ProjKrigSpTi function

#' Spatio temporal interpolation using projected spatial temporal normal model.

ProjKrigSpTi function computes the spatio-temporal prediction for circular space-time data using samples from the posterior distribution of the space-time projected normal model.

ProjKrigSpTi(ProjSpTi_out, coords_obs, coords_nobs, times_obs, times_nobs,
  x_obs)

Arguments

• ProjSpTi_out: the functions takes the output of ProjSpTi function

• coords_obs: coordinates of observed locations (in UTM)

• coords_nobs: coordinates of unobserved locations (in UTM)

• times_obs: numeric vector of observed time coordinates

• times_nobs: numeric vector of unobserved time coordinates

• x_obs: observed values in $[0,2\pi)$

  If they are not in $[0,2\pi)$, the function will tranform the data in the right interval

Returns

a list of 3 elements

• M_out: the mean of the associated linear process on the prediction locations coords_nobs (rows) over all the posterior samples (columns) returned by ProjSpTi
• V_out: the variance of the associated linear process on the prediction locations coords_nobs (rows) over all the posterior samples (columns) returned by ProjSpTi
• Prev_out: are the posterior predicted values at the unobserved locations.

Examples

library(CircSpaceTime)
#### simulated example
## auxiliary 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 using a gneiting covariance function
####
set.seed(1)
n <- 20

coords <- cbind(runif(n, 0, 100), runif(n, 0, 100))
coordsT <- cbind(runif(n, 0, 100))
Dist <- as.matrix(dist(coords))
DistT <- as.matrix(dist(coordsT))

rho <- 0.05
rhoT <- 0.01
sep_par <- 0.1
sigma2 <- 1
alpha <- c(0.5)
SIGMA <- sigma2 * (rhoT * DistT^2 + 1)^(-1) * exp(-rho * Dist / (rhoT * DistT^2 + 1)^(sep_par / 2))
tau <- 0.2

Y <- rmnorm(
  1, rep(alpha, times = n),
  kronecker(SIGMA, matrix(c(sigma2, sqrt(sigma2) * tau, sqrt(sigma2) * tau, 1), nrow = 2))
)
theta <- c()
for (i in 1:n) {
  theta[i] <- atan2(Y[(i - 1) * 2 + 2], Y[(i - 1) * 2 + 1])
}
theta <- theta %% (2 * pi) ## to be sure we have values in (0,2pi)
rose_diag(theta)
################ some useful quantities
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
### validation set 20% of the data
val <- sample(1:n, round(n * 0.2))

set.seed(200)

mod <- ProjSpTi(
  x = theta[-val],
  coords = coords[-val, ],
  times = coordsT[-val],
  start = list(
    "alpha" = c(0.66, 0.38, 0.27, 0.13),
    "rho_sp" = c(0.29, 0.33),
    "rho_t" = c(0.25, 0.13),
    "sep_par" = c(0.56, 0.31),
    "tau" = c(0.71, 0.65),
    "sigma2" = c(0.47, 0.53),
    "r" = abs(rnorm(length(theta[-val])))
  ),
  priors = list(
    "rho_sp" = c(rho_sp.min, rho_sp.max), # Uniform prior in this interval
    "rho_t" = c(rho_t.min, rho_t.max), # Uniform prior in this interval
    "sep_par" = c(1, 1), # Beta distribution
    "tau" = c(-1, 1), ## Uniform prior in this interval
    "sigma2" = c(10, 3), # inverse gamma
    "alpha_mu" = c(0, 0), ## a vector of 2 elements,
    ## the means of the bivariate Gaussian distribution
    "alpha_sigma" = diag(10, 2) # a 2x2 matrix, the covariance matrix of the
    # bivariate Gaussian distribution,
  ),
  sd_prop = list(
    "sep_par" = 0.1, "sigma2" = 0.1, "tau" = 0.1, "rho_sp" = 0.1, "rho_t" = 0.1,
    "sdr" = sample(.05, length(theta), replace = TRUE)
  ),
  iter = 4000,
  BurninThin = c(burnin = 2000, thin = 2),
  accept_ratio = 0.234,
  adapt_param = c(start = 155000, end = 156000, exp = 0.5),
  n_chains = 2,
  parallel = TRUE,
)
check <- ConvCheck(mod)
check$Rhat ### convergence has been reached when the values are close to 1
#### graphical checking
#### recall that it is made of as many lists as the number of chains and it has elements named
#### as the model's parameters
par(mfrow = c(3, 3))
coda::traceplot(check$mcmc)
par(mfrow = c(1, 1))
# once convergence is reached we run the interpolation on the validation set
Krig <- ProjKrigSpTi(
  ProjSpTi_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

Proj_ape <- APEcirc(theta[val], Krig$Prev_out)
Proj_crps <- CRPScirc(theta[val],Krig$Prev_out)

References

G. Mastrantonio, G.Jona Lasinio, A. E. Gelfand, "Spatio-temporal circular models with non-separable covariance structure", TEST 25 (2016), 331–350.

F. Wang, A. E. Gelfand, "Modeling space and space-time directional data using projected Gaussian processes", Journal of the American Statistical Association,109 (2014), 1565-1580

T. Gneiting, "Nonseparable, Stationary Covariance Functions for Space-Time Data", JASA 97 (2002), 590-600

See Also

ProjSpTi to sample the posterior distribution of the spatio-temporal Projected Normal model, WrapSpTi to sample the posterior distribution of the spatio-temporal Wrapped Normal model and WrapKrigSpTi for spatio-temporal interpolation under the same model