - 1)
   for (i in 1:n)
   {
      size = acos(x[i, 1])
      scores[i,] = (size / sin(size)) * x[i, 2:p]
   }
   return(scores)
}

#==================================================================================
pcscore2sphere = function(n.pc, X.hat, S, Tan, V)
{
   d = nrow(Tan)
   n = ncol(Tan)
   W = matrix(NA, d, n)
   for (i in 1:n)
   {
      W[, i] = acos(tr(S[, , i] %*% t(X.hat))) * Tan[, i] / sqrt(sum(Tan[, i] ^
                                                                        2))
   }
   lambda = matrix(NA, n, d)
   for (i in 1:n)
   {
      for (j in 1:d)
      {
         lambda[i, j] = sum(W[, i] * V[, j])
      }
   }
   U = matrix(0, n, d)
   for (i in 1:n)
   {
      for (j in 1:n.pc)
      {
         U[i,] = U[i,] + lambda[i, j] * V[, j]
      }
   }
   S.star = matrix(NA, n, n.pc + 1)
   for (i in 1:n)
   {
      U.norm = sqrt(sum(U[i,] ^ 2))
      S.star[i,] = c(cos(U.norm),
                     sin(U.norm) / U.norm * lambda[i, 1:n.pc])
   }
   return(S.star)
}


pcscore2sphere2 = function(n.pc, X.hat, S, Tan, V)
{
   d = nrow(Tan)
   n = ncol(Tan)
   W = matrix(NA, d, n)

  if (n.pc > min(d,n) )
   {
      stop("Error: n.pc must be <= min(n,d)")
   }


   for (i in 1:n)
   {
      W[, i] = acos(tr(S[, , i] %*% t(X.hat))) * Tan[, i] / sqrt(sum(Tan[, i] ^
                                                                        2))
   }
   lambda = matrix(NA, n, d)
   for (i in 1:n)
   {
      for (j in 1:n.pc)
      {
         lambda[i, j] = sum(W[, i] * V[, j])
      }
   }
   U = matrix(0, n, d)
   for (i in 1:n)
   {
      for (j in 1:n.pc)
      {
         U[i,] = U[i,] + lambda[i, j] * V[, j]
      }
   }
   S.star = matrix(NA, n, n.pc + 1)
   for (i in 1:n)
   {
      U.norm = sqrt(sum(U[i,] ^ 2))
      S.star[i,] = c(cos(U.norm),
                     sin(U.norm) / U.norm * lambda[i, 1:n.pc])
   }
   return(S.star)
}


#==================================================================================
pc2sphere = function(x, n.pc)
{
   k = dim(x)[1]
   m = dim(x)[2]
   n = dim(x)[3]
   if (n.pc < ((k - 1) * m))
   {
      stop("Error: n.pc must be >= (k - 1) * m.")
   }
   GPAout = procGPA(
      x = x,
      scale = TRUE,
      reflect = FALSE,
      tangentcoords = "partial",
      distances = FALSE
   )
   cat(
      "First ",
      n.pc,
      " principal components explain ",
      round(sum(GPAout$percent[1:n.pc])),
      "% of total variance. \n",
      sep = ""
   )
   H = defh(k - 1)
   X.hat = H %*% GPAout$mshape
   S = array(NA, c(k - 1, m, n))
   for (i in 1:n)
   {
      S[, , i] = H %*% GPAout$rotated[, , i]
   }
   T.c = GPAout$tan - apply(GPAout$tan, 1, mean)
   out = pcscore2sphere(
      n.pc = n.pc,
      X.hat = X.hat,
      S = S,
      Tan = T.c,
      V = GPAout$pcar
   )
   return(list(spheredata = out, GPAout = GPAout))
}

#==================================================================================
rot.mat = function(Y,
                   X,
                   reflect = FALSE,
                   center = TRUE)
{
   svd.out = svd(t(X) %*% Y)
   R = svd.out$u %*% t(svd.out$v)
   if (!reflect)
   {
      if (det(R) < 0)
      {
         u = svd.out$u
         v = svd.out$v
         if (det(u) < 0)
         {
            u[, dim(u)[2]] = -u[, dim(u)[2]]
         } else if (det(v) < 0) {
            v[, dim(v)[2]] = -v[, dim(v)[2]]
         }
         R = u %*% t(v)
      }
   }
   return(R)
}

#==================================================================================
Procrustes.dist.full = function(x1, x2)
{
   m = ncol(x1)
   z1 = preshape(x1)
   z2 = preshape(x2)
   Q = t(z1) %*% z2 %*% t(z2) %*% z1
   ev = eigen(Q)$values
   sign = ifelse(det(t(z1) %*% z2) >= 0, 1,-1)
   dF = sqrt(abs(1 - sum(sqrt(abs(
      ev[1:(m - 1)]
   )), sign * sqrt(abs(
      ev[m]
   ))) ^ 2))
   R = rot.mat(
      Y = z2,
      X = z1,
      reflect = FALSE,
      center = FALSE
   )
   scale = sum(svd(t(z1) %*% z2)$d)
   return(list(dF = dF, R = R, scale = scale))
}

#==================================================================================
tangent.coords.partial = function(x, p)
{
   k = nrow(x)
   m = ncol(x)
   if (abs(norm(p, "F") - 1) > 1e-15)
   {
      print("||p|| is not 1. Normalised one is used.")
      p = Enormalize(p)
   }
   tmp = Procrustes.dist.full(x, p)
   R = tmp$R
   scale = tmp$scale
   pre.p = preshape(p)
   pre.x = preshape(x)
   ident = diag(k * m - m)
   tan = (ident - matrix(pre.p) %*% t(c(pre.p))) %*% c(pre.x %*% R)
   tan.scale = (ident - matrix(pre.p) %*% t(c(pre.p))) %*% c(pre.x %*% R * scale)
   return(list(
      tan = c(tan),
      tan.scale = c(tan.scale),
      R = R,
      scale = scale
   ))
}

#==================================================================================
shape.pcscores = function(PCAout, x, tangentcoords = "partial")
{
   if (tangentcoords == "partial")
   {
      if (abs(norm(PCAout$mshape, "F") - 1) > 1e-15)
      {
         print("||PCAout$mshape|| is not 1. Normalised one is used.")
         mshape = Enormalize(PCAout$mshape)
      } else {
         mshape = PCAout$mshape
      }
      if (abs(norm(x, "F") - 1) > 1e-15)
      {
         print("||x|| is not 1. Normalised one is used.")
         x = Enormalize(x)
      }
      opa.out = procOPA(mshape, x, scale = FALSE)
      matched = opa.out$Bhat
      tan.out = tangent.coords.partial(matched, mshape)
      mean.tan = apply(PCAout$tan, 1, mean)
      scores = t(tan.out$tan - mean.tan) %*% PCAout$pcar
      scores.scale = t(tan.out$tan.scale - mean.tan) %*% PCAout$pcar
      return(
         list(
            rotated = matched,
            tan = tan.out$tan,
            tan.scale = tan.out$tan.scale,
            scores = c(scores),
            scores.scale = c(scores.scale)
         )
      )
   }
}

#==================================================================================
shape.pcscores.partial = function(PCAout, x)
{
   n = dim(x)[3]
   scores = c()
   for (i in 1:n)
   {
      s = shape.pcscores(PCAout, x[, , i], tangentcoords = "partial")
      scores = rbind(scores, s$scores)
   }
   return(scores)
}

#==================================================================================
plotshapes3d.pns = function(x,
                            type = "p",
                            col = "black",
                            size = 5,
                            aspect = "iso",
                            joinline = TRUE,
                            col.joinline = "#d4d2d2",
                            lwd.joinline = 0.5,
                            tick = FALSE,
                            labels.tick = FALSE,
                            xlab = "",
                            ylab = "",
                            zlab = "")
{
   k = dim(x)[1]
   n = dim(x)[3]
   aa = c()
   bb = c()
   cc = c()
   for (i in 1:n)
   {
      aa = c(aa, x[, 1, i])
      bb = c(bb, x[, 2, i])
      cc = c(cc, x[, 3, i])
   }
   xlim = range(aa)
   ylim = range(bb)
   zlim = range(cc)
   plot3d(
      x[, , 1],
      type = "n",
      xlab = "",
      ylab = "",
      zlab = "",
      box = FALSE,
      axes = FALSE,
      aspect = aspect,
      xlim = xlim,
      ylim = ylim,
      zlim = zlim
   )
   for (i in 1:n)
   {
      plot3d(
         x[, , i],
         type = type,
         col = col,
         size = size,
         add = TRUE
      )
   }
   if (tick)
   {
      axis3d(
         edge = 'x',
         labels = labels.tick,
         tick = TRUE,
         pos = c(NA, 0, 0),
         cex = 0.6,
         lwd = 0.5
      )
      axis3d(
         edge = 'y',
         labels = labels.tick,
         tick = TRUE,
         pos = c(0, NA, 0),
         cex = 0.6,
         lwd = 0.5
      )
      axis3d(
         edge = 'z',
         labels = labels.tick,
         tick = TRUE,
         pos = c(0, 0, NA),
         cex = 0.6,
         lwd = 0.5
      )
   } else {
      
   }
   r = cbind(xlim, ylim, zlim)
   pos = r[2,] + apply(r, 2, diff) / 20
   text3d(pos[1], 0, 0, texts = xlab, cex = 0.8)
   text3d(0, pos[2], 0, texts = ylab, cex = 0.8)
   text3d(0, 0, pos[3], texts = zlab, cex = 0.8)
   if (joinline)
   {
      for (i in 1:n)
      {
         lines3d(x[, , i], col = col.joinline, lwd = lwd.joinline)
      }
   }
}

#==================================================================================
Plot3D = function(x,
                  type = "s",
                  col = "black",
                  size = 1.2,
                  aspect = "iso",
                  joinline = FALSE,
                  col.joinline = "#d4d2d2",
                  lwd.joinline = 0.5,
                  tick = TRUE,
                  tick.boundary = FALSE,
                  labels.tick = TRUE,
                  xlab = "",
                  ylab = "",
                  zlab = "")
{
   n = nrow(x)
   plot3d(
      x,
      type = "n",
      xlab = "",
      ylab = "",
      zlab = "",
      box = FALSE,
      axes = FALSE,
      aspect = aspect
   )
   plot3d(
      x,
      type = type,
      col = col,
      size = size,
      add = TRUE
   )
   if (tick)
   {
      axis3d(
         edge = 'x',
         labels = labels.tick,
         tick = TRUE,
         pos = c(NA, 0, 0),
         cex = 0.6,
         lwd = 0.5
      )
      axis3d(
         edge = 'y',
         labels = labels.tick,
         tick = TRUE,
         pos = c(0, NA, 0),
         cex = 0.6,
         lwd = 0.5
      )
      axis3d(
         edge = 'z',
         labels = labels.tick,
         tick = TRUE,
         pos = c(0, 0, NA),
         cex = 0.6,
         lwd = 0.5
      )
   }
   if (tick.boundary)
   {
      tks = pretty(x[, 1], n = 10)
      axis3d(
         edge = 'x',
         labels = labels.tick,
         tick = TRUE,
         at = c(tks[1], tks[length(tks)]),
         pos = c(NA, 0, 0),
         cex = 0.6,
         lwd = 0.5
      )
      tks = pretty(x[, 2], n = 10)
      axis3d(
         edge = 'y',
         labels = labels.tick,
         tick = TRUE,
         at = c(tks[1], tks[length(tks)]),
         pos = c(0, NA, 0),
         cex = 0.6,
         lwd = 0.5
      )
      tks = pretty(x[, 3], n = 10)
      axis3d(
         edge = 'z',
         labels = labels.tick,
         tick = TRUE,
         at = c(tks[1], tks[length(tks)]),
         pos = c(0, 0, NA),
         cex = 0.6,
         lwd = 0.5
      )
   }
   r = apply(x, 2, range)
   pos = r[2,] + apply(r, 2, diff) / 20
   text3d(pos[1], 0, 0, texts = xlab, cex = 0.8)
   text3d(0, pos[2], 0, texts = ylab, cex = 0.8)
   text3d(0, 0, pos[3], texts = zlab, cex = 0.8)
   if (joinline)
   {
      lines3d(x, col = col.joinline, lwd = lwd.joinline)
   }
}

#==================================================================================
col2RGB = function(col, alpha = 255)
{
   n = length(col)
   out = c()
   for (i in 1:n)
   {
      out[i] = rgb(
         red = col2rgb(col[i])[1],
         green = col2rgb(col[i])[2],
         blue = col2rgb(col[i])[3],
         alpha = alpha,
         maxColorValue = 255
      )
   }
   return(out)
}

#==================================================================================
project.subsphere = function(x, center, r)
{
   n = ncol(x)
   d = nrow(x)
   x.proj = matrix(NA, d, n)
   for (i in 1:n)
   {
      rho = acos(sum(x[, i] * center))
      x.proj[, i] = (sin(r) * x[, i] + sin(rho - r) * center) / sin(rho)
   }
   return(x.proj)
}


##############################################################end of PNS###########



##### Penalised Euclidean Distance Regression

#==================================================================================
ped <- function(X, Y, method = c("AIC")) {
   if (method == "AIC") {
      aicmin <- 999999999
      for (lam in c(0.2, 0.5, 1.0)) {
         for (cofp in c(0.75, 1, 1.35, 1.5)) {
            out   <-
               pedreg(
                  X,
                  Y,
                  nlambda = 1,
                  constc0 = 1.1,
                  constc1 = cofp,
                  lambdainit = lam
               )
            if (out$aic < aicmin) {
               minout <- out
               mincofp <- cofp
               aicmin <- out$aic
            }
         }
      }
      out <- minout
   }
   
   if (method == "BIC") {
      bicmin <- 999999999
      for (lam in c(0.2, 0.5, 1.0)) {
         for (cofp in c(0.75, 1, 1.35, 1.5)) {
            out   <-
               pedreg(
                  X,
                  Y,
                  nlambda = 1,
                  constc0 = 1.1,
                  constc1 = cofp,
                  lambdainit = lam
               )
            if (out$bic < bicmin) {
               minout <- out
               mincofp <- cofp
               bicmin <- out$bic
            }
         }
      }
      out <- minout
   }
   
   if (method == "khat") {
      aicmin <- 999999999
      for (lam in c(0.2, 0.5, 1.0)) {
         for (cofp in c(0.75, 1, 1.25, 1.5)) {
            out   <-
               pedreg(
                  X,
                  Y,
                  nlambda = 1,
                  constc0 = 1.1,
                  constc1 = cofp,
                  lambdainit = lam
               )
            if (-out$khat < aicmin) {
               minout <- out
               mincofp <- cofp
               aicmin <- -out$khat
            }
         }
      }
      out <- minout
   }
   if (method == "CV") {
      n <- length(Y)
      cvmin <- 999999999
      for (lam in c(0.2, 0.5, 1.0)) {
         for (cofp in c(0.75, 1, 1.25, 1.5)) {
            cverr <- 0
            for (jj in 1:10) {
               subsample <- ((jj - 1) * 10 + 1):((jj - 1) * 10 + 10)
               out   <-
                  pedreg(
                     X[-subsample, ],
                     Y[-subsample],
                     nlambda = 1,
                     constc0 = 1.1,
                     constc1 = cofp,
                     lambdainit = lam
                  )
               cverr <-
                  cverr + Enorm(Y[subsample] - out$intercept - X[subsample, ] %*% out$betahat) **
                  2
            }
            if (cverr < cvmin) {
               minout <- out
               minlam <- lam
               mincofp <- cofp
               cvmin <- cverr
            }
         }
      }
      out <-
         pedreg(
            X,
            Y,
            nlambda = 1,
            constc0 = 1.1,
            constc1 = mincofp,
            lambdainit = minlam
         )
   }
   
   
   out1 <- list(
      betahat = 0,
      yhat = 0,
      lambda = 0,
      coef = 0,
      resid = 0
   )
   out1$intercept <- out$intercept
   out1$coef <- c(out$intercept, out$betahat)
   out1$betahat <- out$betahat
   out1$lambda <- out$lambda
   out1$delta <- mincofp
   out1$yhat <- out$yhat
   out1$resid <- Y - out$yhat
   
   
   
   out1
}

###########################function for PED#####################



#==================================================================================
pedreg <-
   function(X,
            Y,
            constc0 = 1.1,
            constc1 = 1.35,
            alpha = 0.05,
            LMM = 50,
            MIT = 10000,
            NUM_METHOD = 1,
            nlambda = 1,
            lambdamax = 1,
            PLOT = TRUE,
            BIC = FALSE,
            lambdainit = 1) {
      # NUM_METHOD = 1 = L-BFGS-B
      # LMM = Parameter M in L-BFGS method 1
      # MIT = Max iterations for optimization
      
      
      
      p <- dim(X)[2]
      n <- dim(X)[1]
      constc <- constc0
      Ymean <- mean(Y)
      Ysd <- sd(Y)
      Yinit <- Y
      
      pinit <- p
      ans0 <- rep(0, times = pinit)
      
      Xorig <- X
      Yorig <- Y
      
      vm <- rep(0, times = ncol(X))
      vsd <- rep(0, times = ncol(X))
      
      for (i in 1:ncol(X)) {
         vm[i] <- mean(X[, i])
      }
      for (i in 1:ncol(X)) {
         vsd[i] <- sd(X[, i])
      }
      
      
      
      
      
      #standardize to sphere
      
      X <- scale(X) / sqrt(n - 1)
      Y <- scale(Y) / sqrt(n - 1)
      
      
      X0 <- X
      Y0 <- Y
      
      
      
      
      lambdainit1 <- constc / sqrt(n - 1) * sqrt(sqrt(p)) / n * qnorm(1 - alpha /
                                                                         (2 * p))
      if (nlambda == 1) {
         lambdainit1 <- lambdainit
      }
      
      
      METHOD1 <- "L-BFGS-B"
      xi <- -9999999
      c1 <- 1
      
      
      
      nlam <- nlambda
      betamat <- matrix(0, p, nlam)
      betamat.sparse <- betamat
      lambdamat <- rep(0, times = nlam)
      ximat <- rep(0, times = nlam)
      aic <- rep(0, times = nlam)
      bic <- rep(0, times = nlam)
      npar <- rep(0, times = nlam)
      selectmat <- betamat
      #cat(c("Lambda iteration (out of ",nlam,"):"))
      
      for (ilam in (nlam:1)) {
         #cat(c(ilam," "))
         
         if (nlam == 1) {
            lambda <- lambdainit1
         }
         if (nlam > 1) {
            c1 <- sqrt(n) + (ilam - 1) / (nlam - 1) * 1 / lambdainit1
            c1 <-
               sqrt(n) + ((ilam - 1) / (nlam - 1)) * 1 / lambdainit1 * (lambdamax - lambdainit1 *
                                                                           sqrt(n))
            lambda <- lambdainit1 * c1
         }
         
         if (ilam == nlam) {
            x0 <- rep(1 / sqrt(p), times = p)
         }
         
         if (ilam != nlam) {
            x0 <- betahat + rnorm(p) / sqrt(p)
         }
         #x0<-rnorm(p)/sqrt(p)
         
         
         
         
         pedfun <- function(pars, Y = 0, X = 0) {
            p <- length(pars)
            pars <- matrix(pars, p, 1)
            ped <- Enorm(Y - X %*% pars) + lambda * sqrt(Enorm(pars) * sum(abs(pars)))
            ped
         }
         
         
         
         pedgrad <- function(pars, X = 0, Y = 0) {
            GM <- sqrt(Enorm(pars) * sum(abs(pars)))
            gradL <- rep(0, times = p)
            
            gradL <-  -t(X) %*% (Y - X %*% pars) / Enorm(Y - X %*% pars)
            gradL <- gradL + matrix(
               lambda / 2 * pars / Enorm(pars) * sum(abs(pars)) / GM +
                  lambda / 2 * sign(pars) * Enorm(pars) / GM ,
               p,
               1
            )
            
            
            c(gradL)
         }
         
         
         
         
         if (NUM_METHOD == 1) {
            repeat {
               #,ndeps=1e-3,factr=1e-5,pgtol=1e-5
               res2 <-
                  optim(
                     par = x0,
                     fn = pedfun,
                     gr = pedgrad,
                     method = METHOD1,
                     control = list(lmm = LMM, maxit = MIT),
                     X = X,
                     Y = Y
                  )
               betahat <- res2$par
               if (res2$convergence == 0) {
                  break
               }
               x0 <- rnorm(p) / sqrt(p)
            }
         }
         
         
         
         
         
         
         oldxi <- xi
         xi <-
            sqrt(Enorm(betahat) / sum(abs(betahat))) - sqrt(n) / (constc * c1 * p ^
                                                                     (1 / 4))
         
         dif <- (xi - oldxi)
         
         
         
         REGC <- 0.0001
         
         betamat[, ilam] <- betahat / (Enorm(betahat) + REGC)
         
         lambdamat[ilam] <- lambda
         ximat[ilam] <- xi
         
         ximat[ilam] <- sqrt(Enorm(betahat) / sum(abs(betahat)))
         
         
         MM <- constc1 / (sqrt(n))
         
         
         select <- (abs(betahat) / (Enorm(betahat) + REGC) > MM)
         selectmat[, ilam] <- select
         betamat.sparse[, ilam] <- betamat[, ilam]
         betamat.sparse[select == FALSE, ilam] <- 0 * betamat[select == FALSE, ilam]
         
         pp <- sum(select)
         npar[ilam] <- pp
         bic[ilam] <- log(Enorm(Y) ** 2 / n) * (n) + log(n) * (1)
         aic[ilam] <- log(Enorm(Y) ** 2 / n) * (n) + 2 * (1)
         
         if (sum(select) > 0) {
            aa <- lm(Y ~ X[, c(1:p)[select]] - 1)
            pred <- predict(aa)
            
            # Use AIC with finite sample correction
            aic[ilam] <-
               log(Enorm(Y - pred) ** 2 / n) * (n)  + 2 * (pp + 1) + 2 * (pp + 1) * (pp +
                                                                                        2) / (n - pp - 2)
            # Use AIC/BIC
            bic[ilam] <- log(Enorm(Y - pred) ** 2 / n) * (n) + log(n) * (pp + 1)
            aic[ilam] <- log(Enorm(Y - pred) ** 2 / n) * (n) + 2 * (pp + 1)
         }
         
         
         
         
      }
      
      
      best <- 1
      if (nlam > 1) {
         ###########################################choose best via AIC
         
         
         best <- c(1:nlam)[aic == min(aic)][1]
         select <- as.logical(selectmat[, best])
         lambdaaic <- lambdamat[best]
         
         
         ###########################################choose best via Corollary 1 with khat
         best2 <- nlam
         if ((sum(ximat > 0.25)) > 0) {
            best2 <- c(1:nlam)[(ximat) > 0.25][1]
         }
         
         #################### biggest xi from Corollary 1
         
         xism <- (ximat)
         if (sum(diff(xism) < 0.01) > 0) {
            best2 <- c(2:nlam)[diff(xism) < 0.01][1] - 1
         }
         
         ############## biggest sqrt( Enorm(beta) / norm(beta)_1 )
         
         best2 <- c(1:nlam)[ximat == max(ximat)]
         
         selectcor <- as.logical(selectmat[, best2])
         lambdacor1 <- lambdamat[best2]
         
         
         if (BIC == FALSE) {
            best <- best2
            select <- selectcor
         }
      }
      
      ################last part######estimate with reduced p###########
      
      if (sum(select) > 0) {
         X <- as.matrix(X[, select])
         p <- sum(select)
         p14 <- sqrt(sqrt(p))
         final <- c(1:pinit)[select]
      }
      
      
      #######
      
      
      
      
      lambda <- constc / sqrt(sqrt(p)) / sqrt(n) * qnorm(1 - alpha / (2 * p))
      
      
      
      
      if (sum(select) > 0) {
         x0 <- rep(1 / p, times = p)
         
         
         if (NUM_METHOD == 1) {
            repeat {
               res2 <-
                  optim(
                     par = x0,
                     fn = pedfun,
                     gr = pedgrad,
                     method = METHOD1,
                     control = list(lmm = LMM, maxit = MIT),
                     X = X,
                     Y = Y
                  )
               betahat <- res2$par
               
               if (res2$convergence == 0) {
                  break
               }
               
               x0 <- rnorm(p) / p
            }
         }
         
         
         
         
         ans0[final] <- betahat
         
      }
      
      #ind<-which(abs(ans0/Enorm(ans0))<10^(-5))
      #ans0[ind]<-0
      
      
      
      out <- list(betahat = 0,
                  yhat = 0,
                  lambda = 0)
      
      
      out$betahatscale <- ans0
      out$yhatscale <- c(X0 %*% ans0)
      
      if (nlam > 1) {
         out$lambdacor1 <- lambdacor1
         out$lambdaaic <- lambdaaic
         out$betamat.sparse <- betamat.sparse
         out$betamat.rescale <- betamat
         out$betamat <- betamat
         for (i in 1:nlam) {
            out$betamat.rescale[, i] <- c(out$betamat[, i] / vsd) * sd(Yorig)
         }
         out$lambdamat <- lambdamat
         out$ximat <- ximat
         out$MM <- MM
         out$fmax <- res2$value
         out$npar <- npar
         out$selectmat <- selectmat
      }
      
      #use AIC
      out$aic <- aic
      #use BIC
      out$bic <- bic
      
      #use khat
      out$khat <- ximat
      
      out$lambdath3 <- lambdainit1 * sqrt(n)
      
      out$lambda <- out$lambdacor1
      if (BIC == TRUE) {
         out$lambda <- out$lambdaaic
      }
      
      if (nlam == 1) {
         out$lambda <- lambdainit1
         out$constc1 <- constc1
      }
      
      
      
      
      
      sol <- sd(Yorig) * c(ans0 / vsd)
      inter <- drop(mean(Yorig) - sd(Yorig) * (vm / vsd) %*% ans0)
      out$intercept <- drop(mean(Yorig) - sd(Yorig) * (vm / vsd) %*% ans0)
      out$betahat <- sd(Yorig) * c(ans0 / vsd)
      out$best <- best
      out$Yinit <- Yinit
      
      
      
      out$yhat <- Xorig %*% sol + inter
      
      out
   }





############################################################
#
#  FUNCTIONS FOR CALCULATING NON-EUCLIDEAN MEANS AND DISTANCES
#  OF COVARIANCE MATRICES
#
############################################################
# Log Euclidean mean: Sigma_L

#==================================================================================
estLogEuclid <- function(S, weights = 1) {
   M <- dim(S)[3]
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   sum <- S[, , 1] * 0
   for (j in 1:M) {
      eS <- eigen(S[, , j], symmetric = TRUE)
      sum <- sum +
         weights[j] * eS$vectors %*% diag(log(eS$values)) %*% t(eS$vectors) /
         sum(weights)
   }
   ans <- sum
   eL <- eigen(ans, symmetric = TRUE)
   eL$vectors %*% diag(exp(eL$values)) %*% t(eL$vectors)
}

#==================================================================================
estPowerEuclid <- function(S, weights = 1, alpha = 0.5) {
   M <- dim(S)[3]
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   sum <- S[, , 1] * 0
   for (j in 1:M) {
      eS <- eigen(S[, , j], symmetric = TRUE)
      sum <- sum +
         weights[j] * eS$vectors %*% diag(abs(eS$values) ** alpha) %*% t(eS$vectors) /
         sum(weights)
   }
   ans <- sum
   eL <- eigen(ans, symmetric = TRUE)
   eL$vectors %*% diag(abs(eL$values) ** (1 / alpha)) %*% t(eL$vectors)
}




# Riemannian (weighted mean) : Sigma_R
#==================================================================================
estLogRiem2 <- function(S, weights = 1) {
   M <- dim(S)[3]
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   check <- 9
   tau <- 1
   Hold <- 99999
   mu <- estLogEuclid(S, weights)
   while (check > 0.0000000001) {
      ev <- eigen(mu, symmetric = TRUE)
      logmu <- ev$vectors %*% diag(log(ev$values)) %*% t(ev$vectors)
      Hnew <- Re(Hessian2(S, mu, weights))
      logmunew <- logmu + tau * Hnew
      ev <- eigen(logmunew, symmetric = TRUE)
      mu <- ev$vectors %*% diag(exp(ev$values)) %*% t(ev$vectors)
      check <- Re(Enorm(Hold) - Enorm(Hnew))
      if (check < 0) {
         tau <- tau / 2
         check <- 999999
      }
      Hold <- Hnew
   }
   mu
}

# Hessian used in calculating Sigma_R
#==================================================================================
Hessian2 <- function(S, Sigma, weights = 1) {
   M <- dim(S)[3]
   k <- dim(S)[1]
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   ev0 <- eigen(Sigma, symmetric = TRUE)
   shalf <-
      ev0$vectors %*% (diag(sqrt((ev0$values)))) %*% t(ev0$vectors)
   sumit <- matrix(0, k, k)
   for (i in 1:(M)) {
      ev2 <- eigen(shalf %*% solve(S[, , i]) %*% shalf, symmetric = TRUE)
      sumit <-
         sumit + weights[i] * ev2$vectors %*% diag (log((ev2$values))) %*% t(ev2$vectors) /
         sum(weights)
   }
   - sumit
}

# Euclidean : Sigma_E
#==================================================================================
estEuclid <- function(S, weights = 1) {
   M <- dim(S)[3]
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   sum <- S[, , 1] * 0
   for (j in 1:M) {
      sum <- sum + S[, , j] * weights[j] / sum(weights)
   }
   sum
}

# Cholesky mean : Sigma_C
#==================================================================================
estCholesky <- function(S, weights = 1) {
   M <- dim(S)[3]
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   sum <- S[, , 1] * 0
   for (j in 1:M) {
      sum <- sum + t(chol(S[, , j])) * weights[j] / sum(weights)
   }
   cc <- sum
   cc %*% t(cc)
}

#==================================================================================
ild_estSS <- function(S, weights = 1) {
   M <- dim(S)[3]
   k <- dim(S)[1]
   H <- defh(k)
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   Q <- array(0, c(k + 1, k, M))
   for (j in 1:M) {
      Q[, , j] <- t(H) %*% (rootmat(S[, , j]))
   }
   ans <- procWGPA(
      Q,
      fixcovmatrix = diag(k + 1),
      scale = FALSE,
      reflect = TRUE,
      sampleweights = weights
   )
   H %*% ans$mshape %*% t(H %*% ans$mshape)
}

#==================================================================================
ild_estShape <- function(S, weights = 1) {
   M <- dim(S)[3]
   k <- dim(S)[1]
   H <- defh(k)
   if (length(weights) == 1) {
      weights <- rep(1, times = M)
   }
   Q <- array(0, c(k + 1, k, M))
   for (j in 1:M) {
      Q[, , j] <- t(H) %*% (rootmat(S[, , j]))
   }
   ans <- procWGPA(
      Q,
      fixcovmatrix = diag(k + 1),
      scale = TRUE,
      reflect = TRUE,
      sampleweights = weights
   )
   H %*% ans$mshape %*% t(H %*% ans$mshape)
}


#==================================================================================
estRiemLe <- function(S, weights) {
   M <- dim(S)[3]
   k <- dim(S)[1]
   if (M != 2)
      print("Sorry - Calculation not implemented for M>2 yet")
   if (M == 2) {
      P1 <- S[, , 1]
      P2 <- S[, , 2]
      detP1 <- prod(eigen(P1)$values)
      detP2 <- prod(eigen(P2)$values)
      P1 <- P1 / (detP1) ^ (1 / k)
      P2 <- P2 / (detP2) ^ (1 / k)
      P1inv <- solve(P1)
      P12sq <- P1inv %*% P2 %*% P2 %*% P1inv
      tem <- eigen(P12sq, symmetric = TRUE)
      A2 <- tem$vectors %*% diag(log(tem$values)) %*% t(tem$vectors)
      logPs2 <- weights[2] * A2
      tem2 <- eigen(logPs2, symmetric = TRUE)
      Ps2 <- tem2$vectors %*% diag(exp(tem2$values)) %*% t(tem2$vectors)
      P12s <- P1 %*% Ps2 %*% P1
      tem3 <- eigen(P12s, symmetric = TRUE)
      P12sA <-
         tem3$vectors %*% diag(sqrt(tem3$values)) %*% t(tem3$vectors)
      Ptildes <- (detP1 * (detP2 / detP1) ^ weights[2]) ^ (1 / k) * P12sA
      Ptildes
   }
}






##########distances#################################

#==================================================================================
distRiemPennec <- function(P1, P2) {
   eig <- eigen(P1, symmetric = TRUE)
   P1half <- eig$vectors %*% diag(sqrt(eig$values)) %*% t(eig$vectors)
   P1halfinv <- solve(P1half)
   AA <- P1halfinv %*% P2 %*% P1halfinv
   tem <- eigen(AA, symmetric = TRUE)
   A2 <- tem$vectors %*% diag(log(tem$values)) %*% t(tem$vectors)
   dd <- Enorm(A2)
   dd
}

#==================================================================================
distLogEuclidean <- function(P1, P2) {
   eig <- eigen(P1, symmetric = TRUE)
   logP1 <- eig$vectors %*% diag(log(eig$values)) %*% t(eig$vectors)
   tem <- eigen(P2, symmetric = TRUE)
   logP2 <- tem$vectors %*% diag(log(tem$values)) %*% t(tem$vectors)
   dd <- Enorm(logP1 - logP2)
   dd
}

#==================================================================================
distRiemannianLe <-
   function(P1, P2) {
      dd <- distRiemPennec(P1 %*% t(P1), P2 %*% t(P2)) / 2
      dd
   }


#==================================================================================
ild_distProcrustesSizeShape <-
   function (P1, P2)
   {
      H <- defh(dim(P1)[1])
      Q1 <- t(H) %*% rootmat(P1)
      Q2 <- t(H) %*% rootmat(P2)
      ans <- sqrt(
         centroid.size(Q1) ^ 2 + centroid.size(Q2) ^ 2 - 2 *
            centroid.size(Q1) * centroid.size(Q2) * cos(riemdist(Q1,
                                                                 Q2, reflect = TRUE))
      )
      ans
   }

#==================================================================================
ild_distProcrustesFull <- function(P1, P2) {
   H <- defh(dim(P1)[1])
   Q1 <- t(H) %*% rootmat(P1)
   Q2 <- t(H) %*% rootmat(P2)
   ans <- riemdist(Q1, Q2, reflect = TRUE)
   ans
}


#==================================================================================
distPowerEuclidean <- function(P1, P2, alpha = 1 / 2) {
   if (alpha != 0) {
      eS <- eigen(P1, symmetric = TRUE)
      Q1 <- eS$vectors %*% diag(abs(eS$values) ^ alpha) %*% t(eS$vectors)
      eS <- eigen(P2, symmetric = TRUE)
      Q2 <- eS$vectors %*% diag(abs(eS$values) ^ alpha) %*% t(eS$vectors)
      dd <- Enorm(Q1 - Q2) / abs(alpha)
   }
   if (alpha == 0) {
      dd <- distLogEuclidean(P1, P2)
   }
   dd
}

#==================================================================================
ild_distCholesky <- function(P1, P2) {
   H <- defh(dim(P1)[1])
   Q1 <- t(H) %*% t(chol(P1))
   Q2 <- t(H) %*% t(chol(P2))
   ans <- Enorm(Q1 - Q2)
   ans
}

#==================================================================================
distEuclidean <- function(P1, P2) {
   ans <- Enorm(P1 - P2)
   ans
}

##################

#==================================================================================
distcov <- function(S1,
                    S2 ,
                    method = "Riemannian",
                    alpha = 1 / 2) {
   if (method == "Procrustes") {
      dd <- distProcrustesSizeShape(S1, S2)
   }
   if (method == "ProcrustesShape") {
      dd <- distProcrustesFull(S1, S2)
   }
   if (method == "Riemannian") {
      dd <- distRiemPennec(S1, S2)
   }
   if (method == "Cholesky") {
      dd <- distCholesky(S1, S2)
   }
   if (method == "Power") {
      dd <- distPowerEuclidean(S1, S2, alpha)
   }
   if (method == "Euclidean") {
      dd <- distEuclidean(S1, S2)
   }
   if (method == "LogEuclidean") {
      dd <- distLogEuclidean(S1, S2)
   }
   if (method == "RiemannianLe") {
      dd <- distRiemannianLe(S1, S2)
   }
   dd
}

#==================================================================================
estcov <-
   function (S,
             method = "Riemannian",
             weights = 1,
             alpha = 1 / 2,
             MDSk = 2)
   {
      out <- list(
         mean = 0,
         sd = 0,
         pco = 0,
         eig = 0,
         dist = 0
      )
      M <- dim(S)[3]
      if (length(weights) == 1) {
         weights <- rep(1, times = M)
      }
      if (method == "Procrustes") {
         dd <- estSS(S, weights)
      }
      if (method == "ProcrustesShape") {
         dd <- estShape(S, weights)
      }
      if (method == "Riemannian") {
         dd <- estLogRiem2(S, weights)
      }
      if (method == "Cholesky") {
         dd <- estCholesky(S, weights)
      }
      if (method == "Power") {
         dd <- estPowerEuclid(S, weights, alpha)
      }
      if (method == "Euclidean") {
         dd <- estEuclid(S, weights)
      }
      if (method == "LogEuclidean") {
         dd <- estLogEuclid(S, weights)
      }
      if (method == "RiemannianLe") {
         dd <- estRiemLe(S, weights)
      }
      out$mean <- dd
      sum <- 0
      for (i in 1:M) {
         sum <-
            sum + weights[i] * distcov(S[, , i], dd, method = method) ^ 2 / sum(weights)
      }
      out$sd <- sqrt(sum)
      dist <- matrix(0, M, M)
      for (i in 2:M) {
         for (j in 1:(i - 1)) {
            dist[i, j] <- distcov(S[, , i], S[, , j], method = method)
            dist[j, i] <- dist[i, j]
         }
      }
      out$dist <- dist
      if (M > MDSk) {
         ans <-
            cmdscale(
               dist,
               k = MDSk,
               eig = TRUE,
               add = TRUE,
               x.ret = TRUE
            )
         out$pco <- ans$points
         out$eig <- ans$eig
         if (MDSk > 2) {
            shapes3d(out$pco[, 1:min(MDSk, 3)], axes3 = TRUE)
         }
         if (MDSk == 2) {
            plot(out$pco,
                 type = "n",
                 xlab = "MDS1",
                 ylab = "MDS2")
            text(out$pco[, 1], out$pco[, 2], 1:length(out$pco[, 1]))
         }
      }
      out
   }

rootmat <- function(P1) {
   eS <- eigen(P1, symmetric = TRUE)
   if (min(eS$values) < -0.001) {
      print("Not positive-semi definite")
   }
   else{
      Q1 <- eS$vectors %*% diag(sqrt(abs(eS$values))) %*% t(eS$vectors)
      Q1
   }
}



##########################

#==================================================================================
shapes.cva <- function(X ,
                       groups ,
                       scale = TRUE,
                       tangentcoords = "residual", 
                       ncv = 2) {
   g <- dim(table (groups))
   ans <- procGPA(X , tangentcoords=tangentcoords, scale = scale)
   
   if (scale == TRUE)
      pp <- (ans$k - 1) * ans$m - (ans$m * (ans$m - 1) / 2) - 1
   if (scale == FALSE)
      pp <- (ans$k - 1) * ans$m - (ans$m * (ans$m - 1) / 2)
   
   pracdim <- min(pp,  ans$n - g)
   out <- lda(ans$scores[, 1:pracdim] , groups)
   print((out))
   cv <- predict(out, dimen = ncv)$x
   if (dim(cv)[2] == 1) {
      cv <- cbind(cv, rnorm(dim(cv)[1]) / 1000)
   }
   if (ncv == 2) {
      eqscplot(cv,
               type = "n",
               xlab = "CV1",
               ylab = "CV2")
      text(cv, labels = groups)
   }
   if (ncv == 3) {
      shapes3d(cv, color = groups, axes3 = TRUE)
   }
   cv
}


#==================================================================================
groupstack <- function(A1,
                       A2,
                       A3 = 0,
                       A4 = 0,
                       A5 = 0,
                       A6 = 0,
                       A7 = 0,
                       A8 = 0) {
   out <- list(x = 0, groups = "")
   dat <- abind(A1, A2)
   group <- c(rep(1, times = dim(A1)[3]), rep(2, times = dim(A2)[3]))
   if (is.array(A3)) {
      dat <- abind(dat, A3)
      group <- c(group, rep(3, times = dim(A3)[3]))
      if (is.array(A4)) {
         dat <- abind(dat, A4)
         group <- c(group, rep(4, times = dim(A4)[3]))
         if (is.array(A5)) {
            dat <- abind(dat, A5)
            group <- c(group, rep(5, times = dim(A5)[3]))
            if (is.array(A6)) {
               dat <- abind(dat, A6)
               group <- c(group, rep(6, times = dim(A6)[3]))
               if (is.array(A7)) {
                  dat <- abind(dat, A7)
                  group <- c(group, rep(7, times = dim(A7)[3]))
                  if (is.array(A8)) {
                     dat <- abind(dat, A8)
                     group <- c(group, rep(8, times = dim(A8)[3]))
                  }
               }
            }
         }
      }
   }
   out$x <- dat
   out$groups <- group
   out
}





###########################

#==================================================================================
procdist <- function(x, y, type = "full", reflect = FALSE) {
   if (type == "full") {
      out <- sin(riemdist(x, y, reflect = reflect))
   }
   if (type == "partial") {
      out <- sqrt(2) * sqrt(abs(1 - cos(riemdist(x, y, reflect = reflect))))
   }
   if (type == "Riemannian") {
      out <- riemdist(x, y, reflect = reflect)
   }
   if (type == "sizeandshape") {
      out <- ssriemdist(x, y, reflect = reflect)
   }
   out
}



#==================================================================================
transformations <- function(Xrotated, Xoriginal) {
   # outputs the translations, rotations and
   # scalings for ordinary Procrustes rotation
   # of each individual in Xoriginal to the
   # Procrustes rotated individuals in Xrotated
   X1 <- Xrotated
   X2 <- Xoriginal
   n <- dim(X1)[3]
   m <- dim(X1)[2]
   translation <- matrix(0, m, n)
   scale <- rep(0, times = n)
   rotation <- array(0, c(m, m, n))
   for (i in 1:n) {
      translation[, i] <- -apply(X2[, , i] - X1[, , i], 2, mean)
      ans <- procOPA(X1[, , i], X2[, , i])
      scale[i] <- ans$s
      rotation[, , i] <- ans$R
   }
   out <- list(translation = 0,
               scale = 0,
               rotation = 0)
   out$translation <- translation
   out$scale <- scale
   out$rotation <- rotation
   out
}

#==================================================================================
iglogl <- function(x , lam, nlam) {
   gamma <- abs(x[1])
   alpha <- gamma / mean(1 / lam[1:nlam])
   ll <-
      -(gamma + 1) * sum(log(lam[1:nlam])) - alpha * sum (1 / lam[1:nlam]) +
      nlam * gamma * log(alpha) - nlam * lgamma (gamma)
   - ll
}

#==================================================================================
procWGPA <-
   function(x,
            fixcovmatrix = FALSE,
            initial = "Identity",
            maxiterations = 10,
            scale = TRUE,
            reflect = FALSE,
            prior = "Exponential",
            diagonal = TRUE,
            sampleweights = "Equal") {
      X <- x
      priorargument <- prior
      alpha <- "not estimated"
      gamma <- "not estimated"
      k <- dim(X)[1]
      n <- dim(X)[3]
      m <- dim(X)[2]
      
      if (initial[1] == "Identity") {
         Sigmak <- diag(k)
      }
      else{
         if (initial[1] == "Rawdata") {
            tol <- 0.0000000001
            if (m == 2) {
               Sigmak <- diag(diag(var(t(X[, 1, ]))) + diag(var(t(X[, 2, ])))) / 2 + tol
            }
            if (m == 3) {
               Sigmak <-
                  diag(diag(var(t(X[, 1, ]))) + diag(var(t(X[, 2, ]))) + diag(var(t(X[, 3, ])))) /
                  3 + tol
            }
         }
         else
         {
            Sigmak <- initial
         }
      }
      
      mu <- procGPA(X, scale = scale)$mshape
      
      
      
