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#' Finite Mixture of Spherical Laplace Distributions
#'
#' For \eqn{n} observations on a \eqn{(p-1)} sphere in \eqn{\mathbf{R}^p},
#' a finite mixture model is fitted whose components are spherical Laplace distributions via the following model
#' \deqn{f(x; \left\lbrace w_k, \mu_k, \sigma_k \right\rbrace_{k=1}^K) = \sum_{k=1}^K w_k SL(x; \mu_k, \sigma_k)}
#' with parameters \eqn{w_k}'s for component weights, \eqn{\mu_k}'s for component locations, and \eqn{\sigma_k}'s for component scales.
#'
#' @param data data vectors in form of either an \eqn{(n\times p)} matrix or a length-\eqn{n} list. See \code{\link{wrap.sphere}} for descriptions on supported input types.
#' @param k the number of clusters (default: 2).
#' @param same.sigma a logical; \code{TRUE} to use same scale parameter across all components, or \code{FALSE} otherwise.
#' @param variants type of the class assignment methods, one of \code{"soft"},\code{"hard"}, and \code{"stochastic"}.
#' @param ... extra parameters including \describe{
#' \item{maxiter}{the maximum number of iterations (default: 50).}
#' \item{eps}{stopping criterion for the EM algorithm (default: 1e-6).}
#' \item{printer}{a logical; \code{TRUE} to show history of the algorithm, \code{FALSE} otherwise.}
#' }
#'
#' @return a named list of S3 class \code{riemmix} containing
#' \describe{
#' \item{cluster}{a length-\eqn{n} vector of class labels (from \eqn{1:k}).}
#' \item{loglkd}{log likelihood of the fitted model.}
#' \item{criteria}{a vector of information criteria.}
#' \item{parameters}{a list containing \code{proportion}, \code{location}, and \code{scale}. See the section for more details.}
#' \item{membership}{an \eqn{(n\times k)} row-stochastic matrix of membership.}
#' }
#'
#' @examples
#' \donttest{
#' # ---------------------------------------------------- #
#' # FITTING THE MODEL
#' # ---------------------------------------------------- #
#' # Load the 'city' data and wrap as 'riemobj'
#' data(cities)
#' locations = cities$cartesian
#' embed2 = array(0,c(60,2))
#' for (i in 1:60){
#' embed2[i,] = sphere.xyz2geo(locations[i,])
#' }
#'
#' # Fit the model with different numbers of clusters
#' k2 = moSL(locations, k=2)
#' k3 = moSL(locations, k=3)
#' k4 = moSL(locations, k=4)
#'
#' # Visualize
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(embed2, col=k2$cluster, pch=19, main="K=2")
#' plot(embed2, col=k3$cluster, pch=19, main="K=3")
#' plot(embed2, col=k4$cluster, pch=19, main="K=4")
#' par(opar)
#'
#' # ---------------------------------------------------- #
#' # USE S3 METHODS
#' # ---------------------------------------------------- #
#' # Use the same 'locations' data as new data
#' # (1) log-likelihood
#' newloglkd = round(loglkd(k3, locations), 5)
#' fitloglkd = round(k3$loglkd, 5)
#' print(paste0("Log-likelihood for K=3 fitted : ", fitloglkd))
#' print(paste0("Log-likelihood for K=3 predicted : ", newloglkd))
#'
#' # (2) label
#' newlabel = label(k3, locations)
#'
#' # (3) density
#' newdensity = density(k3, locations)
#' }
#'
#' @section Parameters of the fitted model:
#' A fitted model is characterized by three parameters. For \eqn{k}-mixture model on a \eqn{(p-1)}
#' sphere in \eqn{\mathbf{R}^p}, (1) \code{proportion} is a length-\eqn{k} vector of component weight
#' that sums to 1, (2) \code{location} is an \eqn{(k\times p)} matrix whose rows are per-cluster locations, and
#' (3) \code{concentration} is a length-\eqn{k} vector of scale parameters for each component.
#'
#' @section Note on S3 methods:
#' There are three S3 methods; \code{loglkd}, \code{label}, and \code{density}. Given a random sample of
#' size \eqn{m} as \code{newdata}, (1) \code{loglkd} returns a scalar value of the computed log-likelihood,
#' (2) \code{label} returns a length-\eqn{m} vector of cluster assignments, and (3) \code{density}
#' evaluates densities of every observation according ot the model fit.
