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################################################################################
##
## R package sgee by Gregory Vaughan, Kun Chen, and Jun Yan
## Copyright (C) 2016-2018
##
## This file is part of the R package sgee.
##
## The R package sgee is free software: You can redistribute it and/or
## modify it under the terms of the GNU General Public License as published
## by the Free Software Foundation, either version 3 of the License, or
## any later version (at your option). See the GNU General Public License
## at <http://www.gnu.org/licenses/> for details.
##
## The R package sgee is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
##
#################################################################################' Hierarchical Stagewise Estimating Equations Implementation.
#'
#' Function to perform HiSEE, a Bi-Level Boosting / Functional Gradient
#' Descent / Forward Stagewise regression in the grouped covariates
#' setting using Generalized Estimating Equations
#'
#' Function to implement HiSEE, a stagewise regression approach
#' that is designed to perform hierarchical selection in the context of
#' Generalized Estimating Equations. Given A response Y, design matrix X
#' (excluding intercept) HiSEE generates a path of stagewise regression
#' estimates for each covariate based on the provided step size epsilon.
#' First an optimal group of covariates is identified, and then an
#' optimal covariate within that group is selected and then updated in each
#' iterative step.
#'
#' The resulting path can then be analyzed to determine an optimal
#' model along the path of coefficient estimates. The
#' \code{summary.sgee} function provides such functionality based on various
#' possible metrics, primarily focused on the Mean Squared Error.
#' Furthermore, the \code{plot.sgee} function can be used to examine the
#' path of coefficient estimates versus the iteration number, or some
#' desired penalty.
#'
#' @param y Vector of response measures that corresponds with modeling family
#' given in 'family' parameter. 'y' is assumed to be the same length as
#' 'clusterID' and is assumed to be organized into clusters as dictated by
#' 'clusterID'.
#' @param x Design matrix of dimension length(y) x nvars where each row is
#' represents an obersvation of predictor variables. Assumed to be scaled.
#' @param family Modeling family that describes the marginal distribution of
#' the response. Assumed to be an object such as 'gaussian()' or 'poisson()'
#' @param clusterID Vector of integers that identifies the clusters of response
#' measures in 'y'. Data and 'clusterID' are assumed to 1) be of equal lengths,
#' 2) sorted so that observations of a cluster are in contiguous rows, and 3)
#' organized so that 'clusterID' is a vector of consecutive integers.
#' @param groupID Vector of integeres that identifies the groups of the
#' covariates/coefficients (i.e. the columns of 'x'). 'x' and 'groupID' are
#' assumed 1) to be of corresponding dimension, (i.e. ncol(x) ==
#' length(groupID)), 2) sorted so that groups of covariates are in contiguous
#' columns, and 3) organized so that 'groupID' is a vector of consecutive
#' integers.
#' @param corstr A character string indicating the desired working correlation
#' structure. The following are implemented : "independence" (default value),
#' "exchangeable", and "ar1".
#' @param alpha An initial guess for the correlation parameter value
#' between -1 and 1 . If left NULL (the default), the initial estimate is 0.
#' @param intercept Binary value indicating where an intercept term is
#' to be included in the model for estimation. Default is to include an
#' intercept.
#' @param offset Vector of offset value(s) for the linear predictor. 'offset'
#' is assumed to be either of length one, or of the same length as 'y'.
#' Default is to have no offset.
#' @param control A list of parameters used to contorl the path generation
#' process; see \code{sgee.control}.
#' @param standardize A logical parameter that indicates whether or not
#' the covariates need to be standardized before fitting.
#' If standardized before fitting, the unstandardized
#' path is returned as the default, with a \code{standardizedPath} and
#' \code{standardizedX} included
#' separately. Default value is \code{TRUE}.
#' @param verbose Logical parameter indicating whether output should be produced
#' while hisee is running. Default value is FALSE.
#' @param ... Not currently used
#'
#' @return Object of class 'sgee' containing the path of coefficient estimates,
#' the path of scale estimates, the path of correlation parameter
#' estimates, and the iteration at which HiSEE terminated, and initial
#' regression
#' values including \code{x}, \code{y}, code{family}, \code{clusterID},
#' \code{groupID}, \code{offset}, \code{epsilon}, and \code{numIt}.