      #cat("Iteration 1 \n")
      
      if (fixcovmatrix[1] != FALSE) {
         Sigmak <- fixcovmatrix
      }
      
      ans <-
         procWGPA1(
            X,
            mu,
            metric = Sigmak,
            scale = scale,
            reflect = reflect,
            sampleweights = sampleweights
         )
      
      
      if ((maxiterations > 1) && (fixcovmatrix[1] == FALSE)) {
         ans0 <- ans
         dif <- 999999
         it <- 1
         while ((dif > 0.00001) && (it < maxiterations)) {
            it <- it + 1
            if (it == 2) {
               cat("Differences in norm of Sigma estimates... \n ")
            }
            
            if (prior[1] == "Identity") {
               prior <- diag(k)
            }
            
            if (prior[1] == "Inversegamma") {
               lam <- eigen(ans$Sigmak)$values
               nlam <- min(c(n * m - m - 3, k - 3))
               mu <- mean(1 / lam[1:(nlam)])
               alpha <- 1 / mu
               out <- nlm(iglogl,
                          p = c(1) ,
                          lam = lam,
                          nlam = nlam)
               #print(out)
               gamma <- abs(out$estimate[1])
               alpha <- gamma / mean(1 / lam[1:nlam])
               newmetric <-
                  n * m / (n * m + 2 * (1 + gamma)) * (ans$Sigmak + (2 * alpha / (n * m)) *
                                                          diag(k))
               #dif2<-999999
               #while (dif2> 0.000001){
               #old<-alpha
               #lam <- eigen(newmetric)$values
               #out <- nlm( iglogl, p=c(1) ,lam=lam, nlam=nlam)
               #gamma <- abs(out$estimate[1])
               #alpha<- gamma/ mean(1/lam[1:nlam])
               #newmetric <- n*m/(n*m+2*(1+gamma))*( ans$Sigmak + (2*alpha/(n*m))*diag(k) )
               #dif2<- abs(alpha- old)
               #print(dif2)
               #}
            }
            
            
            if (prior[1] == "Exponential") {
               lam <- eigen(ans$Sigmak)$values
               nlam <- min(c(n * m - m - 2, k - 2))
               mu <- mean(1 / lam[1:(nlam)])
               alpha <- 1 / mu
               gamma <- 1
               newmetric <-
                  n * m / (n * m + 2 * (2)) * (ans$Sigmak + (2 * alpha / (n * m)) * diag(k))
               #dif2<-999999
               #while (dif2> 0.000001){
               #old<-alpha
               #newmetric <- n*m/(n*m+2*(2))*( ans$Sigmak + (2*alpha/(n*m))*diag(k) )
               #lam <- eigen(newmetric)$values
               #mu <- mean(1/lam[1:( nlam)])
               #alpha <- 1/mu
               #newmetric <- n*m/(n*m+2*(2))*( ans$Sigmak + (2*alpha/(n*m))*diag(k) )
               #dif2<- abs(alpha- old)
               #}
            }
            
            if (is.double(prior[1])) {
               newmetric <- (ans$Sigmak + prior)
            }
            
            
            
            if (diagonal == TRUE) {
               newmetric <- diag(diag(newmetric))
            }
            
            if (fixcovmatrix[1] != FALSE) {
               newmetric <- fixcovmatrix
            }
            
            ans2 <-
               procWGPA1(
                  X,
                  ans$mshape,
                  metric = newmetric ,
                  scale = scale,
                  sampleweights = sampleweights
               )
            plotshapes(ans2$rotated)
            dif <- Enorm((ans$Sigmak - ans2$Sigmak))
            ans <- ans2
            cat(c(it, " ", dif, " \n"))
         }
      }
      if ((priorargument[1] == "Exponential") ||
          (priorargument[1] == "Inversegamma")) {
         ans$alpha <- alpha
         ans$gamma <- gamma
      }
      cat(" \n")
      ans
   }




#==================================================================================
procWGPA1 <- function(X,
                      mu,
                      metric = "Identity",
                      scale = TRUE,
                      reflect = FALSE,
                      sampleweights = "Equal") {
   k <- dim(X)[1]
   n <- dim(X)[3]
   m <- dim(X)[2]
   
   sum <- 0
   for (i in 1:n) {
      sum <- sum + centroid.size(X[, , i]) ** 2
   }
   size1 <- sqrt(sum)
   
   
   
   
   
   if (sampleweights[1] == "Equal") {
      sampleweights <- rep(1 / n, times = n)
   }
   
   
   if (length(sampleweights) != n) {
      cat("Sample weight vector not of correct length \n")
   }
   
   
   if (metric[1] == "Identity") {
      Sigmak <- diag(k)
   }
   else{
      Sigmak <- metric
   }
   
   eig <- eigen(Sigmak, symmetric = TRUE)
   
   Sighalf <-
      eig$vectors %*% diag (sqrt(abs(eig$values))) %*% t(eig$vectors)
   Siginvhalf <-
      eig$vectors %*% diag(1 / sqrt(abs(eig$values))) %*% t(eig$vectors)
   Siginv <- eig$vectors %*% diag (1 / (eig$values)) %*% t(eig$vectors)
   
   one <- matrix(rep(1, times = k), k, 1)
   
   Xstar <- X
   
   
   
   for (i in 1:n) {
      Xstar[, , i] <- Xstar[, , i] - one %*% t(one) %*% Siginv %*% Xstar[, , i] /
         c(t(one) %*% Siginv %*% one)
      Xstar[, , i] <- Siginvhalf %*% Xstar[, , i]
   }
   
   mu <- mu - one %*% t(one) %*% Siginv %*% mu / c(t(one) %*% Siginv %*% one)
   
   
   ans <- procGPA(Xstar, eigen2d = FALSE)
   ans2 <- ans
   
   dif3 <- 99999999
   while (dif3 > 0.00001) {
      for (i in 1:n) {
         old <- mu
         tem <-
            procOPA(Siginvhalf %*% mu ,
                    Xstar[, , i],
                    scale = scale,
                    reflect = reflect)
         Gammai <- tem$R
         betai <- tem$s
         #ci <- t(one)%*% Siginvhalf %*% X[,,i] %*% Gammai*betai/k
         #Yi <- Sighalf%*% ans$rotated[,,i] + Sighalf%*%one%*% ci
         #Zi <- Yi - one %*% t(one)%*% Siginv %*% Yi / c( t(one)%*%Siginv%*%one )
         
         Zi <- Sighalf %*% Xstar[, , i] %*% Gammai * betai
         ans2$rotated[, , i] <- Zi
      }
      
      sum2 <- 0
      for (i in 1:n) {
         sum2 <- sum2 + centroid.size(ans2$rotated[, , i]) ** 2
      }
      size2 <- sqrt(sum2)
      
      
      
      tem <- ans2$mshape * 0
      for (i in 1:n) {
         ans2$rotated[, , i] <- ans2$rotated[, , i] * size1 / size2
         tem <- tem + ans2$rotated[, , i] * sampleweights[i] / sum(sampleweights)
      }
      
      mu <- tem
      dif3 <- riemdist(old,  mu)
   }
   
   
   
   
   
   z <- ans2
   z$mshape <- tem
   
   tan <- z$rotated[, 1,] - z$mshape[, 1]
   for (i in 2:m) {
      tan <- rbind(tan, z$rotated[, i,] - z$mshape[, i])
   }
   pca <- prcomp1(t(tan))
   z$tan <- tan
   npc <- 0
   for (i in 1:length(pca$sdev)) {
      if (pca$sdev[i] > 1e-07) {
         npc <- npc + 1
      }
   }
   z$scores <- pca$x
   z$rawscores <- pca$x
   z$stdscores <- pca$x
   for (i in 1:npc) {
      z$stdscores[, i] <- pca$x[, i] / pca$sdev[i]
   }
   z$pcar <- pca$rotation
   z$pcasd <- pca$sdev
   z$percent <- z$pcasd ^ 2 / sum(z$pcasd ^ 2) * 100
   
   size <- rep(0, times = n)
   rho <- rep(0, times = n)
   x <- X
   size <- apply(x, 3, centroid.size)
   rho <- apply(x, 3, y <- function(x) {
      riemdist(x, z$mshape)
   })
   
   z$rho <- rho
   z$size <- size
   z$rmsrho <- sqrt(mean(rho ^ 2))
   z$rmsd1 <- sqrt(mean(sin(rho) ^ 2))
   
   z$k <- k
   z$m <- m
   z$n <- n
   
   
   tem <- matrix(0, k, k)
   
   for (i in 1:n) {
      tem <-
         tem +  (z$rotated[, , i] - z$mshape) %*% t((z$rotated[, , i] - z$mshape))
   }
   tem <- tem / (n * m)
   z$Sigmak <-  tem
   
   
   return(z)
}

#==================================================================================
testshapes <-
   function(A,
            B,
            resamples = 1000,
            replace = TRUE,
            scale = TRUE) {
      if (replace == TRUE) {
         out <- bootstraptest(A, B, resamples = resamples, scale = scale)
      }
      if (replace == FALSE) {
         out <- permutationtest(A, B, nperms = resamples, scale = scale)
      }
      out
   }

#==================================================================================
testmeanshapes <-
   function(A,
            B,
            resamples = 1000,
            replace = FALSE,
            scale = TRUE) {
      if (replace == TRUE) {
         out <- bootstraptest(A, B, resamples = resamples, scale = scale)
      }
      if (replace == FALSE) {
         out <- permutationtest(A, B, nperms = resamples, scale = scale)
      }
      
      
      if (resamples > 0) {
         aa <- list(
            H = 0,
            H.pvalue = 0,
            H.table.pvalue = 0,
            G = 0,
            G.pvalue = 0,
            G.table.pvalue = 0,
            J = 0,
            J.pvalue = 0,
            J.table.pvalue = 0
         )
         
         aa$H <- out$H
         aa$H.pvalue <- out$H.pvalue
         aa$H.table.pvalue <- out$H.table.pvalue
         aa$G <- out$G
         aa$G.pvalue <- out$G.pvalue
         aa$G.table.pvalue <- out$G.table.pvalue
         aa$J <- out$J
         aa$J.pvalue <- out$J.pvalue
         aa$J.table.pvalue <- out$J.table.pvalue
         
      }
      
      if (resamples == 0) {
         aa <- list(
            H = 0,
            H.table.pvalue = 0,
            G = 0,
            G.table.pvalue = 0,
            J = 0,
            J.table.pvalue = 0
         )
         
         
         aa$H <- out$H
         aa$H.table.pvalue <- out$H.table.pvalue
         aa$G <- out$G
         aa$G.table.pvalue <- out$G.table.pvalue
         aa$J <- out$J
         aa$J.table.pvalue <- out$J.table.pvalue
         
      }
      
      aa
   }


#==================================================================================
permutationtest2 <- function (A, B, nperms = 1000, scale = scale)
{
   A1 <- A
   A2 <- B
   mdim <- dim(A1)[2]
   B <- nperms
   nsam1 <- dim(A1)[3]
   nsam2 <- dim(A2)[3]
   pool <-
      procGPA(
         abind (A1, A2) ,
         scale = scale,
         tangentcoords = "partial",
         pcaoutput = FALSE
      )
   
   tempool <- pool
   for (i in 1:(nsam1 + nsam2)) {
      tempool$tan[, i] <- pool$tan[, i] / Enorm(pool$tan[, i]) * pool$rho[i]
   }
   pool <- tempool
   
   permpool <- pool
   Gtem <- Goodall(pool, nsam1, nsam2)
   Htem <- Hotelling(pool, nsam1, nsam2)
   Jtem <- James(pool, nsam1, nsam2, table = TRUE)
   Ltem <- Lambdamin(pool, nsam1, nsam2)
   Gumc <- Gtem$F
   Humc <- Htem$F
   Jumc <- Jtem$Tsq
   Lumc <- Ltem$lambdamin
   Gtabpval <- Gtem$pval
   Htabpval <- Htem$pval
   Jtabpval <- Jtem$pval
   Ltabpval <- Ltem$pval
   if (B > 0) {
      Apool <- array(0, c(dim(A1)[1], dim(A1)[2], dim(A1)[3] +
                             dim(A2)[3]))
      Apool[, , 1:nsam1] <- A1
      Apool[, , (nsam1 + 1):(nsam1 + nsam2)] <- A2
      out <-
         list(
            H = 0,
            H.pvalue = 0,
            H.table.pvalue = 0,
            J = 0,
            J.pvalue = 0,
            J.table.pvalue = 0,
            G = 0,
            G.pvalue = 0,
            G.table.pvalue = 0
         )
      Gu <- rep(0, times = B)
      Hu <- rep(0, times = B)
      Ju <- rep(0, times = B)
      Lu <- rep(0, times = B)
      cat("Permutations - sampling without replacement: ")
      cat(c("No of permutations = ", B, "\n"))
      for (i in 1:B) {
         if (i / 100 == trunc(i / 100)) {
            cat(c(i, " "))
         }
         select <- sample(1:(nsam1 + nsam2))
         permpool$tan <- pool$tan[, select]
         Gu[i] <- Goodall(permpool, nsam1, nsam2)$F
         Hu[i] <- Hotelling(permpool, nsam1, nsam2)$F
         Ju[i] <- James(permpool, nsam1, nsam2)$Tsq
         Lu[i] <-
            Lambdamin(permpool, nsam1, nsam2)$lambdamin
      }
      Gu <- sort(Gu)
      numbig <- length(Gu[Gumc < Gu])
      pvalG <- (1 + numbig) / (B + 1)
      Lu <- sort(Lu)
      numbig <- length(Lu[Lumc < Lu])
      pvalL <- (1 + numbig) / (B + 1)
      Ju <- sort(Ju)
      numbig <- length(Ju[Jumc < Ju])
      pvalJ <- (1 + numbig) / (B + 1)
      Hu <- sort(Hu)
      numbig <- length(Hu[Humc < Hu])
      pvalH <- (1 + numbig) / (B + 1)
      cat(" \n")
      out$Hu <- Hu
      out$Ju <- Ju
      out$Gu <- Gu
      out$Lu <- Lu
      out$H <- Humc
      out$H.pvalue <- pvalH
      out$H.table.pvalue <- Htabpval
      out$J <- Jumc
      out$J.pvalue <- pvalJ
      out$J.table.pvalue <- Jtabpval
      out$G <- Gumc
      out$G.pvalue <- pvalG
      out$G.table.pvalue <- Gtabpval
      out$L <- Lumc
      out$L.pvalue <- pvalL
      out$L.table.pvalue <- Ltabpval
   }
   if (B == 0) {
      out <- list(
         H = 0,
         H.table.pvalue = 0,
         G = 0,
         G.table.pvalue = 0
      )
      out$H <- Humc
      out$H.table.pvalue <- Htabpval
      out$J <- Jumc
      out$J.table.pvalue <- Jtabpval
      out$G <- Gumc
      out$G.table.pvalue <- Gtabpval
      out$L <- Lumc
      out$L.table.pvalue <- Ltabpval
   }
   out
}

#==================================================================================
bootstraptest <- function (A,
                           B,
                           resamples = 200,
                           scale = TRUE)
{
   A1 <- A
   A2 <- B
   mdim <- dim(A1)[2]
   B <- resamples
   nsam1 <- dim(A1)[3]
   nsam2 <- dim(A2)[3]
   pool <-
      procGPA(
         abind (A1, A2) ,
         scale = scale ,
         tangentcoords = "partial",
         pcaoutput = FALSE
      )
   tempool <- pool
   for (i in 1:(nsam1 + nsam2)) {
      tempool$tan[, i] <- pool$tan[, i] / Enorm(pool$tan[, i]) * pool$rho[i]
   }
   pool <- tempool
   bootpool <- pool
   Gtem <- Goodall(pool, nsam1, nsam2)
   Htem <- Hotelling(pool, nsam1, nsam2)
   Jtem <- James(pool, nsam1, nsam2, table = TRUE)
   Ltem <- Lambdamin(pool, nsam1, nsam2)
   Gumc <- Gtem$F
   Humc <- Htem$F
   Jumc <- Jtem$Tsq
   Lumc <- Ltem$lambdamin
   Gtabpval <- Gtem$pval
   Htabpval <- Htem$pval
   Jtabpval <- Jtem$pval
   Ltabpval <- Ltem$pval
   if (B > 0) {
      Apool <- array(0, c(dim(A1)[1], dim(A1)[2], dim(A1)[3] +
                             dim(A2)[3]))
      Apool[, , 1:nsam1] <- A1
      Apool[, , (nsam1 + 1):(nsam1 + nsam2)] <- A2
      out <-
         list(
            H = 0,
            H.pvalue = 0,
            H.table.pvalue = 0,
            J = 0,
            J.pvalue = 0,
            J.table.pvalue = 0,
            G = 0,
            G.pvalue = 0,
            G.table.pvalue = 0
         )
      Gu <- rep(0, times = B)
      Hu <- rep(0, times = B)
      Ju <- rep(0, times = B)
      Lu <- rep(0, times = B)
      pool2 <- pool
      pool2$tan[, 1:nsam1] <-
         pool$tan[, 1:nsam1] - apply(pool$tan[, 1:nsam1], 1, mean)
      pool2$tan[, (nsam1 + 1):(nsam1 + nsam2)] <-
         pool$tan[, (nsam1 + 1):(nsam1 + nsam2)] -
         apply(pool$tan[, (nsam1 + 1):(nsam1 + nsam2)], 1, mean)
      
      
      cat("Bootstrap - sampling with replacement within each group under H0: ")
      cat(c("No of resamples = ", B, "\n"))
      for (i in 1:B) {
         if (i / 100 == trunc(i / 100)) {
            cat(c(i, " "))
         }
         select1 <- sample(1:nsam1, replace = TRUE)
         select2 <- sample((nsam1 + 1):(nsam1 + nsam2), replace = TRUE)
         bootpool$tan <- pool2$tan[, c(select1, select2)]
         Gu[i] <- Goodall(bootpool, nsam1, nsam2)$F
         Hu[i] <- Hotelling(bootpool, nsam1, nsam2)$F
         Ju[i] <- James(bootpool, nsam1, nsam2)$Tsq
         Lu[i] <- Lambdamin(bootpool, nsam1, nsam2)$lambdamin
      }
      Gu <- sort(Gu)
      numbig <- length(Gu[Gumc < Gu])
      pvalG <- (1 + numbig) / (B + 1)
      Ju <- sort(Ju)
      numbig <- length(Ju[Jumc < Ju])
      pvalJ <- (1 + numbig) / (B + 1)
      Hu <- sort(Hu)
      numbig <- length(Hu[Humc < Hu])
      pvalH <- (1 + numbig) / (B + 1)
      numbig <- length(Lu[Lumc < Lu])
      pvalL <- (1 + numbig) / (B + 1)
      cat(" \n")
      out$Hu <- Hu
      out$Ju <- Ju
      out$Gu <- Gu
      out$Lu <- Lu
      out$H <- Humc
      out$H.pvalue <- pvalH
      out$H.table.pvalue <- Htabpval
      out$J <- Jumc
      out$J.pvalue <- pvalJ
      out$J.table.pvalue <- Jtabpval
      out$G <- Gumc
      out$G.pvalue <- pvalG
      out$G.table.pvalue <- Gtabpval
      out$L <- Lumc
      out$L.pvalue <- pvalL
      out$L.table.pvalue <- Ltabpval
   }
   if (B == 0) {
      out <-
         list(
            H = 0,
            H.table.pvalue = 0,
            G = 0,
            G.table.pvalue = 0,
            J = 0,
            J.table.pvalue = 0
         )
      out$H <- Humc
      out$H.table.pvalue <- Htabpval
      out$J <- Jumc
      out$J.table.pvalue <- Jtabpval
      out$G <- Gumc
      out$G.table.pvalue <- Gtabpval
      out$L <- Lumc
      out$L.table.pvalue <- Ltabpval
   }
   out
}

#==================================================================================
Lambdamin <-
   function (pool, n1, n2, p = 0)
   {
      censiz <- centroid.size(pool$mshape)
      tan1 <- pool$tan[, 1:n1]
      tan2 <- pool$tan[, (n1 + 1):(n1 + n2)]
      kt <- dim(tan1)[1]
      n <- n1 + n2
      k <- pool$k
      m <- pool$m
      if (p == 0) {
         p <- min(k * m - (m * (m - 1)) / 2 - 1 - m, n1 + n2 - 2)
      }
      
      
      HH <- diag(k)
      
      
      mu1 <- pool$mshape
      
      if (dim(tan1)[1] == k * m - m) {
         HH <- defh(k - 1)
         
         mu1 <- preshape(pool$mshape)
      }
      
      
      if (m == 2) {
         mu <- c(mu1[, 1], mu1[, 2])
         
      }
      if (m == 3) {
         mu <- c(mu1[, 1], mu1[, 2], mu1[,
                                         3])
         
      }
      
      
      
      
      
      dd <- kt
      X1 <- tan1 * 0
      X2 <- tan2 * 0
      S1 <- matrix(0, dd, dd)
      S2 <- matrix(0, dd, dd)
      for (i in 1:n1) {
         X1[, i] <- (mu + tan1[, i]) / Enorm(mu + tan1[, i])
         S1 <- S1 + X1[, i] %*% t(X1[, i])
      }
      for (i in 1:n2) {
         X2[, i] <- (mu + tan2[, i]) / Enorm(mu + tan2[, i])
         S2 <- S2 + X2[, i] %*% t(X2[, i])
      }
      
      sumx1 <- 0
      sumx2 <- 0
      for (i in 1:n1) {
         sumx1 <- sumx1 + X1[, i]
      }
      for (i in 1:n2) {
         sumx2 <- sumx2 + X2[, i]
      }
      
      
      sum1 <- apply(X1, 1, sum)
      sum2 <- apply(X2, 1, sum)
      mean1 <- sum1 / Enorm(sum1)
      mean2 <- sum2 / Enorm(sum2)
      
      
      bb1 <- mean1[1:(dd - 1)]
      cc1 <- mean1[dd]
      bb2 <- mean2[1:(dd - 1)]
      cc2 <- mean2[dd]
      A1 <- cc1 / abs(cc1) * diag(dd - 1) - cc1 / (abs(cc1) + cc1 ^ 2) *
         bb1 %*% t(bb1)
      M1 <- cbind(A1,-bb1)
      A1 <- cc2 / abs(cc2) * diag(dd - 1) - cc2 / (abs(cc2) + cc2 ^ 2) *
         bb2 %*% t(bb2)
      M2 <- cbind(A1,-bb2)
      G1 <- matrix(0, dd - 1, dd - 1)
      G2 <- matrix(0, dd - 1, dd - 1)
      for (iu in 1:(dd - 1)) {
         for (iv in iu:(dd - 1)) {
            G1[iu, iv] <- G1[iu, iv] + t((t(M1))[, iu]) %*% S1 %*%
               (t(M1))[, iv]
            G1[iv, iu] <- G1[iu, iv]
            G2[iu, iv] <- G2[iu, iv] + t((t(M2))[, iu]) %*% S2 %*%
               (t(M2))[, iv]
            G2[iv, iu] <- G2[iu, iv]
         }
      }
      G1 <- G1 / n1 / Enorm(sumx1 / n1) ^ 2
      G2 <- G2 / n2 / Enorm(sumx2 / n2) ^ 2
      #   eva1 <- eigen(G1, symmetric = TRUE, EISPACK = TRUE)
      eva1 <- eigen(G1, symmetric = TRUE)
      pcar1 <- eva1$vectors[, 1:p]
      pcasd1 <- sqrt(abs(eva1$values[1:p]))
      #    eva2 <- eigen(G2, symmetric = TRUE, EISPACK = TRUE)
      eva2 <- eigen(G2, symmetric = TRUE)
      pcar2 <- eva2$vectors[, 1:p]
      pcasd2 <- sqrt(abs(eva2$values[1:p]))
      if ((pcasd1[p] < 1e-06) || (pcasd2[p] < 1e-06)) {
         offset <- 1e-06
         cat("*")
         pcasd1 <- sqrt(pcasd1 ^ 2 + offset)
         pcasd2 <- sqrt(pcasd2 ^ 2 + offset)
      }
      Ahat1 <-
         n1 * t(M1) %*% (pcar1 %*% diag(1 / pcasd1 ^ 2) %*% t(pcar1)) %*%
         M1
      Ahat2 <-
         n2 * t(M2) %*% (pcar2 %*% diag(1 / pcasd2 ^ 2) %*% t(pcar2)) %*%
         M2
      
      
      Ahat <- (Ahat1 +  Ahat2)
      #    eva <- eigen(Ahat, symmetric = TRUE, EISPACK = TRUE)
      eva <- eigen(Ahat, symmetric = TRUE)
      
      
      
      lambdamin <- eva$values[p + 1]
      pval <- 1 - pchisq(lambdamin, p)
      #print(lambdamin)
      #print(pval)
      z <- list()
      z$pval <- pval
      z$df <- p
      z$lambdamin <- lambdamin
      return(z)
   }

#==================================================================================
Goodall <- function(pool , n1, n2, p = 0) {
   tan1 <- pool$tan[, 1:n1]
   tan2 <- pool$tan[, (n1 + 1):(n1 + n2)]
   kt <- dim(tan1)[1]
   n <- n1 + n2
   k <- pool$k
   m <- pool$m
   if (p == 0) {
      p <- min(k * m - (m * (m - 1)) / 2 - 1 - m, n1 + n2 - 2)
   }
   top <-  Enorm(apply(tan1, 1, mean) - apply(tan2, 1, mean)) ** 2
   bot <-
      sum(diag(var(t(tan1)))) * (n1 - 1) +  sum(diag(var(t(tan2)))) * (n2 - 1)
   Fstat <- ((n1 + n2 - 2) / (1 / n1 + 1 / n2) * top) / bot
   pval <- 1 - pf(Fstat, p, (n1 + n2 - 2) * p)
   z <- list()
   z$F <- Fstat
   z$pval <- pval
   z$df1 <- p
   z$df2 <- (n1 + n2 - 2) * p
   return(z)
}

#==================================================================================
Hotelling <- function(pool , n1, n2, p = 0) {
   tan1 <- pool$tan[, 1:n1]
   tan2 <- pool$tan[, (n1 + 1):(n1 + n2)]
   kt <- dim(tan1)[1]
   n <- n1 + n2
   k <- pool$k
   m <- pool$m
   S1 <- var(t(tan1))
   S2 <- var(t(tan2))
   Sw <- ((n1 - 1) * S1 + (n2 - 1) * S2) / (n1 + n2 - 2)
   if (p == 0) {
      p <- min(k * m - (m * (m - 1)) / 2 - 1 - m, n1 + n2 - 2)
   }
   #    eva <- eigen(Sw, symmetric = TRUE,EISPACK=TRUE)
   eva <- eigen(Sw, symmetric = TRUE)
   pcar <- eva$vectors[, 1:p]
   pcasd <- sqrt(abs(eva$values[1:p]))
   if (pcasd[p] < 1e-06) {
      offset <- 1e-06
      cat("*")
      pcasd <- sqrt(pcasd ^ 2 + offset)
   }
   lam <- rep(0, times =  kt)
   lam[1:p] <- 1 / pcasd ^ 2
   Suinv <- eva$vectors %*% diag(lam) %*% t(eva$vectors)
   pcax <- t(pool$tan) %*% pcar
   one1 <- matrix(1 / n1, n1, 1)
   one2 <- matrix(1 / n2, n2, 1)
   oneone <- rbind(one1,-one2)
   vbar <- pool$tan %*% oneone
   scores1 <- matrix(vbar, 1, kt) %*% pcar
   scores <- scores1 / pcasd
   F.partition <- ((scores[1:p] ^ 2) * (n1 * n2 * (n1 + n2 - p -
                                                      1))) / ((n1 + n2) * (n1 + n2 - 2) * p)
   FF <- sum(F.partition)
   pval <- 1 - pf(FF, p, (n1 + n2 - p - 1))
   z <- list()
   z$F.partition <- F.partition
   z$F <- FF
   z$pval <- pval
   z$df1 <- p
   z$T.df1 <- p
   z$df2 <- (n1 + n2 - p - 1)
   mm <- n - 2
   z$T.df2 <- mm
   z$Tsq <- FF * (n1 + n2) * (n1 + n2 - 2) * p / (n1 * n2) / (n1 +
                                                                 n2 - p - 1)
   z$Tsq.partition <- F.partition * (n1 + n2) * (n1 + n2 - 2) *
      p / (n1 * n2) / (n1 + n2 - p - 1)
   return(z)
}

James <- function(pool ,
                  n1,
                  n2,
                  p = 0,
                  table = FALSE) {
   tan1 <- pool$tan[, 1:n1]
   tan2 <- pool$tan[, (n1 + 1):(n1 + n2)]
   kt <- dim(tan1)[1]
   n <- n1 + n2
   k <- pool$k
   m <- pool$m
   S1 <- var(t(tan1))
   S2 <- var(t(tan2))
   Sw <- S1 / n1 + S2 / n2
   if (p == 0) {
      p <- min(k * m - (m * (m - 1)) / 2 - 1 - m, n1 + n2 - 2)
   }
   #    eva <- eigen(Sw, symmetric = TRUE,EISPACK=TRUE)
   eva <- eigen(Sw, symmetric = TRUE)
   pcar <- eva$vectors[, 1:p]
   pcasd <- sqrt(abs(eva$values[1:p]))
   if (pcasd[p] < 1e-06) {
      offset <- 1e-06
      cat("*")
      pcasd <- sqrt(pcasd ^ 2 + offset)
   }
   lam <- rep(0, times =  kt)
   lam[1:p] <- 1 / pcasd ^ 2
   Suinv <- eva$vectors %*% diag(lam) %*% t(eva$vectors)
   pcax <- t(pool$tan) %*% pcar
   one1 <- matrix(1 / n1, n1, 1)
   one2 <- matrix(1 / n2, n2, 1)
   oneone <- rbind(one1,-one2)
   vbar <- pool$tan %*% oneone
   #    scores1 <- matrix(vbar, 1, kt ) %*% pcar
   #    scores <- scores1/pcasd
   #    F.partition <- ((scores[1:p]^2) * (n1 * n2 * (n1 + n2 - p -
   #        1)))/((n1 + n2) * (n1 + n2 - 2) * p)
   #    FF <- sum(F.partition)
   #    pval <- 1 - pf(FF, p, (n1 + n2 - p - 1))
   #########
   #        ginvSw<- pcar%*%diag(1/pcasd**2)%*%t(pcar)
   ginvSw <- Suinv
   pval = 0
   T1 <- sum(diag((ginvSw %*% S1 / n1)))
   T2 <- sum(diag((ginvSw %*% S2 / n2)))
   T1sq <- sum(diag(((ginvSw %*% S1 / n1) %*% ginvSw %*% S1 / n1)))
   T2sq <- sum(diag(((ginvSw %*% S2 / n2) %*% ginvSw %*% S2 / n2)))
   
   Tsq <- (t(vbar) %*% (ginvSw) %*% vbar)[1, 1]
   if (table == TRUE) {
      AA <- 1 + 1 / (2 * p) * (T1 ** 2 / (n1 - 1) + T2 ** 2 / (n2 - 1))
      BB <-
         1 / (p * (p + 2)) * ((T1 ** 2 / 2 + T1sq) / (n1 - 1) + (T2 ** 2 / 2 + T2sq) /
                                 (n2 - 1))
      kk <- rep(0, times = 1000)
      for (i in 0:999) {
         alphai <- i / 1000
         kk[i + 1] <- qchisq(alphai, df = p) * (AA + BB * qchisq(alphai, df = p))
      }
      pval <- 1 - max(c(1:1000)[kk < Tsq]) / 1000
   }
   #######
   z <- list()
   z$pval <- pval
   
   z$Tsq <- Tsq
   
   return(z)
}

#==================================================================================
tpsgrid <-
   function (TT,
             YY,
             xbegin = -999,
             ybegin = -999,
             xwidth = -999,
             opt = 1,
             ext = 0.1,
             ngrid = 22,
             cex = 1,
             pch = 20,
             col = 2,
             zslice = 0,
             mag = 1,
             axes3 = FALSE)
   {
      k <- nrow(TT)
      m <- dim(TT)[2]
      
      YY <- TT + (YY - TT) * mag
      
      bb <- array(TT, c(dim(TT), 1))
      aa <- defplotsize2(bb)
      
      if (xwidth == -999) {
         xwidth <- aa$width
      }
      if (xbegin == -999) {
         xbegin <- aa$xl
      }
      if (ybegin == -999) {
         ybegin <- aa$yl
      }
      
      
      if (m == 3) {
         zup <- max(TT[, 3])
         zlo <- min(TT[, 3])
         
         zpos <- zslice
         for (ii in 1:length(zslice)) {
            zpos[ii] <- (zup + zlo) / 2 + (zup - zlo) / 2 * zslice[ii]
         }
      }
      
      
      
      xstart <- xbegin
      ystart <- ybegin
      ngrid <- trunc(ngrid / 2) * 2
      kx <- ngrid
      ky <- ngrid - 1
      l <- kx * ky
      step <- xwidth / (kx - 1)
      r <- 0
      X <- rep(0, times = kx)
      Y2 <- rep(0, times = ky)
      for (p in 1:kx) {
         ystart <- ybegin
         xstart <- xstart + step
         for (q in 1:ky) {
            ystart <- ystart + step
            r <- r + 1
            X[r] <- xstart
            Y2[r] <- ystart
         }
      }
      
      TPS <- bendingenergy(TT)
      gamma11 <- TPS$gamma11
      gamma21 <- TPS$gamma21
      gamma31 <- TPS$gamma31
      W <- gamma11 %*% YY
      ta <- t(gamma21 %*% YY)
      B <- gamma31 %*% YY
      WtY <- t(W) %*% YY
      trace <- c(0)
      for (i in 1:m) {
         trace <- trace + WtY[i, i]
      }
      benergy <- 16 * pi * trace
      if (m == 3) {
         benergy <- 8 * pi * trace
      }
      l <- kx * ky
      phi <- matrix(0, l, m)
      s <- matrix(0, k, 1)
      
      
      for (islice in 1:length(zslice)) {
         if (m == 3) {
            refc <- matrix(c(X, Y2, rep(zpos[islice], times = kx * ky)), kx * ky, m)
         }
         if (m == 2) {
            refc <- matrix(c(X, Y2), kx * ky, m)
         }
         
         
         for (i in 1:l) {
            s <- matrix(0, k, 1)
            for (im in 1:k) {
               s[im,] <- sigmacov(refc[i,] - TT[im,])
            }
            phi[i,] <- ta + t(B) %*% refc[i,] + t(W) %*% s
         }
         
         if (m == 3) {
            if (opt == 2) {
               shapes3d(TT,
                        color = 2,
                        axes3 = axes3,
                        rglopen = FALSE)
               shapes3d(YY, color = 4, rglopen = FALSE)
               for (i in 1:k) {
                  lines3d(rbind(TT[i, ], YY[i, ]), col = 1)
               }
               for (j in 1:kx) {
                  lines3d(refc[((j - 1) * ky + 1):(ky * j) , ], color = 6)
               }
               for (j in 1:ky) {
                  lines3d(refc[(0:(kx - 1) * ky) + j , ], color = 6)
               }
            }
            
            shapes3d(TT,
                     color = 2,
                     axes3 = axes3,
                     rglopen = FALSE)
            shapes3d(YY, color = 4, rglopen = FALSE)
            for (i in 1:k) {
               lines3d(rbind(TT[i, ], YY[i, ]), col = 1)
            }
            for (j in 1:kx) {
               lines3d(phi[((j - 1) * ky + 1):(ky * j) , ], color = 3)
            }
            for (j in 1:ky) {
               lines3d(phi[(0:(kx - 1) * ky) + j , ], color = 3)
            }
         }
      }
      
      if (m == 2) {
         par(pty = "s")
         if (opt == 2) {
            par(mfrow = c(1, 2))
            order <- linegrid(refc, kx, ky)
            plot(
               order[1:l, 1],
               order[1:l, 2],
               type = "l",
               xlim = c(xbegin -
                           xwidth * ext, xbegin + xwidth * (1 + ext)),
               ylim = c(
                  ybegin -
                     (xwidth * ky) / kx * ext,
                  ybegin + (xwidth * ky) / kx *
                     (1 + ext)
               ),
               xlab = " ",
               ylab = " "
            )
            lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l),
                                                   2], type = "l")
            points(TT,
                   cex = cex,
                   pch = pch,
                   col = col)
         }
         order <- linegrid(phi, kx, ky)
         plot(
            order[1:l, 1],
            order[1:l, 2],
            type = "l",
            xlim = c(xbegin -
                        xwidth * ext, xbegin + xwidth * (1 + ext)),
            ylim = c(ybegin -
                        (xwidth * ext * ky) / kx, ybegin + (xwidth * (1 + ext) *
                                                               ky) / kx),
            xlab = " ",
            ylab = " "
         )
         lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l), 2],
               type = "l")
         points(YY,
                cex = cex,
                pch = pch,
                col = col + 1)
         points(TT,
                cex = cex,
                pch = pch,
                col = col)
         for (i in 1:(k)) {
            arrows(
               TT[i, 1],
               TT[i, 2] ,
               YY[i, 1],
               YY[i, 2] ,
               col = col + 2,
               length = 0.1,
               angle = 20
            )
         }
      }
      
      
   }
#

#==================================================================================
rotatexyz <- function(x, thetax, thetay, thetaz) {
   thetax <- thetax / 180 * pi
   thetay <- thetay / 180 * pi
   thetaz <- thetaz / 180 * pi
   
   Rx <-
      matrix(c(
         1,
         0,
         0,
         0,
         cos(thetax),
         sin(thetax),
         0,
         -sin(thetax),
         cos(thetax)
      ), 3, 3)
   Ry <-
      matrix(c(
         cos(thetay),
         0,
         sin(thetay),
         0,
         1,
         0,
         -sin(thetay),
         0,
         cos(thetay)
      ), 3, 3)
   Rz <-
      matrix(c(
         cos(thetaz),
         sin(thetaz),
         0,
         -sin(thetaz),
         cos(thetaz),
         0,
         0,
         0,
         1
      ), 3, 3)
   
   
   y <- x
   n <- dim(x)[3]
   for (i in 1:n) {
      y[, , i] <- x[, , i] %*% Rx %*% Ry %*% Rz
   }
   y
}

#==================================================================================
rigidbody <-
   function(X,
            transx = 0,
            transy = 0,
            transz = 0,
            thetax = 0,
            thetay = 0,
            thetaz = 0) {
      if (is.matrix(X)) {
         X <- array(X, c(dim(X), 1))
      }
      m <- dim(X)[2]
      n <- dim(X)[3]
      Y <- X
      if (m == 2) {
         #xx<-as.3d(X)
         if (dim(X)[3] < 2) {
            xx <- array(as.3d(X), dim = c(nrow(X), 3, 1))
         } else{
            xx <- as.3d(X)
         }
         for (i in 1:n) {
            for (j in 1:m) {
               xx[j, , i] <- xx[j, , i] - c(transx, transy, transz)
            }
         }
         yy <- rotatexyz(xx, thetax, thetay, thetaz)
         Y <- yy
         if ((sum(abs(yy[, 3, ]))) < 0.000000001) {
            Y <- yy[, 1:2, ]
         }
      }
      if (m == 3) {
         for (i in 1:n) {
            for (j in 1:m) {
               X[j, , i] <- X[j, , i] - c(transx, transy, transz)
            }
         }
         Y <- rotatexyz(X, thetax, thetay, thetaz)
      }
      Y
   }




#==================================================================================
as.3d <- function(X) {
   k <- dim(X)[1]
   
   if (is.matrix(X)) {
      X <- array(X, c(dim(X), 1))
   }
   
   
   n <- dim(X)[3]
   if (dim(X)[2] != 2) {
      print("not 2 dimensional!")
   }
   Y <- array(0, c(k, 3, n))
   Y[, 1:2, ] <- X
   
   if (n == 1) {
      Y <- Y[, , 1]
   }
   
   Y
}

#==================================================================================
abind <- function(X1 , X2) {
   k <- dim(X1)[1]
   m <- dim(X1)[2]
   
   if (is.matrix(X1)) {
      tem <- array(0, c(k, m, 1))
      tem[, , 1] <- X1
      X1 <- tem
   }
   
   if (is.matrix(X2)) {
      tem <- array(0, c(k, m, 1))
      tem[, , 1] <- X2
      X2 <- tem
   }
   
   n1 <- dim(X1)[3]
   n2 <- dim(X2)[3]
   Y <- array(0, c(k, m, n1 + n2))
   Y[, , 1:n1] <- X1
   Y[, , (n1 + 1):(n1 + n2)] <- X2
   Y
}

#==================================================================================
shapes3d <-
   function(x,
            loop = 0,
            type = "p",
            color = 2,
            joinline = c(1:1),
            axes3 = FALSE,
            rglopen = TRUE) {
      if (is.matrix(x)) {
         xt <- array(0, c(dim(x), 1))
         xt[, , 1] <- x
         x <- xt
      }
      
      if (is.array(x) == FALSE) {
         cat("Data not in right format : require an array \n")
      }
      
      if (is.array(x) == TRUE) {
         if (rglopen) {
            open3d()
         }
         
         
         
         if (dim(x)[2] == 2) {
            x <- as.3d(x)
         }
         
         
         
         if (loop == 0) {
            k <- dim(x)[1]
            sz <- centroid.size(x[, , 1]) / sqrt(k) / 30
            plotshapes3d(
               x,
               type = type,
               color = color,
               size = sz,
               joinline = joinline
            )
            if (axes3) {
               axes3d(color = "black")
               title3d(
                  xlab = "x",
                  ylab = "y",
                  zlab = "z",
                  color = "black"
               )
            }
         }
         if (loop > 0) {
            for (i in 1:loop) {
               plotshapestime3d(x, type = type)
               
            }
            
         }
      }
   }

#==================================================================================
plotshapes3d <-
   function (x,
             type = "p",
             rgl = TRUE,
             color = 2,
             size = 1,
             joinline = c(1:1))
   {
      k <- dim(x)[1]
      n <- dim(x)[3]
      y <- matrix(0, k * n, 3)
      for (i in 1:n) {
         y[(i - 1) * k + (1:k),] <- x[, , i]
      }
      if (rgl == FALSE) {
         par(mfrow = c(1, 1))
         out <- defplotsize3(x)
         xl <- out$xl
         xu <- out$xu
         yl <- out$yl
         yu <- out$yu
         zl <- out$zl
         zu <- out$zu
         scatterplot3d(
            y,
            xlim = c(xl, xu),
            ylim = c(yl, yu),
            zlim = c(zl, zu),
            xlab = "x",
            ylab = "y",
            zlab = "z",
            axis = TRUE,
            type = type,
            color = color,
            highlight.3d = TRUE
         )
      }
      if (rgl == TRUE) {
         if (type == "l") {
            points3d(y, col = color, size = size)
            for (j in 1:n) {
               lines3d(x[, , j], col = 8)
            }
         }
         if (type == "dots") {
            points3d(y, col = color, size = size)
         }
         if (type == "p") {
            spheres3d(y, col = color, radius = size)
         }
         if (length(joinline) > 1) {
            for (j in 1:n) {
               lines3d(x[joinline, , j], col = 8)
            }
         }
         
      }
   }


#==================================================================================
plotshapestime3d <- function (x, type = "p")
{
   par(mfrow = c(1, 1))
   out <- defplotsize3(x)
   xl <- out$xl
   xu <- out$xu
   yl <- out$yl
   yu <- out$yu
   zl <- out$zl
   zu <- out$zu
   
   n <- dim(x)[3]
   for (i in 1:n) {
      scatterplot3d(
         x[, , i],
         xlim = c(xl, xu),
         ylim = c(yl, yu),
         zlim = c(zl, zu),
         xlab = "x",
         ylab = "y",
         zlab = "z",
         axis = TRUE,
         type = type,
         highlight.3d = TRUE
      )
      title(i)
      
   }   
}

#==================================================================================
plotPDMnoaxis3 <-
   function (mean, pc, sd, xl, xu, yl, yu, lineorder, i)
   {
      fig <- mean + i * pc * sd
      k <- length(mean) / 2
      figx <- fig[1:k]
      figy <- fig[(k + 1):(2 * k)]
      plot(
         figx,
         figy,
         axes = FALSE,
         xlab = " ",
         ylab = " ",
         ylim = c(yl,
                  yu),
         type = "n",
         xlim = c(xl, xu)
      )
      text(figx, figy, 1:k)
      lines(figx[lineorder], figy[lineorder])
      for (aa in 1:9999) {
         aaa <- 1
      }
   }
#################################