#'
#' @concept sphere
#' @export
moSL <- function(data, k=2, same.sigma=FALSE, variants=c("soft","hard","stochastic"), ...){
## PREPROCESSING -------------------------------------------------------------
spobj = wrap.sphere(data)
x = sp2mat(spobj)
FNAME = "moSL"
pars = list(...)
pnames = names(pars)
if ("maxiter"%in% pnames){
myiter = max(pars$maxiter, 50)
} else {
myiter = 100
}
if ("eps" %in% pnames){
myeps = max(.Machine$double.eps, as.double(pars$eps))
} else {
myeps = 1e-6
}
if ("printer" %in% pnames){
myprint = as.logical(pars$printer)
} else {
myprint = FALSE
}
same.sigma = as.logical(same.sigma)
myn = base::nrow(x)
myp = base::ncol(x)-1
myk = max(1, round(k))
myvers = match.arg(variants)
## INITIALIZATION ------------------------------------------------------------
# label
initlabel = as.vector(stats::kmeans(x, myk, nstart=round(5))$cluster)
# membership
old.eta = array(0,c(myn,myk))
for (i in 1:myn){
old.eta[i,initlabel[i]] = 1 # {0,1} for the initial
}
old.mu = moSL.median(spobj, old.eta)
old.mobj = wrap.sphere(old.mu)
old.d2med = moSL.d2medmat(spobj, old.mobj)
old.sigma = moSL.updatesig(old.eta, old.d2med, myp, homogeneous = same.sigma)
old.pi = as.vector(base::colSums(old.eta)/myn)
old.loglkd = moSL.loglkd(old.d2med, old.sigma, old.pi, myp)
## ITERATION -----------------------------------------------------------------
inc.params = rep(0, 5)
for (it in 1:myiter){
# E-step
new.eta = moSL.eta(old.d2med, old.sigma, old.pi, myp)
# H/S-step by option
if (all(myvers=="hard")){
new.eta = spmix.hard(new.eta)
} else if (all(myvers=="stochastic")){
new.eta = spmix.stochastic(new.eta)
}
# Stop if there is empty cluster
new.label = apply(new.eta, 1, which.max)
if (length(unique(new.label)) < myk){
break
}
# M-step
# M1. mu / centers & d2med
new.mu = moSL.median(spobj, new.eta)
new.mobj = wrap.sphere(new.mu)
new.d2med = moSL.d2medmat(spobj, new.mobj)
# M2. sigma / scales
new.sigma = moSL.updatesig(new.eta, new.d2med, myp, homogeneous = same.sigma)
# M3. proportions
new.pi = as.vector(base::colSums(new.eta)/myn)
# update
new.loglkd = moSL.loglkd(new.d2med, new.sigma, new.pi, myp)
# Incremental changes
inc.params[1] = base::norm(old.mu-new.mu, type = "F")
inc.params[2] = base::sqrt(base::sum((old.sigma - new.sigma)^2))
inc.params[3] = base::sqrt(base::sum((old.pi-new.pi)^2))
inc.params[4] = base::norm(old.d2med-new.d2med, type="F")
inc.params[5] = base::norm(old.eta-new.eta, type="F")
# rule : log-likelihood should increase
if (new.loglkd < old.loglkd){
if (myprint){
print(paste0("* mixsplaplace : terminated at iteration ", it, " : log-likelihood is decreasing."))
}
break
} else {
old.eta = new.eta
old.mu = new.mu
old.mobj = new.mobj
old.d2med = new.d2med
old.sigma = new.sigma
old.pi = new.pi
old.loglkd = new.loglkd
}
if (max(inc.params) < myeps){
if (myprint){
print(paste0("* mixsplaplace : terminated at iteration ", it," : all parameters converged."))
}
break
}
if (myprint){
print(paste0("* mixsplaplace : iteration ",it,"/",myiter," complete with loglkd=",round(old.loglkd,5),"."))