#'
#' @note Function to execute HiSEE Technique. Functionally equivalent
#' to SEE when all elements in groupID are unique.
#' @author Gregory Vaughan
#' @references Vaughan, G., Aseltine, R., Chen, K., Yan, J., (2017). Stagewise
#' Generalized Estimating Equations with Grouped Variables. Biometrics 73,
#' 1332-1342. URL: http://dx.doi.org/10.1111/biom.12669,
#' doi:10.1111/biom.12669.
#'
#' Wolfson, J. (2011). EEBoost: A general method for prediction
#' and variable selection based on estimating equations. Journal of the
#' American Statistical Association 106, 296--305.
#'
#' Tibshirani, R. J. (2015). A general framework for fast stagewise
#' algorithms. Journal of Machine Learning Research 16, 2543--2588.
#' @examples
#'
#' #####################
#' ## Generate test data
#' #####################
#'
#' ## Initialize covariate values
#' p <- 50
#' beta <- c(rep(2,5),
#' c(1, 0, 1.5, 0, .5),
#' rep(0.5,5),
#' rep(0,p-15))
#' groupSize <- 5
#' numGroups <- length(beta)/groupSize
#'
#'
#' generatedData <- genData(numClusters = 50,
#' clusterSize = 4,
#' clusterRho = 0.6,
#' clusterCorstr = "exchangeable",
#' yVariance = 1,
#' xVariance = 1,
#' numGroups = numGroups,
#' groupSize = groupSize,
#' groupRho = 0.3,
#' beta = beta,
#' family = gaussian(),
#' intercept = 1)
#'
#' ## Perform Fitting by providing y and x values
#' coefMat1 <- hisee(y = generatedData$y, x = generatedData$x,
#' family = gaussian(),
#' clusterID = generatedData$clusterID,
#' groupID = generatedData$groupID,
#' corstr="exchangeable",
#' control = sgee.control(maxIt = 50, epsilon = 0.5))
#'
#' ## Perform Fitting by providing formula and data
#' genDF <- data.frame(generatedData$y, generatedData$x)
#' names(genDF) <- c("Y", paste0("Cov", 1:p))
#' coefMat2 <- hisee(formula(genDF), data = genDF,
#' family = gaussian(),
#' subset = Y<1,
#' waves = rep(1:4, 50),
#' clusterID = generatedData$clusterID,
#' groupID = generatedData$groupID,
#' corstr="exchangeable",
#' control = sgee.control(maxIt = 50, epsilon = 0.5))
#'
#' par(mfrow = c(2,1))
#' plot(coefMat1)
#' plot(coefMat2)
#'
#' @export hisee
#' @name hisee
NULL
#' @export
#' @rdname hisee
hisee <- function(y, ...) UseMethod("hisee")
#' @param formula Object of class 'formula'; a symbolic description of
#' the model to be fitted
#' @param data Optional data frame containing the variables in the model.
#' @param waves An integer vector which identifies components in clusters.
#' The length of \code{waves} should be the same as the number of
#' observations. \code{waves} is automatically generated if none is supplied,
#' but when using \code{subset} parameter, the \code{waves} parameter must be
#' provided by the user for proper calculation.
#' @param contrasts An optional list provided when using a formula.
#' similar to \code{contrasts} from \code{glm}.
#' See the \code{contrasts.arg} of \code{model.matrix.default}.
#' @param subset An optional vector specifying a subset of observations to be
#' used in the fitting process.
#'
#' @rdname hisee
#' @export
hisee.formula <- function(formula, data=list(),
clusterID,
waves = NULL,
contrasts = NULL,
subset,
...)
{
mf <- match.call(expand.dots = FALSE)
m <- match(c("formula", "data", "clusterID", "waves", "subset"), names(mf), 0L)
mf <- mf[c(1L, m)]
if (is.null(mf$clusterID)){
mf$clusterID <- as.name("clusterID")
}
if (is.null(mf$waves)){
mf$waves <- NULL
}
mf$drop.unused.levels <- TRUE
mf[[1L]] <- quote(stats::model.frame)
mf <- eval(mf, parent.frame())
clusterID <- model.extract(mf, clusterID)
if(is.null(clusterID)){
stop("clusterID variable not found.")