#==================================================================================
shapepca <-
   function (proc,
             pcno = c(1, 2, 3),
             type = "r",
             mag = 1,
             joinline = c(1, 1),
             project = c(1, 2),
             scores3d = FALSE,
             color = 2,
             axes3 = FALSE,
             rglopen = TRUE,
             zslice = 0)
   {
      if (scores3d == TRUE) {
         axes3 <- TRUE
         sz <-
            max(proc$rawscores[, max(pcno)]) - min(proc$rawscores[, max(pcno)])
         spheres3d(proc$rawscores[, pcno] , radius = sz / 30, col = color)
         if (axes3) {
            axes3d()
         }
      }
      
      m <- dim(proc$mshape)[2]
      k <- dim(proc$mshape)[1]
      
      
      if (scores3d == FALSE) {
         if ((m == 2)) {
            out <- defplotsize2(proc$rotated, project = project)
            xl <- out$xl
            yl <- out$yl
            width <- out$width
            plotpca(proc, pcno, type, mag, xl, yl, width, joinline, project)
         }
         if ((m == 3) && (type == "m")) {
            #        plot3Dmean(proc)
            #        cat("Mean shape \n")
            #        for (i in 1:length(pcno)) {
            #            cat("PC ", pcno[i], " \n")
            #            plot3Dpca(proc, pcno[i])
            #        }
            
            for (i in 1:length(pcno)) {
               cat("PC ", pcno[i], " \n")
               plotpca3d(proc, pcno[i])
            }
            
         }
         
         
         ## correct length of tangent vector if in Helmertized space
            
         h <- defh(k - 1)
         zero <- matrix(0, k - 1, k)
         H <- cbind(h, zero, zero)
         H1 <- cbind(zero, h, zero)
         H2 <- cbind(zero, zero, h)
         H <- rbind(H, H1, H2)
         
         if (dim(proc$pcar)[1] == (3 * (k - 1))) {
            pcarot <- (t(H) %*% proc$pcar)
            proc$pcar <- pcarot
         }
         
         
         if (((m == 3) && (type != "m")) && (type != "g")) {
            if (rglopen) {
               open3d()
            }
            
            
            sz <- centroid.size(proc$mshape) / sqrt(k) / 30
            spheres3d(proc$mshape, radius = sz, col = color)
            if (axes3) {
               axes3d()
            }
            
            for (i in pcno) {
               pc <- proc$mshape + 3 * mag * proc$pcasd[i] * cbind(proc$pcar[1:k, i],
                                                                   proc$pcar[(k + 1):(2 * k), i], proc$pcar[(2 * k + 1):(3 * k), i])
               for (j in 1:k) {
                  lines3d(rbind(proc$mshape[j, ], pc[j, ]), col = i)
               }
               
            }
         }
      }
      
      
      if ((m == 3) && (type == "g")) {
         if (rglopen) {
            open3d()
         }
         
         for (i in pcno) {
            pc <- proc$mshape + 3 * mag * proc$pcasd[i] * cbind(proc$pcar[1:k, i],
                                                                proc$pcar[(k + 1):(2 * k), i], proc$pcar[(2 * k + 1):(3 * k), i])
            tpsgrid(proc$mshape, pc, zslice = zslice)
         }
      }
      
      
      
      
   }


#==================================================================================
plotpca3d <- function (procreg, pcno = 1)
{
   par(mfrow = c(1, 1))
   out <- defplotsize3(procreg$rotated)
   xl <- out$xl
   xu <- out$xu
   yl <- out$yl
   yu <- out$yu
   zl <- out$zl
   zu <- out$zu
   
   k <- dim(procreg$mshape)[1]
   subx <- 1:k
   suby <- (k + 1):(2 * k)
   subz <- (2 * k + 1):(3 * k)
   
   evec <-
      cbind(procreg$pcar[subx, pcno], procreg$pcar[suby, pcno], procreg$pcar[subz, pcno])
   
   
   for (j in 1:10) {
      for (ii in-12:12) {
         mag <- ii / 4
         scatterplot3d(
            procreg$mshape + mag * evec * procreg$pcasd[pcno],
            xlim = c(xl, xu),
            ylim = c(yl, yu),
            zlim = c(zl, zu),
            xlab = "x",
            ylab = "y",
            zlab = "z",
            axis = TRUE,
            highlight.3d = TRUE
         )
         title(pcno)
      }
      for (ii in-11:11) {
         mag <- -ii / 4
         scatterplot3d(
            procreg$mshape + mag * evec * procreg$pcasd[pcno],
            xlim = c(xl, xu),
            ylim = c(yl, yu),
            zlim = c(zl, zu),
            xlab = "x",
            ylab = "y",
            zlab = "z",
            axis = TRUE
         )
         title(pcno)
      }
   }
}


##############################
#==================================================================================
Hotelling2Djames <-
   function (A, B)
   {
      z <- list(Tsq = 0, pval = 0)
      n1 <- dim(A)[3]
      n2 <- dim(B)[3]
      n <- n1 + n2
      k <- dim(A)[1]
      m <- dim(B)[2]
      if (m != 2) {
         print("Data not two dimensional")
         return(z)
      }
      else {
         pool <- array(0, c(k, m, n))
         pool[, , 1:n1] <- A
         pool[, , (n1 + 1):n] <- B
         poolpr <- procrustes2d(pool, 1, 2)
         S1 <- var(t(poolpr$tan[, 1:n1]))
         S2 <- var(t(poolpr$tan[, (n1 + 1):(n1 + n2)]))
         gamma <- realtocomplex(preshape(poolpr$mshape))
         Sw <- (S1 / n1 + S2 / n2)
         p <- 2 * k - 4
         #	TT<-eigen(Sw,symmetric=TRUE,EISPACK=TRUE)
         TT <- eigen(Sw, symmetric = TRUE)
         pcar <- TT$vectors[, 1:p]
         pcasd <- sqrt(abs(TT$values[1:p]))
         ####### add small offset if defecient in rank
         if (pcasd[p] < 0.000001)
         {
            offset <- 0.000001
            cat("*")
            pcasd <- sqrt(pcasd ** 2 + offset ** 2)
         }
         #######################################
         pcax <- t(poolpr$tan) %*% pcar
         h <- defh(k - 1)
         zero <- matrix(0, k - 1, k)
         H <- cbind(h, zero)
         H1 <- cbind(zero, h)
         H <- rbind(H, H1)
         meanxy <- t(H) %*% V(gamma)
         realrot <- t(H) %*% pcar
         one1 <- matrix(1 / n1, n1, 1)
         one2 <- matrix(1 / n2, n2, 1)
         oneone <- rbind(one1,-one2)
         vbar <- poolpr$tan %*% oneone
         scores1 <- matrix(vbar, 1, (2 * k - 2)) %*% pcar
         scores <- scores1 / pcasd
         F.partition <- ((scores[1:p] ^ 2) * (n1 * n2 * (n1 + n2 -
                                                            p - 1))) / ((n1 + n2) * (n1 + n2 - 2) * p)
         FF <- sum(F.partition)
         pval <- 1 - pf(FF, p, (n1 + n2 - p - 1))
         ginvSw <- pcar %*% diag(1 / pcasd ** 2) %*% t(pcar)
         T1 <- sum(diag((ginvSw %*% S1 / n1)))
         T2 <- sum(diag((ginvSw %*% S2 / n2)))
         T1sq <- sum(diag((
            (ginvSw %*% S1 / n1) %*% ginvSw %*% S1 / n1
         )))
         T2sq <- sum(diag((
            (ginvSw %*% S2 / n2) %*% ginvSw %*% S2 / n2
         )))
         
         Tsq <- (t(vbar) %*% (ginvSw) %*% vbar)[1, 1]
         AA <- 1 + 1 / (2 * p) * (T1 ** 2 / (n1 - 1) + T2 ** 2 / (n2 - 1))
         BB <-
            1 / (p * (p + 2)) * ((T1 ** 2 / 2 + T1sq) / (n1 - 1) + (T2 ** 2 / 2 + T2sq) /
                                    (n2 - 1))
         
         kk <- rep(0, times = 1000)
         for (i in 0:999) {
            alphai <- i / 1000
            kk[i + 1] <- qchisq(alphai, df = p) * (AA + BB * qchisq(alphai, df = p))
         }
         pval <- 1 - max(c(1:1000)[kk < Tsq]) / 1000
         
         #        z$F.partition <- F.partition
         #        z$F <- FF
         z$pval <- pval
         z$Tsq <- Tsq
         
         #        z$df1 <- p
         #        z$T.df1 <- p
         #        z$df2 <- (n1 + n2 - p - 1)
         #        mm <- n - 2
         #        z$T.df2 <- mm
         #        z$Tsq <- FF * (n1 + n2) * (n1 + n2 - 2) * p/(n1 * n2)/(n1 +
         #            n2 - p - 1)
         #        z$Tsq.partition <- F.partition * (n1 + n2) * (n1 + n2 -
         #            2) * p/(n1 * n2)/(n1 + n2 - p - 1)
         return(z)
      }
   }





MGM <- function(zst) {
   nsam <- dim(zst)[2]
   k <- dim(zst)[1]
   Mhat <- matrix(0, k - 1, k - 2)
   lamhat <- rep(0, times = (k - 1))
   Sighat <- matrix(0, k - 2, k - 2)
   kk <- k * 2 - 2
   t1 <- reassqpr(preshape(zst)) / nsam
   #	t2 <- eigen(t1,symmetric=TRUE,EISPACK=TRUE)
   t2 <- eigen(t1, symmetric = TRUE)
   reagamma <- (t2$vectors[, 1] + t2$vectors[, 2]) / sqrt(2)
   gamma <- Vinv(reagamma)
   muhat <- gamma
   
   for (i in 1:(k - 2)) {
      Mhat[, i] <- Vinv(t2$vectors[, 1 + (2 * i)])
   }
   for (i in 1:(k - 1)) {
      lamhat[i] <- t2$values[(2 * i) - 1]
   }
   for (j in 2:(k - 1)) {
      for (l in 2:(k - 1)) {
         sum <- 0
         for (i in 1:nsam) {
            zi <- preshape(zst[, i])
            sum <-
               sum + st(Mhat[, j - 1]) %*% zi * st(zi) %*% Mhat[, l - 1] * st(zi) %*% muhat *
               st(muhat) %*% zi
         }
         Sighat[j - 1, l - 1] <-
            1 / (lamhat[1] - lamhat[j]) / (lamhat[1] - lamhat[l]) * sum / nsam
      }
   }
   SR <- Re(Sighat)
   SI <- Im(Sighat)
   S1 <- cbind(SR, SI)
   S2 <- cbind(-(SI), SR)
   S <- rbind(S1, S2)
   
   offset <- 0
   #es<-eigen(S,symmetric=TRUE,EISPACK=TRUE)$values
   es <- eigen(S, symmetric = TRUE)$values
   nn <- length(es)
   if (es[nn] < 0.000001)
   {
      offset <- 0.000001
      #cat("Warning: test: small samples, lambda I added to within group covariance matrix \n")
      cat("*")
   }
   invS <- solve(S + offset * diag(nn))
   
   invSR <- invS[1:(k - 2), 1:(k - 2)]
   invSI <- invS[1:(k - 2), (k - 1):(2 * k - 4)]
   invS <- invSR + 1i * invSI
   Mhat <- st(Mhat)
   MGM <- st(Mhat) %*% invS %*% Mhat
   MGM
}



#======================================================================================
resampletest <- function(A,
                         B,
                         resamples = 200,
                         replace = TRUE) {
   A1 <- A
   A2 <- B
   B <- resamples
   
   k <- dim(A1)[1]
   m <- dim(A1)[2]
   nmin <- min(dim(A1)[3], dim(A2)[3])
   ntot <- dim(A1)[3] + dim(A2)[3]
   M <- (k - 1) * m - m * (m - 1) / 2 - 1
   if (M >= ntot) {
      cat("Warning: Low sample size (n1 + n2 <= p) \n")
   }
   if ((M >= nmin) && (replace == TRUE)) {
      cat(
         "Warning: Low sample sizes : min(n1,n2)<=p : * indicates some regularization carried out \n"
      )
   }
   
   permutation <- !replace
   
   
   
   if (is.complex(A1)) {
      tem <- array(0, c(nrow(A1), 2, ncol(A1)))
      tem[, 1,] <- Re(A1)
      tem[, 2,] <- Im(A1)
      A1 <- tem
   }
   if (is.complex(A2)) {
      tem <- array(0, c(nrow(A2), 2, ncol(A2)))
      tem[, 1,] <- Re(A2)
      tem[, 2,] <- Im(A2)
      A2 <- tem
   }
   m <- dim(A1)[2]
   if (m != 2) {
      print("Data not two dimensional")
      print("Carrying out tests on Procrustes residuals")
      out <-
         testmeanshapes(A1, A2, resamples = resamples, replace = replace)
      return(out)
   }
   
   zst1 <- A1[, 1, ] + 1i * A1[, 2, ]
   zst2 <- A2[, 1, ] + 1i * A2[, 2, ]
   
   
   
   
   nsam1 <- dim(zst1)[2]
   nsam2 <- dim(zst2)[2]
   k <- dim(zst1)[1]
   LL <- (MGM(zst1) + MGM(zst2)) * (nsam1 + nsam2)
   LL1 <- cbind(Re(LL), Im(LL))
   LL2 <- cbind(-Im(LL), Re(LL))
   LL <- rbind(LL1, LL2)
   #Tumc<-min(eigen(LL,symmetric=TRUE,only.values=TRUE,EISPACK=TRUE)$values)
   Tumc <- min(eigen(LL, symmetric = TRUE, only.values = TRUE)$values)
   
   
   m1 <- preshape(procrustes2d(zst1)$mshape)
   m1 <- m1[, 1] + 1i * m1[, 2]
   m2 <- preshape(procrustes2d(zst2)$mshape)
   m2 <- m2[, 1] + 1i * m2[, 2]
   m0 <- preshape(procrustes2d(cbind(zst1, zst2))$mshape)
   m0 <- m0[, 1] + 1i * m0[, 2]
   d <- length(m1)
   H <- defh(k - 1)
   b <- m1
   a <- m0
   bt <- b * c((st(b) %*% a) / Mod(st(b) %*% a))
   abt <- c(Re(st(bt) %*% a))
   ct <- (bt - a * abt)
#   ct <- ct / sqrt(st(ct) %*% ct)
   ct <- ct / c(sqrt(st(as.vector(ct)) %*% as.vector(ct)))
   At <- a %*% st(ct) - ct %*% st(a)
   salph <- sqrt(1 - abt ** 2)
   calph <- abt
   Id <- diag(rep(1, times = d))
   U1 <- Id + salph * At + (calph - 1) * (a %*% st(a) + ct %*% st(ct))
   b <- m2
   a <- m0
   bt <- b * c((st(b) %*% a) / Mod(st(b) %*% a))
   abt <- c(Re(st(bt) %*% a))
   ct <- (bt - a * abt)
 #  ct <- ct / sqrt(st(ct) %*% ct)
   ct <- ct / c(sqrt(st(as.vector(ct)) %*% as.vector(ct)))
   At <- a %*% st(ct) - ct %*% st(a)
   salph <- sqrt(1 - abt ** 2)
   calph <- abt
   Id <- diag(rep(1, times = d))
   U2 <- Id + salph * At + (calph - 1) * (a %*% st(a) + ct %*% st(ct))
   yst1 <- t(H) %*% U1 %*% preshape(zst1)
   yst2 <- t(H) %*% U2 %*% preshape(zst2)
   ybind <- cbind(yst1, yst2)
   
   zr1 <- array(0, c(k, 2, nsam1))
   zr2 <- array(0, c(k, 2, nsam2))
   zr3 <- array(0, c(k, 2, nsam1 + nsam2))
   zr1[, 1, ] <- Re(zst1)
   zr1[, 2, ] <- Im(zst1)
   zr2[, 1, ] <- Re(zst2)
   zr2[, 2, ] <- Im(zst2)
   zr3[, 1, ] <- cbind(Re(zst1), Re(zst2))
   zr3[, 2, ] <- cbind(Im(zst1), Im(zst2))
   yr1 <- array(0, c(k, 2, nsam1))
   yr2 <- array(0, c(k, 2, nsam2))
   yr3 <- array(0, c(k, 2, nsam1 + nsam2))
   yr1[, 1, ] <- Re(yst1)
   yr1[, 2, ] <- Im(yst1)
   yr2[, 1, ] <- Re(yst2)
   yr2[, 2, ] <- Im(yst2)
   yr3[, 1, ] <- cbind(Re(yst1), Re(yst2))
   yr3[, 2, ] <- cbind(Im(yst1), Im(yst2))
   
   
   Gtem <- Goodall2D(zr1, zr2)
   Htem <- Hotelling2D(zr1, zr2)
   Jtem <- Hotelling2Djames(zr1, zr2)
   
   Gumc <- Gtem$F
   Humc <- Htem$F
   Jumc <- Jtem$Tsq
   
   Gtabpval <- Gtem$pval
   Htabpval <- Htem$pval
   Jtabpval <- Jtem$pval
   
   
   if (B > 0) {
      Tu <- rep(0, times = B)
      Gu <- Tu
      Hu <- Tu
      Ju <- Tu
      
      
      cat("Resampling...")
      cat(c("No of resamples = ", B, "\n"))
      
      if (permutation) {
         cat("Permutations - sampling without replacement \n")
      }
      if (permutation == FALSE) {
         cat("Bootstrap - sampling with replacement \n")
      }
      
      for (i in 1:B) {
         cat(c(i, " "))
         
         
         select1 <- sample(1:nsam1, replace = TRUE)
         select2 <- sample(1:nsam2, replace = TRUE)
         zb1 <- yst1[, select1]
         zb2 <- yst2[, select2]
         zbgh1 <- yr1[, , select1]
         zbgh2 <- yr2[, , select2]
         
         if (permutation) {
            select0 <- sample(c(1:(nsam1 + nsam2)), (nsam1 + nsam2), replace = FALSE)
            select1 <- select0[1:nsam1]
            select2 <- select0[(nsam1 + 1):(nsam1 + nsam2)]
            zb1 <- zr3[, 1, select1] + 1i * zr3[, 2, select1]
            zb2 <- zr3[, 1, select2] + 1i * zr3[, 2, select2]
            zbgh1 <- yr3[, , select1]
            zbgh2 <- yr3[, , select2]
         }
         
         
         LL <- (MGM(zb1) + MGM(zb2)) * (nsam1 + nsam2)
         LL1 <- cbind(Re(LL), Im(LL))
         LL2 <- cbind(-Im(LL), Re(LL))
         LL <- rbind(LL1, LL2)
         #lmin<-min(eigen(LL,symmetric=TRUE,only.values=TRUE,EISPACK=TRUE)$values)
         lmin <- min(eigen(LL, symmetric = TRUE, only.values = TRUE)$values)
         Tu[i] <- lmin
         Gu[i] <- Goodall2D(zbgh1, zbgh2)$F
         Hu[i] <- Hotelling2D(zbgh1, zbgh2)$F
         Ju[i] <- Hotelling2Djames(zbgh1, zbgh2)$Tsq
      }
      
      Tu <- sort(Tu)
      numbig <- length(Tu[Tumc < Tu])
      pvalb <- (1 + numbig) / (B + 1)
      Gu <- sort(Gu)
      numbig <- length(Gu[Gumc < Gu])
      pvalG <- (1 + numbig) / (B + 1)
      Hu <- sort(Hu)
      numbig <- length(Hu[Humc < Hu])
      pvalH <- (1 + numbig) / (B + 1)
      Ju <- sort(Ju)
      numbig <- length(Ju[Jumc < Ju])
      pvalJ <- (1 + numbig) / (B + 1)
      
      
      
      
      cat(" \n")
      out <-
         list(
            lambda = 0,
            lambda.pvalue = 0,
            lambda.table.pvalue = 0,
            H = 0,
            H.pvalue = 0,
            H.table.pvalue = 0,
            J = 0,
            J.pvalue = 0,
            J.table.pvalue = 0,
            G = 0,
            G.pvalue = 0,
            G.table.pvalue = 0
         )
      
      out$lambda <- Tumc
      out$lambda.pvalue <- pvalb
      out$lambda.table.pvalue <- 1 - pchisq(Tumc, 2 * k - 4)
      out$H <- Humc
      out$H.pvalue <- pvalH
      out$H.table.pvalue <- Htabpval
      out$J <- Jumc
      out$J.pvalue <- pvalJ
      out$J.table.pvalue <- Jtabpval
      out$G <- Gumc
      out$G.pvalue <- pvalG
      out$G.table.pvalue <- Gtabpval
      
   }
   
   if (resamples == 0) {
      out <-
         list(
            lambda = 0,
            lambda.table.pvalue = 0,
            H = 0,
            H.table.pvalue = 0,
            J = 0,
            J.table.pvalue = 0,
            G = 0,
            G.table.pvalue = 0
         )
      
      out$lambda <- Tumc
      out$lambda.table.pvalue <- 1 - pchisq(Tumc, 2 * k - 4)
      out$H <- Humc
      out$H.table.pvalue <- Htabpval
      out$J <- Jumc
      out$J.table.pvalue <- Jtabpval
      out$G <- Gumc
      out$G.table.pvalue <- Gtabpval
      
      
   }
   
   
   
   out
}


#==================================================================================
prcomp1 <-
   function (x,
             retx = TRUE,
             center = TRUE,
             scale. = FALSE,
             tol = NULL,
             svd = TRUE)
   {
      x <- as.matrix(x)
      x <- scale(x, center = center, scale = scale.)
      if (svd == FALSE) {
         a <- eigen(cov(x))
         r <- list(sdev = 0,
                   rotation = 0,
                   x = 0)
         r$sdev <- sqrt(abs(a$values))
         r$rotation <- a$vectors
         r$x <- x %*% a$vectors
      }
      else
      {
         s <- svd(x, nu = 0)
         if (!is.null(tol)) {
            rank <- sum(s$d > (s$d[1] * tol))
            if (rank < ncol(x))
               s$v <- s$v[, 1:rank, drop = FALSE]
         }
         s$d <- s$d / sqrt(max(1, nrow(x) - 1))
         dimnames(s$v) <-
            list(colnames(x), paste("PC", seq(len = ncol(s$v)),
                                    sep = ""))
         r <- list(sdev = s$d, rotation = s$v)
         if (retx)
            r$x <- x %*% s$v
         class(r) <- "prcomp1"
      }
      r
   }





#==================================================================================
defplotsize3 <- function(Y)
{
   out <- list(
      xl = 0,
      yl = 0,
      zl = 0,
      xu = 0,
      yu = 0,
      zu = 0,
      width = 0
   )
   n <- dim(Y)[3]
   xm <- mean(Y[, 1,])
   ym <- mean(Y[, 2,])
   zm <- mean(Y[, 3,])
   x <- Y
   x[, 1,] <- Y[, 1,] - xm
   x[, 2,] <- Y[, 2,] - ym
   x[, 3,] <- Y[, 3,] - zm
   mn1 <- min(x[, 1, ])
   mn2 <- min(x[, 2, ])
   mn3 <- min(x[, 3, ])
   mx1 <- max(x[, 1, ])
   mx2 <- max(x[, 2, ])
   mx3 <- max(x[, 3, ])
   xl <- -max(-mn1, mx1)
   yl <- -max(-mn2, mx2)
   zl <- -max(-mn3, mx3)
   width <- max(-2 * xl,-2 * yl,-2 * zl)
   out$xl <- -width / 2 * 1.2 + xm
   out$yl <- -width / 2 * 1.2 + ym
   out$zl <- -width / 2 * 1.2 + zm
   out$xu <- width / 2 * 1.2 + xm
   out$yu <- width / 2 * 1.2 + ym
   out$zu <- width / 2 * 1.2 + zm
   out$width <- width * 1.2
   out
}


#==================================================================================
procOPA <- function(A,
                    B,
                    scale = TRUE,
                    reflect = FALSE) {
   out <- list(
      R = 0,
      s = 0,
      Ahat = 0,
      Bhat = 0,
      OSS = 0,
      rmsd = 0
   )
   if (is.complex(sum(A)) == TRUE) {
      k <- length(A)
      Areal <- matrix(0, k, 2)
      Areal[, 1] <- Re(A)
      Areal[, 2] <- Im(A)
      A <- Areal
   }
   if (is.complex(sum(B)) == TRUE) {
      k <- length(B)
      Breal <- matrix(0, k, 2)
      Breal[, 1] <- Re(B)
      Breal[, 2] <- Im(B)
      B <- Breal
   }
   k <- dim(A)[1]
   if (reflect == FALSE) {
      R <- fort.ROTATION(A, B)
   } else
   {
      R <- fort.ROTATEANDREFLECT(A, B)
   }
   s <- 1
   if (scale == TRUE) {
      s <- fos(A, B)
      
      if (reflect == TRUE) {
         s <- fos.REFLECT(A, B)
      }
      
   }
   Ahat <- fcnt(A)
   Bhat <- fcnt(B) %*% R * s
   resid <- Ahat - Bhat
   OSS <- sum(diag(t(resid) %*% resid))
   out$R <- R
   out$s <- s
   out$Ahat <- Ahat
   out$Bhat <- Bhat
   m <- dim(Ahat)[2]
   out$OSS <- OSS
   out$rmsd <- sqrt(OSS / (k))
   out
}

#==================================================================================
defplotsize2 <- function(Y, project = c(1, 2))
{
   out <- list(
      xl = 0,
      yl = 0,
      xu = 0,
      yu = 0,
      width = 0
   )
   n <- dim(Y)[3]
   xm <- mean(Y[, project[1],])
   ym <- mean(Y[, project[2],])
   x <- Y
   x[, project[1],] <- Y[, project[1],] - xm
   x[, project[2],] <- Y[, project[2],] - ym
   out <- list(xl = 0, yl = 0, width = 0)
   mn1 <- min(x[, project[1], ])
   mn2 <- min(x[, project[2], ])
   mx1 <- max(x[, project[1], ])
   mx2 <- max(x[, project[2], ])
   xl <- -max(-mn1, mx1)
   yl <- -max(-mn2, mx2)
   width <- max(-2 * xl,-2 * yl)
   out$xl <- -width / 2 * 1.2 + xm
   out$yl <- -width / 2 * 1.2 + ym
   out$xu <- width / 2 * 1.2 + xm
   out$yu <- width / 2 * 1.2 + ym
   out$width <- width * 1.2
   out
}

#==================================================================================
plotshapes <-
   function(A,
            B = 0,
            joinline = c(1, 1),
            orthproj = c(1, 2),
            color = 1,
            symbol = 1) {
      CHECKOK <- TRUE
      if (is.array(A) == FALSE) {
         if (is.matrix(A) == FALSE) {
            cat("Error !! argument should be an array or matrix \n")
            CHECKOK <- FALSE
         }
      }
      
      if (CHECKOK) {
         k <- dim(A)[1]
         m <- dim(A)[2]
         kk <- k
         if (k >= 15) {
            kk <- 1
         }
         par(pty = "s")
         #if (length(c(B))==1){
         #par(mfrow=c(1,1))
         #}
         if (length(c(B)) != 1) {
            par(mfrow = c(1, 2))
         }
         if (length(dim(A)) == 3) {
            A <- A[, orthproj, ]
         }
         if (is.matrix(A) == TRUE) {
            a <- array(0, c(k, 2, 1))
            a[, , 1] <- A[, orthproj]
            A <- a
         }
         out <- defplotsize2(A)
         width <- out$width
         if (length(c(B)) != 1) {
            if (length(dim(B)) == 3) {
               B <- B[, orthproj, ]
            }
            if (is.matrix(B) == TRUE) {
               a <- array(0, c(k, 2, 1))
               a[, , 1] <- B[, orthproj]
               B <- a
            }
            ans <- defplotsize2(B)
            width <- max(out$width, ans$width)
         }
         n <- dim(A)[3]
         
         lc <- length(color)
         lt <- k * m * n / lc
         color <- rep(color, times = lt)
         lc <- length(symbol)
         lt <- k * m * n / lc
         symbol <- rep(symbol, times = lt)
         
         
         
         plot(
            A[, , 1],
            xlim = c(out$xl, out$xl + width),
            ylim = c(out$yl,
                     out$yl + width),
            type = "n",
            xlab = " ",
            ylab = " "
         )
         for (i in 1:n) {
            select <- ((i - 1) * k * m + 1):(i * k * m)
            points(A[, , i], pch = symbol[select], col = color[select])
            lines(A[joinline, , i])
         }
         if (length(c(B)) != 1) {
            A <- B
            if (is.matrix(A) == TRUE) {
               a <- array(0, c(k, 2, 1))
               a[, , 1] <- A
               A <- a
            }
            out <- defplotsize2(A)
            n <- dim(A)[3]
            plot(
               A[, , 1],
               xlim = c(ans$xl, ans$xl + width),
               ylim = c(ans$yl,
                        ans$yl + width),
               type = "n",
               xlab = " ",
               ylab = " "
            )
            for (i in 1:n) {
               points(A[, , i], pch = symbol[select], col = color[select])
               lines(A[joinline, , i])
            }
         }
      }
   }


#==================================================================================
BoxM <- function(A, B, npc)
{
   #carries out Box's M test
   #(see Mardia, Kent, Bibby 1979, p140)
   #in: data arrays A, B
   #out: z$M   M statistic
   #     z$df degrees of freedom for approx distn of chi-squared statistic
   #    z$pval  p-value
   z <- list(M = 0, df = 0, pval = 0)
   n1 <- dim(A)[3]
   n2 <- dim(B)[3]
   k <- dim(A)[1]
   m <- dim(A)[2]
   if (m > 2) {
      print("Only works for 2D data at the moment!")
   }
   if (m == 2) {
      C <- array(0, c(k, m, n1 + n2))
      C[,  , 1:n1] <- A
      C[,  , (n1 + 1):(n1 + n2)] <- B
      Cpr <- procrustes2d(C, 1, 2)
      p <- npc
      ng <- 2
      n <- n1 + n2
      S1 <- var(t(Cpr$tan[1:npc, 1:n1]))
      S2 <- var(t(Cpr$tan[1:npc, (n1 + 1):(n1 + n2)]))
      Su <- ((n1 - 1) * S1 + (n2 - 1) * S2) / (n1 + n2 - 2)
      S1inv <-
         eigen(S1)$vectors %*% diag(1 / eigen(S1)$values) %*% t(eigen(S1)$vectors)
      S2inv <-
         eigen(S2)$vectors %*% diag(1 / eigen(S2)$values) %*% t(eigen(S2)$vectors)
      logdet1 <- sum(log(eigen(S1inv %*% Su)$values))
      logdet2 <- sum(log(eigen(S2inv %*% Su)$values))
      gam <-
         1 - ((2 * p ^ 2 + 3 * p - 1) / (6 * (p + 1) * (ng - 1))) * (1 / (n1 -
                                                                             1) + 1 / (n2 - 1) - 1 / (n - ng))
      M <- gam * ((n1 - 1) * logdet1 + (n2 - 1) * logdet2)
      df <- (p * (p + 1) * (ng - 1)) / 2
      pval <- 1 - pchisq(M, df)
      z$M <- M
      z$df <- df
      z$pval <- pval
   }
   return(z)
}

#==================================================================================
Goodall2D <- function(A, B)
{
   #Calculates Goodall's two sample F test for 2d data only
   #in: data arrays A, B k x 2 x n data arrays
   #out: z$F  F statistic
   #     z$df1, z$df2  degrees of freedom
   #     z$pval: p-value
   z <- list(
      F = 0,
      pval = 0,
      df1 = 0,
      df2 = 0
   )
   n1 <- dim(A)[3]
   n2 <- dim(B)[3]
   k <- dim(A)[1]
   m <- dim(A)[2]
   if (m != 2) {
      print("Data not two dimensional")
      return(z)
   }
   p <- 2 * k - 4
   Apr <- procrustes2d(A, 1, 2)
   Bpr <- procrustes2d(B, 1, 2)
   top <- sin(riemdist(Apr$mshape, Bpr$mshape)) ^ 2
   bot <- Apr$rmsd1 ^ 2 * n1 + Bpr$rmsd1 ^ 2 * n2
   Fstat <- ((n1 + n2 - 2) / (1 / n1 + 1 / n2) * top) / bot
   pval <- 1 - pf(Fstat, p, (n1 + n2 - 2) * p)
   z$F <- Fstat
   z$pval <- pval
   z$df1 <- p
   z$df2 <- (n1 + n2 - 2) * p
   return(z)
}

#==================================================================================
Goodalltest <- function(A, B, tol1 = 1e-07, tol2 = tol1)
{
   #Calculates Goodall's two sample F test
   #in: data arrays A, B:
   #out: z$F  F statistic
   #     z$df1, z$df2  degrees of freedom
   #     z$pval: p-value
   z <- list(
      F = 0,
      pval = 0,
      df1 = 0,
      df2 = 0
   )
   n1 <- dim(A)[3]
   n2 <- dim(B)[3]
   k <- dim(A)[1]
   m <- dim(A)[2]
   p <- min(k * m - (m * (m - 1)) / 2 - 1 - m, n1 + n2 - 2)
   Apr <- procrustesGPA(A, tol1, tol2)
   Bpr <- procrustesGPA(B, tol1, tol2)
   top <- sin(riemdist(Apr$mshape, Bpr$mshape)) ^ 2
   bot <- Apr$rmsd1 ^ 2 * n1 + Bpr$rmsd1 ^ 2 * n2
   Fstat <- ((n1 + n2 - 2) / (1 / n1 + 1 / n2) * top) / bot
   pval <- 1 - pf(Fstat, p, (n1 + n2 - 2) * p)
   z$F <- Fstat
   z$pval <- pval
   z$df1 <- p
   z$df2 <- (n1 + n2 - 2) * p
   return(z)
}

#==================================================================================
Hotelling2D <- function (A, B)
{
   z <- list(
      Tsq.partition = 0,
      Tsq = 0,
      F.partition = 0,
      F = 0,
      pval = 0,
      df1 = 0,
      df2 = 0,
      T.df1 = 0,
      T.df2 = 0
   )
   n1 <- dim(A)[3]
   n2 <- dim(B)[3]
   n <- n1 + n2
   k <- dim(A)[1]
   m <- dim(B)[2]
   if (m != 2) {
      print("Data not two dimensional")
      return(z)
   }
   else {
      pool <- array(0, c(k, m, n))
      pool[, , 1:n1] <- A
      pool[, , (n1 + 1):n] <- B
      poolpr <- procrustes2d(pool, 1, 2)
      S1 <- var(t(poolpr$tan[, 1:n1]))
      S2 <- var(t(poolpr$tan[, (n1 + 1):(n1 + n2)]))
      gamma <- realtocomplex(preshape(poolpr$mshape))
      Sw <- ((n1 - 1) * S1 + (n2 - 1) * S2) / (n1 + n2 - 2)
      p <- 2 * k - 4
      #        pcar <- eigen(Sw,EISPACK=TRUE)$vectors[, 1:p]
      pcar <- eigen(Sw)$vectors[, 1:p]
      pcasd <- sqrt(abs(eigen(Sw)$values[1:p]))
      ####### add small offset if defecient in rank
      if (pcasd[p] < 0.000001)
      {
         offset <- 0.000001
         cat("*")
         pcasd <- sqrt(pcasd ** 2 + offset ** 2)
      }
      #######################################
      pcax <- t(poolpr$tan) %*% pcar
      h <- defh(k - 1)
      zero <- matrix(0, k - 1, k)
      H <- cbind(h, zero)
      H1 <- cbind(zero, h)
      H <- rbind(H, H1)
      meanxy <- t(H) %*% V(gamma)
      realrot <- t(H) %*% pcar
      one1 <- matrix(1 / n1, n1, 1)
      one2 <- matrix(1 / n2, n2, 1)
      oneone <- rbind(one1,-one2)
      vbar <- poolpr$tan %*% oneone
      scores1 <- matrix(vbar, 1, (2 * k - 2)) %*% pcar
      scores <- scores1 / pcasd
      F.partition <- ((scores[1:p] ^ 2) * (n1 * n2 * (n1 + n2 -
                                                         p - 1))) / ((n1 + n2) * (n1 + n2 - 2) * p)
      FF <- sum(F.partition)
      pval <- 1 - pf(FF, p, (n1 + n2 - p - 1))
      z$F.partition <- F.partition
      z$F <- FF
      z$pval <- pval
      z$df1 <- p
      z$T.df1 <- p
      z$df2 <- (n1 + n2 - p - 1)
      mm <- n - 2
      z$T.df2 <- mm
      z$Tsq <- FF * (n1 + n2) * (n1 + n2 - 2) * p / (n1 * n2) / (n1 +
                                                                    n2 - p - 1)
      z$Tsq.partition <-
         F.partition * (n1 + n2) * (n1 + n2 - 2) * p / (n1 * n2) / (n1 + n2 - p -
                                                                       1)
      return(z)
   }
}


#==================================================================================
Hotellingtest <- function (A, B, tol1 = 1e-07, tol2 = 1e-07)
{
   z <- list(
      Tsq.partition = 0,
      Tsq = 0,
      F.partition = 0,
      F = 0,
      pval = 0,
      df1 = 0,
      df2 = 0,
      T.df1 = 0,
      T.df2 = 0
   )
   n1 <- dim(A)[3]
   n2 <- dim(B)[3]
   n <- n1 + n2
   k <- dim(A)[1]
   m <- dim(B)[2]
   pool <- array(0, c(k, m, n))
   pool[, , 1:n1] <- A
   pool[, , (n1 + 1):n] <- B
   poolpr <- procrustesGPA(pool, tol1, tol2, approxtangent = FALSE)
   S1 <- var(t(poolpr$tan[, 1:n1]))
   S2 <- var(t(poolpr$tan[, (n1 + 1):(n1 + n2)]))
   Sw <- ((n1 - 1) * S1 + (n2 - 1) * S2) / (n1 + n2 - 2)
   p <- min(k * m - (m * (m - 1)) / 2 - 1 - m, n1 + n2 - 2)
   eva <- eigen(Sw, symmetric = TRUE)
   pcar <- eva$vectors[, 1:p]
   pcasd <- sqrt(abs(eva$values[1:p]))
   
   ####### add small offset if defecient in rank
   if (pcasd[p] < 0.000001)
   {
      offset <- 0.000001
      cat("*")
      pcasd <- sqrt(pcasd ** 2 + offset)
   }
   #######################################
   
   lam <- rep(0, times = (k * m - m))
   lam[1:p] <- 1 / pcasd ** 2
   Suinv <- eva$vectors %*% diag(lam) %*% t(eva$vectors)
   #    check <- p
   #    for (i in 1:p) {
   #        if (pcasd[p + 1 - i] < 1e-04) {
   #            check <- p + 1 - i - 1
   #        }
   #    }
   #    p <- check
   pcax <- t(poolpr$tan) %*% pcar
   one1 <- matrix(1 / n1, n1, 1)
   one2 <- matrix(1 / n2, n2, 1)
   oneone <- rbind(one1,-one2)
   vbar <- poolpr$tan %*% oneone
   scores1 <- matrix(vbar, 1, m * k - m) %*% pcar
   scores <- scores1 / pcasd
   #    tem<-c(t(vbar)%*%Suinv%*%vbar)  #(=Dsq)#
   F.partition <- ((scores[1:p] ^ 2) * (n1 * n2 * (n1 + n2 - p -
                                                      1))) / ((n1 + n2) * (n1 + n2 - 2) * p)
   FF <- sum(F.partition)
   pval <- 1 - pf(FF, p, (n1 + n2 - p - 1))
   z$F.partition <- F.partition
   z$F <- FF
   z$pval <- pval
   z$df1 <- p
   z$T.df1 <- p
   z$df2 <- (n1 + n2 - p - 1)
   mm <- n - 2
   z$T.df2 <- mm
   z$Tsq <- FF * (n1 + n2) * (n1 + n2 - 2) * p / (n1 * n2) / (n1 + n2 -
                                                                 p - 1)
   z$Tsq.partition <-
      F.partition * (n1 + n2) * (n1 + n2 - 2) * p / (n1 * n2) / (n1 + n2 - p -
                                                                    1)
   return(z)
}



# Hotellingtest<-function(A, B, tol1=1e05,tol2=1e05)
# OLD VERSION using $tan rather than $tanpartial
#{
#Calculates two sample Hotelling Tsq test for testing whether
#mean shapes are equal (m - Dimensions where m >= 2)
#in: A, B the k x m x n arrays of data for each group
#out: z$F : F-statistic
#     z$df1, z$df2 : dgrees of freedom
#     z$pval: pvalue
#        z <- list(Tsq.partition = 0, Tsq = 0, F.partition = 0, F = 0, pval = 0,
#                df1 = 0, df2 = 0, T.df1 = 0, T.df2 = 0)
#        n1 <- dim(A)[3]
#        n2 <- dim(B)[3]
#        n <- n1 + n2
#        k <- dim(A)[1]
#        m <- dim(B)[2]
#        pool <- array(0, c(k, m, n))
#        pool[,  , 1:n1] <- A
#        pool[,  , (n1 + 1):n] <- B
#        poolpr <- procrustesGPA(pool,tol1,tol2)
#        S1 <- var(t(poolpr$tan[, 1:n1]))
#        S2 <- var(t(poolpr$tan[, (n1 + 1):(n1 + n2)]))
#        Sw <- ((n1 - 1) * S1 + (n2 - 1) * S2)/(n1 + n2 - 2)
#        p <- min(k * m - (m * (m - 1))/2 - 1 - m, n1 + n2 - 2)
#        pcar <- eigen(Sw)$vectors[, 1:p]
#        pcasd <- sqrt(eigen(Sw)$values[1:p])
#        check<-p
## checks to see if rank is reasonable
#        for (i in 1:p){
#        if (pcasd[p+1-i] < 0.0001){
#        check<-p+1-i-1
#        }
#        }
#        p<-check
#        pcax <- t(poolpr$tan) %*% pcar
#        one1 <- matrix(1/n1, n1, 1)
#        one2 <- matrix(1/n2, n2, 1)
#        oneone <- rbind(one1,  - one2)
#        vbar <- poolpr$tan %*% oneone
#        scores1 <- matrix(vbar, 1, m*k) %*% pcar
#        scores <- scores1/pcasd
#        F.partition <- ((scores[1:p]^2) * (n1 * n2 * (n1 + n2 - p - 1)))/((n1 +
#                n2) * (n1 + n2 - 2) * p)
#        FF <- sum(F.partition)
#        pval <- 1 - pf(FF, p, (n1 + n2 - p - 1))
#        z$F.partition <- F.partition
#        z$F <- FF
#        z$pval <- pval
#        z$df1 <- p
#        z$T.df1 <- p
#        z$df2 <- (n1 + n2 - p - 1)
#        mm <- n - 2
#        z$T.df2 <- mm
#        z$Tsq <- (FF * (mm * p))/(mm - p + 1)
#        z$Tsq.partition <- (F.partition * (mm * p))/(mm - p + 1)
#        return(z)
#}

#==================================================================================
I2mat <- function(Be)
{
   zero <- rep(0, times = dim(Be)[1] ^ 2)
   zero <- matrix(zero, dim(Be)[1], dim(Be)[2])
   temp <- cbind(Be, zero)
   temp1 <- cbind(zero, Be)
   tem <- rbind(temp, temp1)
   tem
}

