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 : ...
  23 #'   affec : ...
  24 #'
  25 #' @export
  26 EMGLLF <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  27   X, Y, eps, fast)
  28 {
  29   if (!fast)
  30   {
  31     # Function in R
  32     return(.EMGLLF_R(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  33       X, Y, eps))
  34   }
  35
  36   # Function in C
  37   n <- nrow(X)  #nombre d'echantillons
  38   p <- ncol(X)  #nombre de covariables
  39   m <- ncol(Y)  #taille de Y (multivarié)
  40   k <- length(piInit)  #nombre de composantes dans le mélange
  41   .Call("EMGLLF", phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  42     X, Y, eps, phi = double(p * m * k), rho = double(m * m * k), pi = double(k),
  43     LLF = double(maxi), S = double(p * m * k), affec = integer(n), n, p, m, k,
  44     PACKAGE = "valse")
  45     list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
  46 }
  47
  48 # R version - slow but easy to read
  49 .EMGLLF_R <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  50   X, Y, eps)
  51 {
  52   # Matrix dimensions
  53   n <- nrow(X)
  54   p <- ncol(X)
  55   m <- ncol(Y)
  56   k <- length(piInit)
  57
  58   # Adjustments required when p==1 or m==1 (var.sel. or output dim 1)
  59   if (p==1 || m==1)
  60     phiInit <- array(phiInit, dim=c(p,m,k))
  61   if (m==1)
  62     rhoInit <- array(rhoInit, dim=c(m,m,k))
  63
  64   # Outputs
  65   phi <- phiInit
  66   rho <- rhoInit
  67   pi <- piInit
  68   llh <- -Inf
  69   S <- array(0, dim = c(p, m, k))
  70
  71   # Algorithm variables
  72   gam <- gamInit
  73   Gram2 <- array(0, dim = c(p, p, k))
  74   ps2 <- array(0, dim = c(p, m, k))
  75   X2 <- array(0, dim = c(n, p, k))
  76   Y2 <- array(0, dim = c(n, m, k))
  77
  78   for (ite in 1:maxi)
  79   {
  80     # Remember last pi,rho,phi values for exit condition in the end of loop
  81     Phi <- phi
  82     Rho <- rho
  83     Pi <- pi
  84
  85     # Computations associated to X and Y
  86     for (r in 1:k)
  87     {
  88       for (mm in 1:m)
  89         Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
  90       for (i in 1:n)
  91         X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
  92       for (mm in 1:m)
  93         ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
  94       for (j in 1:p)
  95       {
  96         for (s in 1:p)
  97           Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
  98       }
  99     }
 100
 101     ## M step
 102
 103     # For pi
 104     b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
 105     gam2 <- colSums(gam)
 106     a <- sum(gam %*% log(pi))
 107
 108     # While the proportions are nonpositive
 109     kk <- 0
 110     pi2AllPositive <- FALSE
 111     while (!pi2AllPositive)
 112     {
 113       pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
 114       pi2AllPositive <- all(pi2 >= 0)
 115       kk <- kk + 1
 116     }
 117
 118     # t(m) is the largest value in the grid O.1^k such that it is nonincreasing
 119     while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
 120       # na.rm=TRUE to handle 0*log(0)
 121       -sum(gam2 * log(pi2), na.rm=TRUE)/n + lambda * sum(pi2^gamma * b))
 122     {
 123       pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
 124       kk <- kk + 1
 125     }
 126     t <- 0.1^kk
 127     pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
 128
 129     # For phi and rho
 130     for (r in 1:k)
 131     {
 132       for (mm in 1:m)
 133       {
 134         ps <- 0
 135         for (i in 1:n)
 136           ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm, r])
 137         nY2 <- sum(Y2[, mm, r]^2)
 138         rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
 139       }
 140     }
 141
 142     for (r in 1:k)
 143     {
 144       for (j in 1:p)
 145       {
 146         for (mm in 1:m)
 147         {
 148           S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r] +
 149             sum(phi[-j, mm, r] * Gram2[j, -j, r])
 150           if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma)) {
 151             phi[j, mm, r] <- 0
 152           } else if (S[j, mm, r] > n * lambda * (pi[r]^gamma)) {
 153             phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j, j, r]
 154           } else {
 155             phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j, j, r]
 156           }
 157         }
 158       }
 159     }
 160
 161     ## E step
 162
 163     # Precompute det(rho[,,r]) for r in 1...k
 164     detRho <- sapply(1:k, function(r) gdet(rho[, , r]))
 165     sumLogLLH <- 0
 166     for (i in 1:n)
 167     {
 168       # Update gam[,]; use log to avoid numerical problems
 169       logGam <- sapply(1:k, function(r) {
 170         log(pi[r]) + log(detRho[r]) - 0.5 *
 171           sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)
 172       })
 173
 174       logGam <- logGam - max(logGam) #adjust without changing proportions
 175       gam[i, ] <- exp(logGam)
 176       norm_fact <- sum(gam[i, ])
 177       gam[i, ] <- gam[i, ] / norm_fact
 178       sumLogLLH <- sumLogLLH + log(norm_fact) - log((2 * base::pi)^(m/2))
 179     }
 180
 181     sumPen <- sum(pi^gamma * b)
 182     last_llh <- llh
 183     llh <- -sumLogLLH/n #+ lambda * sumPen
 184     dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
 185     Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
 186     Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
 187     Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
 188     dist2 <- max(Dist1, Dist2, Dist3)
 189
 190     if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
 191       break
 192   }
 193
 194     affec = apply(gam, 1, which.max)
 195   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
 196 }