-#' EMGLLF
+#' EMGLLF
#'
-#' Description de 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 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 ... phi,rho,pi,LLF,S,affec:
-#' phi : parametre de moyenne renormalisé, calculé par l'EM
-#' rho : parametre de variance renormalisé, calculé par l'EM
-#' pi : parametre des proportions renormalisé, calculé par l'EM
-#' LLF : log vraisemblance associée à cet échantillon, pour les valeurs estimées des paramètres
-#' S : ... affec : ...
+#' @return A list (corresponding to the model collection) defined by (phi,rho,pi,LLF,S,affec):
+#' phi : regression mean for each cluster
+#' rho : variance (homothetic) for each cluster
+#' pi : proportion for each cluster
+#' LLF : log likelihood with respect to the training set
+#' S : selected variables indexes
+#' 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 = TRUE)
- {
+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,
+ return(.EMGLLF_R(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
X, Y, eps))
}
-
+
# Function in C
- n <- nrow(X) #nombre d'echantillons
- p <- ncol(X) #nombre de covariables
- m <- ncol(Y) #taille de Y (multivarié)
- k <- length(piInit) #nombre de composantes dans le mélange
- .Call("EMGLLF", phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
- X, Y, eps, phi = double(p * m * k), rho = double(m * m * k), pi = double(k),
- LLF = double(maxi), S = double(p * m * k), affec = integer(n), n, p, m, k,
- PACKAGE = "valse")
+ .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,
- X2, Y, eps)
- {
+.EMGLLF_R <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
+ X, Y, eps)
+{
# Matrix dimensions
- n <- dim(Y)[1]
- if (length(dim(phiInit)) == 2)
- {
- p <- 1
- m <- dim(phiInit)[1]
- k <- dim(phiInit)[2]
- } else
- {
- p <- dim(phiInit)[1]
- m <- dim(phiInit)[2]
- k <- dim(phiInit)[3]
- }
- X <- matrix(nrow = n, ncol = p)
- X[1:n, 1:p] <- X2
+ 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 <- array(NA, dim = c(p, m, k))
- phi[1:p, , ] <- phiInit
+ 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))
- EPS <- 1e-15
-
+
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 (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])
+ for (s in 1:p)
+ Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
}
}
-
- ######### M step #
-
+
+ ## 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
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) < -sum(gam2 * log(pi2))/n +
- lambda * sum(pi2^gamma * b))
- {
+ 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])
+ 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]
+ 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#
-
+
+ ## E step
+
# Precompute det(rho[,,r]) for r in 1...k
- detRho <- sapply(1:k, function(r) det(rho[, , r]))
- gam1 <- matrix(0, nrow = n, ncol = k)
+ detRho <- sapply(1:k, function(r) gdet(rho[, , r]))
+ sumLogLLH <- 0
for (i in 1:n)
{
- # Update gam[,]
- for (r in 1:k) gam1[i, r] <- pi[r] * exp(-0.5 * sum((Y[i, ] %*% rho[,
- , r] - X[i, ] %*% phi[, , r])^2)) * detRho[r]
+ # 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))
}
- gam <- gam1/rowSums(gam1)
- sumLogLLH <- sum(log(rowSums(gam)) - log((2 * base::pi)^(m/2)))
+
sumPen <- sum(pi^gamma * b)
last_llh <- llh
- llh <- -sumLogLLH/n + lambda * sumPen
+ 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)))
+
+ if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
break
}
-
- list(phi = phi, rho = rho, pi = pi, llh = llh, S = S)
+
+ affec = apply(gam, 1, which.max)
+ list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
}
projects / valse.git / blobdiff