#==================================================================================
tpsgrid.old <-
   function (TT,
             YY,
             xbegin = -999,
             ybegin = -999,
             xwidth = -999,
             opt = 2,
             ext = 0.1,
             ngrid = 22,
             cex = 1,
             pch = 20,
             col = 2)
   {
      k <- nrow(TT)
      if (xwidth == -999) {
         bb <- array(TT, c(dim(TT), 1))
         aa <- defplotsize2(bb)
         xwidth <- aa$width
      }
      if (xbegin == -999) {
         bb <- array(TT, c(dim(TT), 1))
         aa <- defplotsize2(bb)
         xbegin <- aa$xl
      }
      if (ybegin == -999) {
         bb <- array(TT, c(dim(TT), 1))
         aa <- defplotsize2(bb)
         ybegin <- aa$yl
      }
      xstart <- xbegin
      ystart <- ybegin
      ngrid <- trunc(ngrid / 2) * 2
      kx <- ngrid
      ky <- ngrid - 1
      l <- kx * ky
      step <- xwidth / (kx - 1)
      r <- 0
      X <- rep(0, times = kx)
      Y2 <- rep(0, times = ky)
      for (p in 1:kx) {
         ystart <- ybegin
         xstart <- xstart + step
         for (q in 1:ky) {
            ystart <- ystart + step
            r <- r + 1
            X[r] <- xstart
            Y2[r] <- ystart
         }
      }
      refc <- matrix(c(X, Y2), kx * ky, 2)
      TPS <- bendingenergy(TT)
      gamma11 <- TPS$gamma11
      gamma21 <- TPS$gamma21
      gamma31 <- TPS$gamma31
      W <- gamma11 %*% YY
      ta <- t(gamma21 %*% YY)
      B <- gamma31 %*% YY
      WtY <- t(W) %*% YY
      trace <- c(0)
      for (i in 1:2) {
         trace <- trace + WtY[i, i]
      }
      benergy <- 16 * pi * trace
      if (m == 3) {
         benergy <- 8 * pi * trace
      }
      l <- kx * ky
      phi <- matrix(0, l, 2)
      s <- matrix(0, k, 1)
      for (i in 1:l) {
         s <- matrix(0, k, 1)
         for (m in 1:k) {
            s[m,] <- sigmacov(refc[i,] - TT[m,])
         }
         phi[i,] <- ta + t(B) %*% refc[i,] + t(W) %*% s
      }
      par(pty = "s")
      if (opt == 2) {
         par(mfrow = c(1, 2))
         order <- linegrid(refc, kx, ky)
         plot(
            order[1:l, 1],
            order[1:l, 2],
            type = "l",
            xlim = c(xbegin -
                        xwidth * ext, xbegin + xwidth * (1 + ext)),
            ylim = c(
               ybegin -
                  (xwidth * ky) / kx * ext,
               ybegin + (xwidth * ky) / kx *
                  (1 + ext)
            ),
            xlab = " ",
            ylab = " "
         )
         lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l),
                                                2], type = "l")
         points(TT,
                cex = cex,
                pch = pch,
                col = col)
      }
      order <- linegrid(phi, kx, ky)
      plot(
         order[1:l, 1],
         order[1:l, 2],
         type = "l",
         xlim = c(xbegin -
                     xwidth * ext, xbegin + xwidth * (1 + ext)),
         ylim = c(ybegin -
                     (xwidth * ext * ky) / kx, ybegin + (xwidth * (1 + ext) *
                                                            ky) / kx),
         xlab = " ",
         ylab = " "
      )
      lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l), 2],
            type = "l")
      points(YY,
             cex = cex,
             pch = pch,
             col = col)
   }
#
#
#==================================================================================
V <- function(z)
{
   #input complex k -vector
   #ouput vectorized 2k vector of real stacked on imaginary components
   x <- c(Re(z), Im(z))
   x
}

#==================================================================================
Vinv <- function(x)
{
   #input vectorized 2k vector of x1 stacked on x2 components
   #input complex k -vector of the form x1 + 1i*x2
   nel <- length(x) / 2
   zx <- x[1:nel]
   zy <- x[(nel + 1):(2 * nel)]
   z <- zx + (1i) * zy
   z
}

#==================================================================================
Vmat <- function(z)
{
   #as Vinv but input is a k x n complex matrix
   # output 2k x n matrix of stacked real then complex components
   x <- rbind(Re(z), Im(z))
   x
}

#==================================================================================
bendingenergy <- function (TT)
{
   z <- list(
      gamma11 = 0,
      gamma21 = 0,
      gamma31 = 0,
      prinwarps = 0,
      prinwarpeval = 0,
      Un = 0
   )
   
   
   k <- nrow(TT)
   m <- dim(TT)[2]
   S <- matrix(0, k, k)
   for (i in 1:k) {
      for (j in 1:k) {
         S[i, j] <- sigmacov(TT[i,] - TT[j,])
      }
   }
   
   one <- matrix(1, k, 1)
   zero <- matrix(0, m + 1, m + 1)
   #    P <- cbind(S, one, TT)
   P <- rbind(S, t(one))
   Q <- rbind(P, t(TT))
   O <- cbind(one, TT)
   U <- rbind(O, zero)
   star <- cbind(Q, U)
   star <- matrix(star, k + m + 1, k + m + 1)
   A <- eigen(star, symmetric = TRUE)
   deltainv <- diag(1 / A$values)
   gamma <- A$vectors
   starinv <- gamma %*% deltainv %*% t(gamma)
   gamma11 <- matrix(0, k, k)
   for (i in 1:k) {
      for (j in 1:k) {
         gamma11[i, j] <- starinv[i, j]
      }
   }
   gamma21 <- matrix(0, 1, k)
   for (i in 1:1) {
      for (j in 1:k) {
         gamma21[i, j] <- starinv[k + 1, j]
      }
   }
   gamma31 <- matrix(0, m, k)
   for (i in 1:(m)) {
      for (j in 1:k) {
         gamma31[i, j] <- starinv[i + k + 1, j]
      }
   }
   prinwarp <- eigen(gamma11, symmetric = TRUE)
   prinwarps <- prinwarp$vectors
   prinwarpeval <- prinwarp$values
   
   ####need to rotate to compute affine components
   Rot <- prcomp(TT)$rotation
   TT <- TT  %*% Rot
   if (m == 2) {
      meanxy <- c(TT[, 1], TT[, 2])
      alpha <- sum(meanxy[1:k] ^ 2)
      beta <- sum(meanxy[(k + 1):(2 * k)] ^ 2)
      u1 <- c(alpha * meanxy[(k + 1):(2 * k)], beta * meanxy[1:k])
      u2 <- c(-beta * meanxy[1:k], alpha * meanxy[(k + 1):(2 *
                                                              k)])
      u1 <- u1 / sqrt(alpha * beta) / sqrt(alpha + beta)
      u2 <- u2 / sqrt(alpha * beta) / sqrt(alpha + beta)
      
      
      Un <- matrix(0, 2 * k, 2)
      Un[, 1] <- u1
      Un[, 2] <- u2
      Vn <- Un
      Vn[, 1] <-  cbind(Un[1:k, 1], Un[(k + 1):(2 * k), 1]) %*% t(Rot)
      Vn[, 2] <-  cbind(Un[1:k, 2], Un[(k + 1):(2 * k), 2]) %*% t(Rot)
      Un <- Vn
   }
   
   if (m == 3) {
      meanxy <- c(TT[, 1], TT[, 2], TT[, 3])
      alpha <- sum(meanxy[1:k] ^ 2)
      beta <- sum(meanxy[(k + 1):(2 * k)] ^ 2)
      gamma <- sum(meanxy[(2 * k + 1):(3 * k)] ^ 2)
      
      mu <- meanxy[1:k]
      nu <- meanxy[(k + 1):(2 * k)]
      omega <- meanxy[(2 * k + 1):(3 * k)]
      ze <- rep(0, times = k)
      u1 <- c(ze , alpha * beta * omega , alpha * gamma * nu) /
         sqrt(alpha ^ 2 * beta ^ 2 * gamma + alpha ^ 2 * gamma ^
                 2 * beta)
      u2 <- c(alpha * beta * omega , ze,  beta * gamma * mu) /
         sqrt(beta ^ 2 * alpha ^ 2 * gamma + beta ^ 2 * gamma ^
                 2 * alpha)
      u3 <- c(alpha * gamma * nu , beta * gamma * mu, ze) /
         sqrt(alpha ^ 2 * gamma ^ 2 * beta  + beta ^ 2 * gamma ^
                 2 * alpha)
      
      u4 <- c(ze , ze , omega) /
         sqrt(gamma)
      
      u5 <- c(-beta * gamma * mu , alpha * gamma * nu, ze) /
         sqrt(alpha * gamma ^ 2 * beta ^ 2  + beta * gamma ^ 2 *
                 alpha ^ 2)
      
      tem <- c(-gamma * beta * mu , ze,  beta * alpha * omega) /
         sqrt(beta ^ 2 * alpha * gamma ^ 2 + beta ^ 2 * gamma *
                 alpha ^ 2)
      tem2 <- tem - u5 * sum(u5 * tem)
      u4 <- tem2 / Enorm(tem2)
      
      
      
      Un <- matrix(0, 3 * k, 5)
      Un[, 1] <- u1
      Un[, 2] <- u2
      Un[, 3] <- u3
      Un[, 4] <- u4
      Un[, 5] <- u5
      Vn <- Un
      Vn[, 1] <-
         cbind(Un[1:k, 1], Un[(k + 1):(2 * k), 1], Un[(2 * k + 1):(3 * k), 1]) %*%
         t(Rot)
      Vn[, 2] <-
         cbind(Un[1:k, 2], Un[(k + 1):(2 * k), 2], Un[(2 * k + 1):(3 * k), 2]) %*%
         t(Rot)
      Vn[, 3] <-
         cbind(Un[1:k, 3], Un[(k + 1):(2 * k), 3], Un[(2 * k + 1):(3 * k), 3]) %*%
         t(Rot)
      Vn[, 4] <-
         cbind(Un[1:k, 4], Un[(k + 1):(2 * k), 4], Un[(2 * k + 1):(3 * k), 4]) %*%
         t(Rot)
      Vn[, 5] <-
         cbind(Un[1:k, 5], Un[(k + 1):(2 * k), 5], Un[(2 * k + 1):(3 * k), 5]) %*%
         t(Rot)
      Un <- Vn
   }
   
   
   z$gamma11 <- gamma11
   z$gamma21 <- gamma21
   z$gamma31 <- gamma31
   z$prinwarps <- prinwarps
   z$prinwarpeval <- prinwarpeval
   z$Un <- Un
   return(z)
}

#==================================================================================
shaperw <- function(proc ,
                    alpha = 1,
                    affine = FALSE) {
   rw <- proc
   
   if ((alpha != 0) || (affine == TRUE)) {
      k <- dim(proc$mshape)[1]
      m <- dim(proc$mshape)[2]
      n <- dim(proc$mshape)[3]
      
      if (dim(proc$tan)[1] == (k * m - m)) {
         if (m == 2) {
            He <- t(defh(k - 1))
            Ze <- He * 0
            HH <- rbind(cbind(He, Ze) , cbind(Ze, He))
            proc$tan <- HH %*% proc$tan
         }
         if (m == 3) {
            He <- t(defh(k - 1))
            Ze <- He * 0
            HH <-
               rbind(cbind(He, Ze, Ze) ,
                     cbind(Ze, He , Ze) ,
                     cbind(Ze, Ze, He))
            proc$tan <- HH %*% proc$tan
         }
      }
      
      
      
      
      nconstr <- m + m * (m - 1) / 2 + 1
      M <- k * m - nconstr
      
      if (m == 2) {
         bb <- bendingenergy(proc$mshape)
         Gamma11 <- bb$gamma11
         Be <-
            rbind(cbind(Gamma11, Gamma11 * 0) , cbind(Gamma11 * 0, Gamma11))
         Un <- bb$Un
         Bedim <- 2
      }
      if (m == 3) {
         bb <- bendingenergy(proc$mshape)
         Gamma11 <- bb$gamma11
         Ze <- Gamma11 * 0
         Be <-
            rbind(cbind(Gamma11, Ze, Ze) ,
                  cbind(Ze, Gamma11, Ze) ,
                  cbind(Ze, Ze, Gamma11))
         Un <- bb$Un
         Bedim <- 5
      }
      
      ev <- eigen(Be, symmetric = TRUE)
      evpw <- eigen(Gamma11, symmetric = TRUE)
      
      Beminusalpha <-
         ev$vectors %*% diag(c(ev$values[1:(M - Bedim)] ** (-alpha / 2), rep(0, times = nconstr + Bedim))) %*%
         t(ev$vectors)
      Bealpha <-
         ev$vectors %*% diag(c(ev$values[1:(M - Bedim)] ** (alpha / 2),  rep(0, times = nconstr + Bedim))) %*%
         t(ev$vectors)
      
      evbe <- ev
      
      SS <- Beminusalpha %*% var(t(proc$tan)) %*% Beminusalpha
      
      ev <- eigen(SS)
      
      
      relw.vec <- ev$vectors
      relw.sd <- sqrt(abs(ev$values))
      
      
      # ratio of eigenvalues of warps (quoted in book)
      rw$percent <- relw.sd ** 2 / sum(relw.sd ** 2) * 100
      
      
      
      
      sgnchange <- sample(c(-1, 1), size = m * k , replace = TRUE)
      
      rw$pcar <- Bealpha %*% relw.vec %*% diag(sgnchange)
      rw$pcasd <- relw.sd
      
      rw$rawscores <-  t(t(relw.vec) %*% Beminusalpha %*% proc$tan)
      
      sd <- sqrt(abs(diag(var((rw$rawscores)
      ))))
      
      rw$scores <- (rw$rawscores) %*% diag(1 / sd)
      rw$stdscores <- rw$scores
      
      rw$scores <- rw$rawscores
      
      
      
      ## partial warp scores
      
      
      
      n <- proc$n
      evbend <- eigen(Gamma11, symmetric = TRUE)
      
      partialwarpscores <- array(0 , c(n , m , k))
      
      for (i in 1:m) {
         partialwarpscores[, i, ] <-
            t(t(evbend$vectors) %*% proc$rotated[, i, ])
      }
      
      rw$principalwarps <- evpw$vectors[, (k - m - 1):1]
      rw$principalwarps.eigenvalues <- evpw$values[(k - m - 1):1]
      rw$partialwarpscores <- partialwarpscores[, , (k - m - 1):1]
      
      sumvar <- rep(0, times = (k - m - 1))
      
      for (i in 1:(k - m - 1)) {
         sumvar[i] <- sum(diag(var(partialwarpscores[, , k - m - i])))
      }
      
      rw$partialwarps.percent <- sumvar / sum(proc$pcasd ** 2) * 100
      
   }
   
   if (affine == TRUE) {
      dimun <- dim(Un)[2]
      rw$pcar <- Un %*% diag(sgnchange[1:(dimun)])
      pcno <- c(1:dimun)
      rw$rawscores <- t(Un) %*% proc$tan
      
      
      sd <- sqrt(abs(diag(var(
         t(rw$rawscores)
      ))))
      rw$pcasd <- sd
      
      rw$percent <- sd ** 2 / sum(proc$pcasd ** 2) * 100
      
      rw$scores <- t(rw$rawscores) %*% diag(1 / sd)
      rw$rawscores <- t(rw$rawscores)
      #######
      
      tem <- prcomp1((rw$rawscores))
      
      npc <- 0
      rw$stdscores <- tem$x
      for (i in 1:length(tem$sdev)) {
         if (tem$sdev[i] > 1e-07) {
            npc <- npc + 1
         }
      }
      for (i in 1:npc) {
         rw$stdscores[, i] <- tem$x[, i] / tem$sdev[i]
      }
      rw$pcasd <- tem$sdev
      rw$percent <- tem$sdev ** 2 / sum(proc$pcasd ** 2) * 100
      rw$pcar <- Un %*% tem$rotation
      rw$rawscores <- tem$x
      rw$scores <- rw$rawscores
      
      
   }
   
   rw
   
}

#==================================================================================
bookstein2d <- function(A, l1 = 1, l2 = 2) {
   #input:  A: k x 2 x n array of 2D data, or  k x n complex matrix
   #l1,l2: baseline choice for sending to (-0.5,0),(0.5,0)
   #output: z$bshpv - Bookstein shape variables array (including baseline)
   # z$mshape - Bookstein mean shape (including baseline points)
   z <- list(
      k = 0,
      n = 0,
      mshape = 0,
      bshpv = 0
   )
   if (is.complex(sum(A)) == TRUE) {
      n <- dim(A)[2]
      k <- dim(A)[1]
      B <- array(0, c(k, 2, n))
      B[, 1, ] <- Re(A)
      B[, 2, ] <- Im(A)
      A <- B
   }
   
   if (is.matrix(A) == TRUE) {
      bb <- array(A, c(dim(A), 1))
      A <- bb
   }
   k <- dim(A)[1]
   m <- 2
   n <- dim(A)[3]
   reorder <- c(l1, l2, c(1:k)[-c(l1, l2)])
   A[, , ] <- A[reorder, , 1:n]
   bshpv <- array(0, c(k, m, n))
   for (i in 1:n)
   {
      bshpv[, , i] <- bookstein.shpv(A[, , i])
   }
   bookmean <- matrix(0, k, m)
   for (i in 1:n)
   {
      bookmean <- bookmean + bshpv[, , i]
   }
   bookmean <- bookmean / n
   bookmean[reorder, ] <- bookmean
   bshpv[reorder, ,] <- bshpv
   glim <- max(-min(bshpv), max(bshpv))
   #par(pty="s")
   #par(mfrow=c(1,1))
   #plot(bshpv[,,1],xlim=c(-glim,glim),ylim=c(-glim,glim),type="n",xlab="u",ylab="v")
   #for (i in 1:n)
   #{
   #for (j in 1:k){
   #text(bshpv[j,1,i],bshpv[j,2,i],as.character(j))
   #}
   #}
   z$mshape <- bookmean
   z$bshpv <- bshpv
   z$k <- k
   z$n <- n
   return(z)
}

#==================================================================================
bookstein.shpv <- function(x)
{
   #input x:  k x 2 matrix or complex k-vector
   #output u: k x 2 matrix of Bookstein shape variables
   #           with baseline sent to (-0.5,0) (0.5,0)
   if (is.complex(x)) {
      x <- complextoreal(x)
   }
   nj <- dim(x)[1]
   j <- rep(1, times = nj)
   w <-
      (x[, 1] + (1i) * x[, 2] - (j * (x[1, 1] + (1i) * x[1, 2]))) / (x[2,
                                                                       1] + (1i) * x[2, 2] - x[1, 1] - (1i) * x[1, 2]) - 0.5
   w <- w[1:nj]
   y <- (Re(w))
   z <- (Im(w))
   u <- cbind(y, z)
   u <- matrix(u, nj, 2)
   u
}

#==================================================================================
bookstein.shpv.complex <- function(z)
{
   #input z:  complex k vector
   #output u: k-2  complex vector of Bookstein shape variables
   #           with baseline sent to (-0.5) (0.5)
   nj <- length(z)
   x <- matrix(cbind(Re(z), Im(z)), nj, 2)
   j <- rep(1, times = nj)
   w <-
      (x[, 1] + (1i) * x[, 2] - (j * (x[1, 1] + (1i) * x[1, 2]))) / (x[2,
                                                                       1] + (1i) * x[2, 2] - x[1, 1] - (1i) * x[1, 2]) - 0.5
   u <- w[3:nj]
   u
}

#==================================================================================
cbevec <- function(z)
{
   t1 <- reassqpr(z)
   #	t2 <- eigen(t1,symmetric=TRUE,EISPACK=TRUE)
   t2 <- eigen(t1, symmetric = TRUE)
   reagamma <- t2$vectors[, 1]
   #	print(t2$values/sum(t2$values))
   
   gamma <- Vinv(reagamma)
   gamma
}

#==================================================================================
cbevectors <- function(z, j)
{
   t1 <- reassqpr(z)
   #	t2 <- eigen(t1,symmetric=TRUE,EISPACK=TRUE)
   t2 <- eigen(t1, symmetric = TRUE)
   reagamma <- t2$vectors[, j]
   gamma <- Vinv(reagamma)
   gamma
}

#==================================================================================
ild_centroid.size <- function(x)
{
   #returns the centroid size of a configuration (or configurations)
   #input: k x m matrix/or a complex k-vector
   # or input a real k x m x n array to get a vector of sizes for a sample
   if ((is.vector(x) == FALSE) && is.complex(x)) {
      k <- nrow(x)
      n <- ncol(x)
      tem <- array(0, c(k, 2, n))
      tem[, 1, ] <- Re(x)
      tem[, 2, ] <- Im(x)
      x <- tem
   }
   {
      if (length(dim(x)) == 3) {
         n <- dim(x)[3]
         sz <- rep(0, times = n)
         k <- dim(x)[1]
         h <- defh(k - 1)
         for (i in 1:n) {
            xh <- h %*% x[, , i]
            sz[i] <- sqrt(sum(diag(t(xh) %*% xh)))
         }
         sz
      }
      else
      {
         if (is.vector(x) && is.complex(x)) {
            x <- cbind(Re(x), Im(x))
         }
         k <- nrow(x)
         h <- defh(k - 1)
         xh <- h %*% x
         size <- sqrt(sum(diag(t(xh) %*% xh)))
         size
      }
   }
}

#==================================================================================
ild_centroid.size.complex <- function(zstar)
{
   #returns the centroid size of a complex vector zstar
   h <- defh(nrow(as.matrix(zstar)) - 1)
   ztem <- h %*% zstar
   size <- sqrt(diag(Re(st(ztem) %*% ztem)))
   size
}

#==================================================================================
ild_centroid.size.mD <- function(x)
{
   #returns the centroid size of a  k x m matrix
   if (is.complex(x)) {
      x <- cbind(Re(x), Im(x))
   }
   k <- nrow(x)
   h <- defh(k - 1)
   xh <- h %*% x
   size <- sqrt(sum(diag(t(xh) %*% xh)))
   size
}

#==================================================================================
complextoreal <- function(z)
{
   #input complex k-vector  - return k x 2 matrix
   nj <- length(z)
   x <- matrix(cbind(Re(z), Im(z)), nj, 2)
   x
}

#==================================================================================
ild_defh <- function(nrow)
{
   #Defines and returns an nrow x (nrow+1) Helmert sub-matrix
   k <- nrow
   h <- matrix(0, k, k + 1)
   j <- 1
   while (j <= k) {
      jj <- 1
      while (jj <= j) {
         h[j, jj] <- -1 / sqrt(j * (j + 1))
         jj <- jj + 1
      }
      h[j, j + 1] <- j / sqrt(j * (j + 1))
      j <- j + 1
   }
   h
}

#==================================================================================
full.procdist <- function(x, y)
{
   #input  k x 2 matrices x, y
   #output full Procrustes distance rho between x,y
   sin(riemdist(x, y))
}

#==================================================================================
genpower <- function(Be, alpha)
{
   k <- dim(Be)[1]
   if (alpha == 0) {
      gen <- diag(rep(1, times = k))
   }
   else {
      l <- k - 3
      #		eb <- eigen(Be, symmetric = TRUE,EISPACK=TRUE)
      eb <- eigen(Be, symmetric = TRUE)
      ev <- c(eb$values[1:l] ^ (-alpha / 2), 0, 0, 0)
      gen <- eb$vectors %*% diag(ev) %*% t(eb$vectors)
      gen
   }
}

#==================================================================================
isotropy.test <- function(sd, p, n)
{
   #LR test for isotropy with Bartlett adjustment
   #in: sd - square roots of eigenvalues of covariance matrix
   #     p - the number of larger eigenvalues to consider
   #     n - sample size
   #out: z$bartlett  - test statistic (e.g. see Mardia, Kent, Bibby, 1979, p235)
   #     z$pval - p-value
   z <- list(bartlett = 0, pval = 0)
   tem <- sd ^ 2
   bartlett <-
      (log(mean(tem[1:p])) - mean(log(tem[1:p]))) * p * (n - (2 *
                                                                 p + 11) / 6)
   pval <- 1 - pchisq(bartlett, ((p + 2) * (p - 1)) / 2)
   z$bartlett <- bartlett
   z$pval <- pval
   return(z)
}

#==================================================================================
linegrid <- function(ref, kx, ky)
{
   n <- ky
   m <- kx
   w <- n * m
   newgrid1 <- matrix(0, w, 2)
   v <- m * 0.5
   k <- 0
   for (l in 1:v) {
      k <- k + 1
      a <- (n + m - 1) * (k - 1) + 1
      b <- n * ((2 * k) - 1)
      d <- 2 * n * k
      for (j in a:b) {
         newgrid1[j,] <- ref[j,]
      }
      for (u in 1:n) {
         down <- d - u + 1
         up <- b + u
         newgrid1[up,] <- ref[down,]
      }
   }
   newgrid2 <- matrix(0, w, 2)
   for (i in 1:v) {
      z <- (2 * i) - 1
      for (x in 1:m) {
         r1 <- m * (z - 1) + x
         e <- n * (x - 1) + z
         newgrid2[r1,] <- ref[e,]
      }
   }
   y <- v - 1
   for (p in 1:y) {
      f <- 2 * p
      for (q in 1:m) {
         r2 <- m * (f - 1) + q
         s <- n * (m - 1) + f - n * (q - 1)
         newgrid2[r2,] <- ref[s,]
      }
   }
   order <- rbind(newgrid1, newgrid2)
   order
}

#==================================================================================
mahpreshapedist <- function(z, m, pcar, pcasdev)
{
   if (is.double(z) == TRUE)
      z <- realtocomplex(z)
   if (is.double(m) == TRUE)
      m <- realtocomplex(m)
   w <- preshape(z)
   y <- preshape(m)
   zp <- project(w, y)
   k <- length(pcasdev) / 2
   if (pcasdev[2 * k - 1] < 1e-07)
      pcasdev[2 * k - 1] <- 1e+22
   if (pcasdev[2 * k] < 1e-07)
      pcasdev[2 * k] <- 1e+22
   Sinv <- (pcar) %*% diag(1 / pcasdev ^ 2) %*% t(pcar)
   Z <- V(zp)
   d2 <- t(Z) %*% Sinv %*% (Z)
   dist <- sqrt(d2)
   dist
}
makearray <- function(x, k, m, n) {
   #makes a k x m x n array from a dataset read in as a table
   tem <- c(t(x))
   tem <- array(tem, c(m, k, n))
   tem <- aperm(tem, c(2, 1, 3))
   tem
}

#==================================================================================
movie <-
   function(mean,
            pc,
            sd,
            xl,
            xu,
            yl,
            yu,
            lineorder,
            movielength = 20)
   {
      k <- length(mean) / 2
      for (i in 1:movielength) {
         plotPDMnoaxis(mean, pc * (-1) ^ i, sd, xl, xu, yl, yu, lineorder)
      }
      plot(
         mean[c(1:k)],
         mean[c((k + 1):(2 * k))],
         xlim = c(xl, xu),
         ylim = c(yl, yu),
         xlab = " ",
         ylab = " ",
         axes = FALSE
      )
   }

#==================================================================================
ild_Enorm <- function(X)
{
   #finds Euclidean/Frobenius norm of a matrix X
   if (is.complex(X)) {
      n <- sqrt(sum(diag(Re(st(X) %*% X))))
   }
   else {
      n <- sqrt(sum(diag(t(X) %*% X)))
   }
   n
}

#==================================================================================
partial.procdist <- function(x, y)
{
   #input  k x 2 matrices x, y
   #output partial Procrustes distance rho between x,y
   sqrt(2) * sqrt(1 - cos(riemdist(x, y)))
}

#==================================================================================
partialwarpgrids <-
   function(TT, YY, xbegin, ybegin, xwidth, nr, nc, mag)
   {
      #
      #affine grid and partial warp grids for the TPS deformation of TT to YY
      #displayed as an nr x nc array of plots
      #mag = magnification effect
      k <- nrow(TT)
      YY <- TT + (YY - TT) * mag
      xstart <- xbegin
      ystart <- ybegin
      kx <- 22
      ky <- 21
      l <- kx * ky
      step <- xwidth / (kx - 1)
      r <- 0
      X <- rep(0, times = 220)
      Y2 <- rep(0, times = 220)
      for (p in 1:kx) {
         ystart <- ybegin
         xstart <- xstart + step
         for (q in 1:ky) {
            ystart <- ystart + step
            r <- r + 1
            X[r] <- xstart
            Y2[r] <- ystart
         }
      }
      refc <- matrix(c(X, Y2), kx * ky, 2)
      TPS <- bendingenergy(TT)
      gamma11 <- TPS$gamma11
      gamma21 <- TPS$gamma21
      gamma31 <- TPS$gamma31
      W <- gamma11 %*% YY
      ta <- t(gamma21 %*% YY)
      B <- gamma31 %*% YY
      WtY <- t(W) %*% YY
      R <- matrix(0, k, 2)
      par(mfrow = c(nr, nc))
      par(pty = "s")  #AFFINEPART
      phi <- matrix(0, l, 2)
      s <- matrix(0, k, 1)
      for (i in 1:l) {
         s <- matrix(0, k, 1)
         for (m in 1:k) {
            s[m,] <- sigmacov(refc[i,] - TT[m,])
         }
         phi[i,] <- ta + t(B) %*% refc[i,]
      }
      newpt <- matrix(0, k, 2)
      for (i in 1:k) {
         s <- matrix(0, k, 1)
         for (m in 1:k) {
            s[m,] <- sigmacov(TT[i,] - TT[m,])
         }
         newpt[i,] <- ta + t(B) %*% TT[i,]
      }
      order <- linegrid(phi, kx, ky)
      plot(
         order[1:l, 1],
         order[1:l, 2],
         type = "l",
         xlim = c(xbegin - xwidth /
                     10, xbegin + (xwidth * 11) / 10),
         ylim = c(ybegin - (xwidth / 10 *
                               ky) / kx, ybegin + ((xwidth * 11) / 10 * ky) / kx),
         xlab = " ",
         ylab
         = " "
      )
      lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l), 2], type = "l")
      points(newpt, cex = 2)
      for (jnw in 1:(k - 3)) {
         nw <- k - 2 - jnw
         phi <- matrix(0, l, 2)
         s <- matrix(0, k, 1)
         for (i in 1:l) {
            s <- matrix(0, k, 1)
            for (m in 1:k) {
               s[m,] <- sigmacov(refc[i,] - TT[m,])
            }
            phi[i,] <- refc[i,] + TPS$prinwarpeval[nw] * t(YY) %*%
               TPS$prinwarps[, nw] %*% t(TPS$prinwarps[, nw]) %*%
               s
         }
         newpt <- matrix(0, k, 2)
         for (i in 1:k) {
            s <- matrix(0, k, 1)
            for (m in 1:k) {
               s[m,] <- sigmacov(TT[i,] - TT[m,])
            }
            newpt[i,] <- TT[i,] + TPS$prinwarpeval[nw] * t(YY) %*%
               TPS$prinwarps[, nw] %*% t(TPS$prinwarps[, nw]) %*%
               s
         }
         R <- newpt - TT + R
         order <- linegrid(phi, kx, ky)
         plot(
            order[1:l, 1],
            order[1:l, 2],
            type = "l",
            xlim = c(xbegin -
                        xwidth / 10, xbegin + (xwidth * 11) / 10),
            ylim = c(ybegin - (xwidth / 10 * ky) / kx, ybegin + ((xwidth * 11) / 10 * ky) /
                        kx),
            xlab = " ",
            ylab = " "
         )
         lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l), 2],
               type = "l")
         points(newpt, cex = 2)
      }
      #percentage  (need to normalize)
      d2 <- sin(riemdist(YY, TT)) ^ 2
      d3 <- sin(riemdist(R + TT, TT)) ^ 2
      percentaff <- (d2 - d3) / d2 * 100
      print("percent affine")
      print(percentaff)
   }

#==================================================================================
partialwarps <- function(mshape, rotated)
{
   #obtain the affine and partial warp scores for a dataset
   #where the reference configuration is mshape and the full procrustes
   #rotated figures are given in the array rotated
   #output: y$pwpwercent percentage of variability (squared Procrustes distance)
   #  in the direction of each of the affine and principal warps
   #        y$pwscores: the affine and partial warps scores
   #
   y <- list(pwpercent = 0,
             pwscores = 0,
             unpercent = 0)
   k <- nrow(mshape)
   n <- dim(rotated)[3]
   msh <- mshape
   rot <- rotated
   TPS <- bendingenergy(msh)
   FX <- rot[, 1,]
   FY <- rot[, 2,]
   U <- TPS$prinwarps[, 1:(k - 3)]
   partialX <- t(U) %*% FX
   partialY <- t(U) %*% FY
   Un <- TPS$Un
   UnXY <- t(Un) %*% rbind(FX, FY)
   scores <- matrix(0, 2 * (k - 3), n)
   for (i in 1:(k - 3)) {
      r <- 2 * i - 1
      scores[r,] <- partialX[k - 2 - i,]
      scores[r + 1,] <- partialY[k - 2 - i,]
   }
   scores <- rbind(UnXY, scores)
   percwarp <- rep(0, times = (k - 2))
   sumev <- sum(eigen(var(t(scores)))$values)
   for (i in 1:(k - 2)) {
      sum1 <- sum(eigen(var(t(scores[(2 * i - 1):(2 * i),])))$values)
      percwarp[i] <- sum1 / sumev
   }
   unpercent <- c(0, 0)
   unpercent[1] <- var(scores[1,]) / sumev
   unpercent[2] <- var(scores[2,]) / sumev
   y$unpercent <- unpercent
   y$pwpercent <- percwarp
   y$pwscores <- t(scores)
   return(y)
}

#==================================================================================
plot2rwscores <- function(rwscores, rw1, rw2, ng1, ng2)
{
   par(pch = "x")
   glim <- max(-min(rwscores), max(rwscores))
   plot(
      rwscores[1:ng1, rw1],
      rwscores[1:ng1, rw2],
      xlim = c(-glim, glim),
      ylim = c(-glim, glim),
      xlab = " ",
      ylab = " "
   )
   par(pch = "+")
   points(rwscores[(ng1 + 1):(ng1 + ng2), rw1], rwscores[(ng1 + 1):(ng1 +
                                                                       ng2), rw2])
}

#==================================================================================
plotPDM <- function(mean, pc, sd, xl, xu, yl, yu, lineorder)
{
   for (i in c(-3, 0, 3)) {
      fig <- mean + i * pc * sd
      k <- length(mean) / 2
      figx <- fig[1:k]
      figy <- fig[(k + 1):(2 * k)]
      plot(
         figx,
         figy,
         axes = TRUE,
         xlab = " ",
         ylab = " ",
         ylim = c(yl,
                  yu),
         xlim = c(xl, xu)
      )  #               par(lty = i + 1)
      lines(figx[lineorder], figy[lineorder])
      if (i == -3)
         title(sub = "mean - c sd")
      if (i == 0)
         title(sub = "mean")
      if (i == 3)
         title(sub = "mean + c sd")
      par(lty = 1)
   }
}

#==================================================================================
plotPDM2 <- function(mean, pc, sd, xl, xu, yl, yu, lineorder)
{
   par(lty = 1)
   k <- length(mean) / 2
   plot(
      mean[1:k],
      mean[(k + 1):(2 * k)],
      axes = TRUE,
      xlab = " ",
      ylab =
         "  ",
      ylim = c(yl, yu),
      xlim = c(xl, xu)
   )
   for (i in c(-3:3)) {
      fig <- mean + i * pc * sd
      figx <- fig[1:k]
      figy <- fig[(k + 1):(2 * k)]    #
      if (i < 0) {
         par(lty = 1)
         par(pch = "*")
      }
      if (i == 0) {
         par(lty = 4)
         par(pch = 1)
      }
      if (i > 0) {
         par(lty = 2)
         par(pch = "+")
      }
      points(figx, figy)
      lines(figx[lineorder], figy[lineorder])
   }
}

#==================================================================================
plotPDM3 <- function(mean, pc, sd, xl, xu, yl, yu, lineorder)
{
   par(lty = 1)
   k <- length(mean) / 2
   figx <- matrix(0, 2 * k, 7)
   figy <- figx
   plot(
      mean[1:k],
      mean[(k + 1):(2 * k)],
      axes = TRUE,
      xlab = " ",
      ylab =
         "  ",
      ylim = c(yl, yu),
      xlim = c(xl, xu)
   )
   for (i in c(-3:3)) {
      fig <- mean + i * pc * sd
      figx[, i + 4] <- fig[1:k]
      figy[, i + 4] <- fig[(k + 1):(2 * k)]
   }
   for (i in 1:k) {
      #               par(lty = 2)
      #               lines(figx[i, 1:4], figy[i, 1:4])
      par(lty = 1)
      lines(figx[i, 4:7], figy[i, 4:7])
   }
}

#==================================================================================
plotPDMbook <- function(mean, pc, sd, xl, xu, yl, yu, lineorder)
{
   par(lty = 1)
   k <- length(mean) / 2
   figx <- matrix(0, 2 * k, 7)
   figy <- figx
   plot(
      bookstein.shpv(cbind(mean[1:k], mean[(k + 1):(2 * k)])),
      axes = TRUE,
      xlab = " ",
      ylab = "  ",
      ylim = c(yl, yu),
      xlim = c(xl, xu)
   )
   for (i in c(-3:3)) {
      fig <- mean + i * pc * sd
      figx[, i + 4] <- fig[1:k]
      figy[, i + 4] <- fig[(k + 1):(2 * k)]
      u <- bookstein.shpv(cbind(figx[, i + 4], figy[, i + 4]))
      figx[, i + 4] <- u[, 1]
      figy[, i + 4] <- u[, 2]
   }
   for (i in 1:k) {
      #               par(lty = 2)
      #               lines(figx[i, 1:4], figy[i, 1:4])
      par(lty = 1)
      lines(figx[i, 4:7], figy[i, 4:7])
   }
}

#==================================================================================
plotPDMnoaxis <- function(mean, pc, sd, xl, xu, yl, yu, lineorder)
{
   for (i in c(-3:3)) {
      fig <- mean + i * pc * sd
      k <- length(mean) / 2
      figx <- fig[1:k]
      figy <- fig[(k + 1):(2 * k)]
      plot(
         figx,
         figy,
         axes = FALSE,
         xlab = " ",
         ylab = " ",
         ylim = c(yl,
                  yu),
         xlim = c(xl, xu)
      )
      lines(figx[lineorder], figy[lineorder])
      for (ii in 1:1000) {
         aa <- 1
      }
   }
}

#==================================================================================
pointsPDMnoaxis3 <-
   function(mean, pc, sd, xl, xu, yl, yu, lineorder, i)
   {
      fig <- mean + i * pc * sd
      k <- length(mean) / 2
      figx <- fig[1:k]
      figy <- fig[(k + 1):(2 * k)]
      points(figx, figy)
      text(figx, figy, 1:k)
      lines(figx[lineorder], figy[lineorder])
   }

#==================================================================================
plotpairscores <- function(scores, nr, nc, ng1, ng2, ch1, ch2)
{
   #plots pairs of scores score 2 vs score 1, score 4 vs score 3 etc
   #in an nr x nc grid of plots
   par(pty = "s")
   par(cex = 2)
   par(mfrow = c(nr, nc))
   k <- ncol(scores) / 2 + 2
   glim <- max(-min(scores), max(scores))
   for (i in 1:(k - 2)) {
      plot(
         scores[1:ng1, (2 * i - 1)],
         scores[1:ng1, (2 * i)],
         pch =
            ch1,
         xlim = c(-glim, glim),
         ylim = c(-glim, glim),
         xlab = " ",
         ylab = " "
      )
      points(scores[(ng1 + 1):(ng1 + ng2), (2 * i - 1)], scores[(ng1 +
                                                                    1):(ng1 + ng2), (2 * i)], pch = ch2)
   }
}
#################################

#==================================================================================
plotpca <-
   function (proc,
             pcno,
             type,
             mag,
             xl,
             yl,
             width,
             joinline = c(1,
                          1),
             project = c(1, 2))
   {
      k <- proc$k
      zero <- matrix(0, k - 1, k)
      h <- defh(k - 1)
      H <- cbind(h, zero)
      H1 <- cbind(zero, h)
      H <- rbind(H, H1)
      if (project[1] == 1) {
         select1 <- 1:k
      }
      if (project[1] == 2) {
         select1 <- (k + 1):(2 * k)
      }
      if (project[1] == 3) {
         select1 <- (2 * k + 1):(3 * k)
      }
      if (project[2] == 1) {
         select2 <- 1:k
      }
      if (project[2] == 2) {
         select2 <- (k + 1):(2 * k)
      }
      if (project[2] == 3) {
         select2 <- (2 * k + 1):(3 * k)
      }
      select <- c(select1, select2)
      meanxy <- c(proc$mshape[, project[1]], proc$mshape[, project[2]])
      
      if (dim(proc$pcar)[1] == (2 * (k - 1))) {
         pcarot <- (t(H) %*% proc$pcar)[select, ]
      }
      if (dim(proc$pcar)[1] != (2 * (k - 1))) {
         pcarot <- proc$pcar[select, ]
      }
      par(pty = "s")
      par(lty = 1)
      np <- length(pcno)
      nr <- trunc((length(pcno) + 1) / 2)
      if (type == "g") {
         par(mfrow = c(nr, 2))
         if (np == 1) {
            par(mfrow = c(1, 1))
         }
         for (i in 1:np) {
            j <- pcno[i]
            fig <- meanxy + pcarot[, j] * 3 * mag * proc$pcasd[j]
            figx <- fig[1:k]
            figy <- fig[(k + 1):(2 * k)]
            YY <- cbind(figx, figy)
            tpsgrid(cbind(proc$mshape[, project[1]], proc$mshape[, project[2]])
                    ,
                    YY,
                    xl,
                    yl,
                    width,
                    1,
                    0.1,
                    22)
         }
      }
      else {
         if (type == "r") {
            par(mfrow = c(np, 3))
            for (i in 1:np) {
               j <- pcno[i]
               plotPDM(meanxy,
                       pcarot[, j],
                       mag * proc$pcasd[j],
                       xl,
                       xl + width,
                       yl,
                       yl + width,
                       joinline)
               title(as.character(
                  paste(
                     "PC ",
                     as.character(pcno[i]),
                     ": ",
                     as.character(round(proc$percent[i], 1)),
                     "%"
                  )
               ))
            }
         }
         else {
            if (type == "v") {
               par(mfrow = c(nr, 2))
               if (np == 1) {
                  par(mfrow = c(1, 1))
               }
               for (i in 1:np) {
                  j <- pcno[i]
                  plotPDM3(meanxy,
                           pcarot[, j],
                           mag * proc$pcasd[j],
                           xl,
                           xl + width,
                           yl,
                           yl + width,
                           joinline)
                  title(as.character(
                     paste(
                        "PC ",
                        as.character(pcno[i]),
                        ": ",
                        as.character(round(proc$percent[i],
                                           1)),
                        "%"
                     )
                  ))
               }
            }
            else {
               if (type == "b") {
                  par(mfrow = c(nr, 2))
                  if (np == 1) {
                     par(mfrow = c(1, 1))
                  }
                  for (i in 1:np) {
                     j <- pcno[i]
                     plotPDMbook(meanxy,
                                 pcarot[, j],
                                 mag * proc$pcasd[j],-0.6,
                                 0.6,
                                 -0.6,
                                 0.6,
                                 joinline)
                     title(as.character(
                        paste(
                           "PC ",
                           as.character(pcno[i]),
                           ": ",
                           as.character(round(proc$percent[i],
                                              1)),
                           "%"
                        )
                     ))
                  }
               }
               else {
                  if (type == "s") {
                     par(mfrow = c(nr, 2))
                     if (np == 1) {
                        par(mfrow = c(1, 1))
                     }
                     for (i in 1:np) {
                        j <- pcno[i]
                        plotPDM2(
                           meanxy,
                           pcarot[, j],
                           mag * proc$pcasd[j],
                           xl,
                           xl + width,
                           yl,
                           yl + width,
                           joinline
                        )
                        title(as.character(
                           paste(
                              "PC ",
                              as.character(pcno[i]),
                              ": ",
                              as.character(round(proc$percent[i],
                                                 1)),
                              "%"
                           )
                        ))
                     }
                  }
                  else {
                     if (type == "m") {
                        par(mfrow = c(1, 1))
                        for (i in 1:np) {
                           j <- pcno[i]
                           cat(paste("PC ", pcno[i], " \n"))
                           movie(
                              meanxy,
                              pcarot[, j],
                              mag * proc$pcasd[j],
                              xl,
                              xl + width,
                              yl,
                              yl + width,
                              joinline,
                              20
                           )
                        }
                     }
                  }
               }
            }
         }
      }
      par(mfrow = c(1, 1))
   }

##############################################

#==================================================================================
plotprinwarp <- function(TT, xbegin, ybegin, xwidth, nr, nc)
{
   #
   #plots the principal warps of TT as perspective plots
   #the plots are displayed  in an nr x nc array of plots
   kx <- 21
   k <- nrow(TT)
   l <- kx ^ 2
   xstart0 <- xbegin
   ystart0 <- ybegin
   xstart <- xstart0
   ystart <- ystart0
   step <- xwidth / kx
   r <- 0
   X <- rep(0, times = l)
   Y2 <- rep(0, times = l)
   for (p in 1:kx) {
      ystart <- ystart0
      xstart <- xstart + step
      for (q in 1:kx) {
         ystart <- ystart + step
         r <- r + 1
         X[r] <- xstart
         Y2[r] <- ystart
      }
   }
   refperp <- matrix(c(X, Y2), l, 2)
   xstart <- xstart0
   xgrid <- rep(0, times = kx)
   for (i in 1:kx) {
      xstart <- xstart + step
      xgrid[i] <- xstart
   }
   ystart <- ystart0
   ygrid <- rep(0, times = kx)
   for (i in 1:kx) {
      ystart <- ystart + step
      ygrid[i] <- ystart
   }
   TPS <- bendingenergy(TT)
   prinwarp <- TPS$prinwarps
   phi <- matrix(0, l, k - 3)
   s <- matrix(0, k, 1)
   for (i in 1:l) {
      s <- matrix(0, k, 1)
      for (m in 1:k) {
         s[m,] <- sigmacov(refperp[i,] - TT[m,])
      }
      phi[i,] <- diag(sqrt(TPS$prinwarpeval[1:(k - 3)])) %*% t(prinwarp[, 1:(k - 3)]) %*% s
   }
   phiTT <- matrix(0, k, k - 3)
   for (i in 1:k) {
      s <- matrix(0, k, 1)
      for (m in 1:k) {
         s[m,] <- sigmacov(TT[i,] - TT[m,])
      }
      phiTT[i,] <- diag(sqrt(TPS$prinwarpeval[1:(k - 3)])) %*% t(prinwarp[, 1:(k - 3)]) %*% s
   }
   par(mfrow = c(nr, nc))
   for (nw in 1:(k - 3)) {
      zgrid <- matrix(0, kx, kx)
      m <- 0
      for (i in 1:kx) {
         for (j in 1:kx) {
            m <- m + 1
            zgrid[i, j] <- phi[m, k - 2 - nw]
         }
      }
      zpersp <- persp(xgrid, ygrid, zgrid, axes = TRUE)
      #  NB the following is an S-Plus function : use trans3d() in R
      #		points(perspp(TT[, 1], TT[, 2], phiTT[, k - 2 - nw], zpersp),
      #			cex = 2)
   }
}

