# R/vbsmm.R In keyuan/ccube: ccube

```############ Sort the model  ############
# Sort model paramters in increasing order of averaged means
# of d variables
sort_components_smm <- function(model) {

idx <- order(apply(model\$normalMean, 2, mean))

model\$normalMean <- model\$normalMean[, idx]

model\$responsibility <- model\$responsibility[, idx]

model\$logResponsibility <- model\$logResponsibility[, idx]

model\$dirichletConcentration <- model\$dirichletConcentration[idx]

model\$normalRelativePrecision <- model\$normalRelativePrecision[idx]

model\$whishartDof <- model\$whishartDof[idx]

model\$invWhishartScale <- model\$invWhishartScale[,,idx]

model
}

#' Variational Bayesian Student-t mixture model (VB-SMM)
#' @param data N x D data matrix
#' @param init k (1 x 1) or label (1 x n, 1<=label(i)<=k) or center (d x k)
#' @param prior prior parameters
#' @param tol VBEM convergence threshold
#' @param maxiter VBEM maximum iteration
#' @param verbose show progress
#' @return a list containing model parameters
#' @export
vbsmm <- function(data, init=2, prior, tol=1e-20, maxiter=1e3, verbose=FALSE) {

data <- as.matrix(data)

n <- nrow(data)
d <- ncol(data)

X <- t(data) # Work with D by N for convenience

message(sprintf("Running VB-SMM on a %d-by-%d data with %d clusters ...\n", n, d, init))

if(missing(prior)) {

if( length(init) == 1 ) {
# more general prior with equal dirichletConcentration
k <-  init
ranged <- range(data)
prior <- list(
dirichletConcentration = rep(1e-2,k),
normalRelativePrecision = 1,
normalMean = t(as.matrix(rep(0.5, k))),
#normalMean = t(as.matrix(seq(ranged[1],ranged[2],length.out = k))),
whishartDof = d,
invWhishartScale = var(data)/k, # M = inv(W)
)
}
} else {

stopifnot(
all(names(prior) %in% c("dirichletConcentration","normalRelativePrecision","normalMean","whishartDof","invWhishartScale")) &
all(sapply(prior, is.numeric)) & nrow(prior\$normalMean) == d &
ncol(prior\$normalMean) == 1 &
nrow(prior\$normalMean) == d & ncol(prior\$invWhishartScale) == d)
}

# lower variational bound (objective function)
L <- rep(-Inf, maxiter)
converged <- FALSE
t <- 1

model <- list()

initParams <- initialization_smm(X, init, prior) # initialize responsibility and hidden scale
model\$responsibility <- initParams\$R
model\$gammaAlpha <- initParams\$gammaAlpha
model\$gammaBeta <- initParams\$gammaBeta
model\$hiddenScale <- initParams\$U

while(!converged & t < maxiter) {
t <- t + 1
model <- VariationalMaximimizationStep_smm(X, model, prior)
model <- VarationalExpectationStep_smm(X, model)
L[t] <- variationalLowerBound_smm(X, model, prior)/n
converged <- abs(L[t] - L[t-1]) < tol * abs(L[t])
if(verbose) message(sprintf("VB-EM-%d: L = %.6f \r", t, L[t]))
}

L <- L[2:t]

model <- sort_components_smm(model)

if (init > 1) {
label <- rep(0, n)
label <- apply(model\$responsibility, 1, which.max)
nk <- colSums(model\$responsibility)
Epi <- (model\$dirichletConcentration + nk) / (k*prior\$dirichletConcentration + n)
model\$Epi <- Epi/sum(Epi)
} else {
label <- rep(1, n)
model\$Epi <- 1
}

# unique to have consecutive label eg 2,3,6 changed to 1,2,3
# label <- match(label, sort(unique(label)))

if(converged) message(sprintf("Converged in %d steps.\n", t-1)) else
warnings(sprintf("Not converged in %d steps.\n", maxiter))

list(label=label, R=model\$responsibility, mu=model\$normalMean, full.model=model, L=L)
}

############ Initialization of responsibility (intialization) ############

initialization_smm <- function(X, init, prior) {

d <- nrow(X)

n <- ncol(X)

stopifnot(length(init) %in% c(1, n) ||
(nrow(init) == d  & ncol(init) == k))

if(length(init) == 1) { # init = k gaussian components

k <- init

res <- kmeans(t(X), k)
label <- res\$cluster
normalMean <- t(res\$centers)

R <- as.matrix(Matrix::sparseMatrix(1:n, label, x=1))

} else {

if(length(init) == n) { # initialize with labels

label <- init
k <- max(label)
R <- as.matrix(sparseMatrix(1:n, label, x=1))

} else {

if(!is.null(dim(init))) {

if(nrow(init) == d  & ncol(init) == k) { # initialize with centers

k <- ncol(init)
m <- init
label <- apply(bsxfun.se("-", crossprod(m, X),
as.matrix(dot.ext(m,m,1)/2)), 2, which.max)
R <- as.matrix(Matrix::sparseMatrix(1:n, label, x=1))

} else stop(message("Invalid init."))
