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### sCorrect-hessian2.R ---
##----------------------------------------------------------------------
## Author: Brice Ozenne
## Created: dec 11 2019 (14:09)
## Version:
## Last-Updated: jan 23 2024 (10:25)
## By: Brice Ozenne
## Update #: 144
##----------------------------------------------------------------------
##
### Commentary:
##
### Change Log:
##----------------------------------------------------------------------
##
### Code:
## * Documentation - hessian2
#' @title Hessian With Small Sample Correction.
#' @description Extract the hessian from a latent variable model, with small sample correction
#' @name hessian2
#'
#' @param object a \code{lvmfit} or \code{lvmfit2} object (i.e. output of \code{lava::estimate} or \code{lavaSearch2::estimate2}).
#' @param indiv [logical] If \code{TRUE}, the hessian relative to each observation is returned. Otherwise the total hessian is returned.
#' @param cluster [integer vector] the grouping variable relative to which the observations are iid.
#' @param as.lava [logical] if \code{TRUE}, uses the same names as when using \code{stats::coef}.
#' @param ssc [character] method used to correct the small sample bias of the variance coefficients: no correction (\code{"none"}/\code{FALSE}/\code{NA}),
#' correct the first order bias in the residual variance (\code{"residual"}), or correct the first order bias in the estimated coefficients \code{"cox"}).
#' Only relevant when using a \code{lvmfit} object.
#' @param ... additional argument passed to \code{estimate2} when using a \code{lvmfit} object.
#'
#' @details When argument object is a \code{lvmfit} object, the method first calls \code{estimate2} and then extract the hessian.
#'
#' @seealso \code{\link{estimate2}} to obtain \code{lvmfit2} objects.
#'
#' @return An array containing the second derivative of the likelihood relative to each sample (dim 3)
#' and each pair of model coefficients (dim 1,2).
#'
#' @examples
#' #### simulate data ####
#' n <- 5e1
#' p <- 3
#' X.name <- paste0("X",1:p)
#' link.lvm <- paste0("Y~",X.name)
#' formula.lvm <- as.formula(paste0("Y~",paste0(X.name,collapse="+")))
#'
#' m <- lvm(formula.lvm)
#' distribution(m,~Id) <- Sequence.lvm(0)
#' set.seed(10)
#' d <- lava::sim(m,n)
#'
#' #### latent variable models ####
#' e.lvm <- estimate(lvm(formula.lvm),data=d)
#' hessian2(e.lvm)
#'
#' @concept small sample inference
#' @export
`hessian2` <-
function(object, indiv, cluster, as.lava, ...) UseMethod("hessian2")
## * hessian2.lvmfit
#' @rdname hessian2
#' @export
hessian2.lvmfit <- function(object, indiv = FALSE, cluster = NULL, as.lava = TRUE, ssc = lava.options()$ssc, ...){
return(hessian2(estimate2(object, ssc = ssc, hessian = TRUE, ...), cluster = cluster, as.lava = as.lava))
}
## * hessian2.lvmfit2
#' @rdname hessian2
#' @export
hessian2.lvmfit2 <- function(object, indiv = FALSE, cluster = NULL, as.lava = TRUE, ...){
dots <- list(...)
