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     p <- 1
  54     m <- dim(phiInit)[1]
  55     k <- dim(phiInit)[2]
  56   } else {
  57     p <- dim(phiInit)[1]
  58     m <- dim(phiInit)[2]
  59     k <- dim(phiInit)[3]
  60   }
  61   X <- matrix(nrow = n, ncol = p)
  62   X[1:n, 1:p] <- X2
  63
  64   # Outputs
  65   phi <- array(NA, dim = c(p, m, k))
  66   phi[1:p, , ] <- phiInit
  67   rho <- rhoInit
  68   pi <- piInit
  69   llh <- -Inf
  70   S <- array(0, dim = c(p, m, k))
  71
  72   # Algorithm variables
  73   gam <- gamInit
  74   Gram2 <- array(0, dim = c(p, p, k))
  75   ps2 <- array(0, dim = c(p, m, k))
  76   X2 <- array(0, dim = c(n, p, k))
  77   Y2 <- array(0, dim = c(n, m, k))
  78   EPS <- 1e-15
  79
  80   for (ite in 1:maxi)
  81   {
  82     # Remember last pi,rho,phi values for exit condition in the end of loop
  83     Phi <- phi
  84     Rho <- rho
  85     Pi <- pi
  86
  87     # Computations associated to X and Y
  88     for (r in 1:k)
  89     {
  90       for (mm in 1:m)
  91         Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
  92       for (i in 1:n)
  93         X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
  94       for (mm in 1:m)
  95         ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
  96       for (j in 1:p)
  97       {
  98         for (s in 1:p)
  99           Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
 100       }
 101     }
 102
 103     ## M step
 104
 105     # For pi
 106     b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
 107     gam2 <- colSums(gam)
 108     a <- sum(gam %*% log(pi))
 109
 110     # While the proportions are nonpositive
 111     kk <- 0
 112     pi2AllPositive <- FALSE
 113     while (!pi2AllPositive)
 114     {
 115       pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
 116       pi2AllPositive <- all(pi2 >= 0)
 117       kk <- kk + 1
 118     }
 119
 120     # t(m) is the largest value in the grid O.1^k such that it is nonincreasing
 121     while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
 122       -sum(gam2 * log(pi2))/n + lambda * sum(pi2^gamma * b))
 123     {
 124       pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
 125       kk <- kk + 1
 126     }
 127     t <- 0.1^kk
 128     pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
 129
 130     # For phi and rho
 131     for (r in 1:k)
 132     {
 133       for (mm in 1:m)
 134       {
 135         ps <- 0
 136         for (i in 1:n)
 137           ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm, r])
 138         nY2 <- sum(Y2[, mm, r]^2)
 139         rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
 140       }
 141     }
 142
 143     for (r in 1:k)
 144     {
 145       for (j in 1:p)
 146       {
 147         for (mm in 1:m)
 148         {
 149           S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r]
 150             + sum(phi[-j, mm, r] * Gram2[j, -j, r])
 151           if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma)) {
 152             phi[j, mm, r] <- 0
 153           } else if (S[j, mm, r] > n * lambda * (pi[r]^gamma)) {
 154             phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j, j, r]
 155           } else {
 156             phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j, j, r]
 157           }
 158         }
 159       }
 160     }
 161
 162     ######## E step#
 163
 164     # Precompute det(rho[,,r]) for r in 1...k
 165     detRho <- sapply(1:k, function(r) det(rho[, , r]))
 166     gam1 <- matrix(0, nrow = n, ncol = k)
 167     for (i in 1:n)
 168     {
 169       # Update gam[,]
 170       for (r in 1:k)
 171       {
 172         gam1[i, r] <- pi[r] * exp(-0.5
 173           * sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)) * detRho[r]
 174       }
 175     }
 176     gam <- gam1 / rowSums(gam1)
 177     sumLogLLH <- sum(log(rowSums(gam)) - log((2 * base::pi)^(m/2)))
 178     sumPen <- sum(pi^gamma * b)
 179     last_llh <- llh
 180     llh <- -sumLogLLH/n + lambda * sumPen
 181     dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
 182     Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
 183     Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
 184     Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
 185     dist2 <- max(Dist1, Dist2, Dist3)
 186
 187     if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
 188       break
 189   }
 190
 191   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S)
 192 }