inst/scripts/fdarm-ch08.R

In fda: Functional Data Analysis

###
### Ramsay, Hooker & Graves (2009)
### Functional Data Analysis with R and Matlab (Springer)
###

#  Remarks and disclaimers

#  These R commands are either those in this book, or designed to
#  otherwise illustrate how R can be used in the analysis of functional
#  data.
#  We do not claim to reproduce the results in the book exactly by these
#  commands for various reasons, including:
#    -- the analyses used to produce the book may not have been
#       entirely correct, possibly due to coding and accuracy issues
#       in the functions themselves
#    -- we may have changed our minds about how these analyses should be
#       done since, and we want to suggest better ways
#    -- the R language changes with each release of the base system, and
#       certainly the functional data analysis functions change as well
#    -- we might choose to offer new analyses from time to time by
#       augmenting those in the book
#    -- many illustrations in the book were produced using Matlab, which
#       inevitably can imply slightly different results and graphical
#       displays
#    -- we may have changed our minds about variable names.  For example,
#       we now prefer "yearRng" to "yearRng" for the weather data.
#    -- three of us wrote the book, and the person preparing these scripts
#       might not be the person who wrote the text
#  Moreover, we expect to augment and modify these command scripts from time
#  to time as we get new data illustrating new things, add functionality
#  to the package, or just for fun.

###
### ch. 8.  Registration: Aligning Features
###         for Samples of Curves
###

#  load the fda package

library(fda)

#  display the data files associated with the fda package

data(package='fda')

#  start the HTML help system if you are connected to the Internet, in
#  order to open the R-Project documentation index page in order to obtain
#  information about R or the fda package.

help.start()

##
## Section 8.1 Amplitude and Phase Variation
##

#  Figure 8.1 in this section requires that we firt do
#  landmark registration first.
#  This figure is therefore plotted in Section 8.3 below.

#  Figure 8.2

#  set up a fine mesh of t-values for plotting, and define mu and sigma

tvec  = seq(-5,5,len=201)
mu    = seq(-1,1,len=  5)
sigma = (1:5)/3

#  Here is function Dgauss that we use below to compute values
#  of the first derivative of a Gaussian density:

DGauss = function(tvec, mu, sigma)
{
  var = as.matrix(sigma)^2

  n = length(tvec)
  m = length(mu)
  if (length(sigma) != m)
    stop('MU and SIGMA not of same length')

  tvec  = as.matrix(tvec)
  mu    = as.matrix(mu)
  onesm = matrix(1,1,m)
  onesn = matrix(1,n,1)

  res   = tvec %*% onesm - onesn %*% t(mu)
  expon = res^2/(2*onesn %*% t(var))
  DpG   = -res*exp(-expon)

  return(DpG)
}

#  Data to be plotted in the left panel

DpGphase = matrix(0,201,5)
for (i in 1:5) DpGphase[,i] = DGauss(tvec+mu[i], 0, 1)
DpGphaseMean = apply(DpGphase,1,mean)

#  Data to be plotted in the right panel

DpGampli = matrix(0,201,5)
for (i in 1:5) DpGampli[,i] = sigma[i]*DGauss(tvec, 0, 1)
DpGampliMean = apply(DpGampli,1,mean)

op = par(mfrow=c(2,1), cex=1, ask=FALSE)
#  top panel
matplot(tvec, DpGphase, "l", lwd=1, col=1, lty=1,
        xlim=c(-5,5), ylim=c(-0.8,0.8),
        xlab="", ylab="")
lines(tvec, DpGphaseMean, col=1, lty=2, lwd=4)
lines(c(-5,5), c(0,0), col=1, lty=3, lwd=1)
# bottom panel
matplot(tvec, DpGampli, "l", lwd=1, col=1, lty=1,
        xlim=c(-5,5), ylim=c(-1.2,1.2),
        xlab="", ylab="")
lines(tvec, DpGampliMean, col=1, lty=2, lwd=4)
lines(c(-5,5), c(0,0), col=1, lty=3, lwd=1)
par(op)