}
waves <- model.extract(mf, waves)
y <- model.response(mf, "numeric")
x <- model.matrix(attr(mf, "terms"), data=mf, contrasts)
## hisee deterimes intercept based on 'intercept' parameter
if(all(x[,1] ==1)){
x <- x[,-1]
}
if(any(colSums(x) == 0)){
cat("######## ERROR! ########\n")
cat(colnames(x)[colSums(x) == 0])
cat("\n")
stop("The above factors are not found in the given observations")
}
results <- hisee.default(y, x,
clusterID = clusterID,
waves = waves,
...)
results$call <- match.call()
results$call <- contrasts
results
}
#' @export
#' @rdname hisee
hisee.default <- function(y, x,
waves = NULL,
...){
results <- hisee.fit(y, x,
waves = waves,
...)
results$call <- match.call()
results
}
#' @export
#' @rdname hisee
hisee.fit <-
function(y, x, family,
clusterID, waves = NULL, groupID = 1:ncol(x),
corstr="independence", alpha = NULL,
intercept = TRUE,
offset = 0,
control = sgee.control(maxIt = 200, epsilon = 0.05,
stoppingThreshold = min(length(y), ncol(x))-intercept,
undoThreshold = 0),
standardize = TRUE,
verbose = FALSE,
...){
#######################
## Preliminaries/set up
#######################
maxIt <- control$maxIt
epsilon <- control$epsilon
undoThreshold <- control$undoThreshold
interceptLimit <- control$interceptLimit
##If the undoThreshold is >= epsilon
## the check wil always trigger;
## so this check is added to prevent
## an infinte loop
if(undoThreshold >= epsilon){
if(verbose){
cat(paste0("****** undoThreshold too large! reducing threshold now **********\n"))
}
undoThreshold <- epsilon/10
}
if(is.null(control$stoppingThreshold)){
stoppingThreshold <- min(length(y), ncol(x))-intercept
} else {
stoppingThreshold <- control$stoppingThreshold
}
p <- ncol(x)
if (is.character(family)){
family <- get(family, mode = "function", envir = parent.frame())
}
if (is.function(family)){
family <- family()
}
if(standardize){
unstandardizedX <- x
if(intercept){
x <- scale(x)
} else{
x <- scale(x, center = FALSE)
}
}
if(is.null(waves)){
clusz <- unlist(sapply(unique(clusterID), function(x) {sum(clusterID == x)}))
waves <- as.integer(unlist(sapply(clusz, function(x) 1:x)))
}
## currently assuming only intercept in estimation of
## of correlation and dispersion
r <- 1 # number of covariates in dispersion modeling
q <- 1 # number of covariates in correlation modeling
## Initail estimates for all parameters
beta <- rep(0,p)
phi <- stats::sd(y)^2
## current initial estimation for correlation parameter
if(is.null(alpha)){
alpha <- 0
}
## Intercept value
beta0 <- 0
meanLink <- family$linkfun
meanLinkInv <- family$linkinv
varianceLink <- family$variance
mu.eta <- family$mu.eta
## Paths of parameter estimates
path <- matrix(rep(0,(maxIt)*(p + intercept)), nrow = maxIt)
## currently assuming only intercept in estimating dispersion
phiPath <- matrix(rep(0,(maxIt)*r), nrow = maxIt)
## currently assuming only intercept in estimating correlation
alphaPath <- matrix(rep(0,(maxIt)*q), nrow = maxIt)
clusterIDs <- unique(clusterID)
numClusters <- length(clusterIDs)
maxClusterSize <- max(waves)
##stoppedOn added to keep track of when the algorithm stops
## it assumes it goes the whole lenght unless stopped prematurely
stoppedOn <- maxIt
# Working correlation matrix
R <- genCorMat(corstr = corstr, rho = alpha, maxClusterSize = maxClusterSize)
RInv <- solve(R)
##################
## Main Algorithim
##################
cat("\n")
oldDelta <- rep(0, length(beta))
it <- 0
while (it <maxIt){
it <- it +1
if(verbose){
cat(paste0("****** Beginning iteration # ", it, " **********\n"))
}
GEEValues <- evaluateGEE(y = y,
x = x,
beta = beta,
beta0 = beta0,
intercept,
phi = phi,
offset = offset,