#==================================================================================
plotproc <- function(proc, xl, yl, width, joinline = c(1, 1))
{
   #provides plots of the full Procrustes rotated objects in proc
   #proc is an S object of the type output from the function procrustes2d
   #xl, yl lower xlimit and ylimit in plot
   #width = width (and height) of the square plotting region
   par(pty = "s")
   plot(
      proc$rotated[,  , 1],
      xlim = c(xl, xl + width),
      ylim = c(yl, yl +
                  width),
      type = "n",
      xlab = "",
      ylab = ""
   )
   for (i in 1:proc$n) {
      points(proc$rotated[,  , i])
      lines(proc$rotated[joinline,  , i])
   }
}

#==================================================================================
plotrelwarp <-
   function(mshape,
            rotsd,
            pcno,
            type,
            mag,
            xl,
            yl,
            width,
            joinline)
   {
      #provides PC plots: similar to plotpca but different argument
      #here rotsd is the rotation x s.d. , and can be from the usual
      # PCA or from using relative warps
      #pcno is a vector of the numbers (index) of PCs to be plotted
      #e.g. pcno<-c(1,2,4,7) will plot the four PCs no. 1,2,4,7
      #type = type of display
      #  "r" : rows along PCs evaluated at c = -3,-2,-1,0,1,2,3 sd's along PC
      #  "v" : vectors drawn from mean to +/- 3 sd's along PC
      #  "b" : vectors drawn as in `v' but using Bookstein shape variables
      #  "s" : plots along c= -3, -2, -1, 0, 1, 2, 3 superimposed
      #  "m" : movie backward and forwards from -3 to +3 sd's along PC
      #
      #mag = magnification of effect (1 = use s.d.'s from the data)
      #xl, yl lower xlimit and ylimit in plot
      #width = width (and height) of the square plotting region
      #joinline = vector of landmark numbers which are joined up in the plot by
      #straight lines: joinline = c(1,1) will give no lines
      #
      k <- nrow(mshape)
      pcarot <- rotsd
      par(pty = "s")
      par(lty = 1)
      meanxy <- c(mshape[, 1], mshape[, 2])
      np <- length(pcno)
      if (type == "g") {
         par(mfrow = c(1, np))
         for (i in 1:np) {
            j <- pcno[i]
            fig <- meanxy + pcarot[, j] * mag * 3
            figx <- fig[1:k]
            figy <- fig[(k + 1):(2 * k)]
            YY <- cbind(figx, figy)
            tpsgrid(mshape, YY, xl, yl, width, 1, 0.1, 22)
         }
      }
      else {
         if (type == "r") {
            par(mfrow = c(np, 7))
            for (i in 1:np) {
               j <- pcno[i]
               plotPDM(meanxy,
                       pcarot[, j],
                       mag,
                       xl,
                       xl +
                          width,
                       yl,
                       yl + width,
                       joinline)
            }
         }
         else {
            if (type == "v") {
               par(mfrow = c(1, np))
               for (i in 1:np) {
                  j <- pcno[i]
                  plotPDM3(meanxy,
                           pcarot[, j],
                           mag,
                           xl,
                           xl +
                              width,
                           yl,
                           yl + width,
                           joinline)
               }
            }
            else {
               if (type == "b") {
                  par(mfrow = c(1, np))
                  for (i in 1:np) {
                     j <- pcno[i]
                     plotPDMbook(meanxy,
                                 pcarot[, j],
                                 mag,
                                 xl,
                                 xl + width,
                                 yl,
                                 yl + width,
                                 joinline)
                  }
               }
               else {
                  if (type == "s") {
                     par(mfrow = c(1, np))
                     for (i in 1:np) {
                        j <- pcno[i]
                        plotPDM2(meanxy,
                                 pcarot[, j],
                                 mag,
                                 xl,
                                 xl +
                                    width,
                                 yl,
                                 yl + width,
                                 joinline)
                     }
                  }
                  else {
                     if (type == "m") {
                        par(mfrow = c(1, 1))
                        for (i in 1:np) {
                           j <- pcno[i]
                           movie(meanxy,
                                 pcarot[, j],
                                 mag,
                                 xl,
                                 xl +
                                    width,
                                 yl,
                                 yl + width,
                                 joinline,
                                 20)
                        }
                     }
                  }
               }
            }
         }
      }
      par(mfrow = c(1, 1))
   }

#==================================================================================
ild_preshape <- function(x)
{
   #input k x m matrix / complex k-vector
   #output k-1 x m matrix / k-1 x 1 complex matrix
   if (is.complex(x)) {
      k <- nrow(as.matrix(x))
      h <- defh(k - 1)
      zstar <- x
      ztem <- h %*% zstar
      size <- sqrt(diag(Re(st(ztem) %*% ztem)))
      if (is.vector(zstar))
         z <- ztem / size
      if (is.matrix(zstar))
         z <- ztem %*% diag(1 / size)
   }
   else {
      if (length(dim(x)) == 3) {
         k <- dim(x)[1]
         h <- defh(k - 1)
         n <- dim(x)[3]
         m <- dim(x)[2]
         z <- array(0, c(k - 1, m, n))
         for (i in 1:n) {
            z[,  , i] <- h %*% x[,  , i]
            size <- centroid.size(x[,  , i])
            z[,  , i] <- z[,  , i] / size
         }
      }
      else {
         k <- nrow(as.matrix(x))
         h <- defh(k - 1)
         ztem <- h %*% x
         size <- centroid.size(x)
         z <- ztem / size
      }
   }
   z
}

#==================================================================================
ild_preshape.mD <- function(x)
{
   #input k x m matrix
   #output k-1 x 1 matrix
   h <- defh(nrow(x) - 1)
   ztem <- h %*% x
   size <- centroid.size.mD(x)
   z <- ztem / size
   z
}

#==================================================================================
ild_preshape.mat <- function(zstar)
{
   h <- defh(nrow(as.matrix(zstar)) - 1)
   ztem <- h %*% zstar
   size <- sqrt(diag(Re(st(ztem) %*% ztem)))
   if (is.vector(zstar))
      z <- ztem / size
   if (is.matrix(zstar))
      z <- ztem %*% diag(1 / size)
   z
}

#==================================================================================
ild_preshapetoicon <- function(z)
{
   #convert a preshape (real or complex) to an icon in configuration space
   h <- defh(nrow(z))
   t(h) %*% z
}
#
#
#
#prcomp1<-function(x, retx = TRUE)
#{
#	s <- svd(scale(x, scale = FALSE), nu = 0)	# remove column means
#	rank <- sum(s$d > 0)
#	if(rank < ncol(x))
#		s$v <- s$v[, 1:rank]
#	s$d <- s$d/sqrt(max(1, nrow(x) - 1))
#	if(retx)
#		list(sdev = s$d, rotation = s$v, x = x %*% s$v)
#	else list(sdev = s$d, rotation = s$v)
#}

#==================================================================================
prinwscoregrids <-
   function(TT,
            TPS,
            score,
            xbegin,
            ybegin,
            xwidth,
            nr,
            nc)
   {
      #grids displaying the effect of each principal warp at `score'
      #along each warp. Grids displayed in an nr x nc array
      par(pty = "s")
      par(mfrow = c(nr, nc))
      k <- nrow(TT)
      xstart <- xbegin
      ystart <- ybegin
      kx <- 22
      ky <- 21
      l <- kx * ky
      step <- xwidth / (kx - 1)
      r <- 0
      X <- rep(0, times = 220)
      Y2 <- rep(0, times = 220)
      for (p in 1:kx) {
         ystart <- ybegin
         xstart <- xstart + step
         for (q in 1:ky) {
            ystart <- ystart + step
            r <- r + 1
            X[r] <- xstart
            Y2[r] <- ystart
         }
      }
      refc <-
         matrix(c(X, Y2), kx * ky, 2)    #       TPS <- bendingenergy(TT)
      for (jnw in 1:(k - 3)) {
         nw <- k - 2 - jnw
         phi <- matrix(0, l, 2)
         s <- matrix(0, k, 1)
         for (i in 1:l) {
            s <- matrix(0, k, 1)
            for (m in 1:k) {
               s[m,] <- sigmacov(refc[i,] - TT[m,])
            }
            phi[i,] <- refc[i,] + sqrt(TPS$prinwarpeval[nw]) *
               score * t(TPS$prinwarps[, nw]) %*% s
         }
         newpt <- matrix(0, k, 2)
         for (i in 1:k) {
            s <- matrix(0, k, 1)
            for (m in 1:k) {
               s[m,] <- sigmacov(TT[i,] - TT[m,])
            }
            newpt[i,] <- TT[i,] + sqrt(TPS$prinwarpeval[nw]) *
               score * t(TPS$prinwarps[, nw]) %*% s
         }
         order <- linegrid(phi, kx, ky)
         plot(
            order[1:l, 1],
            order[1:l, 2],
            type = "l",
            xlim = c(xbegin -
                        xwidth / 10, xbegin + (xwidth * 11) / 10),
            ylim = c(ybegin - (xwidth / 10 * ky) / kx, ybegin + ((xwidth * 11) / 10 * ky) /
                        kx),
            xlab = " ",
            ylab = " "
         )
         lines(order[(l + 1):(2 * l), 1], order[(l + 1):(2 * l), 2],
               type = "l")
         points(newpt, cex = 2)
      }
   }

#==================================================================================
procdistreflect <- function(x, y)
{
   #input  k x m matrices x, y
   #output reflection shape distance (rho*) between them
   #if x, y are not too far apart then (rho*)=rho (Riemannian dist)
   if (sum((x - y) ^ 2) == 0) {
      riem <- 0
   }
   if (sum((x - y) ^ 2) != 0) {
      m <- ncol(x)
      z <- preshape(x)
      w <- preshape(y)
      Q <- t(z) %*% w %*% t(w) %*% z
      ev <- sqrt(eigen(Q, symmetric = TRUE)$values)
      #	riem <- acos(sum(ev))
      riem <- acos(min(sum(ev), 1))
   }
   riem
}

#==================================================================================
procrustes2d <-
   function(x,
            l1 = 1,
            l2 = 2,
            approxtangent = FALSE,
            expomap = FALSE)
   {
      #input k x 2 x n real array, or k x n complex matrix
      #mean shape will have landmarks l1, l2 horizontal (l1 left, l2 right)
      #
      #output:
      # z$k : no of landmarks
      # z$m : no of dimensions (=2 here)
      # z$n : sample size
      # z$tan : the real 2k-2 x n matrix of partial Procrustes tangent coordinates
      #          with pole given by the preshape of the full Procrustes mean
      # z$rotated : the k x m x n array of real full Procrustes rotated data
      # z$pcar : the columns are eigenvectors (PCs) of the sample covariance Sv of z$tan
      # z$pcasd : the square roots of eigenvalues of Sv (s.d.'s of PCs)
      # z$percent : the % of variability explained by the PCs
      # z$scores : PC scores normalised to have unit variance
      # z$rawscores : PC scores (unnormalised)
      # z$size : the centroid sizes of the configurations
      # z$rho : Kendall's Procrustean (Riemannian) distance rho to the mean shape
      # z$rmsrho : r.m.s. of rho
      # z$rmsd1 : r.m.s. of full Procrustes distances to the mean shape d1
      #
      z <- list(
         k = 0,
         m = 0,
         n = 0,
         rotated = 0,
         tan = 0,
         pcar = 0,
         scores = 0,
         rawscores = 0,
         pcasd = 0,
         percent = 0,
         size = 0,
         rho = 0,
         rmsrho = 0,
         rmsd1 = 0,
         mshape = 0
      )
      if (is.complex(x) == FALSE) {
         x <- x[, 1,] + (1i) * x[, 2,]
      }
      #	cat("Procrustes 2D eigenanalysis \n")
      k <- nrow(x)
      n <- ncol(x)
      h <- defh(k - 1)
      zp <- preshape(x)
      gamma <- cbevec(zp)
      cbmean <- t(h) %*% gamma
      theta <- Arg(cbmean[l2] - cbmean[l1])
      cbmeanrot <- exp((-0 - 1i) * theta) * cbmean
      gamma <- h %*% cbmeanrot
      tan <- project(zp, gamma)
      icon <- array(0, c(k, 2, n))
      tanapprox <- matrix(0, 2 * k, n)
      size <- rep(0, times = n)
      rho <- rep(0, times = n)
      mu <- complextoreal(cbmeanrot)
      sum <- 0
      for (i in 1:n) {
         tem <- tanfigurefull(tan[, i], gamma)
         icon[, 1, i] <- Re(tem)
         icon[, 2, i] <- Im(tem)
         sum <- sum + icon[, , i]
         size[i] <- centroid.size(x[, i])
         rho[i] <- riemdist(x[, i], c(cbmeanrot))
      }
      xbar <- sum / n
      
      rv <- Vmat(tan)
      if (approxtangent == TRUE) {
         for (i in 1:n) {
            tanapprox[, i] <- as.vector(icon[, , i]) - as.vector(xbar)
         }
         tanapprox <- tanapprox / centroid.size(xbar)
         pca <- prcomp1(t(tanapprox))
         z$tan <- tanapprox
      }
      
      if (expomap == TRUE) {
         temp <- rv
         for (i in 1:(n)) {
            temp[, i] <- rv[, i] / Enorm(rv[, i]) * rho[i]
         }
         rv <- temp
      }
      
      if (approxtangent == FALSE) {
         pca <- prcomp1(t(rv))
         z$tan <- rv
      }
      z$pcar <- pca$rotation
      z$pcasd <- pca$sdev
      z$percent <- z$pcasd ^ 2 / sum(z$pcasd ^ 2) * 100
      z$rotated <- icon
      npc <- 0
      for (i in 1:length(pca$sdev)) {
         if (pca$sdev[i] > 1e-07) {
            npc <- npc + 1
         }
      }
      z$scores <- pca$x
      z$rawscores <- pca$x
      for (i in 1:npc) {
         z$scores[, i] <- pca$x[, i] / pca$sdev[i]
      }
      z$rho <- rho
      z$size <- size
      z$mshape <- mu
      z$k <- k
      z$m <- 2
      z$n <- n
      z$rmsrho <- sqrt(mean(rho ^ 2))
      z$rmsd1 <- sqrt(mean(sin(rho) ^ 2))
      return(z)
   }

#==================================================================================
testmeanshapes.old <-
   function(A,
            B,
            Hotelling = TRUE,
            tol1 = 1e05,
            tol2 = 1e05) {
      if (is.complex(A)) {
         tem <- array(0, c(nrow(A), 2, ncol(A)))
         tem[, 1,] <- Re(A)
         tem[, 2,] <- Im(A)
         A <- tem
      }
      if (is.complex(B)) {
         tem <- array(0, c(nrow(B), 2, ncol(B)))
         tem[, 1,] <- Re(B)
         tem[, 2,] <- Im(B)
         B <- tem
      }
      m <- dim(A)[2]
      if (Hotelling == TRUE)  {
         if (m == 2) {
            test <- Hotelling2D(A, B)
         }
         if (m > 2) {
            test <- Hotellingtest(A, B, tol1 = tol1, tol2 = tol2)
         }
         cat(
            "Hotelling's T^2 test: ",
            c("Test statistic = ", round(test$F, 2)),
            c("\n p-value = ", round(test$pval, 4)),
            c("Degrees of freedom = ",
              test$df1, test$df2),
            "\n"
         )
      }
      
      if (Hotelling == FALSE)  {
         if (m == 2) {
            test <- Goodall2D(A, B)
         }
         if (m > 2) {
            test <- Goodalltest(A, B, tol1 = tol1, tol2 = tol2)
         }
         cat(
            "Goodall's F test: ",
            c("Test statistic = ", round(test$F, 2)),
            c("\n p-value = ", round(test$pval, 4)),
            c("Degrees of freedom = ",
              test$df1, test$df2),
            "\n"
         )
      }
      test
   }

#==================================================================================
procGPA <- function(x,
                    scale = TRUE,
                    reflect = FALSE,
                    eigen2d = FALSE,
                    tol1 = 1e-05,
                    tol2 = tol1,
                    tangentcoords = "residual",
                    proc.output = FALSE,
                    distances = TRUE,
                    pcaoutput = TRUE,
                    alpha = 0,
                    affine = FALSE)
{
   #
   #
   n <- dim(x)[length(dim(x))]
#   if ((n > 100) & (distances == TRUE)) {
#      print("To speed up use option distances=FALSE")
#   }
#   if ((n > 100) & (pcaoutput == TRUE)) {
#      print("To speed up use option pcaoutput=FALSE")
#   }
   
   
   
   if (scale == TRUE) {
      if (tangentcoords == "residual") {
         tangentresiduals <- TRUE
         expomap <- FALSE
      }
      if (tangentcoords == "partial") {
         tangentresiduals <- FALSE
         expomap <- FALSE
      }
      if (tangentcoords == "expomap") {
         tangentresiduals <- FALSE
         expomap <- TRUE
      }
   }
   if (scale == FALSE) {
      #all three options are equivalent
      if (tangentcoords == "residual") {
         tangentresiduals <- TRUE
         expomap <- FALSE
      }
      if (tangentcoords == "partial") {
         tangentresiduals <- TRUE
         expomap <- FALSE
      }
      if (tangentcoords == "expomap") {
         tangentresiduals <- TRUE
         expomap <- FALSE
      }
   }
   
   approxtangent <- tangentresiduals
   if (is.complex(x)) {
      tem <- array(0, c(nrow(x), 2, ncol(x)))
      tem[, 1,] <- Re(x)
      tem[, 2,] <- Im(x)
      x <- tem
   }
   m <- dim(x)[2]
   n <- dim(x)[3]
   if (reflect == FALSE) {
      if ((m == 2) && (scale == TRUE)) {
         if (eigen2d == TRUE) {
            out <- procrustes2d(x, approxtangent = approxtangent, expomap = expomap)
         }
         else
         {
            out <-
               procrustesGPA(
                  x,
                  tol1,
                  tol2,
                  approxtangent = approxtangent,
                  proc.output = proc.output,
                  distances = distances,
                  pcaoutput = pcaoutput,
                  reflect = reflect,
                  expomap = expomap
               )
         }
      }
      if ((m > 2) && (scale == TRUE)) {
         out <-
            procrustesGPA(
               x,
               tol1,
               tol2,
               approxtangent = approxtangent,
               proc.output = proc.output
               ,
               distances = distances,
               pcaoutput = pcaoutput,
               reflect = reflect,
               expomap = expomap
            )
      }
      if (scale == FALSE) {
         out <- procrustesGPA.rot(
            x,
            tol1,
            tol2,
            approxtangent = approxtangent,
            proc.output = proc.output,
            distances = distances,
            pcaoutput = pcaoutput,
            reflect = reflect,
            expomap = expomap
         )
      }
   }
   if (reflect == TRUE) {
      if (scale == TRUE) {
         out <- procrustesGPA(
            x,
            tol1,
            tol2,
            approxtangent = approxtangent,
            proc.output = proc.output,
            distances = distances,
            pcaoutput = pcaoutput,
            reflect = reflect,
            expomap = expomap
         )
      }
      if (scale == FALSE) {
         out <- procrustesGPA.rot(
            x,
            tol1,
            tol2,
            approxtangent = approxtangent,
            proc.output = proc.output,
            distances = distances,
            pcaoutput = pcaoutput,
            reflect = reflect,
            expomap = expomap
         )
      }
   }
   out$stdscores <- out$scores
   out$scores <- out$rawscores
   
   if (approxtangent == FALSE) {
      out$mshape <- out$mshape / centroid.size(out$mshape)
      for (i in 1:n) {
         out$rotated[, , i] <-
            out$rotated[, , i] / centroid.size(out$rotated[, , i])
      }
      
   }
   
   rw <- out
   rw <- shaperw(out, alpha = alpha , affine = affine)
   rw$GSS <- sum((n - 1) * rw$pcasd ** 2)
   rw
}

#==================================================================================
procrustesGPA <-
   function (x,
             tol1 = 1e-05,
             tol2 = 1e-05,
             distances = TRUE,
             pcaoutput = TRUE,
             approxtangent = TRUE,
             proc.output = FALSE,
             reflect = FALSE,
             expomap = FALSE)
   {
      z <- list(
         k = 0,
         m = 0,
         n = 0,
         rotated = 0,
         tan = 0,
         pcar = 0,
         scores = 0,
         rawscores = 0,
         pcasd = 0,
         percent = 0,
         size = 0,
         rho = 0,
         rmsrho = 0,
         rmsd1 = 0,
         mshape = 0
      )
      if (is.complex(x)) {
         tem <- array(0, c(nrow(x), 2, ncol(x)))
         tem[, 1,] <- Re(x)
         tem[, 2,] <- Im(x)
         x <- tem
      }
      k <- dim(x)[1]
      m <- dim(x)[2]
      n <- dim(x)[3]
      x <- cnt3(x)
      zgpa <-
         fgpa(x, tol1, tol2, proc.output = proc.output, reflect = reflect)
      
      if (distances == TRUE) {
         if (proc.output) {
            cat("Shape distances and sizes calculation ...\n")
         }
         size <- rep(0, times = n)
         rho <- rep(0, times = n)
         size <- apply(x, 3, centroid.size)
         rho <- apply(x, 3, y <- function(x) {
            riemdist(x, zgpa$mshape)
         })
      }
      
      
      
      tanpartial <- matrix(0, k * m - m , n)
      ident <- diag(rep(1, times = (m * k - m)))
      gamma <- as.vector(preshape(zgpa$mshape))
      for (i in 1:n) {
         tanpartial[, i] <- (ident - gamma %*% t(gamma)) %*%
            as.vector(preshape(zgpa$r.s.r[, , i]))
      }
      
      if (expomap == TRUE) {
         temp <- tanpartial
         for (i in 1:(n)) {
            temp[, i] <- tanpartial[, i] / Enorm(tanpartial[, i]) * rho[i]
         }
         tanpartial <- temp
      }
      
      tan <- zgpa$r.s.r[, 1,] - zgpa$mshape[, 1]
      for (i in 2:m) {
         tan <- rbind(tan, zgpa$r.s.r[, i,] - zgpa$mshape[, i])
      }
      
      if (pcaoutput == TRUE) {
         if (proc.output) {
            cat("PCA calculation ...\n")
         }
         if (approxtangent == FALSE) {
            pca <- prcomp1(t(tanpartial))
         }
         if (approxtangent == TRUE) {
            pca <- prcomp1(t(tan))
         }
         npc <- 0
         for (i in 1:length(pca$sdev)) {
            if (pca$sdev[i] > 1e-07) {
               npc <- npc + 1
            }
         }
         z$scores <- pca$x
         z$rawscores <- pca$x
         for (i in 1:npc) {
            z$scores[, i] <- pca$x[, i] / pca$sdev[i]
         }
         z$pcar <- pca$rotation
         z$pcasd <- pca$sdev
         z$percent <- z$pcasd ^ 2 / sum(z$pcasd ^ 2) * 100
      }
      
      
      if (approxtangent == FALSE) {
         z$tan <- tanpartial
      }
      if (approxtangent == TRUE) {
         z$tan <- tan
      }
      
      
      if (distances == TRUE) {
         z$rho <- rho
         z$size <- size
         z$rmsrho <- sqrt(mean(rho ^ 2))
         z$rmsd1 <- sqrt(mean(sin(rho) ^ 2))
      }
      
      z$rotated <- zgpa$r.s.r
      z$mshape <- zgpa$mshape
      z$k <- k
      z$m <- m
      z$n <- n
      
      if (proc.output) {
         cat("Finished.\n")
      }
      
      return(z)
   }

#==================================================================================
procrustesGPA.rot <-
   function (x,
             tol1 = 1e-05,
             tol2 = 1e-05,
             distances = TRUE,
             pcaoutput = TRUE,
             approxtangent = TRUE,
             proc.output = FALSE,
             reflect = FALSE,
             expomap = FALSE)
   {
      z <- list(
         k = 0,
         m = 0,
         n = 0,
         rotated = 0,
         tan = 0,
         pcar = 0,
         scores = 0,
         rawscores = 0,
         pcasd = 0,
         percent = 0,
         size = 0,
         rho = 0,
         rmsrho = 0,
         rmsd1 = 0,
         mshape = 0
      )
      if (is.complex(x)) {
         tem <- array(0, c(nrow(x), 2, ncol(x)))
         tem[, 1,] <- Re(x)
         tem[, 2,] <- Im(x)
         x <- tem
      }
      k <- dim(x)[1]
      m <- dim(x)[2]
      n <- dim(x)[3]
      
      
      
      #    print("GPA (rotation only)")
      x <- cnt3(x)
      zgpa <-
         fgpa.rot(x, tol1, tol2, proc.output = proc.output, reflect = reflect)
      
      if (distances == TRUE) {
         if (proc.output) {
            cat("Shape distances and sizes calculation ...\n")
         }
         size <- rep(0, times = n)
         rho <- rep(0, times = n)
         size <- apply(x, 3, centroid.size)
         rho <- apply(x, 3, y <- function(x) {
            riemdist(x, zgpa$mshape)
         })
      }
      
      
      
      tanpartial <- matrix(0, k * m - m, n)
      ident <- diag(rep(1, times = (m * k - m)))
      gamma <- as.vector(preshape(zgpa$mshape))
      for (i in 1:n) {
         tanpartial[, i] <- (ident - gamma %*% t(gamma)) %*%
            as.vector(preshape(zgpa$r.s.r[, , i]))
      }
      
      if (expomap == TRUE) {
         temp <- tanpartial
         for (i in 1:(n)) {
            temp[, i] <- tanpartial[, i] / Enorm(tanpartial[, i]) * rho[i]
         }
         tanpartial <- temp
      }
      
      
      
      tan <- zgpa$r.s.r[, 1,] - zgpa$mshape[, 1]
      for (i in 2:m) {
         tan <- rbind(tan, zgpa$r.s.r[, i,] - zgpa$mshape[, i])
      }
      
      if (approxtangent == FALSE) {
         z$tan <- tanpartial
      }
      if (approxtangent == TRUE) {
         z$tan <- tan
      }
      
      
      if (pcaoutput == TRUE) {
         if (proc.output) {
            cat("PCA calculation ...\n")
         }
         
         if (approxtangent == FALSE) {
            pca <- prcomp1(t(tanpartial))
         }
         if (approxtangent == TRUE) {
            pca <- prcomp1(t(tan))
         }
         
         
         
         npc <- 0
         for (i in 1:length(pca$sdev)) {
            if (pca$sdev[i] > 1e-07) {
               npc <- npc + 1
            }
         }
         
         z$scores <- pca$x
         z$rawscores <- pca$x
         for (i in 1:npc) {
            z$scores[, i] <- pca$x[, i] / pca$sdev[i]
         }
         
         z$pcar <- pca$rotation
         z$pcasd <- pca$sdev
         z$percent <- z$pcasd ^ 2 / sum(z$pcasd ^ 2) * 100
      }
      
      if (distances == TRUE) {
         z$rho <- rho
         z$size <- size
         z$rmsrho <- sqrt(mean(rho ^ 2))
         z$rmsd1 <- sqrt(mean(sin(rho) ^ 2))
      }
      z$rotated <- zgpa$r.s.r
      z$mshape <- zgpa$mshape
      z$k <- k
      z$m <- m
      z$n <- n
      if (proc.output) {
         cat("Finished.\n")
      }
      return(z)
   }

#==================================================================================
project <- function(z, gamma)
{
   #input z: preshape, gamma: preshape (k-1 x 1 matrices)
   #output Kent's tangent plane coordinates
   #of z at the pole gamma (k-1 complex vector)
   nr <- nrow(z)
   nc <- ncol(z)
   g <- matrix(gamma, nr, 1)
   ident <- diag(nr)
   theta <- diag(c(exp((-0 - 1i) * Arg(st(
      g
   ) %*% z))), nc, nc)
   v <- (ident - g %*% st(g)) %*% z %*% theta
   v
}

#==================================================================================
read.array <- function(name, k, m, n)
{
   #input name : filename, k: no of points, m: no of dimensions, n: sample size
   #output x: k x m x n array of data
   #e.g. for 2D data assume file format x1 y1 x2 y2 .. xn yn for each object
   tem <- scan(name)
   tem <- array(tem, c(m, k, n))
   tem <- aperm(tem, c(2, 1, 3))
   x <- tem
   x
}

#==================================================================================
read.in <- function(name, k, m)
{
   #input name : filename, k: no of points, m: no of dimensions
   #output x: k x m x n array of data ( n: sample size)
   #e.g. for m=2-D data assume file format x1 y1 x2 y2 ... xk yk for each object
   #for m=3-D data: x1 y1 z1 x2 y2 z2 ... xk yk zk
   tem <- scan(name)
   n <- length(tem) / (k * m)
   tem <- array(tem, c(m, k, n))
   tem <- aperm(tem, c(2, 1, 3))
   x <- tem
   x
}

#==================================================================================
realtocomplex <- function(x)
{
   #input k x 2 matrix - return complex k-vector
   k <- nrow(x)
   zstar <- x[, 1] + (1i) * x[, 2]
   zstar
}

#==================================================================================
reassqpr <- function(z)
{
   j <- 1
   nc <- ncol(z)
   nr <- nrow(z)
   stemp <- matrix(0, 2 * nr, 2 * nr)
   repeat {
      t1 <- matrix(z[, j], nr, 1)
      vz <- rbind(Re(t1), Im(t1))
      viz <- rbind(Re((1i) * t1), Im((1i) * t1))
      stemp <- stemp + vz %*% t(vz) + viz %*% t(viz)
      if (j == nc)
         break
      j <- j + 1
   }
   stemp
}

#==================================================================================
relwarps <- function(mshape, rotated, alpha)
{
   #find the relative warps for a dataset with mshape as the reference
   #and `rotated' as the array of Procrustes rotated figures
   #alpha is the power of the bending energy
   #  alpha=+1 : emphasizes large scale
   #  alpha=-1 : emphasizes small scale
   #output:
   #     z$rwarps  : the relative warps
   #     z$rwscores : the relative warp scores
   #     z$rwpercent : the percentage of total variability explained by each            #relative warp
   z <-
      list(
         rwarps = 0,
         rwscores = 0,
         rwpercent = 0,
         ev = 0,
         unif = 0,
         unscores = 0,
         lengths = 0
      )
   k <- nrow(mshape)
   TPS <- bendingenergy(mshape)
   Be <- TPS$gamma11
   stackxy <- rbind(rotated[, 1,], rotated[, 2,])
   n <- dim(rotated)[3]
   msum <- rep(0, times = 2 * k)
   for (i in 1:n) {
      msum <- msum + stackxy[, i]
   }
   msum <- msum / n
   meanxy <- msum
   cstackxy <- matrix(0, 2 * k, n)
   for (i in 1:n) {
      cstackxy[, i] <- stackxy[, i] - meanxy
   }
   Bpow <- genpower(Be, alpha)
   Bpowinv <- genpower(Be,-alpha)
   IBpow <- I2mat(Bpow)
   IBpowinv <- I2mat(Bpowinv)
   if (alpha == 0) {
      IBpow <- diag(rep(1, times = (2 * k)))
      IBpowinv <- diag(rep(1, times = (2 * k)))
   }
   stacknew <- IBpow %*% cstackxy
   gamma <- matrix(0, 2 * k, 2 * k)
   pcarotation <-
      eigen(stacknew %*% t(stacknew) / n, symmetric = TRUE)$vectors
   pcaev <- eigen(stacknew %*% t(stacknew) / n, symmetric = TRUE)$values
   pcasdev <- rep(0, times = 2 * k)
   for (i in 1:(2 * k)) {
      pcasdev[i] <- sqrt(abs(pcaev[i]))
   }
   scores <- t(IBpow %*% pcarotation) %*% cstackxy
   percent <- rep(0, times = 2 * k)
   for (i in 1:(2 * k)) {
      percent[i] <- pcasdev[i] ^ 2
   }
   Un <- TPS$Un
   UnXY <- t(Un) %*% cstackxy
   z$unif <-
      Un %*% t(matrix(c(sqrt(var(
         UnXY[1,]
      )), 0, 0, sqrt(var(
         UnXY[2,]
      ))), 2, 2))
   z$unscores <- t(UnXY)
   z$lengths <- sqrt(abs(percent))
   z$rwarps <- IBpowinv %*% pcarotation %*% diag(pcasdev)
   z$rwscores <- t(scores)
   z$ev <- pcaev
   percentrw <- percent / sum(percent) * 100
   z$rwpercent <- percentrw
   return(z)
}

#==================================================================================
ssriemdist <- function(x, y, reflect = FALSE) {
   sx <- centroid.size(x)
   sy <- centroid.size(y)
   sd <- sx ** 2 + sy ** 2 - 2 * sx * sy * cos(riemdist(x, y, reflect = reflect))
   sqrt(abs(sd))
}

#==================================================================================
riemdist <- function(x, y, reflect = FALSE)
{
   #input two k x m matrices x, y or complex k-vectors
   #output Riemannian distance rho between them
   
   if (sum((x - y) ** 2) == 0) {
      riem <- 0
   }
   if (sum((x - y) ** 2) != 0) {
      if (reflect == FALSE) {
         if (ncol(as.matrix(x)) < 3) {
            if (is.complex(x) == FALSE) {
               x <- realtocomplex(x)
            }
            if (is.complex(y) == FALSE) {
               y <- realtocomplex(y)
            }
            #riem <- c(acos(Mod(st(preshape(x)) %*% preshape(y))))
            riem <- c(acos(min(1, (
               Mod(st(preshape(x)) %*% preshape(y))
            ))))
         }
         else {
            m <- ncol(x)
            z <- preshape(x)
            w <- preshape(y)
            Q <- t(z) %*% w %*% t(w) %*% z
            ev <- eigen(t(z) %*% w)$values
            check <- 1
            for (i in 1:m) {
               check <- check * ev[i]
            }
            ev <- sqrt(abs(eigen(Q, symmetric = TRUE)$values))
            if (Re(check) < 0)
               ev[m] <-  -ev[m]
            riem <- acos(min(sum(ev), 1))
         }
      }
      if (reflect == TRUE) {
         m <- ncol(x)
         z <- preshape(x)
         w <- preshape(y)
         Q <- t(z) %*% w %*% t(w) %*% z
         
         ev <- sqrt(abs(eigen(Q, symmetric = TRUE)$values))
         
         riem <- acos(min(sum(ev), 1))
      }
      
   }
   riem
}

#==================================================================================
riemdist.complex <- function(z, w)
{
   #input complex k-vectors z, w
   #output Riemannian distance rho between them
   c(acos(min(Mod(
      st(preshape(z)) %*% preshape(w)
   ), 1)))
}

#==================================================================================
riemdist.mD <- function(x, y)
{
   #input  k x m matrices x, y
   #output Riemannian distance rho between them
   m <- ncol(x)
   z <- preshape.mD(x)
   w <- preshape.mD(y)
   Q <- t(z) %*% w %*% t(w) %*% z
   ev <- eigen(t(z) %*% w)$values
   check <- 1
   for (i in 1:m) {
      check <- check * ev[i]
   }
   ev <- sqrt(eigen(Q, symmetric = TRUE)$values)
   if (check < 0)
      ev[m] <-  -ev[m]
   riem <- acos(min(sum(ev), 1))
   riem
}

#==================================================================================
rotateaxes <- function(mshapein, rotatedin)
{
   #Rotates a mean shape and the Procrustes rotated data to have
   #horizontal and vertical principal axes
   #output: z$mshape rotated mean shape
   #        z$rotated   rotated procrustes registered data
   #        z$R  the rotation matrix
   #
   z <- list(mshape = 0,
             rotated = 0,
             R = 0)
   n <- dim(rotatedin)[3]
   S <- var(mshapein)
   R <- eigen(S)$vectors
   msh <- mshapein %*% R
   ico <- rotatedin
   for (i in 1:n) {
      ico[,  , i] <- rotatedin[,  , i] %*% R
   }
   z$mshape <- msh
   z$rotated <- ico
   z$R <- R
   return(z)
}

#sigma<-function(x)
#{
#	length <- sqrt(x[1]^2 + x[2]^2)
#	if(length == 0)
#		sig <- 0
#	else sig <- length^2 * log(length^2)
#	sig
#}

#==================================================================================
sigmacov <- function(x)
{
   #  other radial basis functions/covariance functions are possible of course
   
   hh <- Enorm(x)
   if (hh == 0)
      sig <- 0
   else
   {
      if (length(x) == 2) {
         sig <-
            hh ^ 2 * log(hh ^ 2)   # null space includes affine terms (2D data)
      }
      if (length(x) == 3) {
         sig <-  -hh   # null space includes affine terms (3D data)
      }
   }
   sig
}

#==================================================================================
st <- function(zstar)
{
   #input complex matrix
   #output transpose of the complex conjugate
   st <- t(Conj(zstar))
   st
}

#==================================================================================
ild_tanfigure <- function(vv, gamma)
{
   #inverse projection from complex tangent plane coordinates vv, using pole gamma
   #output centred icon
   k <- nrow(gamma) + 1
   h <- defh(k - 1)
   zvv <- tanpreshape(vv, gamma)
   zstvv <- t(h) %*% zvv
   zstvv
}

#==================================================================================
ild_tanfigurefull <- function(vv, gamma)
{
   #inverse projection from complex tangent plane coordinates vv, using pole gamma
   #using Procrustes to with scaling to the pole gamma
   #output centred icon
   k <- nrow(gamma) + 1
   f1 <- tanfigure(vv, gamma)
   h <- defh(k - 1)
   f2 <- t(h) %*% gamma
   beta <- Mod(st(f1) %*% f2)
   f1 <- f1 * c(beta)
   f1
}

#==================================================================================
tanpreshape <- function(vv, gamma)
{
   #inverse projection from tangent plane coordinates vv, using pole gamma
   #output preshape
   z <- c((1 - st(vv) %*% vv) ^ 0.5) * gamma + vv
   z
}

#==================================================================================
plot3Ddata <- function(dna.data,
                       land = 1:k,
                       objects = 1:n,
                       joinline = c(1, 1)) {
   dna <- procGPA(dna.data[, , 1:2])
   w1 <- defplotsize2(dna.data[, 1:2, ])
   w2 <- defplotsize2(dna.data[, c(1, 3), ])
   w3 <- defplotsize2(dna.data[, c(2, 3), ])
   width <- max(c(w1$width, w2$width, w3$width))
   xl <- min(c(w1$xl, w2$xl, w3$xl))
   xu <- xl + width
   yl <- min(c(w1$yl, w2$yl, w3$yl))
   yu <- yl + width
   n <- dim(dna.data)[3]
   k <- dim(dna.data)[1]
   m <- dim(dna.data)[2]
   par(mfrow = c(1, 1))
   par(pty = "s")
   view1 <- 1
   view2 <- 2
   view3 <- 3
   lineorder <- joinline
   for (j in 1:1) {
      for (ii in objects) {
         par(mfrow = c(2, 2))
         mag <- 0
         pcno <- 1
         plotPDMnoaxis3(
            c(dna.data[land, view2, ii], dna.data[land, view3, ii]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
         mag <- 0
         pcno <- 1
         plotPDMnoaxis3(
            c(dna.data[land, view1, ii], dna.data[land, view3, ii]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
         mag <- 0
         pcno <- 1
         plotPDMnoaxis3(
            c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
         plot(
            c(0, 0),
            c(50, 50),
            xlim = c(0, 0),
            ylim = c(0, 0),
            type = "n",
            xlab = " ",
            ylab = " ",
            axes = FALSE
         )
         title(as.character(ii))
      }
   }
}

#==================================================================================
plot3Ddata.static <-
   function(dna.data,
            land = 1:k,
            objects = 1:n,
            joinline = c(1, 1)) {
      dna <- procGPA(dna.data[, , 1:2])
      w1 <- defplotsize2(dna.data[, 1:2, ])
      w2 <- defplotsize2(dna.data[, c(1, 3), ])
      w3 <- defplotsize2(dna.data[, c(2, 3), ])
      width <- max(c(w1$width, w2$width, w3$width))
      xl <- min(c(w1$xl, w2$xl, w3$xl))
      xu <- xl + width
      yl <- min(c(w1$yl, w2$yl, w3$yl))
      yu <- yl + width
      n <- dim(dna.data)[3]
      k <- dim(dna.data)[1]
      m <- dim(dna.data)[2]
      par(mfrow = c(1, 1))
      par(pty = "s")
      
      lineorder <- joinline
      
      par(mfrow = c(2, 2))
      mag <- 0
      pcno <- 1
      ii <- 1
      view1 <- 1
      view2 <- 2
      view3 <- 3
      plotPDMnoaxis3(
         c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
         c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                    (land)), pcno]),
         mag * dna$pcasd[pcno],
         xl,
         xu,
         yl,
         yu,
         lineorder,
         1
      )
      for (ii in objects) {
         pointsPDMnoaxis3(
            c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
      view1 <- 1
      view2 <- 3
      view3 <- 2
      plotPDMnoaxis3(
         c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
         c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                    (land)), pcno]),
         mag * dna$pcasd[pcno],
         xl,
         xu,
         yl,
         yu,
         lineorder,
         1
      )
      for (ii in objects) {
         pointsPDMnoaxis3(
            c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
      view1 <- 2
      view2 <- 3
      view3 <- 1
      plotPDMnoaxis3(
         c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
         c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                    (land)), pcno]),
         mag * dna$pcasd[pcno],
         xl,
         xu,
         yl,
         yu,
         lineorder,
         1
      )
      for (ii in objects) {
         pointsPDMnoaxis3(
            c(dna.data[land, view1, ii], dna.data[land, view2, ii]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
      
      
   }

#==================================================================================
plot3Dmean <- function(dna) {
   land <- 1:dim(dna$mshape)[1]
   w1 <- defplotsize2(dna$rotated[, 1:2, ])
   w2 <- defplotsize2(dna$rotated[, c(1, 3), ])
   w3 <- defplotsize2(dna$rotated[, c(2, 3), ])
   width <- max(c(w1$width, w2$width, w3$width))
   xl <- min(c(w1$xl, w2$xl, w3$xl))
   xu <- xl + width
   yl <- min(c(w1$yl, w2$yl, w3$yl))
   yu <- yl + width
   par(mfrow = c(2, 2))
   par(pty = "s")
   plot(
      dna$mshape[land, 1],
      dna$mshape[land, 2],
      xlim = c(xl, xu),
      ylim = c(yl, yu),
      xlab = " ",
      ylab = " "
   )
   text(dna$mshape[land, 1], dna$mshape[land, 2], land)
   lines(dna$mshape[land, 1], dna$mshape[land, 2])
   plot(
      dna$mshape[land, 1],
      dna$mshape[land, 3],
      xlim = c(xl, xu),
      ylim = c(yl, yu),
      xlab = " ",
      ylab = " "
   )
   text(dna$mshape[land, 1], dna$mshape[land, 3], land)
   lines(dna$mshape[land, 1], dna$mshape[land, 3])
   plot(
      dna$mshape[land, 2],
      dna$mshape[land, 3],
      xlim = c(xl, xu),
      ylim = c(yl, yu),
      xlab = " ",
      ylab = " "
   )
   text(dna$mshape[land, 2], dna$mshape[land, 3], land)
   lines(dna$mshape[land, 2], dna$mshape[land, 3])
   title("Procrustes mean shape estimate")
}