#  get eigenvalues for each panel

eigvecphase = svd(DpGphase)$d^2
(eigvecphase = eigvecphase/sum(eigvecphase))
# First three = 0.55, 0.39, and 0.05

eigvecampli = svd(DpGampli)$d^2
(eigvecampli = eigvecampli/sum(eigvecampli))
# First singular value = 1;
# all others are round-off

##
## Section 8.2 Time-Warping Functions and Registration
##

#  ----------  Registration of the Berkeley female growth data  -----------

#  Figure 8.3 requires landmark registration, and is set up below

##
##  Section 8.3: Landmark registration
##

#  set up ages of measurement and an age mesh

age     = growth$age
nage    = length(age)
ageRng  = range(age)
nfine   = 101
agefine = seq(ageRng[1], ageRng[2], length=nfine)

#  the data

hgtf   = growth$hgtf
ncasef = dim(hgtf)[2]

#  Set up functional data objects for the acceleration curves
#  and their mean.  Suffix UN means "unregistered".

#  This step requires the functional data object hgtfhatfd computed
#  from the monotone smooth of the Berkeley female data.  Refer to
#  Section 5.4.2.2  in the text.  To create it here, we copy
#  relevant lines from the script file fdarm-ch05.R

growbasis = create.bspline.basis(norder=6, breaks=age)
Lfdobj    = 3          #  penalize curvature of acceleration
lambda    = 10^(-0.5)  #  smoothing parameter
cvecf     = matrix(0, growbasis$nbasis, ncol(growth$hgtf))
dimnames(cvecf) = list(growbasis$names, dimnames(growth$hgtf)[[2]])

growfd0   = fd(cvecf, growbasis)
growfdPar = fdPar(growfd0, Lfdobj, lambda)

growthMon = smooth.monotone(age, hgtf, growfdPar)
hgtfhatfd = growthMon$yhatfd

accelfdUN     = deriv.fd(hgtfhatfd, 2)
accelmeanfdUN = mean(accelfdUN)

#  plot unregistered curves

plot(accelfdUN, xlim=c(1,18), ylim=c(-4,3), lty=1, lwd=2,
     cex=2, xlab="Age", ylab="Acceleration (cm/yr/yr)")

#  This is a MANUAL PGS spurt identification procedure requiring
#  a mouse click at the point where the acceleration curve
#  crosses the zero axis with a negative slope during puberty.
#  Here we do this only for the first 10 children.

children = 1:10

PGSctr = rep(0,length(children))
for (icase in children) {
    accveci = eval.fd(agefine, accelfdUN[icase])
    plot(agefine,accveci,"l", ylim=c(-6,4),
         xlab="Year", ylab="Height Accel.",
         main=paste("Case",icase))
    lines(c(1,18),c(0,0),lty=2)
    PGSctr[icase] = locator(1)$x
# **** CLICK ON EACH ACCELERATION CURVE WHERE IT
# **** CROSSES ZERO WITH NEGATIVE SLOPE DURING PUBERTY
}

#  This is an automatic PGS spurt identification procedure.
#  A mouse click advances the plot to the next case.
#  Compute PGS mid point for landmark registration.
#  Downward crossings are computed within the limits defined
#  by INDEX.  Each of the crossings within this interval
#  are plotted.  The estimated PGS center is plotted as a vertical line.

#  The choice of range of argument values (6--18) to consider
#  for a potential mid PGS location is determined by previous
#  analyses, where they have a mean of about 12 and a s.d. of 1.

