pkg/R/EMGLLF.R

   1 #' EMGLLF
   2 #'
   3 #' Description de EMGLLF
   4 #'
   5 #' @param phiInit an initialization for phi
   6 #' @param rhoInit an initialization for rho
   7 #' @param piInit an initialization for pi
   8 #' @param gamInit initialization for the a posteriori probabilities
   9 #' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10
  10 #' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100
  11 #' @param gamma integer for the power in the penaly, by default = 1
  12 #' @param lambda regularization parameter in the Lasso estimation
  13 #' @param X matrix of covariates (of size n*p)
  14 #' @param Y matrix of responses (of size n*m)
  15 #' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4
  16 #'
  17 #' @return A list ... phi,rho,pi,LLF,S,affec:
  18 #'   phi : parametre de moyenne renormalisé, calculé par l'EM
  19 #'   rho : parametre de variance renormalisé, calculé par l'EM
  20 #'   pi : parametre des proportions renormalisé, calculé par l'EM
  21 #'   LLF : log vraisemblance associée à cet échantillon, pour les valeurs estimées des paramètres
  22 #'   S : ... affec : ...
  23 #'
  24 #' @export
  25 EMGLLF <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  26   X, Y, eps, fast = TRUE)
  27 {
  28   if (!fast)
  29   {
  30     # Function in R
  31     return(.EMGLLF_R(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  32       X, Y, eps))
  33   }
  34
  35   # Function in C
  36   n <- nrow(X)  #nombre d'echantillons
  37   p <- ncol(X)  #nombre de covariables
  38   m <- ncol(Y)  #taille de Y (multivarié)
  39   k <- length(piInit)  #nombre de composantes dans le mélange
  40   .Call("EMGLLF", phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  41     X, Y, eps, phi = double(p * m * k), rho = double(m * m * k), pi = double(k),
  42     LLF = double(maxi), S = double(p * m * k), affec = integer(n), n, p, m, k,
  43     PACKAGE = "valse")
  44 }
  45
  46 # R version - slow but easy to read
  47 .EMGLLF_R <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  48   X, Y, eps)
  49 {
  50   # Matrix dimensions: NOTE: phiInit *must* be an array (even if p==1)
  51   n <- dim(Y)[1]
  52   p <- dim(phiInit)[1]
  53   m <- dim(phiInit)[2]
  54   k <- dim(phiInit)[3]
  55
  56   # Outputs
  57   phi <- array(NA, dim = c(p, m, k))
  58   phi[1:p, , ] <- phiInit
  59   rho <- rhoInit
  60   pi <- piInit
  61   llh <- -Inf
  62   S <- array(0, dim = c(p, m, k))
  63
  64   # Algorithm variables
  65   gam <- gamInit
  66   Gram2 <- array(0, dim = c(p, p, k))
  67   ps2 <- array(0, dim = c(p, m, k))
  68   X2 <- array(0, dim = c(n, p, k))
  69   Y2 <- array(0, dim = c(n, m, k))
  70   EPS <- 1e-15
  71
  72   for (ite in 1:maxi)
  73   {
  74     # Remember last pi,rho,phi values for exit condition in the end of loop
  75     Phi <- phi
  76     Rho <- rho
  77     Pi <- pi
  78
  79     # Computations associated to X and Y
  80     for (r in 1:k)
  81     {
  82       for (mm in 1:m)
  83         Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
  84       for (i in 1:n)
  85         X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
  86       for (mm in 1:m)
  87         ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
  88       for (j in 1:p)
  89       {
  90         for (s in 1:p)
  91           Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
  92       }
  93     }
  94
  95     ## M step
  96
  97     # For pi
  98     b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
  99     gam2 <- colSums(gam)
 100     a <- sum(gam %*% log(pi))
 101
 102     # While the proportions are nonpositive
 103     kk <- 0
 104     pi2AllPositive <- FALSE
 105     while (!pi2AllPositive)
 106     {
 107       pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
 108       pi2AllPositive <- all(pi2 >= 0)
 109       kk <- kk + 1
 110     }
 111
 112     # t(m) is the largest value in the grid O.1^k such that it is nonincreasing
 113     while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
 114       -sum(gam2 * log(pi2))/n + lambda * sum(pi2^gamma * b))
 115     {
 116       pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
 117       kk <- kk + 1
 118     }
 119     t <- 0.1^kk
 120     pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
 121
 122     # For phi and rho
 123     for (r in 1:k)
 124     {
 125       for (mm in 1:m)
 126       {
 127         ps <- 0
 128         for (i in 1:n)
 129           ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm, r])
 130         nY2 <- sum(Y2[, mm, r]^2)
 131         rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
 132       }
 133     }
 134
 135     for (r in 1:k)
 136     {
 137       for (j in 1:p)
 138       {
 139         for (mm in 1:m)
 140         {
 141           S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r]
 142             + sum(phi[-j, mm, r] * Gram2[j, -j, r])
 143           if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma)) {
 144             phi[j, mm, r] <- 0
 145           } else if (S[j, mm, r] > n * lambda * (pi[r]^gamma)) {
 146             phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j, j, r]
 147           } else {
 148             phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j, j, r]
 149           }
 150         }
 151       }
 152     }
 153
 154     ## E step
 155
 156     # Precompute det(rho[,,r]) for r in 1...k
 157     detRho <- sapply(1:k, function(r) det(rho[, , r]))
 158     for (i in 1:n)
 159     {
 160       # Update gam[,]
 161       for (r in 1:k)
 162       {
 163         gam[i, r] <- pi[r] * exp(-0.5
 164           * sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)) * detRho[r]
 165       }
 166     }
 167     norm_fact <- rowSums(gam)
 168     gam <- gam / norm_fact
 169     sumLogLLH <- sum(log(norm_fact) - log((2 * base::pi)^(m/2)))
 170     sumPen <- sum(pi^gamma * b)
 171     last_llh <- llh
 172     llh <- -sumLogLLH/n + lambda * sumPen
 173     dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
 174     Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
 175     Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
 176     Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
 177     dist2 <- max(Dist1, Dist2, Dist3)
 178
 179     if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
 180       break
 181   }
 182
 183   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S)
 184 }