# Samples from the posterior distribution of the Wrapped Normal spatial temporal model The `WrapSpTi` function returns samples from the posterior distribution of the spatio-temporal Wrapped Gaussian Model ```r WrapSpTi(x = x, coords = coords, times, start = list(alpha = c(2, 1), rho_sp = c(0.1, 0.5), rho_t = c(0.1, 1), sep_par = c(0.01, 0.1), k = sample(0, length(x), replace = T)), priors = list(alpha = c(pi, 1, -10, 10), rho_sp = c(8, 14), rho_t = c(1, 2), sep_par = c(0.001, 1), sigma2 = c()), sd_prop = list(rho_sp = 0.5, rho_t = 0.5, sep_par = 0.5, sigma2 = 0.5), iter = 1000, BurninThin = c(burnin = 20, thin = 10), accept_ratio = 0.234, adapt_param = c(start = 1, end = 1e+07, exp = 0.9), n_chains = 1, parallel = FALSE, n_cores = 1) ``` ## Arguments - `x`: a vector of n circular data in $[0,2\pi)$. If they are not in $[0,2\pi)$, the function will transform the data into the right interval - `coords`: an nx2 matrix with the sites coordinates - `times`: an n vector with the times of the observations x - `start`: a list of 4 elements giving initial values for the model parameters. Each elements is a vector with `n_chains` elements * alpha the mean which value is in $[0,2\pi)$ * rho_sp the spatial decay parameter, * rho_t the temporal decay parameter, * sigma2 the process variance, * sep_par the separation parameter, * k the vector of `length(x)` winding numbers - `priors`: a list of 5 elements to define priors for the model parameters: - **alpha**: a vector of 2 elements the mean and the variance of a Gaussian distribution, default is mean $\pi$ and variance 1, - **rho_sp**: a vector of 2 elements defining the minimum and maximum of a uniform distribution, - **rho_t**: a vector of 2 elements defining the minimum and maximum of a uniform distribution, - **sep_par**: a vector of 2 elements defining the two parameters of a beta distribution, - **sigma2**: a vector of 2 elements defining the shape and rate of an inverse-gamma distribution, - `sd_prop`: list of 3 elements. To run the MCMC for the rho_sp and sigma2 parameters we use an adaptive metropolis and in sd_prop we build a list of initial guesses for these two parameters and the beta parameter - `iter`: iter number of iterations - `BurninThin`: a vector of 2 elements with the burnin and the chain thinning - `accept_ratio`: it is the desired acceptance ratio in the adaptive metropolis - `adapt_param`: a vector of 3 elements giving the iteration number at which the adaptation must start and end. The third element (exp) must be a number in (0,1) and it is a parameter ruling the speed of changes in the adaptation algorithm, it is recommended to set it close to 1, if it is too small non positive definite matrices may be generated and the program crashes. - `n_chains`: integer, the number of chains to be launched (default is 1, but we recommend to use at least 2 for model diagnostic) - `parallel`: logical, if the multiple chains must be lunched in parallel (you should install doParallel package). Default is FALSE - `n_cores`: integer, required if parallel=TRUE, the number of cores to be used in the implementation. Default value is 1. ## Returns it returns a list of `n_chains` lists each with elements - **`alpha`, `rho_sp`, `rho_t`, `sep_par`, `sigma2`**: vectors with the thinned chains - **`k`**: a matrix with `nrow = length(x)` and `ncol =` the length of thinned chains - **`distribution`**: characters, always "WrapSpTi" ## Implementation Tips To facilitate the estimations, the observations x are centered around pi, and the prior and starting value of alpha are changed accordingly. After the estimations, posterior samples of alpha are changed back to the original scale ## Examples ```r 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)) #### move to the prediction step with WrapKrigSpTi ``` ## References G. Mastrantonio, G. Jona Lasinio, A. E. Gelfand, "Spatio-temporal circular models with non-separable covariance structure", TEST 25 (2016), 331–350. T. Gneiting, "Nonseparable, Stationary Covariance Functions for Space-Time Data", JASA 97 (2002), 590-600 ## See Also `WrapKrigSpTi` for spatio-temporal prediction, `ProjSpTi` to sample from the posterior distribution of the spatio-temporal Projected Normal model and `ProjKrigSpTi` for spatio-temporal prediction under the same model Other spatio-temporal models: `ProjSpTi`

WrapSpTi function