#  We compute landmarks for all 54 children

index  = 1:102  #  wide limits
nindex = length(index)
ageval = seq(8.5,15,len=nindex)
PGSctr = rep(0,ncasef)
op = par(ask=TRUE)
for (icase in 1:ncasef) {
    accveci = eval.fd(ageval, accelfdUN[icase])
    aup     = accveci[2:nindex]
    adn     = accveci[1:(nindex-1)]
    indx    = (1:102)[adn*aup < 0 & adn > 0]
    plot(ageval[2:nindex],aup,"p",
         xlim=c(7.9,18), ylim=c(-6,4))
    lines(c(8,18),c(0,0),lty=2)
    for (j in 1:length(indx)) {
        indxj = indx[j]
        aupj  = aup[indxj]
        adnj  = adn[indxj]
        agej  = ageval[indxj] + 0.1*(adnj/(adnj-aupj))
        if (j == length(indx)) {
            PGSctr[icase] = agej
            lines(c(agej,agej),c(-4,4),lty=1)
        } else {
            lines(c(agej,agej),c(-4,4),lty=3)
        }
    }
    title(paste('Case ',icase))
# ****** CLICK ON EACH PLOT TO ADVANCE TO THE NEXT
# ****** {par(ask=TRUE)}
}
par(op)

#  We use the minimal basis function sufficient to fit 3 points
#  remember that the first coefficient is set to 0, so there
#  are three free coefficients, and the data are two boundary
#  values plus one interior knot.
#  Suffix LM means "Landmark-registered".

PGSctrmean = mean(PGSctr)

#  Define the basis for the function W(t).

wbasisLM = create.bspline.basis(c(1,18), 4, 3, c(1,PGSctrmean,18))
WfdLM    = fd(matrix(0,4,1),wbasisLM)
WfdParLM = fdPar(WfdLM,1,1e-12)

#  Carry out landmark registration.

regListLM = landmarkreg(accelfdUN, PGSctr, PGSctrmean,
                             WfdParLM, TRUE)

accelfdLM     = regListLM$regfd
accelmeanfdLM = mean(accelfdLM)

#  plot registered curves

plot(accelfdLM, xlim=c(1,18), ylim=c(-4,3), lty=1, lwd=1,
     cex=2, xlab="Age", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdLM, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-4,3), lty=2, lwd=1.5)

# Figure 8.1

accelmeanfdUN10 = mean(accelfdUN[children])
accelmeanfdLM10 = mean(accelfdLM[children])

op = par(mfrow=c(2,1))
plot(accelfdUN[children], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=1,
     cex=2, xlab="", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdUN10, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
plot(accelfdLM[children], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=1,
     cex=2, xlab="Age (Years)", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdLM10, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
par(op)

# Figure 8.3

#  plot warping functions for cases 3 and 7

warpfdLM  = regListLM$warpfd
warpmatLM = eval.fd(agefine, warpfdLM)

op = par(mfrow=c(2,2))
plot(accelfdUN[3], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=2,
     xlab="", ylab="")
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
plot(agefine, warpmatLM[,3], "l", lty=1, lwd=2, col=1, cex=1.2,
        xlab="", ylab="")
lines(agefine,  agefine, lty=2, lwd=1.5)
lines(c(PGSctrmean,PGSctrmean), c(1,18), lty=2, lwd=1.5)
text(PGSctrmean+0.1, warpmatLM[61,3]+0.3, "o", lwd=2)

plot(accelfdUN[7], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=2,
     xlab="", ylab="")
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
plot(agefine, warpmatLM[,7], "l", lty=1, lwd=2, col=1, cex=1.2,
        xlab="", ylab="")
lines(c(PGSctrmean,PGSctrmean), c(1,18), lty=2, lwd=1.5)
lines(agefine,  agefine, lty=2, lwd=1.5)
text(PGSctrmean+0.1, warpmatLM[61,7]+0.2, "o", lwd=2)
par(op)

#  Comparing unregistered to landmark registered curves

AmpPhasList = AmpPhaseDecomp(accelfdUN, accelfdLM, warpfdLM, c(3,18))
MS.amp      = AmpPhasList$MS.amp
MS.pha      = AmpPhasList$MS.pha
RSQRLM      = AmpPhasList$RSQR
CLM         = AmpPhasList$C

print(paste("Total MS =",     round(MS.amp+MS.pha,2),
            "Amplitude MS =", round(MS.amp,2),
            "Phase MS =",     round(MS.pha,2)))

