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#' Prediction using wrapped normal spatio-temporal model.
#'
#' \code{WrapKrigSpTi} function computes the spatio-temporal prediction
#' for circular space-time data using samples from the posterior distribution
#' of the space-time wrapped normal model
#'
#' @param WrapSpTi_out the functions takes the output of \code{\link{WrapSpTi}} function
#' @param coords_obs coordinates of observed locations (in UTM)
#' @param coords_nobs coordinates of unobserved locations (in UTM)
#' @param times_obs numeric vector of observed time coordinates
#' @param times_nobs numeric vector of unobserved time coordinates
#' @param x_obs observed values
#' @return a list of 3 elements
#' \describe{
#' \item{\code{M_out}}{ the mean of the associated linear process on the prediction locations coords_nobs (rows) over all the posterior samples (columns) returned by \code{\link{WrapSpTi}}}
#' \item{\code{V_out}}{ the variance of the associated linear process on the prediction locations coords_nobs (rows) over all the posterior samples (columns) returned by \code{\link{WrapSpTi}}}
#' \item{\code{Prev_out}}{ the posterior predicted values at the unobserved locations}
#' }
#' @section Implementation Tips:
#' To facilitate the estimations, the observations x
#' are centered around \eqn{\pi}.
#' Posterior samples of x at the predictive locations and posterior mean are changed back
#' to the original scale
#'
#' @family spatio-temporal estimations
#' @seealso \code{\link{WrapSpTi}} spatio-temporal sampling from
#' Wrapped Normal,
#' \code{\link{ProjSpTi}} for spatio-temporal sampling from
#' Projected Normal and \code{\link{ProjKrigSpTi}} for
#' Kriging estimation
#' @references G. Mastrantonio, G. Jona Lasinio, A. E. Gelfand, "Spatio-temporal circular models with
#' non-separable covariance structure", TEST 25 (2016), 331–350
#' @references T. Gneiting, "Nonseparable, Stationary Covariance Functions for Space-Time
#' Data", JASA 97 (2002), 590-600
#' @examples
#' library(CircSpaceTime)
#' ## functions
#' rmnorm <- function(n = 1, mean = rep(0, d), varcov){
#' d <- if (is.matrix(varcov))
#' ncol(varcov)
#' else 1
#' z <- matrix(rnorm(n * d), n, d) %*% chol(varcov)
#' y <- t(mean + t(z))
#' return(y)
#' }
#'
#' ######################################
#' ## Simulation ##
#' ######################################
#' set.seed(1)
#' n <- 20
#' ### simulate coordinates from a unifrom distribution
#' coords <- cbind(runif(n,0,100), runif(n,0,100)) #spatial coordinates
#' coordsT <- sort(runif(n,0,100)) #time coordinates (ordered)
#' Dist <- as.matrix(dist(coords))
#' DistT <- as.matrix(dist(coordsT))
#'
#' rho <- 0.05 #spatial decay
#' rhoT <- 0.01 #temporal decay
#' sep_par <- 0.5 #separability parameter
#' sigma2 <- 0.3 # variance of the process
#' alpha <- c(0.5)
#' #Gneiting covariance
#' SIGMA <- sigma2 * (rhoT * DistT^2 + 1)^(-1) * exp(-rho * Dist/(rhoT * DistT^2 + 1)^(sep_par/2))
#'
#' Y <- rmnorm(1,rep(alpha, times = n), SIGMA) #generate the linear variable
#' theta <- c()
#' ## wrapping step
#' for(i in 1:n) {
#' theta[i] <- Y[i] %% (2*pi)
#' }
#' ### Add plots of the simulated data
#'
#' rose_diag(theta)
#' ## use this values as references for the definition of initial values and priors
#' rho_sp.min <- 3/max(Dist)
#' rho_sp.max <- rho_sp.min+0.5
