#' EMGrank #' #' Run an generalized EM algorithm developped for mixture of Gaussian regression #' models with variable selection by an extension of the low rank estimator. #' Reparametrization is done to ensure invariance by homothetic transformation. #' It returns a collection of models, varying the number of clusters and the rank of the regression mean. #' #' @param Pi An initialization for pi #' @param Rho An initialization for rho, the variance parameter #' @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 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 rank vector of possible ranks #' #' @return A list (corresponding to the model collection) defined by (phi,LLF): #' phi : regression mean for each cluster #' LLF : log likelihood with respect to the training set #' #' @export EMGrank <- function(Pi, Rho, mini, maxi, X, Y, eps, rank, fast = TRUE) { if (!fast) { # Function in R return(.EMGrank_R(Pi, Rho, mini, maxi, X, Y, eps, rank)) } # Function in C n <- nrow(X) #nombre d'echantillons p <- ncol(X) #nombre de covariables m <- ncol(Y) #taille de Y (multivarie) k <- length(Pi) #nombre de composantes dans le melange .Call("EMGrank", Pi, Rho, mini, maxi, X, Y, eps, as.integer(rank), phi = double(p * m * k), LLF = double(1), n, p, m, k, PACKAGE = "valse") } # helper to always have matrices as arg (TODO: put this elsewhere? improve?) --> # Yes, we should use by-columns storage everywhere... [later!] matricize <- function(X) { if (!is.matrix(X)) return(t(as.matrix(X))) return(X) } # R version - slow but easy to read .EMGrank_R <- function(Pi, Rho, mini, maxi, X, Y, eps, rank) { # matrix dimensions n <- nrow(X) p <- ncol(X) m <- ncol(Y) k <- length(Pi) # init outputs phi <- array(0, dim = c(p, m, k)) Z <- rep(1, n) LLF <- 0 # local variables Phi <- array(0, dim = c(p, m, k)) deltaPhi <- c() sumDeltaPhi <- 0 deltaPhiBufferSize <- 20 # main loop ite <- 1 while (ite <= mini || (ite <= maxi && sumDeltaPhi > eps)) { # M step: update for Beta ( and then phi) for (r in 1:k) { Z_indice <- seq_len(n)[Z == r] #indices where Z == r if (length(Z_indice) == 0) next # U,S,V = SVD of (t(Xr)Xr)^{-1} * t(Xr) * Yr s <- svd(MASS::ginv(crossprod(matricize(X[Z_indice, ]))) %*% crossprod(matricize(X[Z_indice, ]), matricize(Y[Z_indice, ]))) S <- s$d # Set m-rank(r) singular values to zero, and recompose best rank(r) approximation # of the initial product if (rank[r] < length(S)) S[(rank[r] + 1):length(S)] <- 0 phi[, , r] <- s$u %*% diag(S) %*% t(s$v) %*% Rho[, , r] } # Step E and computation of the loglikelihood sumLogLLF2 <- 0 for (i in seq_len(n)) { sumLLF1 <- 0 maxLogGamIR <- -Inf for (r in seq_len(k)) { dotProduct <- tcrossprod(Y[i, ] %*% Rho[, , r] - X[i, ] %*% phi[, , r]) logGamIR <- log(Pi[r]) + log(gdet(Rho[, , r])) - 0.5 * dotProduct # Z[i] = index of max (gam[i,]) if (logGamIR > maxLogGamIR) { Z[i] <- r maxLogGamIR <- logGamIR } sumLLF1 <- sumLLF1 + exp(logGamIR)/(2 * pi)^(m/2) } sumLogLLF2 <- sumLogLLF2 + log(sumLLF1) } LLF <- -1/n * sumLogLLF2 # update distance parameter to check algorithm convergence (delta(phi, Phi)) deltaPhi <- c(deltaPhi, max((abs(phi - Phi))/(1 + abs(phi)))) #TODO: explain? if (length(deltaPhi) > deltaPhiBufferSize) deltaPhi <- deltaPhi[2:length(deltaPhi)] sumDeltaPhi <- sum(abs(deltaPhi)) # update other local variables Phi <- phi ite <- ite + 1 } return(list(phi = phi, LLF = LLF)) }