| 1 | #' EMGrank |
| 2 | #' |
| 3 | #' Run an generalized EM algorithm developped for mixture of Gaussian regression |
| 4 | #' models with variable selection by an extension of the low rank estimator. |
| 5 | #' Reparametrization is done to ensure invariance by homothetic transformation. |
| 6 | #' It returns a collection of models, varying the number of clusters and the rank of the regression mean. |
| 7 | #' |
| 8 | #' @param Pi An initialization for pi |
| 9 | #' @param Rho An initialization for rho, the variance parameter |
| 10 | #' @param mini integer, minimum number of iterations in the EM algorithm, by default = 10 |
| 11 | #' @param maxi integer, maximum number of iterations in the EM algorithm, by default = 100 |
| 12 | #' @param X matrix of covariates (of size n*p) |
| 13 | #' @param Y matrix of responses (of size n*m) |
| 14 | #' @param eps real, threshold to say the EM algorithm converges, by default = 1e-4 |
| 15 | #' @param rank vector of possible ranks |
| 16 | #' |
| 17 | #' @return A list (corresponding to the model collection) defined by (phi,LLF): |
| 18 | #' phi : regression mean for each cluster |
| 19 | #' LLF : log likelihood with respect to the training set |
| 20 | #' |
| 21 | #' @export |
| 22 | EMGrank <- function(Pi, Rho, mini, maxi, X, Y, eps, rank, fast = TRUE) |
| 23 | { |
| 24 | if (!fast) |
| 25 | { |
| 26 | # Function in R |
| 27 | return(.EMGrank_R(Pi, Rho, mini, maxi, X, Y, eps, rank)) |
| 28 | } |
| 29 | |
| 30 | # Function in C |
| 31 | n <- nrow(X) #nombre d'echantillons |
| 32 | p <- ncol(X) #nombre de covariables |
| 33 | m <- ncol(Y) #taille de Y (multivarie) |
| 34 | k <- length(Pi) #nombre de composantes dans le melange |
| 35 | .Call("EMGrank", Pi, Rho, mini, maxi, X, Y, eps, as.integer(rank), phi = double(p * m * k), |
| 36 | LLF = double(1), n, p, m, k, PACKAGE = "valse") |
| 37 | } |
| 38 | |
| 39 | # helper to always have matrices as arg (TODO: put this elsewhere? improve?) --> |
| 40 | # Yes, we should use by-columns storage everywhere... [later!] |
| 41 | matricize <- function(X) |
| 42 | { |
| 43 | if (!is.matrix(X)) |
| 44 | return(t(as.matrix(X))) |
| 45 | return(X) |
| 46 | } |
| 47 | |
| 48 | # R version - slow but easy to read |
| 49 | .EMGrank_R <- function(Pi, Rho, mini, maxi, X, Y, eps, rank) |
| 50 | { |
| 51 | # matrix dimensions |
| 52 | n <- nrow(X) |
| 53 | p <- ncol(X) |
| 54 | m <- ncol(Y) |
| 55 | k <- length(Pi) |
| 56 | |
| 57 | # init outputs |
| 58 | phi <- array(0, dim = c(p, m, k)) |
| 59 | Z <- rep(1, n) |
| 60 | LLF <- 0 |
| 61 | |
| 62 | # local variables |
| 63 | Phi <- array(0, dim = c(p, m, k)) |
| 64 | deltaPhi <- c() |
| 65 | sumDeltaPhi <- 0 |
| 66 | deltaPhiBufferSize <- 20 |
| 67 | |
| 68 | # main loop |
| 69 | ite <- 1 |
| 70 | while (ite <= mini || (ite <= maxi && sumDeltaPhi > eps)) |
| 71 | { |
| 72 | # M step: update for Beta ( and then phi) |
| 73 | for (r in 1:k) |
| 74 | { |
| 75 | Z_indice <- seq_len(n)[Z == r] #indices where Z == r |
| 76 | if (length(Z_indice) == 0) |
| 77 | next |
| 78 | # U,S,V = SVD of (t(Xr)Xr)^{-1} * t(Xr) * Yr |
| 79 | s <- svd(MASS::ginv(crossprod(matricize(X[Z_indice, ]))) %*% |
| 80 | crossprod(matricize(X[Z_indice, ]), matricize(Y[Z_indice, ]))) |
| 81 | S <- s$d |
| 82 | # Set m-rank(r) singular values to zero, and recompose best rank(r) approximation |
| 83 | # of the initial product |
| 84 | if (rank[r] < length(S)) |
| 85 | S[(rank[r] + 1):length(S)] <- 0 |
| 86 | phi[, , r] <- s$u %*% diag(S) %*% t(s$v) %*% Rho[, , r] |
| 87 | } |
| 88 | |
| 89 | # Step E and computation of the loglikelihood |
| 90 | sumLogLLF2 <- 0 |
| 91 | for (i in seq_len(n)) |
| 92 | { |
| 93 | sumLLF1 <- 0 |
| 94 | maxLogGamIR <- -Inf |
| 95 | for (r in seq_len(k)) |
| 96 | { |
| 97 | dotProduct <- tcrossprod(Y[i, ] %*% Rho[, , r] - X[i, ] %*% phi[, , r]) |
| 98 | logGamIR <- log(Pi[r]) + log(gdet(Rho[, , r])) - 0.5 * dotProduct |
| 99 | # Z[i] = index of max (gam[i,]) |
| 100 | if (logGamIR > maxLogGamIR) |
| 101 | { |
| 102 | Z[i] <- r |
| 103 | maxLogGamIR <- logGamIR |
| 104 | } |
| 105 | sumLLF1 <- sumLLF1 + exp(logGamIR)/(2 * pi)^(m/2) |
| 106 | } |
| 107 | sumLogLLF2 <- sumLogLLF2 + log(sumLLF1) |
| 108 | } |
| 109 | |
| 110 | LLF <- -1/n * sumLogLLF2 |
| 111 | |
| 112 | # update distance parameter to check algorithm convergence (delta(phi, Phi)) |
| 113 | deltaPhi <- c(deltaPhi, max((abs(phi - Phi))/(1 + abs(phi)))) #TODO: explain? |
| 114 | if (length(deltaPhi) > deltaPhiBufferSize) |
| 115 | deltaPhi <- deltaPhi[2:length(deltaPhi)] |
| 116 | sumDeltaPhi <- sum(abs(deltaPhi)) |
| 117 | |
| 118 | # update other local variables |
| 119 | Phi <- phi |
| 120 | ite <- ite + 1 |
| 121 | } |
| 122 | return(list(phi = phi, LLF = LLF)) |
| 123 | } |