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   X2, Y, eps)
  49   {
  50   # Matrix dimensions
  51   n <- dim(Y)[1]
  52   if (length(dim(phiInit)) == 2)
  53   {
  54     p <- 1
  55     m <- dim(phiInit)[1]
  56     k <- dim(phiInit)[2]
  57   } else
  58   {
  59     p <- dim(phiInit)[1]
  60     m <- dim(phiInit)[2]
  61     k <- dim(phiInit)[3]
  62   }
  63   X <- matrix(nrow = n, ncol = p)
  64   X[1:n, 1:p] <- X2
  65   # Outputs
  66   phi <- array(NA, dim = c(p, m, k))
  67   phi[1:p, , ] <- phiInit
  68   rho <- rhoInit
  69   pi <- piInit
  70   llh <- -Inf
  71   S <- array(0, dim = c(p, m, k))
  72
  73   # Algorithm variables
  74   gam <- gamInit
  75   Gram2 <- array(0, dim = c(p, p, k))
  76   ps2 <- array(0, dim = c(p, m, k))
  77   X2 <- array(0, dim = c(n, p, k))
  78   Y2 <- array(0, dim = c(n, m, k))
  79   EPS <- 1e-15
  80
  81   for (ite in 1:maxi)
  82   {
  83     # Remember last pi,rho,phi values for exit condition in the end of loop
  84     Phi <- phi
  85     Rho <- rho
  86     Pi <- pi
  87
  88     # Computations associated to X and Y
  89     for (r in 1:k)
  90     {
  91       for (mm in 1:m) Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
  92       for (i in 1:n) X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
  93       for (mm in 1:m) ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
  94       for (j in 1:p)
  95       {
  96         for (s in 1:p) Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
  97       }
  98     }
  99
 100     ######### M step #
 101
 102     # For pi
 103     b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
 104     gam2 <- colSums(gam)
 105     a <- sum(gam %*% log(pi))
 106
 107     # While the proportions are nonpositive
 108     kk <- 0
 109     pi2AllPositive <- FALSE
 110     while (!pi2AllPositive)
 111     {
 112       pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
 113       pi2AllPositive <- all(pi2 >= 0)
 114       kk <- kk + 1
 115     }
 116
 117     # t(m) is the largest value in the grid O.1^k such that it is nonincreasing
 118     while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) < -sum(gam2 * log(pi2))/n +
 119       lambda * sum(pi2^gamma * b))
 120       {
 121       pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
 122       kk <- kk + 1
 123     }
 124     t <- 0.1^kk
 125     pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
 126
 127     # For phi and rho
 128     for (r in 1:k)
 129     {
 130       for (mm in 1:m)
 131       {
 132         ps <- 0
 133         for (i in 1:n) ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm,
 134           r])
 135         nY2 <- sum(Y2[, mm, r]^2)
 136         rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
 137       }
 138     }
 139
 140     for (r in 1:k)
 141     {
 142       for (j in 1:p)
 143       {
 144         for (mm in 1:m)
 145         {
 146           S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r] + sum(phi[-j, mm,
 147           r] * Gram2[j, -j, r])
 148           if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma))
 149           {
 150           phi[j, mm, r] <- 0
 151           } else if (S[j, mm, r] > n * lambda * (pi[r]^gamma))
 152           {
 153           phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j,
 154             j, r]
 155           } else
 156           {
 157           phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j,
 158             j, r]
 159           }
 160         }
 161       }
 162     }
 163
 164     ######## E step#
 165
 166     # Precompute det(rho[,,r]) for r in 1...k
 167     detRho <- sapply(1:k, function(r) det(rho[, , r]))
 168     gam1 <- matrix(0, nrow = n, ncol = k)
 169     for (i in 1:n)
 170     {
 171       # Update gam[,]
 172       for (r in 1:k) gam1[i, r] <- pi[r] * exp(-0.5 * sum((Y[i, ] %*% rho[,
 173         , r] - X[i, ] %*% phi[, , r])^2)) * detRho[r]
 174     }
 175     gam <- gam1/rowSums(gam1)
 176     sumLogLLH <- sum(log(rowSums(gam)) - log((2 * base::pi)^(m/2)))
 177     sumPen <- sum(pi^gamma * b)
 178     last_llh <- llh
 179     llh <- -sumLogLLH/n + lambda * sumPen
 180     dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
 181     Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
 182     Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
 183     Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
 184     dist2 <- max(Dist1, Dist2, Dist3)
 185
 186     if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
 187       break
 188   }
 189
 190   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S)
 191 }