RInv = RInv,
numClusters = numClusters,
clusterID = clusterID,
waves = waves,
meanLinkInv = meanLinkInv,
mu.eta = mu.eta,
varianceLink = varianceLink,
corstr = corstr,
maxClusterSize = maxClusterSize,
interceptLimit = interceptLimit)
## Update Estimates
beta0 <- GEEValues$beta0
phi <- GEEValues$phiHat
alpha <- GEEValues$rhoHat
RInv <- GEEValues$RInv
## Current Values of estimating Equations
sumMean <- GEEValues$sumMean
###################
## Update Selection
###################
a <- rep(0, length(unique(groupID)))
for (j in unique(groupID)){
currentGroupIndex <- groupID == j
aCurrent <- sumMean[currentGroupIndex]
a[j] <- sqrt(sum(aCurrent^2))/sqrt(length(aCurrent))
}
## Identify optimal group
delta <- which(a == max(a))
theGroup <- groupID == delta
aCurrent <- sumMean[theGroup]
maxGradient <- max(abs(aCurrent))
deltaIndex <- abs(aCurrent) == maxGradient
if(verbose){
}
## Check if the update is effectively undoing the last one
if (sum(abs(oldDelta[theGroup] + (deltaIndex * epsilon * sign(aCurrent[deltaIndex]))))<= undoThreshold){
if(verbose){
cat(paste0("****** Step Undone! Reducing Stepsize **********\n"))
}
if(it>2){
if (intercept){
beta <- path[it - 2,-1]
} else {
beta <- path[it - 2,]
}
} else{
beta <- rep(0, length(beta))
}
epsilon <- epsilon/2
it <- it - 2
oldDelta <- rep(0, length(beta))
##If the undoThreshold is >= epsilon
## the check wil always trigger;
## so this check is added to prevent
## an infinte loop
if(undoThreshold >= epsilon){
if(verbose){
cat(paste0("****** undoThreshold too large! reducing threshold now **********\n"))
}
undoThreshold <- epsilon/10
}
## If the check is passed and the update
## is sufficiently different from the previous
## update
} else {
oldDelta <- rep(0, length(beta))
oldDelta[theGroup] <- (deltaIndex * epsilon * sign(aCurrent[deltaIndex]))
## Update estimates
## only the element in the group with the largest L2 norm
## that itself has the largest magnitutde is incremented
beta[theGroup] <- beta[theGroup] + (deltaIndex * epsilon * sign(aCurrent[deltaIndex]))
## Update Paths
if(intercept){
path[it,] <- c(beta0, beta)
}
else{
path[it,] <- beta
}
phiPath[it,] <- phi
alphaPath[it,] <- alpha
###########
## stopping mechanism when the alogrithim has
## reached saturation.
## sum(a) <0.5 threshold added to prevent possible loop
## that can happen with binary data using the adaptive
## step size where the algorithm thinks a step keeps
## being undone, but really the estimating equations
## are all VERY close to 0
if(((sum(beta != 0) >= stoppingThreshold) | sum(a) < 0.5 )& (it< maxIt) ){
print("stopped on")
print(it)
print(a[delta])
path[((it+1):maxIt),] <- matrix(rep(path[it,],(maxIt - it)),
nrow = (maxIt - it),
byrow = TRUE)
phiPath[((it+1):maxIt),] <- matrix(rep(phiPath[it,],(maxIt - it)),
nrow = (maxIt - it),
byrow = TRUE)
alphaPath[((it+1):maxIt),] <- matrix(rep(alphaPath[it,],(maxIt - it)),
nrow = (maxIt - it),
byrow = TRUE)
stoppedOn <- it
break
}
}
}
result <- list(path = path,
gammaPath = phiPath,
alphaPath = alphaPath,
stoppedOn = stoppedOn,
maxIt = maxIt,
y = y,
x = x,
intercept = intercept,
clusterID = clusterID,
groupID = groupID,
family = family,
offset = offset,
epsilon = epsilon)
if(standardize){
result$x <- unstandardizedX
temp <- path
if(intercept){
temp[,-1] <- t(t(path[,-1]) /attr(x, "scaled:scale"))
temp[,1] <- path[,1] - crossprod(t(path[,-1]),
(attr(x, "scaled:center")/attr(x, "scaled:scale")))
} else{
temp <- t(t(path) /attr(x, "scaled:scale"))
}
result$standardizedPath <- path
result$path <- temp
result$standardizedX <- x
}
class(result) <- "sgee"
result
}
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