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#' A Reference Class which contains the parameters of a MRHLP model.
#'
#' ParamMRHLP contains all the parameters of a MRHLP model. The parameters are
#' calculated by the initialization Method and then updated by the Method
#' implementing the M-Step of the EM algorithm.
#'
#' @field mData [MData][MData] object representing the sample (covariates/inputs
#' `X` and observed responses/outputs `Y`).
#' @field K The number of regimes (MRHLP components).
#' @field p The order of the polynomial regression.
#' @field q The dimension of the logistic regression. For the purpose of
#' segmentation, it must be set to 1.
#' @field variance_type Character indicating if the model is homoskedastic
#' (`variance_type = "homoskedastic"`) or heteroskedastic (`variance_type =
#' "heteroskedastic"`). By default the model is heteroskedastic.
#' @field W Parameters of the logistic process. \eqn{\boldsymbol{W} =
#' (\boldsymbol{w}_{1},\dots,\boldsymbol{w}_{K-1})}{W = (w_{1},\dots,w_{K-1})}
#' is a matrix of dimension \eqn{(q + 1, K - 1)}, with `q` the order of the
#' logistic regression. `q` is fixed to 1 by default.
#' @field beta Parameters of the polynomial regressions. \eqn{\boldsymbol{\beta}
#' = (\boldsymbol{\beta}_{1},\dots,\boldsymbol{\beta}_{K})}{\beta =
#' (\beta_{1},\dots,\beta_{K})} is an array of dimension \eqn{(p + 1, d, K)},
#' with `p` the order of the polynomial regression. `p` is fixed to 3 by
#' default.
#' @field sigma2 The variances for the `K` regimes. If MRHLP model is
#' heteroskedastic (`variance_type = "heteroskedastic"`) then `sigma2` is an
#' array of size \eqn{(d, d, K)} (otherwise MRHLP model is homoskedastic
#' (`variance_type = "homoskedastic"`) and `sigma2` is a matrix of size
#' \eqn{(d, d)}).
#' @field nu The degree of freedom of the MRHLP model representing the
#' complexity of the model.
#' @field phi A list giving the regression design matrices for the polynomial
#' and the logistic regressions.
#' @export
ParamMRHLP <- setRefClass(
"ParamMRHLP",
fields = list(
mData = "MData",
phi = "list",
K = "numeric", # Number of regimes
p = "numeric", # Dimension of beta (order of polynomial regression)
q = "numeric", # Dimension of w (order of logistic regression)
variance_type = "character",
nu = "numeric", # Degree of freedom
W = "matrix",
beta = "array",
sigma2 = "array"
),
methods = list(
initialize = function(mData = MData(numeric(1), matrix(1)), K = 1, p = 3, q = 1, variance_type = "heteroskedastic") {
mData <<- mData
phi <<- designmatrix(x = mData$X, p = p, q = q)
K <<- K
p <<- p
q <<- q
variance_type <<- variance_type
if (variance_type == "homoskedastic") {
nu <<- (q + 1) * (K - 1) + mData$d * (p + 1) * K + mData$d * (mData$d + 1) / 2
} else {
nu <<- (q + 1) * (K - 1) + mData$d * (p + 1) * K + K * mData$d * (mData$d + 1) / 2
}
W <<- matrix(0, q + 1, K - 1)
beta <<- array(NA, dim = c(p + 1, mData$d, K))
if (variance_type == "homoskedastic") {
sigma2 <<- matrix(NA, mData$d, mData$d)
} else {
sigma2 <<- array(NA, dim = c(mData$d, mData$d, K))
}
},
initParam = function(try_algo = 1) {
"Method to initialize parameters \\code{W}, \\code{beta} and
\\code{sigma2}.