#==================================================================================
plot3Dpca <- function(dna, pcno, joinline = c(1, 1)) {
   #choose subset
   w1 <- defplotsize2(dna$rotated[, 1:2, ])
   w2 <- defplotsize2(dna$rotated[, c(1, 3), ])
   w3 <- defplotsize2(dna$rotated[, c(2, 3), ])
   width <- max(c(w1$width, w2$width, w3$width))
   xl <- min(c(w1$xl, w2$xl, w3$xl)) - width / 4
   xu <- xl + width * 1.5
   yl <- min(c(w1$yl, w2$yl, w3$yl)) - width / 4
   yu <- yl + width * 1.5
   k <- dim(dna$mshape)[1]
   lineorder <- joinline
   par(mfrow = c(1, 1))
   cat("X-Y view \n")
   view1 <- 1
   view2 <- 2
   view3 <- 3
   land <- c(1:k)
   for (j in 1:10) {
      for (ii in-12:12) {
         mag <- ii / 4
         plotPDMnoaxis3(
            c(dna$mshape[land, view1], dna$mshape[land, view2]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
      for (ii in-11:11) {
         mag <- -ii / 4
         plotPDMnoaxis3(
            c(dna$mshape[land, view1], dna$mshape[land, view2]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
   }
   #choose subset
   par(mfrow = c(1, 1))
   cat("X-Z view \n")
   view1 <- 1
   view2 <- 3
   view3 <- 2
   land <- c(1:k)
   for (j in 1:10) {
      for (ii in-12:12) {
         mag <- ii / 4
         plotPDMnoaxis3(
            c(dna$mshape[land, view1], dna$mshape[land, view2]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
      for (ii in-11:11) {
         mag <- -ii / 4
         plotPDMnoaxis3(
            c(dna$mshape[land, view1], dna$mshape[land, view2]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
   }
   #choose subset
   par(mfrow = c(1, 1))
   cat("Y-Z view \n")
   view1 <- 2
   view2 <- 3
   view3 <- 1
   land <- c(1:k)
   for (j in 1:10) {
      for (ii in-12:12) {
         mag <- ii / 4
         plotPDMnoaxis3(
            c(dna$mshape[land, view1], dna$mshape[land, view2]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
      for (ii in-11:11) {
         mag <- -ii / 4
         plotPDMnoaxis3(
            c(dna$mshape[land, view1], dna$mshape[land, view2]),
            c(dna$pcar[((view1 - 1) * k + (land)), pcno], dna$pcar[((view2 - 1) * k +
                                                                       (land)), pcno]),
            mag * dna$pcasd[pcno],
            xl,
            xu,
            yl,
            yu,
            lineorder,
            1
         )
      }
   }
}

#==================================================================================
banner1 <- function(char)
{
   par(mfrow = c(1, 1))
   plot(
      c(0, 0),
      c(1, 1),
      axes = FALSE,
      type = "n",
      xlab = " ",
      ylab = " "
   )
   a1 <- char
   if (length(a1) == 2)
      a1 <- paste(a1[1], a1[2])
   if (length(a1) == 3)
      a1 <- paste(a1[1], a1[2], a1[3])
   if (is.character(a1) == FALSE)
      char <- as.character(a1)
   title(a1)
}

#==================================================================================
banner4 <- function(a1, a2, a3, a4)
{
   par(mfrow = c(2, 2))
   plot(
      c(0, 0),
      c(1, 1),
      axes = FALSE,
      type = "n",
      xlab = " ",
      ylab = " "
   )
   if (length(a1) == 2)
      a1 <- paste(a1[1], a1[2])
   if (length(a1) == 3)
      a1 <- paste(a1[1], a1[2], a1[3])
   if (is.character(a1) == FALSE)
      a1 <- as.character(a1)
   title(a1)
   plot(
      c(0, 0),
      c(1, 1),
      axes = FALSE,
      type = "n",
      xlab = " ",
      ylab = " "
   )
   if (length(a2) == 2)
      a2 <- paste(a2[1], a2[2])
   if (length(a2) == 3)
      a2 <- paste(a2[1], a2[2], a2[3])
   if (is.character(a2) == FALSE)
      a2 <- as.character(a2)
   title(a2)
   plot(
      c(0, 0),
      c(1, 1),
      axes = FALSE,
      type = "n",
      xlab = " ",
      ylab = " "
   )
   if (length(a3) == 2)
      a3 <- paste(a3[1], a3[2])
   if (length(a3) == 3)
      a3 <- paste(a3[1], a3[2], a3[3])
   if (is.character(a3) == FALSE)
      a3 <- as.character(a3)
   title(a3)
   plot(
      c(0, 0),
      c(1, 1),
      axes = FALSE,
      type = "n",
      xlab = " ",
      ylab = " "
   )
   if (length(a4) == 2)
      a4 <- paste(a4[1], a4[2])
   if (length(a4) == 3)
      a4 <- paste(a4[1], a4[2], a4[3])
   if (is.character(a4) == FALSE)
      a4 <- as.character(a4)
   title(a4)
}




#######
#exact Gaussian MLE - isotropic distribution

#######not fully tested yet
#==================================================================================
isomle <- function(x) {
   if (is.complex(x)) {
      tem <- array(0, c(nrow(x), 2, ncol(x)))
      tem[, 1,] <- Re(x)
      tem[, 2,] <- Im(x)
      x <- tem
   }
   k <- dim(x)[1]
   m <- dim(x)[2]
   n <- dim(x)[3]
   if (m > 2) {
      print("Only valid for 2D data")
   }
   if (m == 2) {
      pm <- rep(0, times = 2 * k - 3)
      tem <- procrustes2d(x)
      tem1 <- bookstein.shpv(tem$mshape)
      sigm <- sum(diag(var(tem$tan))) / (n - 1) / 2
      #cat("Isotropic shape MLE \n")
      pm[1:(k - 2)] <- tem1[3:k, 1]
      pm[(k - 1):(2 * k - 4)] <- tem1[3:k, 2]
      pm[2 * k - 3] <- 10
      ans <- nlm(objfuniso, hessian = TRUE, pm, uu = x)
      #while (ans$code!=1){
      #print("code not equal 1")
      #print(pm)
      #pm<-pm+rnorm(2*k-3,0,0.1)
      #pm[2*k-3]<-abs(pm[2*k-3])
      #ans<-nlm(objfuniso,hessian=TRUE,pm,uu=x) #print(ans)
      #}
      out <- list(
         code = 0,
         mshape = 0,
         tau = 0,
         kappa = 0,
         varcov = 0,
         gradient = 0
      )
      mn <- matrix(0, k, 2)
      mn[1, 1] <- -0.5
      mn[2, 1] <- 0.5
      mn[3:k, 1] <- ans$estimate[1:(k - 2)]
      mn[3:k, 2] <- ans$estimate[(k - 1):(2 * k - 4)]
      out$mshape <- mn
      out$code <- ans$code
      out$loglike <- -ans$minimum
      out$gradient <- ans$gradient
      out$tau <- sqrt(1 / ans$estimate[2 * k - 3] ** 2)
      out$kappa <- centroid.size(mn) ** 2 / (4 * out$tau ** 2)
      out$varcov <- solve(ans$hessian)
      out$se <- c(sqrt(diag(out$varcov)))
      out$se[2 * k - 3] <- out$se[2 * k - 3] * out$tau ** 2
      out
   }
}

#==================================================================================
objfuniso <- function(pm, uu) {
   k <- dim(uu)[1]
   h <- defh(k - 1)
   zero <- matrix(0, k - 1, k)
   L1 <- cbind(h, zero)
   L2 <- cbind(zero, h)
   L <- rbind(L1, L2)
   mustar <- c(-1 / 2, 1 / 2, pm[1:(k - 2)], 0, 0, pm[(k - 1):(2 * k - 4)])
   mu <- L %*% mustar
   obj <- -loglikeiso2(uu, mu, 1 / pm[2 * k - 3])
   obj
}

#==================================================================================
loglikeiso <- function(uu, mu, s) {
   nsam <- dim(uu)[3]
   sum <- 0
   for (i in 1:nsam) {
      sum <- sum + log(isodens(uu[, , i], mu, s))
   }
   sum
}

#==================================================================================
loglikeiso2 <- function(uu, mu, s) {
   nsam <- dim(uu)[3]
   sum <- 0
   for (i in 1:nsam) {
      sum <- sum + isologdens(uu[, , i], mu, s)
   }
   sum
}

#==================================================================================
isodens <- function(usam, mu, s) {
   k <- dim(usam)[1]
   u <- kendall.shpv(usam)
   uuu <- u[, 1]
   vvv <- u[, 2]
   up <- c(1, uuu, 0, vvv)
   vp <- c(0, -vvv, 1, uuu)
   usu <- t(up) %*% up
   beta <- c(t(mu) %*% up, t(mu) %*% vp)
   sin2rho <- 1 - t(beta) %*% beta / (usu * c(t(mu) %*% mu))
   kappa <- c(t(mu) %*% mu) / (4 * s ** 2)
   #finf<-gamma(k-1)*pi/(pi*usu)**(k-1)
   dens <- oneFone(k - 2, 2 * kappa * (1 - sin2rho)) %*% exp(-2 * kappa * sin2rho)
   dens
}

#==================================================================================
isologdens <- function(usam, mu, s) {
   k <- dim(usam)[1]
   u <- kendall.shpv(usam)
   uuu <- u[, 1]
   vvv <- u[, 2]
   up <- c(1, uuu, 0, vvv)
   vp <- c(0, -vvv, 1, uuu)
   usu <- t(up) %*% up
   beta <- c(t(mu) %*% up, t(mu) %*% vp)
   sin2rho <- 1 - t(beta) %*% beta / (usu * c(t(mu) %*% mu))
   kappa <- c(t(mu) %*% mu) / (4 * s ** 2)
   #finf<-lgamma(k-1)+log(pi)-(k-1)*log(pi*usu)
   dens <- loneFone(k - 2, 2 * kappa * (1 - sin2rho)) - 2 * kappa * sin2rho
   c(dens)
}

#==================================================================================
loneFone <- function(r, x) {
   #note this is log 1F1(-r,1,-x)
   if (x > 1) {
      sum1 <- r * log(x)
      sum <- 0
      for (j in 0:r) {
         sum <- sum + choose(r, j) * x ** (j - r) / gamma(j + 1)
      }
      out <- sum1 + log(sum)
   }
   if (x <= 1) {
      sum <- 0
      for (j in 0:r) {
         sum <- sum + choose(r, j) * x ** (j) / gamma(j + 1)
      }
      out <- log(sum)
   }
   out
}

#==================================================================================
ild_kendall.shpv <- function(x) {
   k <- dim(x)[1]
   h <- defh(k - 1)
   zz <- h %*% x
   kendall <- (zz[2:(k - 1), 1] + 1i * zz[2:(k - 1), 2]) / (zz[1, 1] + 1i *
                                                               zz[1, 2])
   kendall <- cbind(Re(kendall), Im(kendall))
   kendall
}

#==================================================================================
oneFone <- function(r, x) {
   #note this is 1F1(-r,1,-x)
   sum <- 0
   for (j in 0:r) {
      sum <- sum + choose(r, j) * x ** j / gamma(j + 1)
   }
   sum
}

#==================================================================================
permutationtest <- function(A, B, nperms = 200) {
   A1 <- A
   A2 <- B
   B <- nperms
   nsam1 <- dim(A1)[3]
   nsam2 <- dim(A2)[3]
   
   
   
   Gtem <- Goodalltest(A1, A2)
   Htem <- Hotellingtest(A1, A2)
   
   
   Gumc <- Gtem$F
   Humc <- Htem$F
   
   Gtabpval <- Gtem$pval
   Htabpval <- Htem$pval
   
   
   if (B > 0) {
      Apool <- array(0, c(dim(A1)[1], dim(A1)[2], dim(A1)[3] + dim(A2)[3]))
      Apool[, , 1:nsam1] <- A1
      Apool[, , (nsam1 + 1):(nsam1 + nsam2)] <- A2
      
      out <-
         list(
            H = 0,
            H.pvalue = 0,
            H.table.pvalue = 0,
            G = 0,
            G.pvalue = 0,
            G.table.pvalue = 0
         )
      
      Gu <- rep(0, times = B)
      Hu <- rep(0, times = B)
      
      
      
      cat("Permutations - sampling without replacement: ")
      cat(c("No of permutations = ", B, "\n"))
      
      for (i in 1:B) {
         cat(c(i, " "))
         
         
         select <- sample(1:(nsam1 + nsam2))
         
         
         Gu[i] <-
            Goodalltest(Apool[, , select[1:nsam1]] , Apool[, , select[(nsam1 + 1):(nsam2 +
                                                                                      nsam1)]])$F
         Hu[i] <-
            Hotellingtest(Apool[, , select[1:nsam1]], Apool[, , select[(nsam1 + 1):(nsam1 +
                                                                                       nsam2)]])$F
         
      }
      
      Gu <- sort(Gu)
      numbig <- length(Gu[Gumc < Gu])
      pvalG <- (1 + numbig) / (B + 1)
      Hu <- sort(Hu)
      numbig <- length(Hu[Humc < Hu])
      pvalH <- (1 + numbig) / (B + 1)
      cat(" \n")
      
      out$H <- Humc
      out$H.pvalue <- pvalH
      out$H.table.pvalue <- Htabpval
      
      out$G <- Gumc
      out$G.pvalue <- pvalG
      out$G.table.pvalue <- Gtabpval
      
   }
   
   if (B == 0) {
      out <- list(
         H = 0,
         H.table.pvalue = 0,
         G = 0,
         G.table.pvalue = 0
      )
      
      out$H <- Humc
      out$H.table.pvalue <- Htabpval
      
      out$G <- Gumc
      out$G.table.pvalue <- Gtabpval
   }
   out
}

#==================================================================================
permutationtest <- permutationtest2

#==================================================================================
frechet <- function(x, mean = "intrinsic") {
   if (mean == "intrinsic") {
      option <- 1
   }
   if (mean == "partial.procrustes") {
      option <- 2
   }
   if (mean == "full.procrustes") {
      option <- 3
   }
   if (mean == "mle") {
      option <- 4
   }
   if (is.double(mean)) {
      if (mean > 0) {
         option <- -mean
      }
   }
   n <- dim(x)[3]
   for (i in 1:n) {
      x[, , i] <- x[, , i] / centroid.size(x[, , i])
   }
   if (option < 4) {
      pm <- procGPA(x, scale = FALSE, tol1 = 10 ^ (-8))$mshape
      m <- dim(x)[2]
      k <- dim(x)[1]
      ans <- list(
         mshape = 0,
         var = 0,
         code = 0,
         gradient = 0
      )
      out <-
         nlm(
            objfun,
            hessian = TRUE,
            c(pm),
            uu = x,
            option = option,
            iterlim = 1000
         )
      B <- matrix(out$estimate, k, m)
      ans$mshape <- procOPA(pm, B)$Bhat
      ans$var <- out$minimum
      ans$code <- out$code
      ans$gradient <- out$gradient
   }
   if (option == 4) {
      pm <- procGPA(x, scale = FALSE, tol1 = 10 ^ (-8))$mshape
      m <- dim(x)[2]
      k <- dim(x)[1]
      if (m == 2) {
         theta <- c(log(centroid.size(pm) ** 2 / (4 * 0.1 ** 2)), pm)
         ans <- list(
            mshape = 0,
            kappa = 0,
            code = 0,
            gradient = 0
         )
         out <- nlm(
            objfun4,
            hessian = TRUE,
            theta,
            uu = x,
            iterlim = 1000
         )
         B <- matrix(out$estimate[-1], k, m)
         ans$mshape <- procOPA(pm, B)$Bhat
         ans$kappa <- exp(out$estimate[1])
         ans$loglike <- -out$minimum
         ans$code <- out$code
         ans$gradient <- out$gradient
      }
      if (m != 2) {
         print("MLE is only appropriate for planar shapes")
      }
   }
   ans
}

#==================================================================================
objfun <- function(pm, uu, option) {
   m <- dim(uu)[2]
   k <- dim(uu)[1]
   pm <- matrix(pm, k, m)
   sum <- 0
   for (i in 1:dim(uu)[3]) {
      if (option == 1) {
         sum <- sum + (riemdist(pm, uu[, , i])) ** 2
      }
      if (option == 2) {
         sum <- sum + 4 * sin(riemdist(pm, uu[, , i]) / 2) ** 2
      }
      if (option == 3) {
         sum <- sum + sin(riemdist(pm, uu[, , i])) ** 2
      }
      if (option < 0) {
         h <- -option
         sum <- sum +  ((1 - cos(riemdist(pm, uu[, , i])) ** (2 * h)) / h)
      }
   }
   sum
}

#==================================================================================
objfun4 <- function(pm, uu) {
   m <- dim(uu)[2]
   k <- dim(uu)[1]
   n <- dim(uu)[3]
   kappa <- exp(pm[1])
   pm <- matrix(pm[-1], k, m)
   sum <- 0
   for (i in 1:n) {
      sin2rho <- sin(riemdist(pm, uu[, , i])) ** 2
      sum <- sum + loneFone(k - 2, 2 * kappa * (1 - sin2rho)) - 2 * kappa * sin2rho
   }
   - sum
}

#==================================================================================
MDSshape <- function(x,
                     alpha = 1,
                     projalpha = 1 / 2) {
   mu <- procGPA(x)$mshape
   k <- dim(x)[1]
   n <- dim(x)[3]
   m <- dim(x)[2]
   H <- defh(k - 1)
   sum <- matrix(0, k - 1, k - 1)
   for (i in 1:n) {
      Z <- preshape(x[, , i])
      if (alpha == 1) {
         sum <- sum + (Z) %*% t((Z))
      }
      if (alpha == 1 / 2) {
         ee <- eigen((Z) %*% t((Z)), symmetric = TRUE)
         sum <- sum + ee$vectors %*% diag(sqrt(abs(ee$values))) %*% t(ee$vectors)
      }
   }
   eig <- eigen(sum / n, symmetric = TRUE)
   lam <- eig$values
   
   if (m == 2) {
      if (projalpha == 1 / 2) {
         meanshape <-
            cbind(
               t(H) %*% (sqrt(lam[1]) * eig$vectors[, 1]) / sqrt(lam[1] + lam[2]) ,
               -t(H) %*% (sqrt(lam[2]) * eig$vectors[, 2]) / sqrt(lam[1] + lam[2])
            )
      }
      if (projalpha == 1) {
         lambar <- (lam[1] + lam[2]) / 2
         meanshape <-
            cbind(t(H) %*% (sqrt(lam[1] - lambar + 1 / m) * eig$vectors[, 1]) ,
                  -t(H) %*% (sqrt(lam[2] - lambar + 1 / m) * eig$vectors[, 2]))
      }
   }
   
   if (m == 3) {
      if (projalpha == 1 / 2) {
         meanshape <-
            cbind(
               t(H) %*% (sqrt(lam[1]) * eig$vectors[, 1]) / sqrt(lam[1] + lam[2] + lam[3]) ,
               t(H) %*% (sqrt(lam[2]) * eig$vectors[, 2]) / sqrt(lam[1] + lam[2] + lam[3]),
               t(H) %*% (sqrt(lam[3]) * eig$vectors[, 3]) / sqrt(lam[1] + lam[2] + lam[3])
            )
      }
      if (projalpha == 1) {
         lambar <- (lam[1] + lam[2] + lam[3]) / 3
         meanshape <-
            cbind(
               t(H) %*% (sqrt(abs(
                  lam[1] - lambar + 1 / m
               )) * eig$vectors[, 1]) ,
               t(H) %*% (sqrt(abs(
                  lam[2] - lambar + 1 / m
               )) * eig$vectors[, 2]) ,
               t(H) %*% (sqrt(abs(
                  lam[3] - lambar + 1 / m
               )) * eig$vectors[, 3])
            )
      }
   }
   if (riemdist(meanshape, mu) > riemdist(meanshape, mu, reflect = TRUE)) {
      meanshape[, m] <- -meanshape[, m]
   }
   meanshape
}



################################################################################################

#The Procrustes routines in the next part were initially
# written by Mohammad Faghihi (University of Leeds) 1993, although many improvements, corrections,
# and speed-ups have been done since then.

# add(a3) compute the summation of a3[,,i]'s
# bgpa(a3) compute the scaling coefficients (bi's)
# close1(a) adds one additional row to matrix a that is the same as the first row
# cnt3(a3) replace each a3[ , , i] by fcnt(a3[ , , i])
# del(po, w1) plots point of po and joins them by contiguity matrix w1.
# dif(a3) compute sum( tr (xi-xj)'(xi-xj) )/n^2 for i<j and ( xi=a3[ , , i] )
#NB RETURN$Gpa is  now GSS/n  (CHANGED to this in version 0.92!)

# dis(a, b, c) compute distances between three points a, b and c. Each 
#  point should contain two co-ordinates

# fJ(n) function makes a nxn matrix as (I - (1/n) * J(n)) such that J(n) is a nxn  
#  matrix with all entries 1.# fcel(n,d) generates n points such that the triangles between them have 
#  equal edges

# fcnt(a) = fJ(no. of rows of matrix "a")*a

# fgpa(a3,tol1,tol2) compute rotated and scaled shapes, Gpa statistics (dif.), and number of
#  iterations.

# fopa(a,b) function computes the ordinary procrustes statistics for two matrices  a
#  and b.

# fort(a,b) computes an orthogonal matrix to rotate matrix b such that the  
#  difference between a and b be minimum. 

# fos(a,b)  computes a scalar to scale matrix b such that the difference between a 
#  and b be minimum.

# ftrsq(a,b) tr{(b'aa'b)^(1/2)}
 
# graf(a3) plot a3[ , , i] for all i's in one plot

# msh(a3) compute the mean shape of a3[ , , i]'s

# Enorm(a) compute || a ||=sqrt{ trace( x'x ) }

# rgpa(a3,p) find the new rotated data till dif(old)-dif(new)<p

# sgpa(a3) scaling a3 by bgpa coefficients

# sh(a) compute the co-ordinates of shape point for triangle a. a should be a 3x2 
#  matrix.

# sim1(n, d, s) simulated n points with normal distribution 
#  (mean=fcel(n,d) sd=s )

# vec1(a3) vectorizes matrices a3[ , , i]


#=========================================================================


#==================================================================================
add <- function(a3)
{
   s <- 0
   for (i in 1:dim(a3)[3]) {
      s <- s + a3[,  , i]
   }
   return(s)
}

#==================================================================================
bgpa <- function(a3, proc.output = FALSE)
{
   #assumes a3 is centred
   h <- 0
   #	zd <- cnt3(a3)
   zd <- a3
   
   s <- 0
   n <- dim(a3)[3]
   #	for(j in 1:dim(a3)[3]) {
   #		s <- s + (Enorm(zd[,  , j])^2)
   #	}
   aa <- apply(zd, c(3), Enorm) ^ 2
   s <- sum(aa)
   #	for(i in 1:dim(a3)[3]) {
   #		h[i] <- sqrt(s/(Enorm(zd[,  , i])^2)) * eigen(zz)$vectors[i, 1]
   #	}
   
   
   #try to speed it up!
   
   omat <- t(vec1(zd))
   kk <- dim(omat)[2]
   nn <- dim(omat)[1]
   if (nn > kk) {
      #      qq<-diag(cov(vec1(zd)))
      qq <- rep(0, times = nn)
      
      for (i in 1:n) {
         qq[i] <- var(omat[i, ]) * (n - 1) / n
         omat[i, ] <- omat[i, ] - mean(omat[i, ])
      }
      omat <- diag(sqrt(1 / qq)) %*% omat
      n <- kk
      Lmat <- t(omat) %*% omat / n
      
      
      eig <- eigen(Lmat, symmetric = TRUE)
      U <- eig$vectors
      lambda <- eig$values
      
      V <- omat %*% U
      
      vv <- rep(0, times = n)
      for (i in 1:n) {
         vv[i] <- sqrt(t(V[, i]) %*% V[, i])
         V[, i] <- V[, i] / vv[i]
      }
      
      delta <- sqrt(abs(lambda / n)) * vv
      od <- order(delta, decreasing = TRUE)
      delta <- delta[od]
      V <- V[, od]
      
      h <- sqrt(s / aa) * V[, 1]
   }
   if (kk >= nn) {
      zz <- cor(vec1(zd))
      h <- sqrt(s / aa) * eigen(zz)$vectors[, 1]
   }
   
   h <- abs(h)
   return(h)
}

#==================================================================================
close1 <- function(a)
{
   a1 <- matrix(0:0, nrow = dim(a)[1] + 1, ncol = dim(a)[2])
   for (i in 1:dim(a)[1]) {
      a1[i,] <- a[i,]
   }
   a1[dim(a)[1] + 1,] <- a[1,]
   a1
}

#==================================================================================
cnt3 <- function(a3)
{
   #zz <- array(c(0:0), dim = c(dim(a3)[1], dim(a3)[2], dim(a3)[3]))
   #for(i in 1:dim(a3)[3]) {
   #zz[,  , i] <- fcnt(a3[,  , i])
   #}
   
   zz <- apply(a3, 3, fcnt)
   zz <- array(zz, dim(a3))
   return(zz)
}

#==================================================================================
del <- function(po, w1)
{
   plot(po,
        type = "n",
        xlab = "x",
        ylab = "y")
   text(po)
   n <- dim(po)[1]
   for (i in 1:n) {
      for (j in i:n) {
         if (w1[i, j] > 0) {
            a1 <- c(po[i, 1], po[j, 1])
            b1 <- c(po[i, 2], po[j, 2])
            lines(a1, b1)
         }
      }
   }
}

#==================================================================================
dis <- function(a, b, c)
{
   d <- 0
   d[1] <- sqrt((a[1] - b[1]) ^ 2 + (a[2] - b[2]) ^ 2)
   d[2] <- sqrt((a[1] - c[1]) ^ 2 + (a[2] - c[2]) ^ 2)
   d[3] <- sqrt((c[1] - b[1]) ^ 2 + (c[2] - b[2]) ^ 2)
   d
}

#==================================================================================
dif.old <- function(a3)
{
   s <- 0
   for (i in 1:(dim(a3)[3] - 1)) {
      for (j in (i + 1):dim(a3)[3]) {
         s <- s + ((Enorm(a3[,  , i] - a3[,  , j])) ^ 2)
      }
   }
   return(s)
}


#dif<-function(a3)
#original (slow) version
#{
#        s <- 0
#n<-dim(a3)[3]
#mshape<-add(a3)/n
#psum<-0
#for (i in 1:n){
#x<-a3[,,i]-mshape
#psum<-psum+sum(diag(t(x)%*%x))
#}
#psum*n
#}

#dif<-function(a3)
##faster version
#{
#x<-sweep(a3,c(1,2),apply(a3,c(1,2),mean))
#z<-Enorm(as.vector(x))^2/dim(a3)[3]
#z
#}

#==================================================================================
dif <- function (a3)
{
   #version that does not depend on scale of original measurements
   # assumes already centred
   cc <- centroid.size(add(a3) / dim(a3)[3])
   x <- sweep(a3, c(1, 2), apply(a3, c(1, 2), mean))
   z <- Enorm(as.vector(x) / cc) ^ 2 / dim(a3)[3]
   z
}

#==================================================================================
fJ <- function(n)
{
   zz <- matrix(1:1, n, n)
   H <- diag(n) - (1 / n) * zz
   H
}

#==================================================================================
fcel <- function(n, d)
{
   v <- ceiling(sqrt(n))
   p <- matrix(c(0:0), n, 2)
   for (i in 1:v) {
      for (j in 1:v) {
         if ((v * (i - 1) + j) < (n + 1)) {
            p[(v * (i - 1) + j), 1] <- (d / 4) * (-1) ^ i + (d *
                                                                j)
            p[(v * (i - 1) + j), 2] <- i * ((d * sqrt(3)) / 2)
         }
      }
   }
   p
}

#==================================================================================
fcnt <- function(a)
{
   aa <- fJ(dim(a)[1]) %*% a
   aa
}

#==================================================================================
fgpa.singleiteration <- function(a3, p)
{
   # Note this is an approximation to GPA -
   # It carries out an initial match by optimally rotating all the data,
   # the rescaling the observations, then rotating the observations
   # NB it does not repeat this until convergence, but in practice
   # for many real datasets this gives an excellent registration
   #
   zd <- list(
      rot. = 0,
      r.s.r. = 0,
      Gpa = 0,
      I.no. = 0,
      mshape = 0
   )
   zd$rot. <- rgpa(a3, p)
   zz <- rgpa(sgpa(zd$rot.$rotated), p)
   zd$r.s.r. <- zz$rotated
   zd$Gpa <- zz$dif
   zd$I.no. <- zz$r.no.
   zd$mshape <- msh(zd$r.s.r.)
   return(zd)
}

#==================================================================================
fgpa <- function(a3,
                 tol1,
                 tol2,
                 proc.output = FALSE,
                 reflect = FALSE)
{
   #
   #  Fully iterative fgpa (now assumes a3 is already centred)
   #
   #
   zd <- list(
      rot. = 0,
      r.s.r. = 0,
      Gpa = 0,
      I.no. = 0,
      mshape = 0
   )
   p <- tol1
   if (proc.output) {
      cat(" Step             | Objective function | change \n")
   }
   if (proc.output) {
      cat("---------------------------------------------------\n")
   }
   x1 <- dif(a3)
   if (proc.output) {
      cat("Initial objective fn", x1, "  -  \n")
   }
   if (proc.output) {
      cat("-----------------------------------------\n")
   }
   zz <- rgpa(a3, p, proc.output = proc.output, reflect = reflect)
   x2 <- dif(zz$rotated)
   if (proc.output) {
      cat("Rotation step      0", x2, x1 - x2, " \n")
   }
   if (proc.output) {
      cat("-----------------------------------------\n")
   }
   ii <- 1
   zz <- rgpa(
      sgpa(zz$rotated, proc.output = proc.output),
      p,
      proc.output = proc.output,
      reflect = reflect
   )
   x1 <- x2
   x2 <- dif(zz$rotated)
   rho <- x1 - x2
   if (proc.output) {
      cat("Scale/rotate step ", ii, x2, rho, " \n")
   }
   if (proc.output) {
      cat("-----------------------------------------\n")
   }
   
   if (rho > tol2) {
      while (rho > tol2) {
         x1 <- x2
         ii <- ii + 1
         zz <- rgpa(
            sgpa(zz$rotated, proc.output = proc.output),
            p,
            proc.output = proc.output,
            reflect = reflect
         )
         x2 <- dif(zz$rotated)
         rho <- x1 - x2
         if (proc.output) {
            cat("Scale/rotate step ", ii, x2, rho, " \n")
         }
         if (proc.output) {
            cat("-----------------------------------------\n")
         }
         
      }
   }
   zd$r.s.r. <- zz$rotated
   zd$Gpa <- zz$dif
   zd$I.no. <- ii
   zd$mshape <- msh(zd$r.s.r.)
   return(zd)
}

#==================================================================================
fgpa.rot <- function(a3,
                     tol1,
                     tol2,
                     proc.output = FALSE,
                     reflect = FALSE)
{
   # Assumes that a3 has been centred already
   zd <- list(
      rot. = 0,
      r.s.r. = 0,
      Gpa = 0,
      I.no. = 0,
      mshape = 0
   )
   p <- tol1
   zz <- rgpa(a3, p, proc.output = proc.output, reflect = reflect)
   x1 <- msh(zz$rotated)
   ii <- zz$r.no.
   #	zz <- rgpa(zz$rotated, p,proc.output=proc.output,reflect=reflect)
   #x2<-msh(zz$rotated)
   #rho<-riemdist(x1,x2)
   #        while (rho > tol2){
   #print(rho)
   #x1<-x2
   #ii<-ii+1
   #	zz <- rgpa(zz$rotated, p,proc.output=proc.output)
   #     x2<-msh(zz$rotated)
   #rho<-riemdist(x1,x2)
   #        }
   zd$r.s.r. <- zz$rotated
   zd$Gpa <- zz$dif
   zd$I.no. <- ii
   zd$mshape <- msh(zd$r.s.r.)
   return(zd)
}

#==================================================================================
fopa <- function(a, b)
{
   abar <- fcnt(a)
   bbar <- fcnt(b)
   q1 <- sum(diag(abar %*% t(abar)))
   q2 <- fos(a, b) ^ 2 * sum(diag(bbar %*% t(bbar)))
   q3 <- 2 * fos(a, b) * sum(diag(fort(a, b) %*% t(abar) %*% bbar))
   gs <- q1 + q2 - q3
   gs
}

#==================================================================================
fort.ROTATEANDREFLECT <- function(a, b)
{
   x <- t(fcnt(a)) %*% fcnt(b)
   xsvd <- svd(x)
   t <- xsvd$v %*% t(xsvd$u)
   return(t)
}

#==================================================================================
fos.REFLECT <- function(a, b)
{
   abar <- fcnt(a)
   bbar <- fcnt(b)
   z <- ftrsq(abar, bbar) / sum(diag(t(bbar) %*% bbar))
   z
}

#==================================================================================
fos <- function (a, b)
{
   z <- cos(riemdist(a, b)) * centroid.size(a) / centroid.size(b)
   z
}

#==================================================================================
ftrsq <- function(a, b)
{
   z <- sum(sqrt(abs(eigen(
      t(b) %*% a %*% t(a) %*% b
   )$values)))
   z
}

#==================================================================================
graf <- function(a3)
{
   l <- 0
   xmin <- 0
   xmax <- 0
   ymin <- 0
   ymax <- 0
   for (i in 1:dim(a3)[3]) {
      xmin[i] <- min(a3[, 1, i])
      xmax[i] <- max(a3[, 1, i])
      ymin[i] <- min(a3[, 2, i])
      ymax[i] <- max(a3[, 2, i])
   }
   l <- c(min(xmin), min(ymin), max(xmax), max(ymax))
   plot((min(l) - 1):(max(l) + 1), (min(l) - 1):(max(l) + 1), type = "n")
   for (i in 1:dim(a3)[3]) {
      lines(close1(a3[,  , i]))
   }
}

#==================================================================================
msh <- function(a3)
{
   s <- 0
   #	print("finding mean shape")
   m <- apply(a3, c(1, 2), mean)
   #	print("found mean shape")
   #	for(i in 1:dim(a3)[3]) {
   #		s <- s + a3[,  , i]
   #	}
   #	m <- (1/dim(a3)[3]) * s
   return(m)
}


#Enorm<-function(a)
#{
#	return(sqrt(sum(diag(t(a) %*% a))))
#}

#==================================================================================
rgpa <- function(a3,
                 p,
                 reflect = FALSE,
                 proc.output = FALSE)
{
   # assumes a3 already centred now
   if (reflect == TRUE)
   {
      fort <- fort.ROTATEANDREFLECT
   }
   zd <- list(
      rotated = 0,
      dif = 0,
      r.no. = 0,
      inc = 0
   )
   l <- dim(a3)[3]
   a <- 0
   d <- 0
   n <- 0
   #	zz <- cnt3(a3)
   zz <- a3
   #        print("Rotations ...")
   #       print("Iteration,meanSS before,meanSS after,difference,tolerance")
   d[1] <- 10 ^ 12
   d[2] <- dif(zz)
   a[1] <- d[2]
   s <- add(zz)
   #       print(c(d[1],d[2]))
   if (dif(zz) > p) {
      while (d[1] - d[2] > p) {
         n <- n + 1
         d[1] <- d[2]
         for (i in 1:l) {
            old <- zz[,  , i]
            zz[,  , i] <- old %*% fort(((1 / (l - 1)) * (s - old)),
                                       old)
            s <- s - old + zz[,  , i]
         }
         d[2] <- dif(zz)
         a[n + 1] <- d[2]
         #		print(c(n,d[1],d[2],d[1]-d[2],p))
         if (proc.output) {
            cat("  Rotation iteration  ", n, d[2], d[1] - d[2], " \n")
         }
      }
   }
   zd$rotated <- zz
   zd$dif <- a
   zd$r.no. <- n
   zd$inc <- d[1] - d[2]
   if (proc.output) {
      cat("-----------------------------------------\n")
   }
   
   fort <- fort.ROTATION
   return(zd)
}


# sgpa<-function(a3)
#{
#	zz <- a3
#	a <- bgpa(zz)
#	for(i in 1:dim(a3)[3]) {
#		zz[,  , i] <- a[i] * a3[,  , i]
#	}
#	return(zz)
#}

#==================================================================================
sgpa <- function(a3, proc.output = FALSE)
{
   #assumes a3 is centred
   zz <- a3
   di <- dim(a3)
   a <- bgpa(zz, proc.output = proc.output)
   i <- rep(dim(a3)[1] * dim(a3)[2], dim(a3)[3])
   sequen <- rep(a, i)
   
   zz <- array(as.vector(a3) * sequen, di)
   if (proc.output) {
      cat("  Scaling updated \n")
   }
   
   return(zz)
}


#==================================================================================
sh <- function(a)
{
   u1 <- (a[2, 1] - a[1, 1]) / sqrt(2)
   u2 <- (a[2, 2] - a[1, 2]) / sqrt(2)
   v1 <- (2 * a[3, 1] - a[2, 1] - a[1, 1]) / sqrt(6)
   v2 <- (2 * a[3, 2] - a[2, 2] - a[1, 2]) / sqrt(6)
   d <- c(0, 0)
   d[1] <- (u1 * v1 + u2 * v2) / (u1 ^ 2 + u2 ^ 2)
   d[2] <- (u1 * v2 - u2 * v1) / (u1 ^ 2 + u2 ^ 2)
   d
}

#==================================================================================
sim1 <- function(n, d, s)
{
   a <- fcel(n, d)
   sig <- matrix(c(1:1), n, 1)[, 1]
   sig <- sig * s
   b <- a
   b[, 1] <- rnorm(n, mean = a[, 1], sd = sig)
   b[, 2] <- rnorm(n, mean = a[, 2], sd = sig)
   b
}

#==================================================================================
vec1 <- function(a3)
{
   #zz <- array(c(0:0), dim = c((dim(a3)[1] * dim(a3)[2]), dim(a3)[3]))
   #for(i in 1:dim(a3)[3]) {
   #for(j in 1:dim(a3)[2]) {
   #for(k in 1:dim(a3)[1]) {
   #zz[((j - 1) * dim(a3)[1] + k), i] <- a3[k, j, i
   #]
   #}
   #}
   #}
   
   zz <- matrix(a3, dim(a3)[1] * dim(a3)[2], dim(a3)[3])
   return(zz)
}

#==================================================================================
fort.ROTATION <- function(a, b)
{
   x <- t(fcnt(a)) %*% fcnt(b)
   xsvd <- svd(x)
   v <- xsvd$v
   u <- xsvd$u
   tt <- v %*% t(u)
   chk1 <- Re(prod(eigen(v)$values))
   chk2 <- Re(prod(eigen(u)$values))
   if ((chk1 < 0) && (chk2 > 0))
   {
      v[, dim(v)[2]] <- v[, dim(v)[2]] * (-1)
      tt <- v %*% t(u)
   }
   if ((chk2 < 0) && (chk1 > 0))
   {
      u[, dim(u)[2]] <- u[, dim(u)[2]] * (-1)
      tt <- v %*% t(u)
   }
   return(tt)
}

############end of Mohammad Faghihi's (adapted) routines

#alias functions (all lower-case)

hotelling2d <- Hotelling2D
hotellingtest <- Hotellingtest
procrustesgpa <- procrustesGPA
goodall2d <- Goodall2D
goodalltest <- Goodalltest

# alias
TPSgrid <- tpsgrid

#if you wish the default to *not* include reflection
#invariance (as is normal in shape analysis) then you need the line below.

fort <- fort.ROTATION


################################################################################
#
#  Datasets
#
################################################################################


#==================================================================================
# Gorillas
#==================================================================================
gorf.dat<-array(c(5,193,53,-27,0,0,0,33,-2,105,18,176,72,114,92,38
                  ,51,191,55,-31,0,0,0,33,25,106,56,171,98,105,99,15
                  ,36,187,59,-31,0,0,0,36,12,102,38,171,91,103,100,19
                  ,23,202,48,-30,0,0,0,39,3,103,33,180,84,112,94,28
                  ,30,185,62,-25,0,0,0,32,11,101,37,168,85,106,96,21
                  ,4,195,65,-21,0,0,0,34,-4,100,15,180,69,120,102,34
                  ,37,195,62,-32,0,0,0,35,20,101,50,173,102,105,105,22
                  ,41,191,58,-34,0,0,0,34,15,100,47,175,93,105,99,18
                  ,40,190,52,-33,0,0,0,38,13,107,44,176,88,113,102,31
                  ,-4,179,62,-21,0,0,0,29,1,89,9,164,70,111,100,36
                  ,41,206,53,-25,0,0,0,39,11,104,47,177,95,111,95,26
                  ,33,197,55,-30,0,0,0,35,7,106,39,175,89,111,95,24
                  ,-12,205,52,-15,0,0,0,38,-10,111,4,187,66,129,80,44
                  ,13,186,56,-32,0,0,0,34,8,101,25,166,80,105,97,26
                  ,20,186,45,-31,0,0,0,34,10,96,31,165,84,104,90,19
                  ,29,183,55,-31,0,0,0,32,10,98,39,163,82,106,95,17
                  ,11,203,57,-28,0,0,0,39,-2,106,23,182,77,122,100,36
                  ,37,187,54,-27,0,0,0,34,11,100,43,171,84,106,93,28
                  ,49,191,53,-31,0,0,0,35,21,102,54,172,94,98,99,18
                  ,-8,191,57,-34,0,0,0,32,-7,93,6,173,71,119,101,30
                  ,43,184,49,-32,0,0,0,33,14,100,49,165,91,99,98,20
                  ,57,185,62,-37,0,0,0,35,22,103,61,169,96,100,104,24
                  ,-10,196,55,-20,0,0,0,38,-10,107,5,181,73,123,88,46
                  ,20,195,60,-28,0,0,0,32,6,101,33,173,84,114,100,30
                  ,35,202,59,-27,0,0,0,34,6,108,41,182,83,117,99,31
                  ,1,188,60,-19,0,0,0,35,-2,99,12,170,70,119,93,45
                  ,24,194,52,-24,0,0,0,39,8,105,34,174,80,115,95,32
                  ,25,204,55,-27,0,0,0,34,7,108,35,185,83,118,92,32
                  ,36,198,47,-30,0,0,0,39,14,110,43,177,92,105,98,25
                  ,8,198,53,-35,0,0,0,34,4,101,22,175,82,111,100,24),c(2,8,30))
gorf.dat<-aperm(gorf.dat,c(2,1,3))


gorm.dat<-array(c(53,220,46,-35,0,0,0,37,12,122,58,204,93,117,103,28
                  ,57,219,50,-43,0,0,0,37,13,119,61,198,102,110,104,20
                  ,89,227,52,-47,0,0,0,32,35,120,93,201,131,92,104,4
                  ,46,222,51,-45,0,0,0,30,11,113,54,196,101,117,101,16
                  ,85,220,48,-38,0,0,0,39,28,125,87,203,121,106,103,7
                  ,64,208,43,-39,0,0,0,36,22,111,67,191,104,102,101,18
                  ,67,216,51,-37,0,0,0,35,17,119,68,191,108,109,94,15
                  ,35,236,61,-42,0,0,0,33,2,119,43,211,90,126,104,30
                  ,116,218,40,-38,0,0,0,41,41,124,116,201,133,94,103,12
                  ,56,234,60,-34,0,0,0,34,12,121,58,215,109,119,112,28
                  ,40,223,58,-36,0,0,0,34,9,113,46,202,97,120,112,24
                  ,94,223,49,-57,0,0,0,33,31,122,94,206,136,99,113,-1
                  ,68,222,59,-41,0,0,0,30,18,119,68,204,104,114,98,11
                  ,65,224,56,-33,0,0,0,35,15,130,67,205,108,115,95,20
                  ,67,214,52,-47,0,0,0,36,26,115,74,192,114,105,104,11
                  ,110,213,52,-46,0,0,0,37,42,121,109,190,133,97,108,-8
                  ,46,219,50,-42,0,0,0,36,11,121,56,199,104,108,102,21
                  ,79,209,66,-43,0,0,0,35,24,114,84,193,108,115,109,14
                  ,58,244,74,-22,0,0,0,37,7,131,64,219,98,128,100,29
                  ,43,236,64,-43,0,0,0,33,12,124,52,215,110,121,105,7
                  ,70,226,54,-37,0,0,0,39,28,121,74,204,122,105,107,7
                  ,68,224,55,-37,0,0,0,35,18,121,71,207,109,108,98,13
                  ,34,247,63,-35,0,0,0,35,4,124,45,225,104,135,110,29
                  ,49,236,59,-40,0,0,0,38,19,127,59,219,105,121,109,25
                  ,98,195,44,-44,0,0,0,36,30,116,98,177,121,89,105,10
                  ,109,208,49,-40,0,0,0,36,29,125,105,189,120,102,102,12
                  ,61,224,51,-35,0,0,0,41,15,122,67,206,107,121,103,25
                  ,43,213,49,-57,0,0,0,28,20,111,58,194,112,106,108,6
                  ,26,249,67,-14,0,0,0,38,-11,130,33,225,87,148,97,53),c(2,8,29))
gorm.dat<-aperm(gorm.dat,c(2,1,3))