if(length(dots)>0){
warning("Argument(s) \'",paste(names(dots),collapse="\' \'"),"\' not used by ",match.call()[1],". \n")
}
## ** define cluster
if(is.null(cluster)){
n.cluster <- object$sCorrect$cluster$n.cluster
cluster.index <- 1:n.cluster
}else{
if(!is.numeric(cluster)){
data <- object$sCorrect$data
if(length(cluster)==1){
if(cluster %in% names(data) == FALSE){
stop("Invalid \'cluster\' argument \n",
"Could not find variable \"",cluster,"\" in argument \'data\' \n")
}
cluster <- data[[cluster]]
}
cluster.index <- as.numeric(factor(cluster, levels = unique(cluster)))
}else{
cluster.index <- as.numeric(factor(cluster, levels = unique(cluster)))
}
n.cluster <- length(unique(cluster.index))
}
## ** get hessian
hessian <- object$sCorrect$hessian
if(is.null(hessian)){
hessian <- .hessian2(dmu = object$sCorrect$dmoment$dmu,
dOmega = object$sCorrect$dmoment$dOmega,
d2mu = object$sCorrect$d2moment$d2mu,
d2Omega = object$sCorrect$d2moment$d2Omega,
epsilon = object$sCorrect$residuals,
OmegaM1 = object$sCorrect$moment$OmegaM1.missing.pattern,
missing.pattern = object$sCorrect$missing$pattern,
unique.pattern = object$sCorrect$missing$unique.pattern,
name.pattern = object$sCorrect$missing$name.pattern,
grid.mean = object$sCorrect$skeleton$grid.dmoment$mean,
grid.var = object$sCorrect$skeleton$grid.dmoment$var,
grid.hybrid = object$sCorrect$skeleton$grid.dmoment$hybrid,
name.param = object$sCorrect$skeleton$Uparam,
leverage = object$sCorrect$leverage,
n.cluster = object$sCorrect$cluster$n.cluster,
weights = object$sCorrect$weights)
hessian.name <- stats::setNames(names(object$sCorrect$skeleton$originalLink2param),object$sCorrect$skeleton$originalLink2param)[object$sCorrect$skeleton$Uparam]
dimnames(hessian) <- list(as.character(hessian.name),
as.character(hessian.name),
NULL)
}
if(!is.null(cluster)){ ## aggregate hessian by cluster
hessianSave <- hessian
hessian <- array(0, dim = dim(hessian),
dimnames = dimnames(hessian))
for(i in 1:length(cluster)){
hessian[,,cluster[i]] <- hessian[,,cluster[i]] + hessianSave[,,i]
}
}
## ** export
hessian <- hessian[names(object$sCorrect$skeleton$originalLink2param),
names(object$sCorrect$skeleton$originalLink2param),
,
drop=FALSE]
if(as.lava == FALSE){
dimnames(hessian) <- list(as.character(object$sCorrect$skeleton$originalLink2param),
as.character(object$sCorrect$skeleton$originalLink2param),
NULL)
}
if(indiv==FALSE){
hessian <- apply(hessian, 1:2, sum)
}else if(!is.null(cluster)){
index2.cluster <- tapply(1:length(cluster),cluster,list)
attr(hessian,"cluster") <- names(index2.cluster)
}
return(hessian)
}
## * .hessian2
#' @title Compute the Hessian Matrix From the Conditional Moments
#' @description Compute the Hessian matrix from the conditional moments.
#' @name hessian2-internal
#'
#' @details \code{calc_hessian} will perform the computation individually when the
#' argument \code{index.Omega} is not null.
#'
#' @keywords internal
.hessian2 <- function(dmu, dOmega, d2mu, d2Omega, epsilon, OmegaM1,
missing.pattern, unique.pattern, name.pattern,
grid.mean, grid.var, grid.hybrid, name.param,
leverage, n.cluster, weights){
if(lava.options()$debug){cat(".hessian2\n")}
## ** Prepare
n.grid.mean <- NROW(grid.mean)
n.grid.var <- NROW(grid.var)
n.grid.hybrid <- NROW(grid.hybrid)
n.param <- length(name.param)
n.pattern <- length(name.pattern)
hessian <- array(NA, dim = c(n.param, n.param, n.cluster),
dimnames = list(name.param,name.param,NULL))
if(length(dmu)>0){
index.mean <- 1:n.grid.mean
}else{
index.mean <- NULL
}
if(length(dOmega)>0){
index.var <- 1:n.grid.var
}else{
index.var <- NULL
}
if(length(dmu)>0 && length(dOmega)>0){
index.hybrid <- 1:n.grid.hybrid
