pkg/R/EMGrank.R

   1 #' EMGrank
   2 #'
   3 #' Run an generalized EM algorithm developped for mixture of Gaussian regression
   4 #' models with variable selection by an extension of the low rank estimator.
   5 #' Reparametrization is done to ensure invariance by homothetic transformation.
   6 #' It returns a collection of models, varying the number of clusters and the rank of the regression mean.
   7 #'
   8 #' @param Pi An initialization for pi
   9 #' @param Rho An initialization for rho, the variance parameter
  10 #' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10
  11 #' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100
  12 #' @param X matrix of covariates (of size n*p)
  13 #' @param Y matrix of responses (of size n*m)
  14 #' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4
  15 #' @param rank vector of possible ranks
  16 #'
  17 #' @return A list (corresponding to the model collection) defined by (phi,LLF):
  18 #'   phi : regression mean for each cluster
  19 #'   LLF : log likelihood with respect to the training set
  20 #'
  21 #' @export
  22 EMGrank <- function(Pi, Rho, mini, maxi, X, Y, eps, rank, fast = TRUE)
  23 {
  24   if (!fast)
  25   {
  26     # Function in R
  27     return(.EMGrank_R(Pi, Rho, mini, maxi, X, Y, eps, rank))
  28   }
  29
  30   # Function in C
  31   n <- nrow(X)  #nombre d'echantillons
  32   p <- ncol(X)  #nombre de covariables
  33   m <- ncol(Y)  #taille de Y (multivarié)
  34   k <- length(Pi)  #nombre de composantes dans le mélange
  35   .Call("EMGrank", Pi, Rho, mini, maxi, X, Y, eps, as.integer(rank), phi = double(p * m * k),
  36     LLF = double(1), n, p, m, k, PACKAGE = "valse")
  37 }
  38
  39 # helper to always have matrices as arg (TODO: put this elsewhere? improve?)  -->
  40 # Yes, we should use by-columns storage everywhere... [later!]
  41 matricize <- function(X)
  42 {
  43   if (!is.matrix(X))
  44     return(t(as.matrix(X)))
  45   return(X)
  46 }
  47
  48 # R version - slow but easy to read
  49 .EMGrank_R <- function(Pi, Rho, mini, maxi, X, Y, eps, rank)
  50 {
  51   # matrix dimensions
  52   n <- nrow(X)
  53   p <- ncol(X)
  54   m <- ncol(Y)
  55   k <- length(Pi)
  56
  57   # init outputs
  58   phi <- array(0, dim = c(p, m, k))
  59   Z <- rep(1, n)
  60   LLF <- 0
  61
  62   # local variables
  63   Phi <- array(0, dim = c(p, m, k))
  64   deltaPhi <- c()
  65   sumDeltaPhi <- 0
  66   deltaPhiBufferSize <- 20
  67
  68   # main loop
  69   ite <- 1
  70   while (ite <= mini || (ite <= maxi && sumDeltaPhi > eps))
  71   {
  72     # M step: update for Beta ( and then phi)
  73     for (r in 1:k)
  74     {
  75       Z_indice <- seq_len(n)[Z == r] #indices where Z == r
  76       if (length(Z_indice) == 0)
  77         next
  78       # U,S,V = SVD of (t(Xr)Xr)^{-1} * t(Xr) * Yr
  79       s <- svd(MASS::ginv(crossprod(matricize(X[Z_indice, ]))) %*%
  80                  crossprod(matricize(X[Z_indice, ]), matricize(Y[Z_indice, ])))
  81       S <- s$d
  82       # Set m-rank(r) singular values to zero, and recompose best rank(r) approximation
  83       # of the initial product
  84       if (rank[r] < length(S))
  85         S[(rank[r] + 1):length(S)] <- 0
  86       phi[, , r] <- s$u %*% diag(S) %*% t(s$v) %*% Rho[, , r]
  87     }
  88
  89     # Step E and computation of the loglikelihood
  90     sumLogLLF2 <- 0
  91     for (i in seq_len(n))
  92     {
  93       sumLLF1 <- 0
  94       maxLogGamIR <- -Inf
  95       for (r in seq_len(k))
  96       {
  97         dotProduct <- tcrossprod(Y[i, ] %*% Rho[, , r] - X[i, ] %*% phi[, , r])
  98         logGamIR <- log(Pi[r]) + log(gdet(Rho[, , r])) - 0.5 * dotProduct
  99         # Z[i] = index of max (gam[i,])
 100         if (logGamIR > maxLogGamIR)
 101         {
 102           Z[i] <- r
 103           maxLogGamIR <- logGamIR
 104         }
 105         sumLLF1 <- sumLLF1 + exp(logGamIR)/(2 * pi)^(m/2)
 106       }
 107       sumLogLLF2 <- sumLogLLF2 + log(sumLLF1)
 108     }
 109
 110     LLF <- -1/n * sumLogLLF2
 111
 112     # update distance parameter to check algorithm convergence (delta(phi, Phi))
 113     deltaPhi <- c(deltaPhi, max((abs(phi - Phi))/(1 + abs(phi)))) #TODO: explain?
 114     if (length(deltaPhi) > deltaPhiBufferSize)
 115       deltaPhi <- deltaPhi[2:length(deltaPhi)]
 116     sumDeltaPhi <- sum(abs(deltaPhi))
 117
 118     # update other local variables
 119     Phi <- phi
 120     ite <- ite + 1
 121   }
 122   return(list(phi = phi, LLF = LLF))
 123 }