If \\code{try_algo = 1} then \\code{beta} and \\code{sigma2} are
initialized by segmenting the time series \\code{Y} uniformly into
\\code{K} contiguous segments. Otherwise, \\code{W}, \\code{beta} and
\\code{sigma2} are initialized by segmenting randomly the time series
\\code{Y} into \\code{K} segments."
if (try_algo == 1) { # Uniform segmentation into K contiguous segments, and then a regression
# Initialization of W
W <<- zeros(q + 1, K - 1)
zi <- round(mData$m / K) - 1
s <- 0
for (k in 1:K) {
i <- (k - 1) * zi + 1
j <- k * zi
yk <- mData$Y[i:j,]
Xk <- phi$XBeta[i:j, , drop = FALSE]
beta[, , k] <<- solve(t(Xk) %*% Xk, tol = 0) %*% t(Xk) %*% yk
muk <- Xk %*% beta[, , k]
sk <- t(yk - muk) %*% (yk - muk)
if (variance_type == "homoskedastic") {
s <- s + sk
sigma2 <<- s / mData$m
} else{
sigma2[, , k] <<- sk / length(yk)
}
}
}
else{# Random segmentation into K contiguous segments, and then a regression on each segment
# Initialization of W
W <<- rand(q + 1, K - 1)
Lmin <- 2 # Minimum number of points in a segment
tk_init <- zeros(K + 1, 1)
K_1 <- K
for (k in 2:K) {
K_1 <- K_1 - 1
temp <- tk_init[k - 1] + Lmin:(mData$m - (K_1 * Lmin) - tk_init[k - 1])
ind <- sample(length(temp))
tk_init[k] <- temp[ind[1]]
}
tk_init[K + 1] <- mData$m
s <- 0
for (k in 1:K) {
i <- tk_init[k] + 1
j <- tk_init[k + 1]
yk <- mData$Y[i:j,]
Xk <- phi$XBeta[i:j, , drop = FALSE]
beta[, , k] <<- solve(t(Xk) %*% Xk, tol = 0) %*% t(Xk) %*% yk
muk <- Xk %*% beta[, , k]
sk <- t(yk - muk) %*% (yk - muk)
if (variance_type == "homoskedastic") {
s <- s + sk
sigma2 <<- s / mData$m
}
else{
sigma2[, , k] <<- sk / length(yk)
}
}
}
},
MStep = function(statMRHLP, verbose_IRLS) {
"Method which implements the M-step of the EM algorithm to learn the
parameters of the MRHLP model based on statistics provided by the object
\\code{statMRHLP} of class \\link{StatMRHLP} (which contains the
E-step)."
# Maximization w.r.t betak and sigmak (the variances)
if (variance_type == "homoskedastic") {
s = 0
}
for (k in 1:K) {
weights <- statMRHLP$tau_ik[, k] # Post probabilities of each component k (dimension nx1)
nk <- sum(weights) # Expected cardinal numnber of class k
Xk <- phi$XBeta * (sqrt(weights) %*% ones(1, p + 1)) # [m*(p+1)]
yk <- mData$Y * (sqrt(weights) %*% ones(1, mData$d))
M <- t(Xk) %*% Xk
epps <- 1e-9
M <- M + epps * diag(p + 1)
beta[, , k] <<- solve(M, tol = FALSE) %*% t(Xk) %*% yk # Maximization w.r.t betak
z <- (mData$Y - phi$XBeta %*% beta[, , k]) * (sqrt(weights) %*% ones(1, mData$d))
# Maximisation w.r.t sigmak (the variances)
sk <- t(z) %*% z
if (variance_type == "homoskedastic") {
s <- s + sk
sigma2 <<- s / mData$m
} else {
sigma2[, , k] <<- sk / nk
}
}
# Maximization w.r.t W
# IRLS : Iteratively Reweighted Least Squares (for IRLS, see our IJCNN 2009 paper for example)
res_irls <- IRLS(phi$Xw, statMRHLP$tau_ik, ones(nrow(statMRHLP$tau_ik), 1), W, verbose_IRLS)
W <<- res_irls$W
piik <- res_irls$piik
reg_irls <- res_irls$reg_irls
}
)
)
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