APEcirc function

Average Prediction Error for circular Variables.

APEcirc computes the average prediction error (APE), defined as the average circular distance across pairs

APEcirc(real, sim, bycol = F)

Arguments

• real: a vector of the values of the process at the test locations
• sim: a matrix with nrow = the test locations and ncol = the number of posterior samples from the posterior distributions by WrapKrigSp WrapKrigSpTi, ProjKrigSp, ProjKrigSpTi
• bycol: logical. It is TRUE if the columns of sim represent the observations and the rows the posterior samples, the default value is FALSE.

Returns

a list of two elements

• ApePoints: a vector of APE, one element for each test point
• Ape: the overall mean

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)

References

G. Jona Lasinio, A. Gelfand, M. Jona-Lasinio, "Spatial analysis of wave direction data using wrapped Gaussian processes", The Annals of Applied Statistics 6 (2013), 1478-1498

See Also

ProjKrigSp and WrapKrigSp for posterior spatial estimations, ProjKrigSpTi and WrapKrigSpTi for posterior spatio-temporal estimations

Other model performance indices: CRPScirc