}
}
}

gammaBeta <- array(0, dim=c(n,k))
invWhishartScale <- prior\$invWhishartScale
whishartDof <- prior\$whishartDof
normalRelativePrecision <- prior\$normalRelativePrecision

for(i in 1:k) {

U <- chol(invWhishartScale)

Q <- solve(t(U), bsxfun.se("-", X, normalMean[,i]))

QQ <- dot.ext(Q, Q, 1)

gammaBeta[,i] <- (d/(normalRelativePrecision) + whishartDof *QQ + gammaDof[i])/2
}

U = bsxfun.se("*", 1/gammaBeta, gammaAlpha)

return(list(R = R, gammaAlpha = gammaAlpha, gammaBeta = gammaBeta, U = U))
}

############ Variational-Maximimization ############
VariationalMaximimizationStep_smm <- function(X, model, prior) {

dirichletConcentration0 <- prior\$dirichletConcentration
normalRelativePrecision0 <- prior\$normalRelativePrecision
normalMean0 <- prior\$normalMean
whishartDof0 <- prior\$whishartDof
invWhishartScale0 <- prior\$invWhishartScale
responsibility <- model\$responsibility
hiddenScale <- model\$hiddenScale

scaledResponsibility <- responsibility * hiddenScale

dataSumResponsibility <- colSums(responsibility) # 10.51
dataSumScaledResponsibility <- colSums(scaledResponsibility)
dirichletConcentration <- dirichletConcentration0 + dataSumResponsibility # 10.58
scaledResponsibilityWeightedX <- X %*% scaledResponsibility
normalRelativePrecision <- normalRelativePrecision0 + dataSumScaledResponsibility # 10.60

normalMean <- bsxfun.se("*", bsxfun.se("+", scaledResponsibilityWeightedX, normalRelativePrecision0 * normalMean0), 1/normalRelativePrecision) # 10.61
whishartDof <- whishartDof0 + dataSumResponsibility # 10.63 (NB: no 1 in the matlab code)

dimensionOfData <- nrow(normalMean)
numberOfComponents <- ncol(normalMean)

invWhishartScale <- array(0, c(dimensionOfData, dimensionOfData, numberOfComponents))
sqrtScaledResponsibility <- sqrt(scaledResponsibility)

averageScaledResponsibilityWeightedX <- bsxfun.se("*", scaledResponsibilityWeightedX, 1/dataSumScaledResponsibility) # 10.52
averageScaledResponsibilityWeightedXNormalMean0 <- bsxfun.se("-", averageScaledResponsibilityWeightedX, normalMean0)
rescaledRelativePrecision <- (normalRelativePrecision0 * dataSumScaledResponsibility) / normalRelativePrecision

for(i in 1:numberOfComponents) {

Xs <- bsxfun.se("*", bsxfun.se("-", X, averageScaledResponsibilityWeightedX[,i]), t(sqrtScaledResponsibility[,i]))

averageScaledResponsibilityWeightedXNormalMean0i <- averageScaledResponsibilityWeightedXNormalMean0[,i] # 10.62

invWhishartScale[,,i] <- invWhishartScale0 +  Matrix::tcrossprod(Xs, Xs) +
rescaledRelativePrecision[i] *
Matrix::tcrossprod(averageScaledResponsibilityWeightedXNormalMean0i, averageScaledResponsibilityWeightedXNormalMean0i)
}

gammaAlpha <- model\$gammaAlpha
gammaBeta <- model\$gammaBeta
expectedlogHiddenScale <- bsxfun.se("+", -log(gammaBeta), digamma(gammaAlpha))
tmp <- colSums(responsibility*(expectedlogHiddenScale - hiddenScale)) / dataSumResponsibility

for (i in 1:numberOfComponents) {
function(x) { log(x/2) + 1 - digamma(x/2) + tmp[i] },
