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   EPS <- 1e-15
  78
  79   for (ite in 1:maxi)
  80   {
  81     # Remember last pi,rho,phi values for exit condition in the end of loop
  82     Phi <- phi
  83     Rho <- rho
  84     Pi <- pi
  85
  86     # Computations associated to X and Y
  87     for (r in 1:k)
  88     {
  89       for (mm in 1:m)
  90         Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
  91       for (i in 1:n)
  92         X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
  93       for (mm in 1:m)
  94         ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
  95       for (j in 1:p)
  96       {
  97         for (s in 1:p)
  98           Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
  99       }
 100     }
 101
 102     ## M step
 103
 104     # For pi
 105     b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
 106     gam2 <- colSums(gam)
 107     a <- sum(gam %*% log(pi))
 108
 109     # While the proportions are nonpositive
 110     kk <- 0
 111     pi2AllPositive <- FALSE
 112     while (!pi2AllPositive)
 113     {
 114       pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
 115       pi2AllPositive <- all(pi2 >= 0)
 116       kk <- kk + 1
 117     }
 118
 119     # t(m) is the largest value in the grid O.1^k such that it is nonincreasing
 120     while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
 121       # na.rm=TRUE to handle 0*log(0)
 122       -sum(gam2 * log(pi2), na.rm=TRUE)/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) gdet(rho[, , r]))
 166     sumLogLLH <- 0
 167     for (i in 1:n)
 168     {
 169       # Update gam[,]; use log to avoid numerical problems
 170       logGam <- sapply(1:k, function(r) {
 171         log(pi[r]) + log(detRho[r]) - 0.5 *
 172           sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)
 173       })
 174
 175       logGam <- logGam - max(logGam) #adjust without changing proportions
 176       gam[i, ] <- exp(logGam)
 177       norm_fact <- sum(gam[i, ])
 178       gam[i, ] <- gam[i, ] / norm_fact
 179       sumLogLLH <- sumLogLLH + log(norm_fact) - log((2 * base::pi)^(m/2))
 180     }
 181
 182     sumPen <- sum(pi^gamma * b)
 183     last_llh <- llh
 184     llh <- -sumLogLLH/n #+ lambda * sumPen
 185     dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
 186     Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
 187     Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
 188     Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
 189     dist2 <- max(Dist1, Dist2, Dist3)
 190
 191     if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
 192       break
 193   }
 194
 195     affec = apply(gam, 1, which.max)
 196   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
 197 }