}else{
index.hybrid <- NULL
}
## ** loop over missing data pattern
for(iP in 1:n.pattern){ ## iP <- 1
iPattern <- name.pattern[iP]
iIndex <- missing.pattern[[iPattern]]
iY <- which(unique.pattern[iP,]==1)
iOmegaM1 <- OmegaM1[[iPattern]]
iEpsilon <- epsilon[iIndex,iY,drop=FALSE]
idmu <- .subsetList(dmu, indexRow = iIndex, indexCol = iY)
idOmega <- .subsetList(dOmega, indexRow = iY, indexCol = iY)
id2mu <- .subsetList2(d2mu, indexRow = iIndex, indexCol = iY)
id2Omega <- .subsetList2(d2Omega, indexRow = iY, indexCol = iY)
if(!is.null(leverage)){
iLeverage <- leverage[iIndex,iY,drop=FALSE]
}
hessian[,,iIndex] <- 0 ## ## initialize (keep NA for missing values)
## *** second derivative relative to the mean parameters
for(iG in index.mean){ # iG <- 1
iP1 <- grid.mean[iG,1]
iP2 <- grid.mean[iG,2]
if(grid.mean[iG,"d2.12"]){
term1 <- rowSums((id2mu[[iP1]][[iP2]] %*% iOmegaM1) * iEpsilon)
}else if(grid.mean[iG,"d2.21"]){
term1 <- rowSums((id2mu[[iP2]][[iP1]] %*% iOmegaM1) * iEpsilon)
}else{
term1 <- 0
}
term2 <- -rowSums((idmu[[iP1]] %*% iOmegaM1) * idmu[[iP2]])
hessian[iP1,iP2,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE] + term1 + term2
hessian[iP2,iP1,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE]
}
## *** second derivative relative to the variance parameters
for(iG in index.var){ # iG <- 2
iP1 <- grid.var[iG,1]
iP2 <- grid.var[iG,2]
term1a <- - diag(iOmegaM1 %*% idOmega[[iP1]] %*% iOmegaM1 %*% idOmega[[iP2]])
term2 <- - rowSums((iEpsilon %*% iOmegaM1 %*% idOmega[[iP2]] %*% iOmegaM1 %*% idOmega[[iP1]] %*% iOmegaM1) * iEpsilon)
if(grid.var[iG,"d2.12"]){
term1b <- diag(iOmegaM1 %*% id2Omega[[iP1]][[iP2]])
term3 <- 1/2 * rowSums((iEpsilon %*% iOmegaM1 %*% id2Omega[[iP1]][[iP2]] %*% iOmegaM1) * iEpsilon)
}else if(grid.var[iG,"d2.21"]){
term1b <- diag(iOmegaM1 %*% id2Omega[[iP2]][[iP1]])
term3 <- 1/2 * rowSums((iEpsilon %*% iOmegaM1 %*% id2Omega[[iP2]][[iP1]] %*% iOmegaM1) * iEpsilon)
}else{
term1b <- 0
term3 <- 0
}
if(is.null(leverage)){
hessian[iP1,iP2,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE] - 1/2 * rep(sum(term1a + term1b), length(iIndex)) + term2 + term3
}else{
hessian[iP1,iP2,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE] - 1/2 * rowSums( sweep(1-iLeverage, FUN = "*", STATS = term1a + term1b, MARGIN = 2) ) + term2 + term3
}
hessian[iP2,iP1,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE]
}
## *** second derivative relative to the mean and variance parameters
for(iG in index.hybrid){ # iG <- 1
iP1 <- grid.hybrid[iG,1]
iP2 <- grid.hybrid[iG,2]
if(!is.null(idmu[[iP1]]) && !is.null(idOmega[[iP2]])){
term1 <- - rowSums((idmu[[iP1]] %*% iOmegaM1 %*% idOmega[[iP2]] %*% iOmegaM1) * iEpsilon)
}else{
term1 <- 0
}
if(!is.null(idmu[[iP2]]) && !is.null(idOmega[[iP1]])){
term2 <- - rowSums((idmu[[iP2]] %*% iOmegaM1 %*% idOmega[[iP1]] %*% iOmegaM1) * iEpsilon)
}else{
term2 <- 0
}
hessian[iP1,iP2,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE] + term1 + term2
hessian[iP2,iP1,iIndex] <- hessian[iP1,iP2,iIndex,drop=FALSE]
}
}
## ** export
if(!is.null(weights)){
for(iI in 1:length(weights)){
hessian[,,iI] <- weights[iI] * hessian[,,iI]
}
}
return(hessian)
}
## * .subsetList
.subsetList <- function(object, indexRow, indexCol){
if(length(object)==0){
return(object)
}else{
return(lapply(object, FUN = function(iL){iL[indexRow,indexCol,drop=FALSE]}))
}
}
## * .subsetList2
.subsetList2 <- function(object, indexRow, indexCol){
if(length(object)==0){
return(object)
}else{
return(lapply(object, FUN = .subsetList, indexRow = indexRow, indexCol = indexCol))
}
}
######################################################################
### sCorrect-hessian2.R ends here
|