#==================================================================================
# mice
#==================================================================================
qset2.dat<-array(c(117.98,219.62,114.52,41.93,166.15,113.59,206.54,121.79,165.11,142.92,62.07,136.58
                   ,105.52,235.08,109.96,57.31,165.44,126.42,223.05,140.88,169.58,156.83,59.5,138.02
                   ,142.83,222,132.08,40.2,188.8,115.39,236.9,125.08,193.18,142.22,83.37,141.47
                   ,126.35,204.27,113.2,36.04,171.77,102.43,228.93,111.75,173.55,131.8,66.26,126.65
                   ,99.11,231.52,119.58,44.52,169.28,129.51,219.93,141.98,167.65,154.41,56.83,141.64
                   ,134.14,228.48,129.09,56.98,175.85,125.39,217.57,138.01,179.29,152.67,73.84,153.08
                   ,119.8,219.48,122.36,48.52,171.44,115.58,216.57,131.13,170.86,148.36,65.8,138.8
                   ,105.62,222.74,96.27,51.3,152.13,119.07,205.02,131.95,157.36,141.61,52.35,142.73
                   ,122.15,202.41,128.75,32.09,176.88,102.84,220.81,119.41,177.78,131.04,76.65,120.61
                   ,127.78,223.48,117.65,53.94,183.45,123.87,236.96,133.84,184.24,149.04,77.96,149.95
                   ,123.4,200.6,116.57,28.43,172.67,97.33,221.34,106.24,175.29,122.05,64.69,115.53
                   ,132.4,227.96,127.46,50.4,171.57,118.67,203.72,131.91,175.25,152.38,71.98,138.5
                   ,123.39,216.23,113.96,29.9,167.68,105.58,202.15,114.18,172.01,133.8,65.78,125.86
                   ,136.83,207.84,117.37,33.67,178.1,100.65,233.17,111.44,184.05,124.85,69.8,129.69
                   ,143.31,219.71,122.1,48.08,177.23,113.52,221.38,122.32,180.86,140.87,80.11,151.72
                   ,105.96,218.06,100.46,48.73,155.23,119.51,217.76,130.01,159.11,145.49,51.29,139.66
                   ,115.73,234.14,115.74,56.11,168.42,128.73,220.32,135.6,169.4,154.65,66.02,152.32
                   ,101.73,215.74,102.67,35.73,151.69,110.19,199.12,120.24,151.6,136.36,48.82,132.42
                   ,124.93,222.42,130.09,46.89,163.86,118.81,197.76,137.59,165.11,148.92,67.45,139.67
                   ,104.68,231.95,88.17,53.4,150.87,128.76,203.38,143.96,151.59,152.11,41.85,151.15
                   ,123.93,242.1,121.99,64.08,175.47,143.77,211.56,155.4,173.07,164.91,68.98,158.26
                   ,137.21,207.27,130.76,34.56,188.92,105.07,242.85,109.74,187.67,127.43,80.29,127.14
                   ,107.35,212.21,89.5,38.88,151.63,105.73,196.56,113.82,152.69,133.73,46.53,135.99),c(2,6,23))
qset2.dat<-aperm(qset2.dat,c(2,1,3))
qcet2.dat<-array(c(168.59,35.66,159.99,215.17,104.93,141.07,49.01,115.13,101.69,110.87,227.11,120.51
                   ,165,38.35,163.89,214.32,108.26,144.79,50.17,124.02,103.5,113.13,220.43,122.65
                   ,166.97,39.44,163.63,221.62,108.18,147.07,54.11,137.43,109.15,118.11,227.89,128.32
                   ,164.02,44.22,171.76,237.29,95.93,161.44,32,131.85,95.91,126.54,222.54,133.51
                   ,153.46,40.67,156.62,225.12,103.81,152.44,43.65,139.01,99.95,120.7,216.62,132.9
                   ,148.3,52.66,147.61,239.16,90.37,164.39,32.39,145.13,88.1,130.85,209.22,145.88
                   ,141.33,32.7,128.49,215.09,71.39,135.31,18.32,115.62,77.48,102.92,186.28,125.49
                   ,130.21,18.71,136.38,201.17,68.85,125.19,11.43,94.64,64.22,90.83,184.64,107
                   ,130,26.8,134.24,217.41,77.07,141.93,14.66,122.68,74.56,105.14,189.9,109.31
                   ,134.99,22.44,116.08,205.87,74.69,130.05,22.18,96.09,68.08,95.91,185.97,115.99
                   ,146.87,22.6,111.5,201.74,77.67,124.1,23.04,97.03,80.91,85.93,192.2,118.4
                   ,146.26,23.38,119.94,209.46,75.94,125.31,19.72,100.92,82.97,87.47,192.22,109.63
                   ,138.32,25.44,119.25,208.88,71.18,133.17,16.68,103.45,69.21,98.08,178.43,123.71
                   ,142.57,19.25,99.1,197.59,62.7,111.06,3.94,86.78,69.17,79.06,180.48,117.38
                   ,144.57,21.44,129,204.76,80.93,121.11,18.96,103.58,82.35,90.59,191.23,111.98
                   ,137.6,19.71,120.16,214.71,77.7,128.06,16.08,102.22,73.54,90.19,193.21,118.14
                   ,123.81,22.67,118.93,207.44,73.63,129.11,4.94,104.21,71.72,100.35,191.62,119.61
                   ,131.1,28.64,94.82,211.86,59.44,129.47,3.89,102.71,70.28,95.52,178.71,131.69
                   ,155.84,22.41,112.53,207.45,68.64,118.65,8.37,97.65,78.78,85.28,184.6,114.92
                   ,174.04,27.44,108.13,202.49,80.74,111.51,20.1,79.49,96.21,80.84,205.25,127.23
                   ,127.85,20.44,119.59,208.01,61.98,125.83,4.2,112.66,80.06,90.68,182.45,108.83
                   ,146.48,28.36,113.82,214.92,75.14,127.12,14.06,98.64,81.57,96.68,198.33,125.63
                   ,139.46,33.99,99.48,220.24,73.24,135.58,8.99,105,75.01,103.38,191.51,135.63
                   ,152.27,30.86,138.67,226.92,91.78,137.46,24.93,121.51,92.28,107.03,216.37,131.07
                   ,139.15,28.78,133.63,210.56,88.84,131.16,26.42,112.92,87.97,99.71,201.01,119.24
                   ,153.32,20.49,109.35,201.5,76.58,109.87,10.91,91.13,74.45,85.03,183.14,119.64
                   ,166.02,23.04,140.31,215.68,91.82,128.43,26.01,109.78,94.61,99.14,212.09,119.48
                   ,127.79,22.26,136.79,202.76,89.87,131.73,26.63,113.72,88.85,90.81,198.88,105.03
                   ,169.22,26.47,158.71,210.32,102.37,134.39,30.64,108.87,100.56,97.55,220.44,125.16
                   ,146.88,23.83,116.78,218.55,84.03,132.22,25.08,111.84,89.31,94.31,194.7,129.03), 
                 c(2,6,30))
qcet2.dat<-aperm(qcet2.dat,c(2,1,3))
qlet2.dat<-array(c(134.26,224.28,122.34,36.86,174.35,111.24,235.79,127.9,174.49,142.43,51.5,141.89
                   ,139.29,231.38,80.82,47.82,159.37,105.97,236.93,120.91,162.59,139.69,39.92,163.16
                   ,151.57,219.95,104.42,49.26,162.95,105.68,241.12,123.23,181.45,133.49,60.47,151.83
                   ,146.16,231.16,95.78,46.87,170.85,111.77,234.72,109.48,178.8,140.05,56.79,157.8
                   ,150.16,222.81,81.59,53.08,161.66,102.76,238.7,94,173.74,131.99,42.42,166.01
                   ,134.04,218.32,84.43,40.37,155.91,104.39,227.07,112.75,163.69,135.45,45.04,154
                   ,141,221.6,98.86,46.69,160.43,112.77,218.15,114.61,162.83,136.17,37.06,153.86
                   ,123.63,231.23,66.42,46.17,140.02,108.31,217.82,118,147.98,137.78,25.35,159.24
                   ,137.62,226.64,96.5,39.64,164.59,105.39,235.04,102.87,169.37,134.5,46.83,150.8
                   ,173.79,206.17,90.39,27.9,161.84,84.83,234.85,91.6,180.59,110.28,55.57,160.27
                   ,117.39,233.75,114.47,42.63,167.1,124.53,229.45,135.47,167.66,150.42,40.92,142.56
                   ,131.55,225.48,102.05,26.81,161.73,103.06,244.48,115.44,170.49,133.86,38.65,144.13
                   ,134.78,226.28,110.56,41.28,170.17,112.5,231.45,114.1,176.5,139.26,51.28,145.88
                   ,115.95,227.77,85.99,46.33,156.48,111.17,220.59,117.43,161.6,140.85,43.06,154.56
                   ,133.09,226.95,99.38,40.42,160.22,111.1,236.36,117.74,168.23,138.63,44.79,149.37
                   ,125.36,216.11,93.92,36.95,161.42,103.33,220.04,104.98,165.19,129.44,43.48,133.74
                   ,123.1,202.86,99.27,42.82,157.22,98.93,206.46,111.18,162.05,129.3,64.39,123.39
                   ,121.37,217.11,108.09,34.09,155.47,105.03,214.66,121.61,160.24,135.04,48.64,133.8
                   ,120.54,232.1,85.37,53.23,157.81,109.98,213.66,117.86,169.01,146.01,38.88,151.63
                   ,126.42,222.64,82.96,46.57,157.34,100.52,218.67,101.9,165.27,131.51,42.51,147.56
                   ,126.91,220.11,105.38,35.76,158.15,109.6,207.12,117.23,160.11,138.72,40.17,142.75
                   ,117.87,227.17,113.96,49.9,163.19,119.44,228.16,124.54,163.3,152.93,42.17,141.99
                   ,125.23,224.48,93.4,39.47,166.22,109.39,234.83,108.47,165.65,139.89,42.06,144.54), 
                 c(2,6,23))
qlet2.dat<-aperm(qlet2.dat,c(2,1,3))

#==================================================================================
digit3.dat<-c(9,27,12,31,17,36,26,39,34,37,36,33,38,27,35,19,30,15,21,14,21,8,16,6,8,5
              ,17,40,21,38,26,36,27,32,25,28,22,27,19,29,24,25,26,20,28,16,26,13,18,14,15,17
              ,19,38,24,38,29,33,30,29,27,24,21,25,17,26,27,24,30,22,31,19,31,16,27,15,24,15
              ,9,40,15,43,24,41,29,36,24,30,20,26,12,22,20,22,24,20,21,16,18,14,13,12,9,10
              ,14,41,21,42,29,42,35,37,32,33,26,30,16,26,25,26,29,24,33,20,30,16,23,11,16,12
              ,24,39,28,40,35,38,38,35,34,30,29,27,22,24,27,24,29,22,31,19,28,15,20,11,13,12
              ,9,39,15,39,21,40,25,36,23,31,21,27,19,25,21,25,23,24,25,22,22,19,15,17,8,17
              ,8,38,14,41,25,43,29,38,25,33,18,29,8,28,12,27,16,25,18,23,13,21,7,21,1,22
              ,4,34,12,39,22,42,31,36,27,30,23,28,11,25,20,25,22,24,22,22,19,19,13,18,8,18
              ,21,36,25,37,31,36,33,32,32,28,29,25,27,22,29,21,31,20,31,18,28,16,24,16,20,16
              ,14,40,20,39,25,37,27,31,26,28,20,29,16,31,21,28,25,23,28,16,25,13,17,15,13,18
              ,12,40,20,42,30,42,36,33,31,24,23,22,16,23,25,22,31,18,33,13,31,9,24,8,17,8
              ,9,35,17,36,26,34,30,31,26,27,20,25,13,27,19,25,23,21,26,15,22,12,12,12,7,13
              ,17,38,24,39,30,37,34,34,31,28,22,25,16,28,21,26,27,24,30,20,26,15,18,14,10,17
              ,21,35,27,36,36,35,39,28,38,22,34,18,28,19,31,18,33,17,31,15,26,15,20,17,14,20
              ,16,40,20,43,25,39,27,31,24,24,19,21,17,23,19,22,21,21,23,21,22,18,19,16,15,16
              ,15,41,21,45,34,44,40,39,36,35,26,30,16,29,24,25,28,20,31,16,28,14,21,14,12,12
              ,11,42,22,42,32,39,35,34,32,29,25,26,20,27,25,26,31,23,35,19,31,14,21,12,16,15
              ,5,44,15,43,24,41,29,36,22,28,13,28,5,29,14,28,24,26,29,22,26,19,17,17,10,20
              ,14,37,19,39,25,38,28,32,25,26,20,22,14,23,17,23,21,20,23,17,21,15,16,15,11,15
              ,16,35,22,38,30,36,32,29,29,23,23,20,17,20,20,19,24,17,26,14,21,11,16,12,12,15
              ,14,38,17,40,25,42,28,38,27,32,24,28,20,25,23,25,26,24,28,21,24,18,18,17,10,18
              ,7,40,13,43,22,45,31,42,27,38,21,34,13,32,18,31,24,30,27,27,23,23,15,22,6,22
              ,14,35,21,36,26,34,31,30,28,26,25,22,21,18,21,17,22,16,23,15,20,12,13,10,5,10
              ,10,46,17,47,27,43,29,36,26,30,22,29,16,28,20,27,21,25,23,21,21,19,15,20,9,20
              ,18,39,24,42,33,41,38,35,37,30,32,28,28,27,33,22,37,18,41,15,37,13,29,11,21,12
              ,18,38,22,42,30,42,34,36,33,32,29,30,22,28,25,26,28,24,28,20,27,19,22,18,18,18
              ,9,41,17,43,30,40,34,31,30,23,23,19,11,19,15,17,18,13,21,10,17,8,12,7,5,7
              ,8,36,12,42,20,43,25,38,24,35,23,33,21,32,20,31,20,30,20,27,16,25,9,24,2,25
              ,19,41,24,45,33,45,38,38,36,31,28,27,21,23,24,22,26,20,28,17,26,14,20,13,14,11)
digit3.dat<-array(digit3.dat,c(2,13,30))
digit3.dat<-aperm(digit3.dat,c(2,1,3))
digit3.dat[,2,]<- -digit3.dat[,2,]

#==================================================================================
dna.dat<-array(c(23.825,12.021,13.002,25.742,18.923,13.225,22.784,25.153,15.445,17.418
                 ,28.811,17.309,12.468,29.084,21.876,8.439,26.152,26.442,5.606,21.087
                 ,29.894,6.200,14.944,33.112,8.585,12.034,38.280,14.445,8.903,39.890
                 ,20.638,10.881,42.477,21.716,27.837,44.502,24.662,23.765,40.767,25.795
                 ,17.985,37.076,24.130,12.686,33.111,20.240,10.380,28.222,14.758,7.779
                 ,24.003,9.126,10.368,21.474,7.514,14.143,16.620,5.636,20.061,13.796
                 ,9.181,24.591,9.975,15.219,27.372,7.792
                 
                 ,24.426,12.320,12.815,25.129,18.383,14.600,22.555,25.151,16.215,17.277
                 ,28.975,17.620,12.170,28.852,22.133,7.779,25.623,26.239,5.167,20.176
                 ,30.154,6.302,14.201,32.961,8.863,11.842,38.282,15.019,9.028,40.102
                 ,21.051,10.202,42.586,22.535,27.544,45.013,24.570,23.693,40.375,25.299
                 ,18.777,35.736,23.716,12.495,33.049,19.815,9.500,29.441,13.961,7.614
                 ,25.021,9.187,10.748,21.303,7.266,14.440,16.882,5.328,20.518,13.417
                 ,9.450,24.724,9.887,16.128,27.037,7.757
                 
                 ,24.824,12.355,12.721,25.459,18.787,14.482,22.545,24.678,16.606,17.068
                 ,28.216,18.062,11.696,28.643,22.449,7.256,25.495,26.539,4.443,20.008
                 ,29.999,5.302,14.541,33.584,8.475,12.111,38.538,14.696,9.392,40.333
                 ,21.118,10.517,42.136,23.203,27.877,44.420,25.594,23.483,39.833,25.459
                 ,18.596,35.894,23.239,12.499,33.283,19.753,10.123,28.758,14.685,7.357
                 ,24.571,9.730,10.141,20.369,7.686,14.172,15.883,5.535,20.148,13.033
                 ,9.722,25.009,10.287,16.126,27.525,8.623
                 
                 ,24.562,12.271,12.907,25.641,18.820,14.322,22.811,24.763,17.048,17.248
                 ,28.619,18.489,11.519,27.989,22.598,8.070,25.157,27.032,3.894,19.924
                 ,29.213,5.220,15.143,33.490,8.348,12.542,38.506,14.119,9.535,40.897
                 ,20.276,10.987,42.136,23.192,28.071,44.539,25.274,23.775,39.509,25.630
                 ,18.350,35.879,23.652,12.505,32.860,19.781,9.775,28.642,14.454,6.712
                 ,24.583,9.440,9.756,20.462,7.865,14.225,16.493,5.837,20.340,13.680
                 ,9.425,25.287,10.594,15.637,27.792,8.320
                 
                 ,24.498,12.655,13.535,26.012,19.043,14.582,22.839,24.695,17.030,17.673
                 ,28.305,18.898,12.275,28.466,22.641,7.960,25.669,26.958,4.000,20.256
                 ,29.476,5.319,15.623,33.821,8.755,12.476,38.043,14.771,9.263,39.758
                 ,20.522,10.340,41.694,22.720,27.780,45.260,25.492,23.558,40.577,25.904
                 ,18.294,35.791,23.664,13.116,32.898,19.581,9.743,29.357,14.351,7.149
                 ,25.092,10.135,9.427,20.128,7.067,13.562,15.964,5.322,20.148,13.918
                 ,9.684,24.717,10.987,15.626,27.045,8.706
                 
                 ,24.253,12.608,13.053,25.931,19.065,13.470,23.190,24.682,16.697,17.355
                 ,28.266,18.178,11.941,28.445,21.624,8.302,26.165,26.289,4.645,20.902
                 ,29.472,5.834,15.445,33.346,9.014,13.039,38.182,14.821,8.706,39.192
                 ,20.416,10.356,42.138,21.676,27.355,45.520,25.249,22.900,41.575,26.118
                 ,18.221,36.773,23.771,12.865,32.905,19.117,9.404,29.920,14.366,7.244
                 ,25.218,9.684,9.810,21.100,7.052,13.603,16.361,5.871,20.211,14.759
                 ,9.332,24.692,10.714,14.984,27.381,8.114
                 
                 ,23.794,12.783,13.221,26.116,19.069,14.260,22.944,24.986,16.310,17.113
                 ,28.409,17.881,11.562,28.432,21.504,7.479,25.690,25.928,4.418,20.599
                 ,30.049,5.580,15.584,34.067,9.173,12.544,38.631,15.060,9.018,38.984
                 ,20.102,10.329,42.574,22.345,27.701,44.920,25.689,22.778,41.294,26.502
                 ,17.802,36.716,23.573,12.346,33.446,19.038,9.376,29.982,14.359,7.379
                 ,25.213,9.768,10.069,20.982,7.294,13.751,16.439,5.559,20.199,13.972
                 ,9.213,24.830,10.359,15.581,27.031,7.719
                 
                 ,23.617,12.350,13.813,25.639,19.130,14.870,22.756,25.155,16.183,16.854
                 ,28.676,17.591,11.233,28.507,21.359,7.934,26.050,26.376,5.116,20.641
                 ,29.944,5.718,15.029,32.991,8.692,12.395,37.840,14.732,8.728,39.201
                 ,19.511,9.906,43.019,22.376,27.740,44.419,25.920,23.168,41.030,26.208
                 ,18.416,36.239,23.729,12.432,33.529,19.435,9.740,29.434,14.798,7.666
                 ,24.759,9.937,10.295,20.754,7.938,13.711,15.998,5.671,20.047,14.131
                 ,9.114,24.716,10.494,14.954,26.951,7.747
                 
                 ,24.012,12.503,13.953,25.912,19.342,14.449,23.029,25.100,16.549,17.484
                 ,28.586,18.903,11.667,28.481,21.397,7.785,26.260,26.607,4.911,20.648
                 ,29.762,5.753,15.350,33.300,8.908,12.434,37.994,14.578,8.736,39.552
                 ,19.979,10.470,43.119,22.662,27.821,43.727,25.443,22.420,41.221,26.247
                 ,17.848,36.494,24.167,12.379,32.929,19.582,9.488,29.216,14.702,7.711
                 ,24.595,9.384,9.829,20.604,7.131,13.683,16.123,5.060,20.085,13.915
                 ,8.908,25.200,11.222,14.847,27.059,8.911
                 
                 ,23.851,12.509,13.804,25.990,19.002,14.469,22.931,24.771,16.634,17.508
                 ,28.373,18.286,11.915,28.212,21.270,7.770,25.930,26.098,5.343,20.386
                 ,29.828,5.846,15.077,33.966,9.031,12.428,38.933,15.099,8.924,39.810
                 ,20.518,10.602,43.163,21.689,28.245,42.375,25.115,22.774,40.248,25.516
                 ,17.567,36.126,23.642,11.533,33.571,19.658,9.291,29.426,14.590,7.776
                 ,24.159,9.043,9.992,20.933,7.292,14.245,16.582,5.694,20.509,13.985
                 ,8.699,25.401,10.715,15.055,27.404,8.281
                 
                 ,23.695,12.829,13.504,25.764,19.437,13.969,22.597,25.450,15.948,17.092
                 ,28.423,18.022,11.790,28.676,21.803,7.059,25.934,26.430,5.091,20.793
                 ,30.234,5.589,15.473,34.066,8.997,11.769,38.160,14.696,8.522,40.353
                 ,19.819,10.878,44.196,22.554,28.318,43.074,25.597,23.197,40.337,25.806
                 ,17.780,36.375,23.787,11.741,33.154,19.557,8.962,29.841,14.286,7.574
                 ,25.083,9.196,9.944,20.748,7.313,14.236,16.045,5.646,20.541,13.471
                 ,9.889,25.682,11.896,15.361,27.634,8.380
                 
                 ,24.570,13.140,12.786,26.102,19.946,13.576,22.686,25.498,15.318,17.702
                 ,28.165,18.723,12.523,28.503,22.326,7.233,26.220,26.025,4.907,21.266
                 ,30.023,5.678,15.793,33.796,9.233,12.043,37.348,14.858,8.771,40.203
                 ,19.898,10.903,44.171,21.444,28.378,42.556,25.142,23.227,39.449,25.636
                 ,17.412,36.446,23.332,11.752,33.264,18.848,9.020,29.643,13.468,7.620
                 ,24.944,9.220,9.848,21.160,7.691,13.422,15.760,6.713,19.640,13.470
                 ,9.683,25.976,12.855,14.939,27.417,8.986
                 
                 ,24.627,13.046,12.380,25.895,19.776,12.833,23.067,25.155,15.595,18.103
                 ,27.715,18.877,13.546,27.639,23.414,7.305,26.190,26.006,4.724,20.975
                 ,29.586,5.140,15.664,33.846,9.195,11.925,37.119,15.034,8.789,39.473
                 ,20.075,10.553,43.273,21.398,28.628,42.349,25.145,23.406,40.183,25.822
                 ,18.112,36.110,23.461,12.655,32.655,18.873,9.201,29.785,13.440,7.458
                 ,25.414,8.322,9.906,21.597,7.603,13.366,16.335,6.232,19.783,14.282
                 ,9.705,25.919,12.853,14.983,27.636,9.035
                 
                 ,24.787,13.151,13.411,26.120,19.950,13.492,22.909,25.361,15.449,17.724
                 ,28.036,18.727,13.426,27.879,23.848,7.001,25.988,26.517,5.155,20.872
                 ,30.584,5.420,15.382,34.616,9.694,11.348,37.645,15.821,8.699,39.613
                 ,20.186,10.669,43.251,21.809,28.473,42.674,25.263,23.078,39.879,25.499
                 ,17.890,35.848,23.637,12.841,32.064,18.802,8.835,28.782,13.400,7.644
                 ,24.509,8.059,10.687,21.628,7.038,13.879,16.268,6.444,19.486,12.936
                 ,9.772,25.582,11.871,15.490,27.474,8.755
                 
                 ,24.879,12.862,13.363,25.855,19.829,14.067,22.454,25.488,15.988,16.932
                 ,28.288,18.371,13.232,28.098,23.747,7.640,26.351,27.493,5.631,20.657
                 ,30.620,5.432,14.891,34.055,10.077,11.331,37.364,15.912,8.673,39.810
                 ,20.365,10.935,43.227,22.006,28.874,42.468,25.377,23.089,40.458,25.434
                 ,18.097,36.265,23.154,12.725,32.439,18.303,8.809,29.316,12.224,7.154
                 ,25.247,8.256,10.648,21.434,6.867,13.846,16.261,6.495,19.229,12.785
                 ,9.444,25.660,11.445,16.015,27.106,9.083
                 
                 ,24.569,13.132,13.356,26.118,20.059,13.598,22.943,25.493,16.332,17.156
                 ,28.220,18.641,12.640,27.993,23.201,6.940,26.026,26.789,5.822,20.299
                 ,30.624,5.577,14.426,34.013,9.260,11.671,38.441,15.405,9.037,40.045
                 ,20.732,11.190,43.389,22.255,28.629,41.898,25.166,23.030,39.331,25.211
                 ,17.886,34.989,22.927,11.813,32.321,18.034,7.986,30.112,12.507,7.039
                 ,25.429,7.760,10.850,22.086,6.699,14.098,16.608,6.578,19.622,12.735
                 ,9.516,25.603,10.904,15.494,27.452,8.388
                 
                 ,24.374,13.368,13.547,25.968,20.247,13.246,23.008,25.503,15.953,17.295
                 ,28.819,18.144,12.941,28.253,23.238,7.486,25.753,26.647,5.568,20.694
                 ,30.577,5.546,14.753,34.301,9.336,11.829,38.770,14.971,8.889,41.320
                 ,20.733,10.856,43.703,22.427,28.706,42.112,25.228,23.369,39.109,25.342
                 ,17.710,35.477,23.173,11.708,32.903,18.173,7.920,30.467,12.415,7.036
                 ,25.282,8.608,10.579,22.073,6.742,14.416,16.828,6.456,19.579,12.318
                 ,9.521,25.792,11.092,15.795,27.217,8.544
                 
                 ,24.523,13.575,13.730,26.283,20.256,14.113,22.742,25.391,16.837,16.910
                 ,28.361,18.931,12.169,28.452,22.750,7.157,25.331,26.758,5.413,20.606
                 ,30.489,5.381,15.019,34.458,9.238,12.654,38.882,15.055,9.116,40.835
                 ,20.612,11.073,44.104,22.835,28.368,41.865,25.681,23.191,38.900,25.820
                 ,17.319,35.189,23.205,11.268,33.056,18.536,8.055,29.949,12.768,7.279
                 ,25.385,7.923,10.343,21.699,7.260,13.681,16.484,6.793,20.112,13.511
                 ,9.567,26.174,11.279,15.334,27.686,8.372
                 
                 ,24.006,13.405,12.592,26.309,20.061,13.619,23.040,25.554,16.139,17.822
                 ,28.432,19.405,12.417,28.345,23.022,7.234,25.262,26.351,5.410,20.248
                 ,30.270,5.563,14.632,34.272,9.276,12.834,39.126,14.824,9.049,40.659
                 ,19.558,11.043,44.656,22.847,28.845,41.482,25.470,23.095,38.692,25.220
                 ,16.932,36.035,22.990,11.096,33.874,18.029,7.900,30.415,12.609,6.544
                 ,25.900,8.550,9.951,21.714,7.448,13.457,16.513,6.870,19.847,13.544
                 ,8.872,25.764,11.318,14.641,27.537,7.839
                 
                 ,23.936,13.085,12.376,25.865,19.427,13.664,23.213,25.058,16.283,18.112
                 ,28.607,19.261,12.743,28.401,22.584,7.267,25.067,26.234,6.259,20.058
                 ,30.295,5.223,14.693,34.091,8.846,12.606,38.941,14.581,8.891,40.367
                 ,19.148,10.977,44.551,22.080,28.672,41.808,25.625,23.181,39.061,25.517
                 ,17.658,35.650,23.555,11.141,33.784,18.422,7.879,30.872,13.176,6.986
                 ,25.093,9.595,10.402,21.787,7.283,14.052,17.155,6.272,19.831,13.550
                 ,8.912,25.658,10.834,15.066,27.919,8.719
                 
                 ,24.228,13.221,11.812,25.933,19.620,13.289,23.382,25.319,15.563,18.068
                 ,28.246,18.967,13.613,28.286,22.944,8.360,25.533,27.070,6.114,20.845
                 ,30.299,5.328,14.990,34.186,8.986,11.480,38.107,14.578,9.012,40.742
                 ,18.999,11.541,45.355,21.695,28.682,41.301,25.220,23.429,38.858,25.281
                 ,17.872,35.522,23.677,11.484,33.448,19.107,8.155,29.984,12.684,7.155
                 ,25.575,10.248,10.120,21.324,7.475,14.113,16.755,5.670,20.340,13.921
                 ,9.459,25.472,10.983,15.771,28.089,9.318
                 
                 ,24.316,12.458,11.843,25.508,18.965,13.747,23.278,24.916,16.394,17.774
                 ,28.402,19.187,12.378,28.542,22.605,7.252,25.365,26.306,6.069,20.368
                 ,29.967,5.496,14.233,33.573,8.681,11.885,38.394,14.356,9.525,41.153
                 ,19.296,11.200,44.990,21.956,28.698,41.790,25.477,23.507,38.898,25.134
                 ,17.916,35.541,23.580,11.891,33.423,18.965,7.818,30.491,13.199,7.030
                 ,25.775,9.665,9.849,21.557,7.725,13.950,16.492,6.011,20.248,14.478
                 ,9.515,25.732,11.493,16.071,28.073,8.995
                 
                 ,24.069,12.514,11.576,26.193,19.281,13.560,23.049,25.025,15.906,17.225
                 ,28.166,19.122,12.145,27.688,22.386,7.635,25.248,26.821,5.438,20.677
                 ,30.447,5.051,15.003,34.500,9.197,12.486,38.468,14.699,9.325,40.616
                 ,19.694,10.810,44.394,21.784,28.486,41.204,25.422,23.303,39.170,25.769
                 ,17.606,36.130,23.125,11.743,33.599,18.675,8.250,30.057,12.399,6.446
                 ,25.964,9.671,9.713,21.051,8.200,13.572,16.156,6.096,20.125,14.059
                 ,9.421,25.580,11.507,15.633,28.168,9.245
                 
                 ,24.382,13.415,12.005,26.227,19.807,13.778,23.018,25.170,16.477,17.422
                 ,28.183,18.648,12.599,27.936,22.634,7.930,25.547,26.937,5.015,20.995
                 ,30.642,5.322,15.062,34.391,9.387,12.104,38.000,14.870,9.037,40.766
                 ,20.116,11.114,44.270,22.390,28.993,40.814,25.331,22.929,39.238,25.429
                 ,17.057,36.390,22.793,10.993,34.070,18.588,7.965,30.069,11.850,6.919
                 ,26.147,9.683,9.913,20.736,7.799,13.653,15.823,5.725,20.032,14.331
                 ,9.347,25.125,11.310,15.548,27.697,8.797
                 
                 ,25.071,13.359,12.219,25.922,20.228,13.700,22.969,25.646,16.479,17.218
                 ,28.753,18.370,12.285,28.254,22.454,8.012,26.006,26.558,5.684,20.677
                 ,30.618,5.143,14.471,34.316,8.841,12.183,39.055,14.522,8.799,41.346
                 ,19.438,10.811,44.824,22.870,28.797,40.740,25.663,23.231,38.089,25.721
                 ,17.236,35.440,22.782,11.278,33.869,18.262,8.303,30.135,11.771,6.803
                 ,25.902,9.355,9.961,21.065,8.367,13.499,15.573,5.697,19.805,14.183
                 ,8.787,25.293,11.196,14.928,27.345,8.810
                 
                 ,24.555,13.167,12.945,26.062,19.730,14.230,23.077,25.395,16.617,17.485
                 ,28.522,18.766,12.427,28.226,22.678,7.346,25.153,26.424,5.141,20.778
                 ,31.057,5.181,14.573,34.397,9.210,12.140,38.451,14.821,8.758,40.470
                 ,20.120,10.521,44.182,22.452,28.993,40.984,25.474,23.382,38.705,25.796
                 ,17.536,35.590,23.012,12.041,32.760,18.130,9.197,29.415,11.158,6.757
                 ,26.335,9.712,9.912,20.939,7.884,13.871,15.872,5.667,20.538,14.465
                 ,9.330,25.550,11.486,15.120,27.100,8.018
                 
                 ,25.137,12.870,12.482,26.450,19.304,14.192,22.839,25.036,16.498,17.098
                 ,27.988,18.477,12.516,27.594,23.342,6.652,25.339,26.480,5.038,20.611
                 ,30.998,5.596,14.238,33.760,9.367,12.383,38.383,14.898,8.935,40.587
                 ,20.080,11.064,43.809,22.269,28.515,41.425,25.532,22.994,39.101,25.689
                 ,17.577,35.290,23.238,12.103,32.263,18.131,8.696,29.280,11.572,6.737
                 ,25.943,9.747,10.059,20.420,7.506,14.235,15.951,5.520,20.764,14.883
                 ,9.477,25.594,12.102,14.816,27.279,8.341
                 
                 ,24.811,13.177,12.248,26.359,19.647,13.791,22.819,25.076,15.935,17.259
                 ,28.137,18.816,12.031,27.817,22.524,7.139,25.285,26.476,5.233,20.844
                 ,30.918,5.023,14.625,34.304,8.951,12.113,38.641,14.903,9.183,40.263
                 ,19.489,10.837,44.214,22.943,28.849,41.934,25.785,23.491,38.818,25.458
                 ,17.501,35.809,23.084,12.499,32.507,18.711,8.415,29.596,11.995,6.648
                 ,25.674,9.782,9.834,20.682,7.330,14.037,16.282,5.679,20.535,14.148
                 ,9.529,25.973,12.347,14.857,27.412,8.891
                 
                 ,25.461,13.475,12.459,26.240,19.803,14.240,22.439,25.184,16.246,17.018
                 ,28.482,18.729,12.138,27.810,22.628,6.935,25.500,26.392,5.181,21.303
                 ,30.774,5.269,15.405,34.596,9.057,12.253,38.675,14.733,9.098,40.950
                 ,18.928,10.930,44.884,22.702,28.490,41.516,25.756,22.649,39.048,26.150
                 ,16.724,36.124,23.328,11.649,32.925,18.644,8.006,30.218,12.586,6.889
                 ,25.838,9.876,10.168,20.938,7.544,13.875,16.115,5.352,20.441,14.098
                 ,9.147,25.602,11.691,14.869,27.495,8.845
                 
                 ,24.646,13.201,12.498,26.020,19.910,14.081,22.700,25.203,16.348,16.921
                 ,28.147,18.532,12.219,27.992,22.139,7.297,25.197,26.330,5.477,21.167
                 ,31.099,5.289,15.822,35.342,8.801,12.200,39.234,14.381,9.267,41.504
                 ,19.133,11.253,44.763,22.661,28.604,41.983,25.373,23.002,39.097,25.724
                 ,16.979,36.597,23.274,11.205,33.696,18.583,7.895,30.512,12.581,6.908
                 ,26.393,10.461,9.886,20.595,7.139,14.317,16.469,5.308,21.069,14.978
                 ,8.870,25.450,10.652,14.988,27.488,7.901)
               ,c(3,22,30))
dna.dat<-aperm(dna.dat,c(2,1,3))
#
#
#
#==================================================================================
macf.dat<-c(54.33203,24.10905,69.5
            ,141.80250,21.59643,69.5
            ,132.23880,62.78124,69.5
            ,88.22106,52.28123,69.5
            ,147.10890,26.61518,90.0
            ,107.42800,27.77578,101.0
            ,99.74427,46.85715,97.0
            ,58.35540,29.57223,67.0
            ,134.87930,26.38988,67.0
            ,120.57150,66.43083,67.0
            ,79.14922,52.19386,67.0
            ,138.17850,37.67144,86.0
            ,101.76780,34.47324,97.5
            ,90.83484,54.19082,92.0
            ,50.04349,15.22191,71.5
            ,139.76850,21.33820,71.5
            ,124.26710,63.79000,71.5
            ,80.94720,51.47673,71.5
            ,147.78980,32.26446,94.0
            ,102.32870,25.23133,105.5
            ,90.64163,47.04449,98.5
            ,41.93115,24.83244,70.5
            ,138.72930,22.35828,70.5
            ,122.68840,65.04043,70.5
            ,74.09913,53.69709,70.5
            ,142.63240,35.36625,92.0
            ,97.04822,33.37946,111.5
            ,89.51760,56.40900,96.5
            ,48.44877,35.75250,68.0
            ,134.68970,33.55962,68.0
            ,120.88830,73.77755,68.0
            ,77.94841,65.53058,68.0
            ,135.42950,42.75692,91.0
            ,101.92880,40.64505,99.5
            ,87.04530,59.04715,92.5
            ,44.05272,36.38397,70.0
            ,133.47320,45.07240,70.0
            ,114.88450,83.74655,70.0
            ,70.13158,71.71946,70.0
            ,139.94810,54.24338,87.0
            ,97.87708,47.19924,105.5
            ,80.04594,65.13947,96.5
            ,53.94042,32.50219,69.0
            ,136.19530,33.46588,69.0
            ,120.75410,74.12678,69.0
            ,78.24210,60.88469,69.0
            ,142.38210,44.35960,89.5
            ,100.75480,38.09180,101.0
            ,87.92361,55.80280,94.5
            ,45.11740,11.62884,68.5
            ,132.08950,17.75877,68.5
            ,112.66980,57.22899,68.5
            ,71.76939,47.64127,68.5
            ,136.93780,26.69818,88.5
            ,96.43125,21.78416,101.0
            ,84.39475,41.89694,95.0
            ,42.09966,16.11359,69.0
            ,131.92440,19.70687,69.0
            ,121.73400,57.71608,69.0
            ,72.94730,49.99466,69.0
            ,136.93340,26.92302,89.0
            ,96.84825,26.58288,103.0
            ,84.30907,48.96717,94.5)
macf.dat<-array(macf.dat,c(3,7,9))
macf.dat<-aperm(macf.dat,c(2,1,3))
macm.dat<-c(34.82811,16.50834,77.5
            ,138.91980,15.13858,77.5
            ,125.15760,58.60464,77.5
            ,72.28854,49.79207,77.5
            ,146.19080,22.68885,100.0
            ,99.30268,24.86908,117.0
            ,91.79910,46.49960,107.0
            ,40.40179,3.73932,73.0
            ,132.23560,7.56574,73.0
            ,114.63210,53.28955,73.0
            ,70.66502,33.57051,73.0
            ,139.58480,21.61227,90.5
            ,97.93692,8.49867,108.5
            ,79.90506,28.91153,100.5
            ,40.54510,9.51130,75.0
            ,136.61260,15.82863,75.0
            ,106.90960,63.82611,75.0
            ,76.19816,46.63517,75.0
            ,145.02210,30.40421,94.5
            ,101.72660,19.45746,113.5
            ,86.97967,43.98130,105.5
            ,21.11454,16.57673,75.0
            ,131.52700,23.12809,75.0
            ,109.44810,63.03707,75.0
            ,61.73774,53.69610,75.0
            ,135.91480,34.78890,101.5
            ,90.65395,25.30813,117.5
            ,75.66082,49.38123,105.5
            ,30.79976,19.21503,73.5
            ,134.92160,32.11148,73.5
            ,115.81510,69.88405,73.5
            ,67.15240,57.06633,73.5
            ,139.56950,44.82271,97.5
            ,95.38217,25.95223,112.0
            ,78.97741,47.89584,107.0
            ,18.88770,10.47136,74.5
            ,130.35790,12.40497,74.5
            ,114.85390,57.63774,74.5
            ,63.47649,48.13175,74.5
            ,138.25830,25.62929,97.5
            ,89.01810,18.95535,117.0
            ,75.67622,43.31009,104.5
            ,40.28789,14.90687,69.0
            ,134.29020,14.66977,69.0
            ,125.54870,56.83236,69.0
            ,75.68020,53.65364,69.0
            ,142.36350,24.22211,92.5
            ,99.44497,25.41932,106.0
            ,87.63929,45.45810,99.5
            ,25.38359,10.64805,72.5
            ,130.99770,9.63434,72.5
            ,118.65580,54.78021,72.5
            ,68.79280,48.67834,72.5
            ,139.77820,20.83856,97.0
            ,91.36346,16.29169,111.5
            ,75.55544,44.28398,101.0
            ,27.93545,5.21197,71
            ,130.98990,4.76235,71
            ,103.16230,51.99304,71
            ,70.59641,43.71388,71
            ,136.03820,13.90246,92
            ,91.75840,14.31955,109
            ,79.95213,35.70748,95)
macm.dat<-array(macm.dat,c(3,7,9))
macm.dat<-aperm(macm.dat,c(2,1,3))

#==================================================================================
sooty.dat<- c(-1426,-310.4167
              ,-1424,-160.4167
              ,-1117,320.5833
              ,-755,854.5833
              ,1238,1363.5833
              ,2330,471.5833
              ,1435,-748.4167
              ,771,-557.4167
              ,433,-395.4167
              ,-176,-299.4167
              ,-376,-290.4167
              ,-933,-248.4167
              ,-1000.20254,-1601.5969
              ,-1076.57007,-1266.4282
              ,-1124.65334,-635.6890
              ,-1193.94980,147.7853
              ,-61.16474,1895.7533
              ,1484.57069,2113.5422
              ,1649.32657,746.7048
              ,1212.33458,388.9087
              ,730.79486,166.1701
              ,156.62415,-383.9590
              ,-88.03479,-533.8656
              ,-689.07556,-1037.3256)
sooty.dat<-array(sooty.dat,c(2,12,2))
sooty.dat<-aperm(sooty.dat,c(2,1,3))