#  [1] "Total MS = 7.06 Amplitude MS = 2.12 Phase MS = 4.95"

print(paste("R-squared =", round(RSQRLM,3), ",  C =", round(CLM,3)))

#  "R-squared = 0.7 ,  C = 0.984"

##
##  Section 8.4: Continuous registration
##

#  Set up a cubic spline basis for continuous registration

nwbasisCR = 15
norderCR  =  5
wbasisCR  = create.bspline.basis(c(1,18), nwbasisCR, norderCR)
Wfd0CR    = fd(matrix(0,nwbasisCR,ncasef),wbasisCR)
lambdaCR  = 1
WfdParCR  = fdPar(Wfd0CR, 1, lambdaCR)

#  carry out the registration

registerlistCR = register.fd(accelmeanfdLM, accelfdLM, WfdParCR)

accelfdCR = registerlistCR$regfd
warpfdCR  = registerlistCR$warpfd
WfdCR     = registerlistCR$Wfd

#  plot landmark and continuously registered curves for the
#  first 10 children

accelmeanfdLM10 = mean(accelfdLM[children])
accelmeanfdCR10 = mean(accelfdCR[children])

op = par(mfrow=c(2,1))
plot(accelfdLM[children], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=1,
     cex=2, xlab="Age (Years)", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdLM10, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
plot(accelfdCR[children], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=1,
     cex=2, xlab="Age (Years)", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdCR10, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
par(op)

#  plot all landmark and continuously registered curves

accelmeanfdCR = mean(accelfdCR)

op = par(mfrow=c(2,1))
plot(accelfdLM, xlim=c(1,18), ylim=c(-4,3), lty=1, lwd=1,
     cex=2, xlab="Age (Years)", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdLM, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-4,3), lty=2, lwd=1.5)
plot(accelfdCR, xlim=c(1,18), ylim=c(-4,3), lty=1, lwd=1,
     cex=2, xlab="Age (Years)", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdCR, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-4,3), lty=2, lwd=1.5)
par(op)

# Figure 8.4

par(mfrow=c(1,1))
plot(accelfdCR[children], xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=1,
     cex=2, xlab="Age (Years)", ylab="Acceleration (cm/yr/yr)")
lines(accelmeanfdCR10, col=1, lwd=2, lty=2)
lines(c(PGSctrmean,PGSctrmean), c(-3,1.5), lty=2, lwd=1.5)
par(op)

# Figure 8.5

accelmeanfdUN = mean(accelfdUN)
accelmeanfdLM = mean(accelfdLM)
accelmeanfdCR = mean(accelfdCR)

plot(accelmeanfdCR, xlim=c(1,18), ylim=c(-3,1.5), lty=1, lwd=2,
     cex=1.2, xlab="Years", ylab="Height Acceleration")
lines(accelmeanfdLM, lwd=1.5, lty=1)
lines(accelmeanfdUN, lwd=1.5, lty=2)

##
## Section 8.5 A Decomposition into Amplitude and Phase Sums of Squares
##

#  Comparing landmark to continuously registered curves

AmpPhasList = AmpPhaseDecomp(accelfdLM, accelfdCR, warpfdCR, c(3,18))
MS.amp      = AmpPhasList$MS.amp
MS.pha      = AmpPhasList$MS.pha
RSQRCR      = AmpPhasList$RSQR
CCR         = AmpPhasList$C

print(paste("Total MS =",     round(MS.amp+MS.pha,2),
            "Amplitude MS =", round(MS.amp,2),
            "Phase MS =",     round(MS.pha,2)))

#  "Total MS = 1.5 Amplitude MS = 1.6 Phase MS = -0.1"

print(paste("R-squared =", round(RSQRCR,3), ",  C =", round(CCR,3)))

#  "R-squared = -0.067 ,  C = 1.001"

##
## 8.6 Registering the Chinese Handwriting Data
##

#  No code for this section

##
## 8.7 Details for Functions landmarkreg and register.fd
##

help(landmarkreg)
help(register.fd)

##
## Section 8.8 Some Things to Try
##
# (exercises for the reader)

##
## Section 8.8  More to Read
##