c(2,200), extendInt = "yes")\$root )
}

model\$dirichletConcentration <- dirichletConcentration
model\$normalRelativePrecision <- normalRelativePrecision
model\$normalMean <- normalMean
model\$whishartDof <- whishartDof
model\$invWhishartScale <- invWhishartScale # Whishart: M = inv(W)

model
}

############ Variational-Expectation ############

VarationalExpectationStep_smm <- function(X, model, no.weights = FALSE) {

dirichletConcentration <- model\$dirichletConcentration	# Dirichlet
normalRelativePrecision <- model\$normalRelativePrecision	# Gaussian
normalMean <- model\$normalMean			# Gasusian
whishartDof <- model\$whishartDof			# Whishart
invWhishartScale <- model\$invWhishartScale	# Whishart: inv(W) = V'*V

if (!is.array(normalMean) ){
normalMean <- array(normalMean, c(1, length(normalMean)))
}

if (!is.array(invWhishartScale) ){
invWhishartScale <- array(invWhishartScale, c(1, 1, length(invWhishartScale)))
}

n <- ncol(X)
d <- nrow(normalMean)
k <- ncol(normalMean)

logW <- array(0, dim=c(1,k))
EQ <- array(0, dim=c(n,k))
gammaBeta <- array(0, dim=c(n,k))

for(i in 1:k) {

U <- chol(invWhishartScale[,,i])

logW[i] <- -2 * sum(log(diag(U)))

Q <- solve(t(U), bsxfun.se("-", X, normalMean[,i]))

QQ <- dot.ext(Q, Q, 1)

EQ[,i] <- d/(normalRelativePrecision[i]* gammaDof[i]) + whishartDof[i]/gammaDof[i] * QQ	# eq (19)  bishop 10.64

gammaBeta[,i] <- (d/(normalRelativePrecision[i]) + whishartDof[i] *QQ + gammaDof[i])/2
}

vd <- bsxfun.se("-", matrix(rep(whishartDof+1, d),nrow=d,byrow=T), as.matrix(1:d))/2

ElogLambda <- colSums(digamma(vd)) + d*log(2) + logW	# 10.65
Elogpi <- digamma(dirichletConcentration) - digamma(sum(dirichletConcentration))			# 10.66

if (no.weights) {
logRho <- 0.5 * bsxfun.se("+", logRho1,
+ ElogLambda - d*log(gammaDof * pi) +
} else {

logRho <- 0.5 * bsxfun.se("+", logRho1,
2*Elogpi + ElogLambda - d*log(gammaDof * pi) +

}

# ke: add bound to avoid numerical issue
#mapper <- logRho < -700
#logRho <- mapper * -700 + (!mapper) * logRho
if (n==k) {
logR <- bsxfun.se("-", logRho, logsumexp(logRho, 1), expandByRow = F)	# 10.49
} else {
logR <- bsxfun.se("-", logRho, logsumexp(logRho, 1))	# 10.49
}

R <- exp(logR)

model\$logResponsibility <- logR
model\$responsibility<- R

hiddenScale <- bsxfun.se("*", 1/gammaBeta, gammaAlpha)

model\$gammaAlpha <- gammaAlpha
model\$gammaBeta <- gammaBeta
model\$hiddenScale <- hiddenScale

model
}

############ Variational-(lower)-Bound Evaluation ############

variationalLowerBound_smm <- function(X, model, prior) {

dirichletConcentration0 <- prior\$dirichletConcentration
normalRelativePrecision0 <- prior\$normalRelativePrecision
m0 <- prior\$normalMean
v0 <- prior\$whishartDof
M0 <- prior\$invWhishartScale

dirichletConcentration <- model\$dirichletConcentration	# Dirichlet
normalRelativePrecision <- model\$normalRelativePrecision	# Gaussian
m <- model\$normalMean			# Gasusian
v <- model\$whishartDof			# Whishart
