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 #' @param fast boolean to enable or not the C function call
  20 #'
  21 #' @return A list (corresponding to the model collection) defined by (phi,rho,pi,llh,S,affec):
  22 #'   phi : regression mean for each cluster, an array of size p*m*k
  23 #'   rho : variance (homothetic) for each cluster, an array of size m*m*k
  24 #'   pi : proportion for each cluster, a vector of size k
  25 #'   llh : log likelihood with respect to the training set
  26 #'   S : selected variables indexes, an array of size p*m*k
  27 #'   affec : cluster affectation for each observation (of the training set)
  28 #'
  29 #' @export
  30 EMGLLF <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  31   X, Y, eps, fast)
  32 {
  33   if (!fast)
  34   {
  35     # Function in R
  36     return(.EMGLLF_R(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  37       X, Y, eps))
  38   }
  39
  40   # Function in C
  41   .Call("EMGLLF", phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  42     X, Y, eps, PACKAGE = "valse")
  43 }
  44
  45 # R version - slow but easy to read
  46 .EMGLLF_R <- function(phiInit, rhoInit, piInit, gamInit, mini, maxi, gamma, lambda,
  47   X, Y, eps)
  48 {
  49   # Matrix dimensions
  50   n <- nrow(X)
  51   p <- ncol(X)
  52   m <- ncol(Y)
  53   k <- length(piInit)
  54
  55   # Adjustments required when p==1 or m==1 (var.sel. or output dim 1)
  56   if (p==1 || m==1)
  57     phiInit <- array(phiInit, dim=c(p,m,k))
  58   if (m==1)
  59     rhoInit <- array(rhoInit, dim=c(m,m,k))
  60
  61   # Outputs
  62   phi <- phiInit
  63   rho <- rhoInit
  64   pi <- piInit
  65   llh <- -Inf
  66   S <- array(0, dim = c(p, m, k))
  67
  68   # Algorithm variables
  69   gam <- gamInit
  70   Gram2 <- array(0, dim = c(p, p, k))
  71   ps2 <- array(0, dim = c(p, m, k))
  72   X2 <- array(0, dim = c(n, p, k))
  73   Y2 <- array(0, dim = c(n, m, k))
  74
  75   for (ite in 1:maxi)
  76   {
  77     # Remember last pi,rho,phi values for exit condition in the end of loop
  78     Phi <- phi
  79     Rho <- rho
  80     Pi <- pi
  81
  82     # Computations associated to X and Y
  83     for (r in 1:k)
  84     {
  85       for (mm in 1:m)
  86         Y2[, mm, r] <- sqrt(gam[, r]) * Y[, mm]
  87       for (i in 1:n)
  88         X2[i, , r] <- sqrt(gam[i, r]) * X[i, ]
  89       for (mm in 1:m)
  90         ps2[, mm, r] <- crossprod(X2[, , r], Y2[, mm, r])
  91       for (j in 1:p)
  92       {
  93         for (s in 1:p)
  94           Gram2[j, s, r] <- crossprod(X2[, j, r], X2[, s, r])
  95       }
  96     }
  97
  98     ## M step
  99
 100     # For pi
 101     b <- sapply(1:k, function(r) sum(abs(phi[, , r])))
 102     gam2 <- colSums(gam)
 103     a <- sum(gam %*% log(pi))
 104
 105     # While the proportions are nonpositive
 106     kk <- 0
 107     pi2AllPositive <- FALSE
 108     while (!pi2AllPositive)
 109     {
 110       pi2 <- pi + 0.1^kk * ((1/n) * gam2 - pi)
 111       pi2AllPositive <- all(pi2 >= 0)
 112       kk <- kk + 1
 113     }
 114
 115     # t(m) is the largest value in the grid O.1^k such that it is nonincreasing
 116     while (kk < 1000 && -a/n + lambda * sum(pi^gamma * b) <
 117       # na.rm=TRUE to handle 0*log(0)
 118       -sum(gam2 * log(pi2), na.rm=TRUE)/n + lambda * sum(pi2^gamma * b))
 119     {
 120       pi2 <- pi + 0.1^kk * (1/n * gam2 - pi)
 121       kk <- kk + 1
 122     }
 123     t <- 0.1^kk
 124     pi <- (pi + t * (pi2 - pi))/sum(pi + t * (pi2 - pi))
 125
 126     # For phi and rho
 127     for (r in 1:k)
 128     {
 129       for (mm in 1:m)
 130       {
 131         ps <- 0
 132         for (i in 1:n)
 133           ps <- ps + Y2[i, mm, r] * sum(X2[i, , r] * phi[, mm, r])
 134         nY2 <- sum(Y2[, mm, r]^2)
 135         rho[mm, mm, r] <- (ps + sqrt(ps^2 + 4 * nY2 * gam2[r]))/(2 * nY2)
 136       }
 137     }
 138
 139     for (r in 1:k)
 140     {
 141       for (j in 1:p)
 142       {
 143         for (mm in 1:m)
 144         {
 145           S[j, mm, r] <- -rho[mm, mm, r] * ps2[j, mm, r] +
 146             sum(phi[-j, mm, r] * Gram2[j, -j, r])
 147           if (abs(S[j, mm, r]) <= n * lambda * (pi[r]^gamma)) {
 148             phi[j, mm, r] <- 0
 149           } else if (S[j, mm, r] > n * lambda * (pi[r]^gamma)) {
 150             phi[j, mm, r] <- (n * lambda * (pi[r]^gamma) - S[j, mm, r])/Gram2[j, j, r]
 151           } else {
 152             phi[j, mm, r] <- -(n * lambda * (pi[r]^gamma) + S[j, mm, r])/Gram2[j, j, r]
 153           }
 154         }
 155       }
 156     }
 157
 158     ## E step
 159
 160     # Precompute det(rho[,,r]) for r in 1...k
 161     detRho <- sapply(1:k, function(r) gdet(rho[, , r]))
 162     sumLogLLH <- 0
 163     for (i in 1:n)
 164     {
 165       # Update gam[,]; use log to avoid numerical problems
 166       logGam <- sapply(1:k, function(r) {
 167         log(pi[r]) + log(detRho[r]) - 0.5 *
 168           sum((Y[i, ] %*% rho[, , r] - X[i, ] %*% phi[, , r])^2)
 169       })
 170
 171       logGam <- logGam - max(logGam) #adjust without changing proportions
 172       gam[i, ] <- exp(logGam)
 173       norm_fact <- sum(gam[i, ])
 174       gam[i, ] <- gam[i, ] / norm_fact
 175       sumLogLLH <- sumLogLLH + log(norm_fact) - log((2 * base::pi)^(m/2))
 176     }
 177
 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     affec = apply(gam, 1, which.max)
 192   list(phi = phi, rho = rho, pi = pi, llh = llh, S = S, affec=affec)
 193 }