pkg/R/EMGLLF.R

   1 #' EMGLLF
   2 #'
   3 #' Run a generalized EM algorithm developped for mixture of Gaussian regression
   4 #' models with variable selection by an extension of the Lasso estimator (regularization parameter lambda).
   5 #' Reparametrization is done to ensure invariance by homothetic transformation.
   6 #' It returns a collection of models, varying the number of clusters and the sparsity in the regression mean.
   7 #'
   8 #' @param phiInit an initialization for phi
   9 #' @param rhoInit an initialization for rho
  10 #' @param piInit an initialization for pi
  11 #' @param gamInit initialization for the a posteriori probabilities
  12 #' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10
  13 #' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100
  14 #' @param gamma integer for the power in the penaly, by default = 1
  15 #' @param lambda regularization parameter in the Lasso estimation
  16 #' @param X matrix of covariates (of size n*p)
  17 #' @param Y matrix of responses (of size n*m)
  18 #' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4
  19 #'
  20 #' @return A list (corresponding to the model collection) defined by (phi,rho,pi,LLF,S,affec):
  21 #'   phi : regression mean for each cluster
  22 #'   rho : variance (homothetic) for each cluster
  23 #'   pi : proportion for each cluster
  24 #'   LLF : log likelihood with respect to the training set
  25 #'   S : selected variables indexes
  26 #'   affec : cluster affectation for each observation (of the training set)
  27 #'
  28 #' @export
  29 EMGLLF <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  30   X, Y, eps, fast)
  31 {
  32   if (!fast)
  33   {
  34     # Function in R
  35     return(.EMGLLF_R(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  36       X, Y, eps))
  37   }
  38
  39   # Function in C
  40   n <- nrow(X)  #nombre d'echantillons
  41   p <- ncol(X)  #nombre de covariables
  42   m <- ncol(Y)  #taille de Y (multivarie)
  43   k <- length(piInit)  #nombre de composantes dans le melange
  44   .Call("EMGLLF", phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  45     X, Y, eps, phi = double(p * m * k), rho = double(m * m * k), pi = double(k),
  46     llh = double(1), S = double(p * m * k), affec = integer(n), n, p, m, k,
  47     PACKAGE = "valse")
  48 }
  49
  50 # R version - slow but easy to read
  51 .EMGLLF_R <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  52   X, Y, eps)
  53 {
  54   # Matrix dimensions
  55   n <- nrow(X)
  56   p <- ncol(X)
  57   m <- ncol(Y)
  58   k <- length(piInit)
  59
  60   # Adjustments required when p==1 or m==1 (var.sel. or output dim 1)
  61   if (p==1 || m==1)
  62     phiInit <- array(phiInit, dim=c(p,m,k))
  63   if (m==1)
  64     rhoInit <- array(rhoInit, dim=c(m,m,k))
  65
  66   # Outputs
  67   phi <- 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
  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       # na.rm=TRUE to handle 0*log(0)
 123       -sum(gam2 * log(pi2), na.rm=TRUE)/n + lambda * sum(pi2^gamma * b))
 124     {
 125       pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
 126       kk <- kk + 1
 127     }
 128     t <- 0.1^kk
 129     pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
 130
 131     # For phi and rho
 132     for (r in 1:k)
 133     {
 134       for (mm in 1:m)
 135       {
 136         ps <- 0
 137         for (i in 1:n)
 138           ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm, r])
 139         nY2 <- sum(Y2[, mm, r]^2)
 140         rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
 141       }
 142     }
 143
 144     for (r in 1:k)
 145     {
 146       for (j in 1:p)
 147       {
 148         for (mm in 1:m)
 149         {
 150           S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r] +
 151             sum(phi[-j, mm, r] * Gram2[j, -j, r])
 152           if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma)) {
 153             phi[j, mm, r] <- 0
 154           } else if (S[j, mm, r] > n * lambda * (pi[r]^gamma)) {
 155             phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j, j, r]
 156           } else {
 157             phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j, j, r]
 158           }
 159         }
 160       }
 161     }
 162
 163     ## E step
 164
 165     # Precompute det(rho[,,r]) for r in 1...k
 166     detRho <- sapply(1:k, function(r) gdet(rho[, , r]))
 167     sumLogLLH <- 0
 168     for (i in 1:n)
 169     {
 170       # Update gam[,]; use log to avoid numerical problems
 171       logGam <- sapply(1:k, function(r) {
 172         log(pi[r]) + log(detRho[r]) - 0.5 *
 173           sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)
 174       })
 175
 176       logGam <- logGam - max(logGam) #adjust without changing proportions
 177       gam[i, ] <- exp(logGam)
 178       norm_fact <- sum(gam[i, ])
 179       gam[i, ] <- gam[i, ] / norm_fact
 180       sumLogLLH <- sumLogLLH + log(norm_fact) - log((2 * base::pi)^(m/2))
 181     }
 182
 183     sumPen <- sum(pi^gamma * b)
 184     last_llh <- llh
 185     llh <- -sumLogLLH/n #+ lambda * sumPen
 186     dist <- ifelse(ite == 1, llh, (llh - last_llh)/(1 + abs(llh)))
 187     Dist1 <- max((abs(phi - Phi))/(1 + abs(phi)))
 188     Dist2 <- max((abs(rho - Rho))/(1 + abs(rho)))
 189     Dist3 <- max((abs(pi - Pi))/(1 + abs(Pi)))
 190     dist2 <- max(Dist1, Dist2, Dist3)
 191
 192     if (ite >= mini && (dist >= eps || dist2 >= sqrt(eps)))
 193       break
 194   }
 195
 196     affec = apply(gam, 1, which.max)
 197   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
 198 }