M <- model\$invWhishartScale			# Whishart: inv(W) = V'*V
R <- model\$responsibility
logR <- model\$logResponsibility
hiddenScale <- model\$hiddenScale

d <- nrow(m)
k <- ncol(m)
nk <- colSums(R)									# 10.51
RU <- R*hiddenScale
omegak <- colSums(RU)

Elogpi <- digamma(dirichletConcentration) - digamma(sum(dirichletConcentration))		# 10.66

Epz = pracma::dot(nk, Elogpi)								# 10.72
Eqz = pracma::dot(as.numeric(R), as.numeric(logR))			# 10.75
# logCalpha0 = lgamma(k * alpha0) - k * lgamma(alpha0) # for scalar alpha0
logCdirichletConcentration0 = lgamma(sum(dirichletConcentration0)) - sum(lgamma(dirichletConcentration0))
# Eppi <- logCalpha0+(alpha0-1)*sum(Elogpi) # for scalar alpha0
Eppi <- logCdirichletConcentration0+pracma::dot(dirichletConcentration0-1, Elogpi)			# 10.73
logCdirichletConcentration <- lgamma(sum(dirichletConcentration))-sum(lgamma(dirichletConcentration))
Eqpi = pracma::dot(dirichletConcentration-1, Elogpi) + logCdirichletConcentration				# 10.76

# part of 10.70
L <- Epz - Eqz + Eppi - Eqpi

U0 <- chol(M0)
sqrtR <- sqrt(R)
sqrtRU <- sqrt(RU)
xbar <- bsxfun.se("*", X %*% (R*hiddenScale), 1/omegak)				# 10.52

logW <- array(0, dim = c(1, k))
trSW <- array(0, dim = c(1, k))
trM0W <- array(0, dim = c(1, k))
xbarmWxbarm <- array(0, dim = c(1, k))
mm0Wmm0 <- array(0, dim = c(1, k))

for(i in 1:k) {

U <- chol(M[,,i])
logW[i] <- -2 * sum(log(diag(U)))

Xs <- bsxfun.se("*", bsxfun.se("-", X, as.matrix(xbar[,i,drop=F])), t(sqrtRU[,i,drop=F]))
V <- chol(Matrix::tcrossprod(Xs, Xs)/omegak[i])
Q <- solve(U, V)
# equivalent to tr(SW)=trace(S/M)
trSW[i] <- pracma::dot(as.numeric(Q), as.numeric(Q))
Q <- solve(U, U0)
trM0W[i] <- pracma::dot(as.numeric(Q), as.numeric(Q))

q <- solve(t(U), xbar[,i,drop=F]-m[,i,drop=F])
xbarmWxbarm[i] = pracma::dot(q, q)
#q <- solve(t(U), m[,i,drop=F]-m0)
q <- solve(t(U), m[,i,drop=F]-m0[,i,drop=F]) #ke: allow multivariate prior
mm0Wmm0[i] <- pracma::dot(q, q)
}

vd <- bsxfun.se("-", matrix(rep(v+1, d),nrow=d,byrow=T), as.matrix(1:d))/2
ElogLambda <- colSums(digamma(vd)) + d*log(2) + logW	# 10.65

# first half of 10.74
Epmu <- sum(d*log(normalRelativePrecision0/(2*pi))+ElogLambda-d*normalRelativePrecision0/normalRelativePrecision-normalRelativePrecision0*(v*mm0Wmm0))/2
logB0 <- v0*sum(log(diag(U0)))-0.5*v0*d*log(2)-logmvgamma(0.5*v0,d) # B.79
# second half of 10.74
EpLambda <- k*logB0+0.5*(v0-d-1)*sum(ElogLambda)-0.5*pracma::dot(v,trM0W)

Eqmu <- 0.5*sum(ElogLambda+d*log(normalRelativePrecision/(2*pi)))-0.5*d*k	# 10.77 (1/2)
logB <- (-v) * (logW+d*log(2))/2 - logmvgamma(0.5*v, d)	# B.79
EqLambda <- 0.5*sum((v-d-1)*ElogLambda-v*d)+sum(logB) 	# 10.77 (2/2)

gammaAlpha <- model\$gammaAlpha
gammaBeta <- model\$gammaBeta
ElogU <- bsxfun.se("+", -log(gammaBeta), digamma(gammaAlpha))
Uterm1 <- colSums(R * ElogU) / nk
Uterm2 <- omegak/nk

EpX <- 0.5*pracma::dot(nk, ElogLambda +
d*Uterm1 - d*Uterm2/normalRelativePrecision - v*Uterm2*trSW -v*xbarmWxbarm/nk-d*log(2*pi))	# eq (41) 10.71