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#' Find Scale Parameter for Generalised Beta Prime (Half-Cauchy) Hyperprior
#'
#' This function implements a optimisation routine that computes the scale parameter \eqn{\theta}
#' of the gamma prior for \eqn{\tau^2} (corresponding to a half cauchy for \eqn{\tau})
#' for a given design matrix and prior precision matrix
#' such that approximately \eqn{P(|f(x_{k}|\le c,k=1,\ldots,p)\ge 1-\alpha}
#'
#' @param alpha denotes the 1-\eqn{\alpha} level.
#' @param method with \code{integrate} as default.
#' Currently no further method implemented.
#' @param Z the design matrix.
#' @param c denotes the expected range of the function.
#' @param eps denotes the error tolerance of the result, default is \code{.Machine$double.eps}.
#' @param Kinv the generalised inverse of K.
#' @return an object of class \code{list} with values from \code{\link{uniroot}}.
#' @author Nadja Klein
#' @references Nadja Klein and Thomas Kneib (2015). Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression.
#' \emph{Working Paper}.
#'
#' Andrew Gelman (2006). Prior Distributions for Variance Parameters in Hierarchical Models.
#' \emph{Bayesian Analysis}, \bold{1}(3), 515--533.
#' @import splines
#' @import MASS
#' @import GB2
#' @import stats
#' @export
#' @examples
#' set.seed(123)
#' require(MASS)
#' # prior precision matrix (second order differences)
#' # of a spline of degree l=3 and with m=20 inner knots
#' # yielding dim(K)=m+l-1=22
#' K <- t(diff(diag(22), differences=2))%*%diff(diag(22), differences=2)
#' # generalised inverse of K
#' Kinv <- ginv(K)
#' # covariate x
#' x <- runif(1)
#' Z <- matrix(DesignM(x)$Z_B,nrow=1)
#' theta <- get_theta_gbp(alpha = 0.01, method = "integrate", Z = Z,
#' c = 3, eps = .Machine$double.eps, Kinv = Kinv)$root
#'
#' \dontrun{
#'
#' set.seed(91179)
#' library(BayesX)
#' library(MASS)
#' # prior precision matrix to zambia data set
#' K <- read.gra(system.file("examples/zambia.gra", package="sdPrior"))
#' # generalised inverse of K
#' Kinv <- ginv(K)
#'
#' # read data
#' dat <- read.table(system.file("examples/zambia_height92.raw", package="sdPrior"), header = TRUE)
#'
#' # design matrix for spatial component
#' Z <- t(sapply(dat$district, FUN=function(x){1*(x==rownames(K))}))
#'
#' # get scale parameter
#' theta <- get_theta_gbp(alpha = 0.01, method = "integrate", Z = Z,
#' c = 3, eps = .Machine$double.eps, Kinv = Kinv)$root
#' }
#'
get_theta_gbp <- function(alpha = 0.01, method = "integrate", Z, c = 3, eps = .Machine$double.eps, Kinv)
{
if(method != "integrate")
stop("method not existing")
ztKz <- diag(Z%*%Kinv%*%t(Z))
#number of grids of x are given by the rows of Z
if(NROW(Z) == 1 | NCOL(Z) == 1)
{
nknots <- 1
} else {
nknots <- NROW(Z)
}
#weights such that sum(weights) = nknots
weights <- rep(1, nknots)
#alpha-level for each point x (alphax = weight * alpha / nknots)
alphafx <- alpha * weights / nknots
eps2 <- eps3 <- eps4 <- eps
marginal_df <- function(f, lambda, ztz)
{
integrand <- function(tau2)
{
dnorm(f, mean = 0, sd = sqrt(tau2 * ztz)) * dgb2(tau2, shape1=1, scale=lambda^2, shape2=0.5, shape3=0.5) #dgenbetaII(tau2, scale = lambda^2, shape1.a=1, shape2.p=0.5, shape3.q=0.5)
}
res <- try(integrate(integrand, 0, Inf)$value, TRUE)
while(inherits(res, "try-error"))
{
eps2 <- eps2 * 10
res <- try(integrate(integrand, eps2, Inf)$value, TRUE)
}
return(res)
}
marginal_Pf <- function(lambda, Cov, alpha)
{
if(method == "integrate")
{
tempvar <- 0
for(countnknots in 1:NROW(Z)) {
contri <- try(2*integrate(Vectorize(marginal_df), -c, 0, lambda = lambda, ztz = Cov[countnknots])$value, TRUE)
while(inherits(contri, "try-error")) {
eps3 <- eps3 * 10
contri <- try(2*integrate(Vectorize(marginal_df), -c, eps3, lambda = lambda, ztz = Cov[countnknots])$value, TRUE)
}
tempvar <- tempvar + contri
}
NROW(Z) - alpha - tempvar
} else {
stop("selected method not implemented.")
}
}
# findlowercands <- .Machine$double.eps
# vals <- (2*NROW(Z)*integrate(Vectorize(marginal_df), -c, 0, lambda = findlowercands[length(findlowercands)], ztz = ztKz[1])$value-(NROW(Z)-alpha))
# while(findlowercands[length(findlowercands)]<100)
# {
# findlowercands <- c(findlowercands,10*findlowercands[length(findlowercands)])
# vals <- c(vals,(2*NROW(Z)*integrate(Vectorize(marginal_df), -c, 0, lambda = findlowercands[length(findlowercands)], ztz = ztKz[1])$value-(NROW(Z)-alpha)))
# }
# findlower <- findlowercands[which.min(abs(vals))]
# while((2*NROW(Z)*integrate(Vectorize(marginal_df), -c, 0, lambda = findlower, ztz = ztKz[1])$value-(NROW(Z)-alpha))<=0)
# {
# findlower <- findlower*0.99
# }
# result <- try(uniroot(marginal_Pf, interval = c(findlower, 1000), Cov = ztKz, alpha = alpha), TRUE)
# while(inherits(result, "try-error"))
# {
# findlower <- findlower*1.01
# result <- try(uniroot(marginal_Pf, interval = c(eps4, 1000), Cov = ztKz, alpha = alpha), TRUE)
# }
result <- try(uniroot(marginal_Pf, interval = c(0.008, 1000), Cov = ztKz, alpha = alpha), TRUE)
if(inherits(result, "try-error"))
{
if(grepl("not of opposite sign",as.character(result,"condition")))
result <- try(uniroot(marginal_Pf, interval = c(0.008, 1000), flower=0.000001, Cov = ztKz, alpha = alpha), TRUE)
}
trials <- 0
while(inherits(result, "try-error"))
{
if(trials <= 10)
{
eps4 <- 0.008*1.01
result <- try(uniroot(marginal_Pf, interval = c(eps4, 1000), Cov = ztKz, alpha = alpha), TRUE)
trials <- trials+1
} else {
findlowercands <- seq(0.007,0.008,length=10)
vals <- c()
for(counti in 1:length(findlowercands))
{
vals <- c(vals,(2*NROW(Z)*integrate(Vectorize(marginal_df), -c, 0, lambda = findlowercands[counti], ztz = ztKz[1])$value-(NROW(Z)-alpha)))
}
result2 <- list(warns="finding the root is not possible", root=findlowercands[which.min(abs(vals))])
warning("finding the root is not possible")
return(result2)
}
}
return(result)
}
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