#==================================================================================
panf.dat<-array(c(47,-23,0,0,0,32,12,87,21,156,31,133,66,92,83,20,
                  63,-22,0,0,0,29,6,89,14,157,23,136,62,101,95,24,
                  56,-11,0,0,0,31,2,89,4,159,17,141,54,107,86,34,
                  51,-27,0,0,0,29,12,89,30,156,39,135,71,94,86,18,
                  51,-19,0,0,0,35,8,97,23,169,36,143,67,101,85,25,
                  54,-23,0,0,0,34,14,84,36,155,44,133,76,87,89,19,
                  53,-21,0,0,0,30,5,90,24,162,31,139,63,99,86,21,
                  56,-23,0,0,0,33,16,92,30,156,40,131,75,95,95,15,
                  55,-20,0,0,0,33,12,89,13,157,31,136,63,97,91,22,
                  35,-23,0,0,0,26,9,81,11,153,26,131,64,92,84,18,
                  48,-25,0,0,0,30,23,89,44,160,49,139,81,95,95,18,
                  35,-34,0,0,0,30,10,84,23,153,36,128,68,95,94,13,
                  46,-23,0,0,0,28,4,92,14,163,27,137,60,101,86,27,
                  42,-23,0,0,0,30,19,88,33,163,45,130,77,95,93,20,
                  50,-19,0,0,0,32,7,90,14,157,25,139,68,101,87,23,
                  54,-19,0,0,0,29,6,92,9,163,20,140,64,104,92,26,
                  46,-31,0,0,0,25,3,89,2,167,24,136,63,95,85,19,
                  47,-19,0,0,0,29,7,84,6,148,22,126,58,89,80,24,
                  46,-27,0,0,0,31,12,86,23,156,35,130,72,93,89,18,
                  49,-23,0,0,0,29,14,88,25,159,36,133,76,90,90,16,
                  50,-23,0,0,0,32,9,92,14,167,31,141,71,97,87,28,
                  40,-23,0,0,0,30,13,92,30,167,40,140,74,99,91,21,
                  32,-25,0,0,0,36,7,96,12,170,22,147,61,108,81,35,
                  41,-30,0,0,0,29,18,87,31,160,43,135,74,87,89,10,
                  46,-25,0,0,0,29,15,88,23,163,37,134,70,91,85,22,
                  43,-22,0,0,0,33,-1,96,10,178,24,153,66,105,87,32),c(2,8,26))
panf.dat<-aperm(panf.dat,c(2,1,3))
select<-c(5,1,2,3,4,6,7,8)
panf.dat<-panf.dat[select,,]

panm.dat<-array(c(43,-21,0,0,0,34,14,101,25,179,40,150,75,104,90,31,
                  48,-23,0,0,0,31,5,92,11,166,24,144,63,99,82,24,
                  43,-23,0,0,0,29,13,92,21,161,33,138,68,100,84,21,
                  45,-32,0,0,0,30,8,100,14,163,28,143,74,102,97,21,
                  40,-27,0,0,0,29,7,93,12,166,23,147,69,102,90,25,
                  49,-19,0,0,0,33,3,94,11,165,23,144,64,108,89,30,
                  55,-17,0,0,0,31,6,97,17,168,29,144,69,101,85,23,
                  49,-27,0,0,0,26,11,92,16,178,30,152,78,100,89,23,
                  48,-23,0,0,0,29,10,96,32,166,42,139,69,96,87,21,
                  49,-19,0,0,0,29,3,100,14,172,25,152,63,109,87,32,
                  49,-26,0,0,0,34,8,91,22,168,34,140,74,93,93,17,
                  52,-26,0,0,0,32,7,92,10,172,28,143,66,100,93,23,
                  36,-26,0,0,0,28,11,92,25,165,34,140,71,98,90,18,
                  46,-26,0,0,0,33,12,94,20,174,35,145,71,106,93,24,
                  47,-22,0,0,0,31,10,88,29,156,34,138,68,93,91,20,
                  47,-25,0,0,0,30,0,97,2,172,19,147,62,102,82,26,
                  53,-19,0,0,0,29,6,80,2,142,13,123,54,100,91,31,
                  49,-24,0,0,0,29,5,90,14,168,31,144,77,96,89,22,
                  52,-25,0,0,0,30,6,93,15,167,25,147,74,99,92,24,
                  42,-27,0,0,0,32,11,87,26,159,37,137,76,95,87,21,
                  37,-25,0,0,0,29,-2,87,-2,175,16,152,63,107,91,33,
                  41,-27,0,0,0,25,6,87,13,167,29,141,70,99,89,28,
                  46,-24,0,0,0,35,12,98,15,175,31,152,76,106,92,27,
                  45,-22,0,0,0,29,8,90,12,176,24,151,70,104,88,24,
                  46,-27,0,0,0,29,2,88,6,166,23,138,67,100,86,24,
                  43,-20,0,0,0,33,7,94,17,165,30,143,70,93,81,24,
                  44,-20,0,0,0,29,1,87,0,154,19,133,64,98,80,16,
                  52,-28,0,0,0,28,2,84,18,155,25,135,63,97,93,21),c(2,8,28))
panm.dat<-aperm(panm.dat,c(2,1,3))
select<-c(5,1,2,3,4,6,7,8)
panm.dat<-panm.dat[select,,]


pongof.dat<-array(c(43,-31,0,0,0,29,13,90,45,150,48,126,74,80,91,19,
                    49,-31,0,0,0,33,28,93,70,152,72,130,86,78,85,-5,
                    51,-36,0,0,0,32,34,100,74,152,74,131,87,73,92,-2,
                    48,-26,0,0,0,32,10,89,35,154,40,136,65,88,86,14,
                    56,-29,0,0,0,24,11,87,44,155,48,133,72,82,91,12,
                    49,-30,0,0,0,29,25,96,72,159,69,137,85,78,93,9,
                    51,-28,0,0,0,26,11,94,50,162,49,132,75,87,89,13,
                    57,-24,0,0,0,35,10,98,55,164,53,138,77,88,93,21,
                    37,-29,0,0,0,38,30,93,63,161,68,129,86,68,87,6,
                    53,-30,0,0,0,29,5,88,39,147,38,131,64,84,90,14,
                    58,-24,0,0,0,30,0,88,14,151,24,133,59,91,88,21,
                    54,-26,0,0,0,36,11,106,41,173,48,146,78,94,99,24,
                    59,-25,0,0,0,29,4,102,28,177,34,156,63,100,90,25,
                    32,-36,0,0,0,25,26,90,71,144,69,124,82,72,91,3,
                    52,-30,0,0,0,35,21,99,55,164,59,143,79,90,95,22,
                    51,-27,0,0,0,35,11,92,37,152,42,132,69,88,95,26,
                    47,-24,0,0,0,35,7,98,36,163,38,138,66,89,87,21,
                    60,-23,0,0,0,23,-2,82,28,158,32,135,56,90,89,25,
                    46,-31,0,0,0,25,4,87,34,145,37,120,66,80,89,20,
                    46,-28,0,0,0,36,29,94,73,148,71,123,82,74,92,11,
                    32,-37,0,0,0,36,32,88,81,140,81,117,91,63,90,-5,
                    43,-27,0,0,0,25,2,90,32,159,37,131,62,91,87,22,
                    38,-27,0,0,0,30,4,93,30,160,36,136,60,92,86,27,
                    38,-27,0,0,0,34,14,92,53,155,48,129,71,82,84,24),c(2,8,24))
pongof.dat<-aperm(pongof.dat,c(2,1,3))
select<-c(5,1,2,3,4,6,7,8)
pongof.dat<-pongof.dat[select,,]


pongom.dat<-array(c(49,-45,0,0,0,26,25,106,68,190,68,151,84,80,100,14,
                    64,-28,0,0,0,31,10,106,46,185,53,156,77,95,99,14,
                    55,-31,0,0,0,33,23,113,72,186,72,160,92,95,102,21,
                    64,-25,0,0,0,36,5,109,35,188,42,165,69,102,101,33,
                    46,-39,0,0,0,31,36,106,97,155,89,133,95,74,92,1,
                    53,-28,0,0,0,33,17,111,54,185,59,159,87,88,98,16,
                    47,-36,0,0,0,36,35,114,72,183,74,150,94,81,105,15,
                    44,-35,0,0,0,37,15,110,69,183,74,153,84,89,94,11,
                    55,-43,0,0,0,26,24,105,71,172,75,146,91,81,104,1,
                    49,-33,0,0,0,35,11,113,58,188,56,159,79,92,93,24,
                    45,-32,0,0,0,29,20,107,67,184,67,155,85,82,93,11,
                    48,-34,0,0,0,32,33,116,89,192,91,160,100,81,99,5,
                    41,-51,0,0,0,29,50,100,127,139,115,119,105,53,96,-19,
                    60,-45,0,0,0,32,36,102,108,163,97,129,98,75,98,-6,
                    65,-35,0,0,0,30,24,112,65,188,72,158,88,87,103,18,
                    54,-28,0,0,0,33,7,114,33,206,42,172,74,98,94,28,
                    41,-39,0,0,0,34,24,115,79,188,79,154,93,79,99,2,
                    42,-40,0,0,0,39,41,114,112,187,102,147,112,70,97,-14,
                    65,-27,0,0,0,30,17,109,62,187,66,151,83,82,96,18,
                    54,-36,0,0,0,30,29,122,65,204,70,176,93,87,96,9,
                    55,-37,0,0,0,32,25,116,75,190,71,155,88,84,98,8,
                    50,-35,0,0,0,26,8,101,39,172,49,148,75,87,98,18,
                    78,-31,0,0,0,28,15,119,56,204,60,179,91,94,103,8,
                    42,-32,0,0,0,34,37,117,102,181,99,148,102,83,97,5,
                    52,-39,0,0,0,38,15,111,58,201,63,158,89,91,95,-8,
                    47,-37,0,0,0,23,37,98,85,160,83,136,89,75,94,-5,
                    49,-37,0,0,0,34,37,115,105,179,98,151,96,80,97,5,
                    48,-32,0,0,0,36,10,113,53,189,57,166,79,100,99,21,
                    41,-24,0,0,0,39,4,128,42,209,47,178,73,102,93,21,
                    50,-32,0,0,0,39,27,121,73,198,75,166,95,89,101,15),c(2,8,30))
pongom.dat<-aperm(pongom.dat,c(2,1,3))
select<-c(5,1,2,3,4,6,7,8)
pongom.dat<-pongom.dat[select,,]

#==================================================================================
schizophrenia.dat<-c( 0.345632   , -0.0360314
                      , -0.356301   , 0.0234333
                      , 0.0119311     , 0.17692
                      , 0.37789    , 0.480402
                      , 0.719631     , -0.41189
                      , 0.397921    , -0.140558
                      , 0.351751    , -0.385748
                      , 0.333756    , -0.655051
                      , 0.032181    , -0.275235
                      , 0.112563    , -0.506533
                      , -0.233126     , -0.28334
                      , -0.667337   , 0.0522613
                      , 0.188945    , -0.142714
                      , 0.237198    , 0.048306
                      , -0.340236   , 0.0997385
                      , -0.0161814    , 0.201017
                      , 0.301584    , 0.516546
                      , 0.510795    , -0.323537
                      , 0.269407   , -0.0562209
                      , 0.239301    , -0.253218
                      , 0.229339     , -0.48236
                      , 0.0201328    , -0.122625
                      , 0.048306    , -0.341874
                      , -0.200997    , -0.122698
                      , -0.62316    , 0.120534
                      , 0.124688   , -0.0262477
                      , 0.341616    , 0.048306
                      , -0.408509    , 0.119819
                      , -0.00814926    , 0.277322
                      , 0.39797     , 0.62498
                      , 0.591116    , -0.299441
                      , 0.35776   , -0.0361405
                      , 0.27143    , -0.249202
                      , 0.273516    , -0.530553
                      , 0.032181    , -0.110577
                      , 0.0724024     , -0.34589
                      , -0.188949    , -0.138762
                      , -0.707498    , 0.140615
                      , 0.124688   , -0.0262477
                      , 0.329567     , -0.10832
                      , -0.359859   , 0.0341931
                      , -0.056342    , 0.152824
                      , 0.377668    , 0.474464
                      , 0.673363    , -0.462009
                      , 0.373825    , -0.228912
                      , 0.323639    , -0.450005
                      , 0.34179    , -0.735372
                      , 0.0924219    , -0.295315
                      , 0.100555    , -0.540522
                      , -0.181195    , -0.260518
                      , -0.651361   , 0.0963372
                      , 0.174798    , -0.178741
                      , 0.193021   , 0.0683864
                      , -0.539516    , 0.210918
                      , -0.16074    , 0.333555
                      , 0.293246    , 0.639288
                      , 0.520753    , -0.301366
                      , 0.245311   , -0.0281084
                      , 0.142916     , -0.28133
                      , 0.128938    , -0.510473
                      , -0.152558    , -0.118609
                      , -0.0761516    , -0.379879
                      , -0.414127    , -0.127988
                      , -0.848201    , 0.246974
                      , -0.00592517  , -0.00203414
                      , 0.337599    , -0.076192
                      , -0.356431    , 0.146356
                      , 0.068156    , 0.201017
                      , 0.520571    , 0.464342
                      , 0.601074    , -0.518234
                      , 0.321616    , -0.208831
                      , 0.251349    , -0.409844
                      , 0.197211    , -0.695211
                      , -0.00797966    , -0.251139
                      , -0.00787855    , -0.500361
                      , -0.26726    , -0.201009
                      , -0.719602     , 0.25526
                      , 0.110557    , -0.140611
                      , 0.223261    , 0.228767
                      , -0.431293    , 0.230133
                      , -0.0439952    , 0.389752
                      , 0.23941    , 0.819652
                      , 0.641234    , -0.088515
                      , 0.377841    , 0.140566
                      , 0.339703     , -0.10864
                      , 0.393998    , -0.345813
                      , 0.0201328   , -0.0583678
                      , 0.172844    , -0.283494
                      , -0.268058   , -0.0890055
                      , -0.777652    , 0.251188
                      , 0.154806     , 0.12832
                      , 0.243341    , -0.052358
                      , -0.451374    , 0.145795
                      , -0.0480112    , 0.297382
                      , 0.408185    , 0.574616
                      , 0.56291    , -0.538339
                      , 0.31561    , -0.190749
                      , 0.251349    , -0.437957
                      , 0.169098    , -0.715291
                      , 0.00005         , -0.194914
                      , 0.000153631    , -0.492329
                      , -0.278085    , -0.117118
                      , -0.797732    , 0.255204
                      , 0.122674   , -0.0905492
                      , 0.287518   , 0.0600918
                      , -0.503583    , 0.121699
                      , -0.0861578    , 0.307424
                      , 0.355977    , 0.614776
                      , 0.603071    , -0.277295
                      , 0.303562   , -0.0100258
                      , 0.259382    , -0.293379
                      , 0.241387    , -0.538584
                      , -0.0521564    , -0.114593
                      , 0.0443303    , -0.343735
                      , -0.274068   , -0.0930217
                      , -0.805764    , 0.138738
                      , 0.12669  , 0.00182023
                      , 0.319646     , 0.20467
                      , -0.431294    , 0.153827
                      , -0.0660775    , 0.391762
                      , 0.29172    , 0.793493
                      , 0.771746  , -0.00420256
                      , 0.407979    , 0.166681
                      , 0.387896     , -0.12872
                      , 0.393998    , -0.394006
                      , 0.0864079   , -0.0382915
                      , 0.168828    , -0.279477
                      , -0.225876    , -0.109086
                      , -0.779633    , 0.126677
                      , 0.190947    , 0.126318
                      , 0.303582    , 0.208686
                      , -0.395149    , 0.302422
                      , -0.00985258    , 0.387746
                      , 0.388105     , 0.72522
                      , 0.611103    , -0.289343
                      , 0.327658   , 0.0743116
                      , 0.283478    , -0.229121
                      , 0.201226    , -0.494408
                      , 0.00608663   , -0.0141951
                      , 0.0362983    , -0.279477
                      , -0.24194  , -0.00466831
                      , -0.699312    , 0.339529
                      , 0.12669   , 0.0901736
                      , 0.287518   , -0.0362937
                      , -0.428786    , 0.133507
                      , -0.0801519    , 0.233129
                      , 0.374063    , 0.614792
                      , 0.671344    , -0.478098
                      , 0.383883    , -0.134524
                      , 0.315606    , -0.417876
                      , 0.265483    , -0.675131
                      , 0.0342193     , -0.19491
                      , 0.0443223    , -0.466226
                      , -0.266036    , -0.201455
                      , -0.735453    , 0.202987
                      , 0.176577   , -0.0906634
                      , 0.251373    , 0.212702
                      , -0.42477    , 0.149571
                      , -0.124329    , 0.345579
                      , 0.145148    , 0.767403
                      , 0.671344  , -0.00821852
                      , 0.424044    , 0.138569
                      , 0.407976    , -0.116671
                      , 0.44219    , -0.377942
                      , 0.122573   , -0.0583636
                      , 0.202939    , -0.283647
                      , -0.14957   , -0.0970378
                      , -0.755533    , 0.114633
                      , 0.180593   , 0.0940755
                      , 0.29555   , -0.0362937
                      , -0.348465 , 0.000976592
                      , -0.0480233    , 0.148792
                      , 0.273662    , 0.458166
                      , 0.65528    , -0.341552
                      , 0.371835    , -0.106411
                      , 0.327655    , -0.377715
                      , 0.333757     , -0.65505
                      , 0.0984763    , -0.247119
                      , 0.118601    , -0.500514
                      , -0.181699    , -0.253664
                      , -0.675212   , 0.0182479
                      , 0.192642    , -0.122792
                      , 0.309487    , 0.265173
                      , -0.3438    , 0.289185
                      , 0.0199632    , 0.482141
                      , 0.355444    , 0.735605
                      , 0.613122    , -0.156788
                      , 0.357761    , 0.128518
                      , 0.315607    , -0.132736
                      , 0.305644     , -0.36991
                      , 0.0121007   , 0.0380178
                      , 0.100555    , -0.223253
                      , -0.182924   , 0.0134128
                      , -0.623245    , 0.301185
                      , 0.162814    , 0.142714
                      , 0.193021    , 0.305334
                      , -0.472766    , 0.163996
                      , -0.172808     , 0.38174
                      , 0.0847168    , 0.829799
                      , 0.538908    , 0.114214
                      , 0.249327    , 0.273096
                      , 0.199141   , 0.0319229
                      , 0.273516    , -0.165091
                      , -0.0360921   , 0.0340017
                      , 0.0563381    , -0.141071
                      , -0.257222   , -0.0263121
                      , -0.791835   , 0.0884059
                      , -0.00382644    , 0.178572
                      , 0.39784    , -0.011935
                      , -0.440185    , 0.132558
                      , -0.00814926    , 0.309451
                      , 0.470037    , 0.542738
                      , 0.580994    , -0.385704
                      , 0.389889    , -0.212847
                      , 0.283478    , -0.401812
                      , 0.237372    , -0.675131
                      , 0.0201328    , -0.267203
                      , 0.0644106    , -0.508393
                      , -0.23742    , -0.196261
                      , -0.711598    , 0.184719
                      , 0.0944764    , -0.130548
                      , 0.269326   , 0.0844506
                      , -0.327735    , 0.048221
                      , -0.0242135    , 0.253226
                      , 0.257186    , 0.643139
                      , 0.601074      , -0.1849
                      , 0.325632   , 0.0160683
                      , 0.307574    , -0.217073
                      , 0.301629    , -0.442199
                      , 0.0803737    , -0.126641
                      , 0.112603    , -0.351767
                      , -0.153083     , -0.13602
                      , -0.635293   , 0.0280923
                      , 0.158733  , -0.00605034
                      , 0.217118    , 0.212965
                      , -0.416089    , 0.192799
                      , -0.104535    , 0.353627
                      , 0.229073    , 0.727476
                      , 0.601074      , -0.1849
                      , 0.293503    , 0.112454
                      , 0.267414    , -0.112655
                      , 0.285565    , -0.402039
                      , 0.0281649   , -0.0101749
                      , 0.0844909    , -0.259397
                      , -0.221356   , -0.0275863
                      , -0.743727    , 0.188735
                      , 0.0944764    , 0.118448
                      , 0.293423    , -0.076192
                      , -0.327736   , 0.0442049
                      , -0.0121653    , 0.160856
                      , 0.357587    , 0.394143
                      , 0.572961    , -0.457993
                      , 0.281455    , -0.228912
                      , 0.263398    , -0.470085
                      , 0.189179    , -0.679147
                      , 0.00005         , -0.259171
                      , 0.02425    , -0.524458
                      , -0.193244    , -0.208309
                      , -0.583084   , 0.0883332
                      , 0.142669    , -0.178741
                      , 0.289407   , 0.0201935
                      , -0.351832    , 0.164687
                      , 0.00389893    , 0.265274
                      , 0.40578    , 0.510609
                      , 0.49264    , -0.433896
                      , 0.3176    , -0.120478
                      , 0.235285    , -0.329523
                      , 0.201227    , -0.554649
                      , 0.00406852     , -0.17885
                      , 0.0081857    , -0.403976
                      , -0.245452    , -0.107908
                      , -0.647341    , 0.228896
                      , 0.0583318   , -0.0622752
                      , 0.301455   , 0.0603542
                      , -0.391993    , 0.224928
                      , -0.0121653    , 0.341579
                      , 0.441925    , 0.594946
                      , 0.45248    , -0.389719
                      , 0.29752   , -0.0963815
                      , 0.17906     , -0.30141
                      , 0.140986    , -0.570714
                      , -0.02806    , -0.106561
                      , -0.0560713    , -0.391927
                      , -0.245452   , -0.0556988
                      , -0.683486    , 0.293153
                      , 0.0583318  , -0.00605033
                      , 0.237198   , -0.0922563
                      , -0.396005  , 0.00807245
                      , -0.116583     , 0.11668
                      , 0.29333    , 0.402175
                      , 0.47256    , -0.502169
                      , 0.22523     , -0.22088
                      , 0.17906    , -0.433941
                      , 0.157051    , -0.634971
                      , -0.0320761    , -0.271219
                      , -0.0400071    , -0.500361
                      , -0.257501    , -0.260518
                      , -0.69955   , 0.0521888
                      , 0.0382515    , -0.158661
                      , 0.321535   , -0.0641438
                      , -0.379941    , 0.108474
                      , -0.0201975    , 0.192985
                      , 0.445941    , 0.450368
                      , 0.49264    , -0.506185
                      , 0.325632    , -0.184735
                      , 0.203157    , -0.365667
                      , 0.124922    , -0.675131
                      , -0.0119958    , -0.206962
                      , -0.0279589    , -0.476265
                      , -0.285613    , -0.192245
                      , -0.651357    , 0.176687
                      , 0.0944764    , -0.126532
                      , 0.317519   , -0.0761919
                      , -0.219297   , -0.0300639
                      , 0.072172    , 0.140776
                      , 0.40578    , 0.430287
                      , 0.580994    , -0.457993
                      , 0.345712    , -0.216863
                      , 0.287494    , -0.417876
                      , 0.2695    , -0.663083
                      , 0.096438    , -0.259171
                      , 0.0764588    , -0.496345
                      , -0.165131     , -0.24847
                      , -0.518831  , -0.00808046
                      , 0.162749    , -0.178741
                      , 0.363819    , -0.132663
                      , -0.295482    , 0.140679
                      , 0.178874    , 0.174894
                      , 0.568759     , 0.29153
                      , 0.593042    , -0.550362
                      , 0.345712    , -0.297185
                      , 0.267414     , -0.50623
                      , 0.136969     , -0.74742
                      , -0.0039636    , -0.275235
                      , 0.0724427    , -0.520442
                      , -0.239946    , -0.153262
                      , -0.524623    , 0.229146
                      , 0.194967    , -0.196981
                      , 0.255389    , 0.160493
                      , -0.435309    , 0.214068
                      , -0.124316    , 0.357623
                      , 0.267522     , 0.65901
                      , 0.597058    , -0.136708
                      , 0.305552   , 0.0763094
                      , 0.255365    , -0.140768
                      , 0.257451    , -0.402038
                      , 0.032181   , -0.0382875
                      , 0.084491    , -0.279478
                      , -0.243962   , -0.0167163
                      , -0.761587    , 0.231107
                      , 0.0945651   , 0.0881595
                      , 0.279486  , 0.00788297
                      , -0.330892    , 0.141779
                      , 0.0222759    , 0.267264
                      , 0.386119    , 0.484248
                      , 0.574959    , -0.365648
                      , 0.33569    , -0.102395
                      , 0.263398    , -0.345587
                      , 0.233355    , -0.566697
                      , -0.0180098    , -0.178854
                      , 0.0965392    , -0.387911
                      , -0.233908     , -0.12515
                      , -0.627023    , 0.206999
                      , 0.0681437   , -0.0866473)
schizo.dat<-schizophrenia.dat
schizophrenia.dat<-array(schizophrenia.dat,c(2,13,28))
schizophrenia.dat<-aperm(schizophrenia.dat,c(2,1,3))
schizo.dat<-array(schizo.dat,c(2,13,28))
schizo.dat<-aperm(schizo.dat,c(2,1,3))
braincon.dat<-schizo.dat[,,1:14]
brainscz.dat<-schizo.dat[,,15:28]


###################### Additional functions by other authors ##################

#
# =============================================================================
# Authors
# =============================================================================
# Gregorio Quintana-Orti
#   Depto. de Ingenieria y Ciencia de Computadores,
#   Universitat Jaume I,
#   12.071 Castellon, Spain
# Amelia Simo
#   Depto. de Matematicas,
#   Universitat Jaume I,
#   12.071 Castellon, Spain
#
# =============================================================================
# Copyright
# =============================================================================
# Copyright (C) 2018,
#   Universitat Jaume I.
#
# =============================================================================
# Disclaimer
# =============================================================================
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
# This module contains several modifications of some functions provided by
# the noteworthy "shapes" package by Ian L. Dryden. These new implementations 
# have been accelerated to be much faster for medium and large datasets 
# than the original codes. All the other functions in the library that employ
# the accelerated ones will also take advantage of this performance 
# improvement.
#
# The new code includes the original code in commented lines with four "#" 
# chars as a reference. 
#
# =============================================================================


# =============================================================================
uji_preshape = function( x ) {
   #
   # It computes the preshape in a faster way on medium and large datasets
   # on real (non-complex) data.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   if ( is.complex( x ) ) {
      #
      # Complex case.
      #
      k <- nrow( as.matrix( x ) )
      h <- uji_defh( k - 1 )
      zstar <- x
      ztem <- h %*% zstar
      size <- sqrt( diag( Re( st( ztem ) %*% ztem ) ) )
      if ( is.vector( zstar ) )
         z <- ztem / size
      if ( is.matrix(zstar ) )
         z <- ztem %*% diag( 1.0 / size )
   } else {
      #
      # Real case.
      #
      if (length(dim(x)) == 3) {
         #
         # Argument X is a 3D array.
         #
         k <- dim( x )[ 1 ]
         #### h <- uji_defh( k - 1 )
         n <- dim( x )[ 3 ]
         m <- dim( x )[ 2 ]
         z <- array( 0, c( k - 1, m, n ) )
         for ( i in 1 : n ) {
            #### z[, , i] <- h %*% x[, , i]
            
            # Accelerated code.
            z[ , , i ] <- multiply_by_helmert( x[ , , i ] )
            
            size <- uji_centroid.size( x[ , , i ] )
            z[ , , i ] <- z[ , , i ] / size
         }
      } else {
         #
         # Argument X is not a 3D array.
         #
         k <- nrow( as.matrix( x ) )
         
         #### h <- defh(k - 1)
         #### ztem <- h %*% x
         
         # Accelerated code.
         ztem <- multiply_by_helmert( x )
         
         size <- uji_centroid.size( x )
         z <- ztem / size
      }
   }
   return( z )
}

# =============================================================================
uji_centroid.size = function( x ) {
   #
   # It returns the centroid size of a configuration (or configurations).
   # Input: 
   #   k x m matrix, or
   #   a complex k-vector, or
   #   a real k x m x n array to get a vector of sizes for a sample
   #
   # It computes the centroid size in a faster way on medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   if ((is.vector(x) == FALSE) && is.complex(x)) {
      k <- nrow(x)
      n <- ncol(x)
      tem <- array(0, c(k, 2, n))
      tem[, 1, ] <- Re(x)
      tem[, 2, ] <- Im(x)
      x <- tem
   }
   {
      if (length( dim( x ) ) == 3 ) {
         #
         # Argument x is a 3D array.
         #
         n <- dim( x )[ 3 ]
         sz <- rep( 0, times = n )
         k <- dim( x )[ 1 ]
         #### h <- defh( k - 1 )
         for ( i in 1 : n ) {
            #### xh <- h %*% x[, , i]
            #### sz[ i ] <- sqrt( sum( diag( t( xh ) %*% xh ) ) )
            
            # Accelerated code.
            xh <- multiply_by_helmert( x[ , , i ] )
            sz[ i ] <- uji_Enorm( xh )
         }
         sz
      } else {
         if ( is.vector( x ) && is.complex( x ) ) {
            x <- cbind( Re( x ), Im( x ) )
         }
         k <- nrow( x )
         
         #### h <- defh(k - 1)
         #### xh <- h %*% x
         #### size <- sqrt( sum( diag( t( xh ) %*% xh ) ) )
         
         #cat( "pepe\n" )
         # Accelerated code.
         xh <- multiply_by_helmert( x )
         size <- uji_Enorm( xh )
         
         size
      }
   }
}

# =============================================================================
uji_defh = function( nrow ) {
   #
   # It generates a Helmert matrix in a faster way on medium and large datasets.
   # The use of this function should be avoided when the Helmert matrix is
   # just built to multiply another matrix or vector.
   # In this case, the "multiply_by_helmert_implicitly" and
   # "multiply_by_transpose_of_helmert_implicitly" should be employed since 
   # this approach is much faster.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   k <- nrow
   h <- matrix( 0, k, k + 1 )
   if( nrow > 0 ) {
      for( j in seq( 1, k ) ) {
         val = -1 / sqrt( j * ( j + 1 ) )
         h[ j, seq( 1, j ) ] = val
         h[ j, j+1 ]         = - j * val
      }
   }
   h
}

# =============================================================================
uji_Enorm = function( X ) {
#
# Accelerated version of the original function for medium and large datasets.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 if( is.complex( X ) ) {
   #### n <- sqrt( sum( diag( Re( st(X) %*% X ) ) ) )
   n <- sqrt( sum( Re( X )^2 + Im( X )^2 ) )
 } else {
   #### n <- sqrt(sum(diag(t(X) %*% X)))
   n <- sqrt( sum( X^2 ) )
 }
 return( n )
}

# =============================================================================
uji_distProcrustesFull = function( P1, P2 ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   #### H <- defh( dim( P1 )[ 1 ] )
   #### Q1 <- t( H ) %*% rootmat( P1 )
   #### Q2 <- t( H ) %*% rootmat( P2 )
   
   # Accelerated code.
   Q1 <- multiply_by_transpose_of_helmert( rootmat( P1 ) )
   Q2 <- multiply_by_transpose_of_helmert( rootmat( P2 ) )
   
   ans <- riemdist( Q1, Q2, reflect = TRUE )
   ans
}

# =============================================================================
uji_distProcrustesSizeShape = function( P1, P2 ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   #### H <- defh( dim( P1 )[ 1 ] )
   #### Q1 <- t( H ) %*% rootmat( P1 )
   #### Q2 <- t( H ) %*% rootmat( P2 )
   
   # Accelerated code.
   Q1 <- multiply_by_transpose_of_helmert( rootmat( P1 ) )
   Q2 <- multiply_by_transpose_of_helmert( rootmat( P2 ) )
   
   ans <- sqrt(centroid.size(Q1)^2 + centroid.size(Q2)^2 - 2 * 
                  centroid.size(Q1) * centroid.size(Q2) * cos(riemdist(Q1, 
                                                                       Q2, reflect = TRUE)))
   ans
}

# =============================================================================
uji_distCholesky = function( P1, P2 ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   #### H <- defh( dim( P1 )[ 1 ] )
   #### Q1 <- t( H ) %*% t( chol( P1 ) )
   #### Q2 <- t( H ) %*% t( chol( P2 ) )
   
   # Accelerated code.
   Q1 <- multiply_by_transpose_of_helmert( t( chol( P1 ) ) )
   Q2 <- multiply_by_transpose_of_helmert( t( chol( P2 ) ) )
   
   ans <- Enorm( Q1 - Q2 )
   ans
}

# =============================================================================
uji_estSS = function( S, weights = 1 ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   M <- dim( S )[ 3 ]
   k <- dim( S )[ 1 ]
   #### H <- defh( k )
   if ( length( weights ) == 1 ) {
      weights <- rep( 1, times = M )
   }
   Q <- array( 0, c( k+1, k, M ) )
   for ( j in 1 : M ){
      #### Q[,,j]<-t(H)%*%(rootmat(S[,,j]))
      
      # Accelerated code.
      Q[ , , j ] <- multiply_by_transpose_of_helmert( 
         rootmat( S[ , , j ] ) )
   }
   
   ans <- procWGPA( Q, fixcovmatrix = diag( k + 1 ), scale = FALSE, 
                    reflect = TRUE, sampleweights = weights )
   
   #### H%*%ans$mshape%*%t(H%*%ans$mshape)
   
   # Accelerated code.
   auxMat = multiply_by_helmert( ans$mshape )
   return( auxMat %*% t( auxMat ) )
}

# =============================================================================
uji_estShape = function( S, weights = 1 ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   M <- dim( S )[ 3 ]
   k <- dim( S )[ 1 ]
   H <- defh( k )
   if ( length( weights ) == 1 ) {
      weights <- rep( 1, times = M )
   }
   Q <- array( 0, c( k+1, k, M ) )
   for ( j in 1 : M ) {
      #### Q[,,j]<-t(H)%*%(rootmat(S[,,j]))
      
      # Accelerated code.
      Q[ , , j ] <- multiply_by_transpose_of_helmert( 
         rootmat( S[ , , j ] ) )
   }
   ans <- procWGPA( Q, fixcovmatrix = diag( k + 1 ), scale = TRUE,
                    reflect = TRUE, sampleweights = weights)
   
   #### H%*%ans$mshape%*%t(H%*%ans$mshape)
   
   # Accelerated code.
   auxMat = multiply_by_helmert( ans$mshape )
   return( auxMat %*% t( auxMat ) )
}

# =============================================================================
uji_centroid.size.complex = function( zstar ) {
   #
   # It returns the centroid size of a complex vector zstar.
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   #### h <- defh( nrow( as.matrix( zstar ) ) - 1 )
   #### ztem <- h %*% zstar
   
   # Accelerated code.
   ztem <- multiply_by_helmert( zstar )
   
   size <- sqrt( diag( Re( st( ztem ) %*% ztem ) ) )
   size
}

# =============================================================================
uji_centroid.size.mD = function( x ) {
   #
   # It returns the centroid size of a  k x m matrix.
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   if( is.complex( x ) ) {
      x <- cbind( Re( x ), Im( x ) )
   }
   
   #### k <- nrow( x )
   #### h <- defh( k - 1 )
   #### xh <- h %*% x
   #### size <- sqrt( sum( diag( t( xh ) %*% xh ) ) )
   
   # Accelerated code.
   xh <- multiply_by_helmert( x )
   size <- uji_Enorm( xh )
   
   return( size )
}

# =============================================================================
uji_preshape.mD = function( x ) {
   #
   # Input:   k x m matrix
   # Output:  k-1 x 1 matrix 
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   #### h <- defh( nrow( x ) - 1 )
   #### ztem <- h %*% x
   #### size <- centroid.size.mD( x )
   
   # Accelerated code.
   ztem <- multiply_by_helmert( x )
   size <- uji_centroid.size.mD( x )
   
   z <- ztem / size
   return( z )
}

# =============================================================================
uji_preshape.mat = function( zstar ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   #### h <- defh( nrow( as.matrix( zstar ) ) - 1 )
   #### ztem <- h %*% zstar
   
   # Accelerated code.
   ztem <- multiply_by_helmert( zstar )
   
   size <- sqrt( diag( Re( st( ztem ) %*% ztem ) ) )
   if( is.vector( zstar ) )
      z <- ztem / size
   if( is.matrix( zstar ) )
      z <- ztem %*% diag( 1.0 / size )
   return( z )
}

# =============================================================================
uji_tanfigure = function( vv, gamma ) {
   #
   # Inverse projection from complex tangent plane coordinates vv, using pole 
   # gamma.
   # Output: centred icon
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   zvv <- tanpreshape(vv, gamma)
   
   #### k <- nrow( gamma ) + 1
   #### h <- defh( k - 1 )
   #### zstvv <- t( h ) %*% zvv
   
   # Accelerated code.
   zstvv <- multiply_by_transpose_of_helmert( zvv )
   
   return( zstvv )
}

# =============================================================================
uji_tanfigurefull = function( vv, gamma ) {
   #
   # Inverse projection from complex tangent plane coordinates vv, using pole 
   # gamma 
   # Using Procrustes to with scaling to the pole gamma.
   # Output: centred icon 
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   f1 <- uji_tanfigure( vv, gamma )
   
   #### k <- nrow( gamma ) + 1
   #### h <- defh( k - 1 )
   #### f2 <- t(h) %*% gamma
   
   # Accelerated code.
   f2 <- multiply_by_transpose_of_helmert( gamma )
   
   beta <- Mod( st( f1 ) %*% f2 )
   f1 <- f1 * c( beta )
   f1
}

# =============================================================================
uji_kendall.shpv = function( x ) {
   #
   # Accelerated version of the original function for medium and large datasets.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   k <- dim( x )[ 1 ]
   #### h <- defh( k - 1 )
   #### zz <- h %*% x
   
   # Accelerated code.
   zz <- multiply_by_helmert( x )
   
   kendall <- ( zz[2:(k-1),1] + 1i*zz[2:(k-1),2] ) / ( zz[1,1] + 1i*zz[1,2] )
   kendall <- cbind( Re( kendall ), Im( kendall ) )
   kendall
}

# =============================================================================
multiply_by_helmert = function( x ) {
#
# This code multiplies the "x" argument by the transpose of the Helmert matrix
# of the corresponding size.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 #
 # threshold chosen as 30
 #
 if( nrow( x ) < 30 ) {
   xh = multiply_by_helmert_explicitly( x )
 } else {
   xh = multiply_by_helmert_implicitly( x )
 }
 xh
}

# =============================================================================
multiply_by_helmert_explicitly = function( x ) {
#
# This code multiplies the "x" argument by the transpose of the Helmert matrix
# of the corresponding size by explicitly building the matrix.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 k <- nrow( x )
 h <- defh( k - 1 )
 xh <- h %*% x
 xh
}

# =============================================================================
multiply_by_helmert_implicitly = function( x ) {
   #
   # This code multiplies the "x" argument by the Helmert matrix of the 
   # corresponding size without explicitly building the matrix in order to 
   # increase performances.
   #
   # Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
   # This code is distributed in the hope that it will be useful, but 
   # WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
   #
   m = dim( x )[ 1 ] - 1
   n = dim( x )[ 2 ]
   
   result <- matrix( 0, m, n )
   vsum   <- rep( 0, n )
   if( m > 0 ) {
      for( i in seq( 1, m ) ) {
         val  = -1 / sqrt( i * ( i + 1 ) )
         hi   = val
         hip1 = - i * val
         
         vsum = vsum + x[ i, ]
         result[ i, ] = vsum *  hi + x[ i + 1, ] * hip1
      }
   }
   return( result )
}

# =============================================================================
multiply_by_transpose_of_helmert = function( x ) {
#
# This code multiplies the "x" argument by the transpose of the Helmert matrix
# of the corresponding size.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 k = nrow( x )
 if( k < 30 ) {
   result = multiply_by_transpose_of_helmert_explicitly( x )
 } else {
   result = multiply_by_transpose_of_helmert_implicitly( x )
 }
 return( result )
}

# ==============================================================================
multiply_by_transpose_of_helmert_explicitly = function( x ) {
#
# This code multiplies the "x" argument by the transpose of the Helmert matrix
# of the corresponding size by explicitly building the matrix.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 #### m = dim( x )[ 1 ]
 m = nrow( x )
 h = defh( m )
 result = t( h ) %*% x
 return( result )
}

# =============================================================================
multiply_by_transpose_of_helmert_implicitly = function( x ) {
#
# This code multiplies the "x" argument by the transpose of the Helmert matrix
# of the corresponding size without explicitly building the matrix in order to
# increase performances.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 #### m = dim( x )[ 1 ] + 1
 #### n = dim( x )[ 2 ]
 m = nrow( x ) + 1
 n = ncol( x )

 result   <- matrix( 0, m, n )
 rowAccum <- rep( 0, n )
 if( m > 0 ) {
   for( i in seq( m, 1, by = -1 ) ) {
     val  = - 1 / sqrt( ( i - 1 ) * i )
     hi   = val
     hip1 = - ( i - 1 ) * val

     if( i == 1 ) {
       result[ i, ] = rowAccum
     } else {
       result[ i, ] = hip1 * x[ i - 1, ] + rowAccum
       rowAccum = rowAccum + hi * x[ i - 1, ]
     }
   }
 }
 return( result )
}

# =============================================================================
# =============================================================================
uji2_centroid.size = function( x ) {
#
# It returns the centroid size of a configuration (or configurations).
# Input:
#   k x m matrix, or
#   a complex k-vector, or
#   a real k x m x n array to get a vector of sizes for a sample
#
# It computes the centroid size in a faster way on medium and large datasets.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 if ((is.vector(x) == FALSE) && is.complex(x)) {
   k <- nrow(x)
   n <- ncol(x)
   tem <- array(0, c(k, 2, n))
   tem[, 1, ] <- Re(x)
   tem[, 2, ] <- Im(x)
   x <- tem
 }
 {
   if (length( dim( x ) ) == 3 ) {
     #
     # Argument x is a 3D array.
     #
     n <- dim( x )[ 3 ]
     k <- dim( x )[ 1 ]
     sz <- rep( 0, times = n )
     for ( i in 1 : n ) {
       xh <- multiply_by_helmert( x[ , , i ] )
       sz[ i ] <- Enorm( xh )
     }
     sz
   } else {
     if ( is.vector( x ) && is.complex( x ) ) {
        x <- cbind( Re( x ), Im( x ) )
     }
     #### k <- nrow( x )
     #### h <- defh(k - 1)
     #### xh <- h %*% x
     #### size <- sqrt( sum( diag( t( xh ) %*% xh ) ) )
     # Accelerated code.
     xh <- multiply_by_helmert( x )
     size <- Enorm( xh )
     size
   }
 }
}




uji3_centroid.size = function( x ) {
#
# It returns the centroid size of a configuration (or configurations).
# Input:
#   k x m matrix, or
#   a complex k-vector, or
#   a real k x m x n array to get a vector of sizes for a sample
#
# It computes the centroid size in a faster way on medium and large datasets.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 if ((is.vector(x) == FALSE) && is.complex(x)) {
   k <- nrow(x)
   n <- ncol(x)
   tem <- array(0, c(k, 2, n))
   tem[, 1, ] <- Re(x)
   tem[, 2, ] <- Im(x)
   x <- tem
 }
 {
   if (length( dim( x ) ) == 3 ) {
     #
     # Argument x is a 3D array.
     #
     n <- dim( x )[ 3 ]
     k <- dim( x )[ 1 ]
     z <- multiply_by_helmert_implicitly_3d( x )
     sz <- rep( 0, times = n )
     for ( i in 1 : n ) {
       sz[ i ] <- Enorm( z[ , , i ] )
     }
     sz
   } else {
     if ( is.vector( x ) && is.complex( x ) ) {
        x <- cbind( Re( x ), Im( x ) )
     }

     #### k <- nrow( x )
     #### h <- defh(k - 1)
     #### xh <- h %*% x
     #### size <- sqrt( sum( diag( t( xh ) %*% xh ) ) )

     # Accelerated code.
     xh <- multiply_by_helmert( x )
     size <- Enorm( xh )

     size
   }
 }
}


# =============================================================================
multiply_by_helmert_implicitly_3d = function( x ) {
#
# This code multiplies the "x" argument by the Helmert matrix of the
# corresponding size without explicitly building the matrix in order to
# increase performances.
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#

 # Initialize result and vsum.
 k <- dim( x )[ 1 ] - 1
 m <- dim( x )[ 2 ]
 n <- dim( x )[ 3 ]
 result <- array( 0, c( k, m, n ) )
 vsum <- matrix( 0, m, n )
 if( m > 0 ) {
   for( i in seq( 1, k ) ) {
     val  = -1 / sqrt( i * ( i + 1 ) )
     hi   = val
     hip1 = - i * val

     vsum = vsum + x[ i, , ]
     result[ i, , ] = vsum *  hi + x[ i + 1, , ] * hip1
   }
 }
 return( result )
}


# ===========================
# Replace original functions  
# ===========================

# =============================================================================
defh = function( nrow ) {
#
# Written by G. Quintana-Orti and Amelia Simo, University Jaume I, Spain.
# This code is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY EXPRESSED OR IMPLIED.
#
 if ( nrow < 100 ) {
   return( ild_defh( nrow ) )
 } else {
   return( uji_defh( nrow ) )
 }
}


centroid.size <- function(x){
if (is.vector(x)==FALSE){
k<-dim(x)[1]
m<-dim(x)[2]
if ( ( m == 2 ) | ( m == 3 ) ) {
 # Matrices with 2D or 3D landmarks.
 if ( k < 40 ) {
   return( ild_centroid.size(x) )
 } else {
   return( uji3_centroid.size(x) )
 }
} else {
# Often square or nearly-square matrices where m is larger
 if ( k < 85 ) {
   return( ild_centroid.size(x) )
 } else {
   return( uji2_centroid.size(x) )
 }
}
}
else{
 return(ild_centroid.size(x))
    }
}



Enorm<-uji_Enorm
preshapetoicon<-ild_preshapetoicon 
preshape<-ild_preshape
distProcrustesFull <- uji_distProcrustesFull
distProcrustesSizeShape <- uji_distProcrustesSizeShape
distCholesky <- uji_distCholesky
estSS <- ild_estSS
estShape <- ild_estShape
centroid.size.complex <- uji_centroid.size.complex
centroid.size.mD <- uji_centroid.size.mD
preshape.mD <- uji_preshape.mD
preshape.mat <- uji_preshape.mat
tanfigure <- uji_tanfigure
tanfigurefull <- uji_tanfigurefull
kendall.shpv <- uji_kendall.shpv



##########################################################################