#' rho_t.min <- 3/max(DistT)
#' rho_t.max <- rho_t.min+0.5
#' val <- sample(1:n,round(n*0.2)) #validation set
#' set.seed(100)
#' mod <- WrapSpTi(
#' x = theta[-val],
#' coords = coords[-val,],
#' times = coordsT[-val],
#' start = list("alpha" = c(.79, .74),
#' "rho_sp" = c(.33,.52),
#' "rho_t" = c(.19, .43),
#' "sigma2" = c(.49, .37),
#' "sep_par" = c(.47, .56),
#' "k" = sample(0,length(theta[-val]), replace = TRUE)),
#' priors = list("rho_sp" = c(0.01,3/4), ### uniform prior on this interval
#' "rho_t" = c(0.01,3/4), ### uniform prior on this interval
#' "sep_par" = c(1,1), ### beta prior
#' "sigma2" = c(5,5),## inverse gamma prior with mode=5/6
#' "alpha" = c(0,20) ## wrapped gaussian with large variance
#' ) ,
#' sd_prop = list( "sigma2" = 0.1, "rho_sp" = 0.1, "rho_t" = 0.1,"sep_par"= 0.1),
#' iter = 7000,
#' BurninThin = c(burnin = 3000, thin = 10),
#' accept_ratio = 0.234,
#' adapt_param = c(start = 1, end = 1000, exp = 0.5),
#' n_chains = 2 ,
#' parallel = FALSE,
#' n_cores = 1
#' )
#' check <- ConvCheck(mod,startit = 1 ,thin = 1)
#' check$Rhat ## convergence has been reached
#' ## when plotting chains remember that alpha is a circular variable
#' par(mfrow = c(3,2))
#' coda::traceplot(check$mcmc)
#' par(mfrow = c(1,1))
#'
#'
#' ############## Prediction on the validation set
#' Krig <- WrapKrigSpTi(
#' WrapSpTi_out = mod,
#' coords_obs = coords[-val,],
#' coords_nobs = coords[val,],
#' times_obs = coordsT[-val],
#' times_nobs = coordsT[val],
#' x_obs = theta[-val]
#' )
#' ### checking the prediction
#' Wrap_Ape <- APEcirc(theta[val], Krig$Prev_out)
#' Wrap_Crps <- CRPScirc(theta[val], Krig$Prev_out)
#' @export
WrapKrigSpTi <- function(
WrapSpTi_out,
coords_obs,
coords_nobs,
times_obs,
times_nobs,
x_obs
)
{
## ## ## ## ## ## ##
## Posterior samples
## ## ## ## ## ## ##
pp <- unlist(WrapSpTi_out)
sigma2 <- as.numeric(pp[regexpr("sigma2",names(pp)) == 1])
alpha <- as.numeric(pp[regexpr("alpha",names(pp)) == 1])
rho_sp <- as.numeric(pp[regexpr("rho_sp",names(pp)) == 1])
rho_t <- as.numeric(pp[regexpr("rho_t",names(pp)) == 1])
sep_par <- as.numeric(pp[regexpr("sep_par",names(pp)) == 1])
row.k <- nrow(WrapSpTi_out[[1]]$k)
pp2 <- as.numeric(pp[regexpr("k",names(pp)) == 1])
k <- matrix(pp2,nrow = row.k)
rm(pp,pp2)
## ## ## ## ## ## ##
## Observations are centerer around pi, and the posterior values of
## alpha are changed accordingly.
## ## ## ## ## ## ##
MeanCirc <- atan2(sum(sin(x_obs)),sum(cos(x_obs)))
x_obs <- (x_obs - MeanCirc + pi) %% (2*pi)
alpha <- (alpha + MeanCirc - pi) %% (2*pi)
## ## ## ## ## ## ##
## Indices
## ## ## ## ## ## ##
n <- nrow(k)
nprev <- nrow(coords_nobs)
nsample <- ncol(k)
## ## ## ## ## ## ##
## Distance matrix for observed and non observed data
## ## ## ## ## ## ##
H_tot <- as.matrix(stats::dist(rbind(coords_obs,coords_nobs)))
Ht_tot <- as.matrix(stats::dist(c(times_obs,times_nobs)))
## ## ## ## ## ## ##
## Model estimation
## ## ## ## ## ## ##
out <- WrapKrigSpTiCpp(sigma2, alpha, rho_sp, rho_t,sep_par, k,
n, nsample, H_tot, Ht_tot, nprev, x_obs)
## ## ## ## ## ## ##
## Posterior samples of x and posterior mean are changed back to
## the original scale
## ## ## ## ## ## ##
out$Prev_out <- (out$Prev_out - pi + MeanCirc) %% (2*pi)
out$M_out <- (out$M_out - pi + MeanCirc) %% (2*pi)
return(out)
}
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