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#' EMGLLF
#'
#' Run a generalized EM algorithm developped for mixture of Gaussian regression
#' models with variable selection by an extension of the Lasso estimator (regularization parameter lambda).
#' Reparametrization is done to ensure invariance by homothetic transformation.
#' It returns a collection of models, varying the number of clusters and the sparsity in the regression mean.
#'
#' @param phiInit an initialization for phi
#' @param rhoInit an initialization for rho
#' @param piInit an initialization for pi
#' @param gamInit initialization for the a posteriori probabilities
#' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10
#' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100
#' @param gamma integer for the power in the penaly, by default = 1
#' @param lambda regularization parameter in the Lasso estimation
#' @param X matrix of covariates (of size n*p)
#' @param Y matrix of responses (of size n*m)
#' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4
#' @param fast boolean to enable or not the C function call
#'
#' @return A list (corresponding to the model collection) defined by (phi,rho,pi,llh,S,affec):
#' phi : regression mean for each cluster, an array of size p*m*k
#' rho : variance (homothetic) for each cluster, an array of size m*m*k
#' pi : proportion for each cluster, a vector of size k
#' llh : log likelihood with respect to the training set
#' S : selected variables indexes, an array of size p*m*k
#' affec : cluster affectation for each observation (of the training set)
#'
#' @export
EMGLLF <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
X, Y, eps, fast)
{
if (!fast)
{
# Function in R
return(.EMGLLF_R(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
X, Y, eps))
}
# Function in C
.Call("EMGLLF", phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
X, Y, eps, PACKAGE = "valse")
}
# R version - slow but easy to read
.EMGLLF_R <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
X, Y, eps)
{
# Matrix dimensions
n <- nrow(X)
p <- ncol(X)
m <- ncol(Y)
k <- length(piInit)
# Adjustments required when p==1 or m==1 (var.sel. or output dim 1)
if (p==1 || m==1)
phiInit <- array(phiInit, dim=c(p,m,k))
if (m==1)
rhoInit <- array(rhoInit, dim=c(m,m,k))
# Outputs
phi <- phiInit
rho <- rhoInit
pi <- piInit
llh <- -Inf
S <- array(0, dim = c(p, m, k))
# Algorithm variables
gam <- gamInit
Gram2 <- array(0, dim = c(p, p, k))
ps2 <- array(0, dim = c(p, m, k))
X2 <- array(0, dim = c(n, p, k))
Y2 <- array(0, dim = c(n, m, k))
for (ite in 1:maxi)
{
# Remember last pi,rho,phi values for exit condition in the end of loop
Phi <- phi
Rho <- rho
Pi <- pi
# Computations associated to X and Y
for (r in 1:k)
{
for (mm in 1:m)
Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
for (i in 1:n)
X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
for (mm in 1:m)
ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
for (j in 1:p)
{
for (s in 1:p)
Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
}
}
## M step
# For pi
b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
gam2 <- colSums(gam)
a <- sum(gam %*% log(pi))
# While the proportions are nonpositive
kk <- 0
pi2AllPositive <- FALSE
while (!pi2AllPositive)
{
pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
pi2AllPositive <- all(pi2 >= 0)
kk <- kk + 1
}
# t(m) is the largest value in the grid O.1^k such that it is nonincreasing
while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
# na.rm=TRUE to handle 0*log(0)
-sum(gam2 * log(pi2), na.rm=TRUE)/n + lambda * sum(pi2^gamma * b))
{
pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
kk <- kk + 1
}
t <- 0.1^kk
pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
# For phi and rho
for (r in 1:k)
{
for (mm in 1:m)
{
ps <- 0
for (i in 1:n)
ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm, r])
nY2 <- sum(Y2[, mm, r]^2)
rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
}
}
for (r in 1:k)
{
for (j in 1:p)
{
for (mm in 1:m)
{
S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r] +
sum(phi[-j, mm, r] * Gram2[j, -j, r])
if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma)) {
phi[j, mm, r] <- 0
} else if (S[j, mm, r] > n * lambda * (pi[r]^gamma)) {
phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j, j, r]
} else {
phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j, j, r]
}
}
}
}
## E step
# Precompute det(rho[,,r]) for r in 1...k
detRho <- sapply(1:k, function(r) gdet(rho[, , r]))
sumLogLLH <- 0
for (i in 1:n)
{
# Update gam[,]; use log to avoid numerical problems
logGam <- sapply(1:k, function(r) {
log(pi[r]) + log(detRho[r]) - 0.5 *
sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)
})
logGam <- logGam - max(logGam) #adjust without changing proportions
gam[i, ] <- exp(logGam)
norm_fact <- sum(gam[i, ])
gam[i, ] <- gam[i, ] / norm_fact
sumLogLLH <- sumLogLLH + log(norm_fact) - log((2 * base::pi)^(m/2))
}
sumPen <- sum(pi^gamma * b)
last_llh <- llh
llh <- -sumLogLLH/n #+ lambda * sumPen
dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
dist2 <- max(Dist1, Dist2, Dist3)
if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
break
}
affec = apply(gam, 1, which.max)
list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
}
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