}
}
## INFORMATION CRITERION -----------------------------------------------------
if (!same.sigma){
par.k = myp*myk + myk + (myk-1)
} else {
par.k = myp*myk + 1 + (myk-1)
}
AIC = -2*old.loglkd + 2*par.k
BIC = -2*old.loglkd + par.k*log(myn)
HQIC = -2*old.loglkd + 2*par.k*log(log(myn))
AICc = AIC + (2*(par.k^2) + 2*par.k)/(myn-par.k-1)
infov = c(AIC, AICc, BIC, HQIC)
names(infov) = c("AIC","AICc","BIC","HQIC")
## RETURN --------------------------------------------------------------------
output = list()
output$cluster = spmix.getcluster(old.eta)
output$loglkd = old.loglkd
output$criteria = infov
output$parameters = list(proportion=old.pi, location=old.mu, scale=old.sigma)
output$membership = old.eta
return(structure(output, class=c("moSL","riemmix")))
}
# auxiliary functions -----------------------------------------------------
# 1. moSL.median : weighted Frechet medians
# 2. moSL.d2medmat : compute data-median pairwise distance
# 3. moSL.solvesig : minimize C/sigma + log(C(sigma))
# 4. moSL.updatesig : update sigma (scale) parameters
# 5. moSL.loglkd : log-likelihood
# 6. moSL.eta : compute soft membership
# 1. compute weighted Frechet median : deals with Riemdata object
#' @keywords internal
#' @noRd
moSL.median <- function(spobj, membership){
N = base::length(spobj$data)
P = base::length(as.vector(spobj$data[[1]]))
K = base::ncol(membership)
output = array(0,c(K,P))
for (k in 1:K){
partweight = as.vector(membership[,k])
if (sum(partweight==1)==1){
output[k,] = as.vector(spobj$data[[which(partweight==1)]])
} else {
sel_id = (partweight > .Machine$double.eps)
sel_data = spobj$data[sel_id]
sel_weight = partweight[sel_id]
sel_weight = sel_weight/base::sum(sel_weight)
output[k,] = as.vector(inference_median_intrinsic("sphere", sel_data, sel_weight, 100, 1e-6)$median)
}
}
return(output)
}
# 2. moSL.d2medmat
#' @keywords internal
#' @noRd
moSL.d2medmat <- function(obj.data, obj.medians){
d2sqmat = as.matrix(basic_pdist2("sphere", obj.data$data, obj.medians$data, "intrinsic"))
return(d2sqmat)
}
# 3. moSL.solvesig : minimize C/sigma + log(C(sigma))
#' @keywords internal
#' @noRd
moSL.solvesig <- function(C, p){
# cost function
fun_cost <- function(sigma){
# term : first
out1 = C/sigma
# term : second
myfunc <- function(r){
return(exp(-r/sigma)*(sin(r)^(p-1)))
}
t1 = 2*(pi^(p/2))/(gamma(p/2))
t2 = stats::integrate(myfunc, lower=sqrt(.Machine$double.eps), upper=pi, rel.tol=sqrt(.Machine$double.eps))$value
out2 = log(t1) + log(t2)
# return the output
return(out1+out2)
}
# minimization
myint = c(0.01, 100)*C
output = as.double(stats::optimize(fun_cost, interval=myint, maximum=FALSE, tol=1e-6)$minimum)
return(output)
}
# 4. moSL.updatesig : update sigma (scale) parameters
#' @keywords internal
#' @noRd
moSL.updatesig <- function(membership, d2mat, p, homogeneous=TRUE){
N = base::nrow(membership)
K = base::ncol(membership)
mysigs = rep(0, K)
if (homogeneous){ # homogeneous
A = base::sum(d2mat*membership)
B = base::sum(membership)
C = (A/B)
sig.single = moSL.solvesig(C, p)
for (k in 1:K){
mysigs[k] = sig.single
}
} else { # heterogeneous
for (k in 1:K){
A = sum(as.vector(d2mat[,k])*as.vector(membership[,k]))
B = sum(as.vector(membership[,k]))
C = (A/B)
mysigs[k] = moSL.solvesig(C, p)
}
}
return(mysigs)
}
# 5. moSL.loglkd : log-likelihood
#' @keywords internal
#' @noRd
moSL.loglkd <- function(d2med, sigmas, props, p){
N = base::nrow(d2med)
K = base::ncol(d2med)
# function to evaluate normalizing constant
eval_constant <- function(sigma){
myfunc <- function(r){
return(exp(-r/sigma)*(sin(r)^(p-1)))
}
t1 = 2*(pi^(p/2))/(gamma(p/2))
t2 = stats::integrate(myfunc, lower=sqrt(.Machine$double.eps), upper=pi, rel.tol=sqrt(.Machine$double.eps))$value
return(t1*t2)
}
# normalizing constants per class
vecCsig = rep(0, K)
for (k in 1:K){
vecCsig[k] = eval_constant(sigmas[k])
}
# evaluate the density
mixdensity = array(0,c(N,K))
for (n in 1:N){
for (k in 1:K){
mixdensity[n,k] = props[k]*exp(-(d2med[n,k])/sigmas[k])/vecCsig[k]
}
}
# evaluate the output
return(base::sum(base::log(base::rowSums(mixdensity))))
}
# 6. moSL.eta
#' @keywords internal
#' @noRd
moSL.eta <- function(d2med, sigmas, props, p){
N = base::nrow(d2med)
K = base::ncol(d2med)
# function to evaluate normalizing constant
eval_constant <- function(sigma){
myfunc <- function(r){
return(exp(-r/sigma)*(sin(r)^(p-1)))
}
t1 = 2*(pi^(p/2))/(gamma(p/2))
t2 = stats::integrate(myfunc, lower=sqrt(.Machine$double.eps), upper=pi, rel.tol=sqrt(.Machine$double.eps))$value
return(t1*t2)
}
# normalizing constants per class
vecCsig = rep(0, K)
for (k in 1:K){
vecCsig[k] = eval_constant(sigmas[k])
}
# evaluate
output = array(0,c(N,K))
for (k in 1:K){
tgtdvec = as.vector(d2med[,k])
output[,k] = exp((-tgtdvec/sigmas[k]) - base::log(vecCsig[k]) + base::log(props[k]))
}
for (n in 1:N){
tgtrow = as.vector(output[n,])
output[n,] = tgtrow/base::sum(tgtrow)
}
return(output)
}
# Methods -----------------------------------------------------------------
# () loglkd : compute the log-likelihood
# () label : predict the labels
# () density : evaluate the density
# S3 METHOD : LOGLKD
#' @param object a fitted \code{moSL} model from the \code{\link{moSL}} function.
