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 #' @param fast boolean to enable or not the C function call
  17 #'
  18 #' @return A list (corresponding to the model collection) defined by (phi,LLF):
  19 #'   phi : regression mean for each cluster, an array of size p*m*k
  20 #'   LLF : log likelihood with respect to the training set
  21 #'
  22 #' @export
  23 EMGrank <- function(Pi, Rho, mini, maxi, X, Y, eps, rank, fast)
  24 {
  25   if (!fast)
  26   {
  27     # Function in R
  28     return(.EMGrank_R(Pi, Rho, mini, maxi, X, Y, eps, rank))
  29   }
  30
  31   # Function in C
  32   .Call("EMGrank", Pi, Rho, mini, maxi, X, Y, eps, as.integer(rank), PACKAGE = "valse")
  33 }
  34
  35 # helper to always have matrices as arg (TODO: put this elsewhere? improve?)  -->
  36 # Yes, we should use by-columns storage everywhere... [later!]
  37 matricize <- function(X)
  38 {
  39   if (!is.matrix(X))
  40     return(t(as.matrix(X)))
  41   X
  42 }
  43
  44 # R version - slow but easy to read
  45 .EMGrank_R <- function(Pi, Rho, mini, maxi, X, Y, eps, rank)
  46 {
  47   # matrix dimensions
  48   n <- nrow(X)
  49   p <- ncol(X)
  50   m <- ncol(Y)
  51   k <- length(Pi)
  52
  53   # init outputs
  54   phi <- array(0, dim = c(p, m, k))
  55   Z <- rep(1, n)
  56   LLF <- 0
  57
  58   # local variables
  59   Phi <- array(0, dim = c(p, m, k))
  60   deltaPhi <- c()
  61   sumDeltaPhi <- 0
  62   deltaPhiBufferSize <- 20
  63
  64   # main loop
  65   ite <- 1
  66   while (ite <= mini || (ite <= maxi && sumDeltaPhi > eps))
  67   {
  68     # M step: update for Beta ( and then phi)
  69     for (r in 1:k)
  70     {
  71       Z_indice <- seq_len(n)[Z == r] #indices where Z == r
  72       if (length(Z_indice) == 0)
  73         next
  74       # U,S,V = SVD of (t(Xr)Xr)^{-1} * t(Xr) * Yr
  75       s <- svd(MASS::ginv(crossprod(matricize(X[Z_indice, ]))) %*%
  76                crossprod(matricize(X[Z_indice, ]), matricize(Y[Z_indice, ])))
  77       S <- s$d
  78       # Set m-rank(r) singular values to zero, and recompose best rank(r) approximation
  79       # of the initial product
  80       if (rank[r] < length(S))
  81         S[(rank[r] + 1):length(S)] <- 0
  82       phi[, , r] <- s$u %*% diag(S) %*% t(s$v) %*% Rho[, , r]
  83     }
  84
  85     # Step E and computation of the loglikelihood
  86     sumLogLLF2 <- 0
  87     for (i in seq_len(n))
  88     {
  89       sumLLF1 <- 0
  90       maxLogGamIR <- -Inf
  91       for (r in seq_len(k))
  92       {
  93         dotProduct <- tcrossprod(Y[i, ] %*% Rho[, , r] - X[i, ] %*% phi[, , r])
  94         logGamIR <- log(Pi[r]) + log(gdet(Rho[, , r])) - 0.5 * dotProduct
  95         # Z[i] = index of max (gam[i,])
  96         if (logGamIR > maxLogGamIR)
  97         {
  98           Z[i] <- r
  99           maxLogGamIR <- logGamIR
 100         }
 101         sumLLF1 <- sumLLF1 + exp(logGamIR)/(2 * pi)^(m/2)
 102       }
 103       sumLogLLF2 <- sumLogLLF2 + log(sumLLF1)
 104     }
 105
 106     LLF <- -1/n * sumLogLLF2
 107
 108     # update distance parameter to check algorithm convergence (delta(phi, Phi))
 109     deltaPhi <- c(deltaPhi, max((abs(phi - Phi))/(1 + abs(phi)))) #TODO: explain?
 110     if (length(deltaPhi) > deltaPhiBufferSize)
 111       deltaPhi <- deltaPhi[2:length(deltaPhi)]
 112     sumDeltaPhi <- sum(abs(deltaPhi))
 113
 114     # update other local variables
 115     Phi <- phi
 116     ite <- ite + 1
 117   }
 118   list(phi = phi, LLF = LLF)
 119 }