#' @param newdata data vectors in form of either an \eqn{(m\times p)} matrix or a length-\eqn{m} list. See \code{\link{wrap.sphere}} for descriptions on supported input types.
#' @rdname moSL
#' @concept sphere
#' @export
loglkd.moSL <- function(object, newdata){
# PREPARE
if (!inherits(object, "moSL")){
stop("* loglkd.moSL : input is not an object of 'moSL' class.")
}
spobj = wrap.sphere(newdata)
myp = base::ncol(object$parameters$location)-1
# INTERMEDIATE VALUES
old.mu = object$parameters$location
old.mobj = wrap.sphere(old.mu)
old.sigma = object$parameters$scale
old.pi = object$parameters$proportion
old.d2med = moSL.d2medmat(spobj, old.mobj)
# COMPUTE AND RETURN
return(moSL.loglkd(old.d2med, old.sigma, old.pi, myp))
}
# S3 METHOD : LABEL
#' @rdname moSL
#' @concept sphere
#' @export
label.moSL <- function(object, newdata){
# PREPARE
if (!inherits(object, "moSL")){
stop("* label.moSL : input is not an object of 'moSL' class.")
}
spobj = wrap.sphere(newdata)
myp = base::ncol(object$parameters$location)-1
# INTERMEDIATE VALUES
old.mu = object$parameters$location
old.mobj = wrap.sphere(old.mu)
old.sigma = object$parameters$scale
old.pi = object$parameters$proportion
old.d2med = moSL.d2medmat(spobj, old.mobj)
# COMPUTE, EXTRACT, AND RETURN
fin.eta = moSL.eta(old.d2med, old.sigma, old.pi, myp)
output = spmix.getcluster(fin.eta)
return(output)
}
# S3 METHOD : DENSITY
#' @rdname moSL
#' @concept sphere
#' @export
density.moSL <- function(object, newdata){
# PREPARE
if (!inherits(object, "moSL")){
stop("* density.moSL : input is not an object of 'moSL' class.")
}
spobj = wrap.sphere(newdata)
myp = base::ncol(object$parameters$location)-1
# INTERMEDIATE VALUES
old.mu = object$parameters$location
old.mobj = wrap.sphere(old.mu)
old.sigma = object$parameters$scale
old.pi = object$parameters$proportion
old.d2med = moSL.d2medmat(spobj, old.mobj)
# COMPUTE AND RETURN
evaldensity = density_mixture_SL(old.d2med, old.sigma, old.pi, myp)
return(evaldensity)
}
#' @keywords internal
#' @noRd
density_mixture_SL <- function(d2med, sigmas, props, p){
N = base::nrow(d2med)
K = base::ncol(d2med)
# normalizing constant
eval_constant <- function(sigma){
myfunc <- function(r){
return(exp(-r/sigma)*(sin(r)^(p-1)))
}
t1 = 2*(pi^(p/2))/(gamma(p/2))
t2 = stats::integrate(myfunc, lower=sqrt(.Machine$double.eps), upper=pi, rel.tol=sqrt(.Machine$double.eps))$value
return(t1*t2)
}
vecCsig = rep(0,K)
for (k in 1:K){
vecCsig[k] = eval_constant(sigmas[k])
}
# evaluate the density
mixdensity = array(0,c(N,K))
for (n in 1:N){
for (k in 1:K){
mixdensity[n,k] = props[k]*exp(-d2med[n,k]/sigmas[k])/vecCsig[k]
}
}
# evaluate the output
output = as.vector(base::rowSums(mixdensity))
return(output)
}
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