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BayesianGridSpacing.R
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BayesianGridSpacing.R
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library(BayesianTools)
library(sp)
library(gstat) #for MoM estimation of variogram
library(ggplot2)
exponential <- function(D, pars) {
C <- pars[1] * pars[2] * (exp(-D/pars[3]))
diag(C) <- pars[1]
C
}
spherical <- function(D, pars) {
C <- ifelse(D<pars[3], pars[1] * pars[2] * (1 - (1.5 * D/pars[3]) + 0.5 * (D/pars[3])^3), 0)
diag(C) <- pars[1]
C
}
# Define loglikelihood
ll <-function(pars) {
C<-spherical(D, pars)
cholC <- try(chol(C), silent = TRUE)
if (!is.character(cholC)){
C_inv <- chol2inv(chol(C))
### estimate the unknown mean
XC <- crossprod(X, C_inv)
XCy <- XC %*% y
XCX <- XC %*% X
betahat <- solve(XCX , XCy)
mu <- as.numeric(t(betahat) %*% t(X))
e <- y - mu
eC <- crossprod(e, C_inv)
eCe <- eC %*% e
logDetC <- determinant(x = C, logarithm = TRUE)$modulus
logLik <- as.numeric(-0.5 * (logDetC + eCe +length(y) * log(2 * pi)))
return(logLik)
}else{return(-Inf)}
}
# Load existing sampling points
load(file = "Data/DataThreeWoredasEthiopia.RData")
# compute matrix with Euclidian distances between sampling points
D <- spDists(priordataEthiopia)
# Estimate variogram by method-of-moments
vg <- variogram(SOC~1, data = priordataEthiopia)
plot(vg)
# vgfit <- fit.variogram(vg, model = vgm("Exp", psill = 0.6, range = 30, nugget = 0.6))
vgfit <- fit.variogram(vg, model = vgm("Sph", psill = 0.6, range = 30, nugget = 0.6))
plot(vg, vgfit)
print(vgfit)
# estimate variogram by maximum likelihood. Use MoM variogram parameters as initial values
sigma2.ini <- vgfit$psill[1]+vgfit$psill[2]
xi.ini <- vgfit$psill[2]/sigma2.ini #proportion spatially structured variance
phi.ini <- vgfit$range[2]
pars <- c(sigma2.ini, xi.ini, phi.ini)
X <- matrix(1, nrow(d), 1)
y <- d$SOC
vgML <- optim(pars, ll, control = list(fnscale = -1), hessian = TRUE)
#invfisher_info <- solve(-vgML$hessian)
#diag(invfisher_info)
sigma2.ini <- vgML$par[1]
xi.ini <- vgML$par[2]
phi.ini <- vgML$par[3]
priors <- createUniformPrior(lower = c(0, 0, 0), upper = c(5, 1, 100))
setup <- createBayesianSetup(likelihood = ll, prior = priors, best = c(sigma2.ini, xi.ini, phi.ini), names = c("sigma2", "xi", "phi"))
set.seed(314)
DEzs.out <- runMCMC(setup, sampler = "DEzs")
save(DEzs.out, file = "DEzs_Ethiopia.RData")
#load(file = "DEzs_Ethiopia.RData")
summary(DEzs.out)
plot(DEzs.out)
correlationPlot(DEzs.out)
marginalPlot(DEzs.out)
mcmcsam <- getSample(DEzs.out, start = 1000, numSamples = 1000)
mcmcsample <- data.frame(mcmcsam)
# Select a simple random sample of size 1000 for evaluating the square grids.
# Add a small number to the x-coordinates and y-coordinates by drawing from a uniform distribution with lower and upper limit equal to -cellsize/2 and +cellsize/2, respectively. This can be done with function jitter.
# Load file with discretisation grid
load("Data/CovariatesThreeWoredasEthiopia.RData")
set.seed(314)
ids <- sample.int(nrow(grdEthiopia), size = 1000)
mysample <- grdEthiopia[ids, ]
# Shift the randomly selected points to random points within the cells (cellsize is 1 km x 1 km)
mysample$s1 <- jitter(mysample$s1, 0.5)
mysample$s2 <- jitter(mysample$s2, 0.5)
coordinates(mysample) <- ~s1+s2
coordinates(grdEthiopia) <- ~s1+s2
gridded(grdEthiopia) <- TRUE
# Define grid spacings
spacing <- seq(from = 1, to = 12, by = 1)
MKV <- matrix(nrow = length(spacing), ncol = nrow(mcmcsample))
for (i in 1:length(spacing)) {
mygridxy <- spsample(x = grdEthiopia, cellsize = spacing[i], type = "regular", offset = c(0.5, 0.5))
mygrid <- data.frame(s1 = mygridxy$x1, s2 = mygridxy$x2, dummy = 1)
coordinates(mygrid) <- ~s1+s2
for (j in 1:nrow(mcmcsample)) {
#Use gstat for ordinary kriging predictions
vgfit$psill[1] <- (1-mcmcsample$xi[j]) * mcmcsample$sigma2[j]
vgfit$psill[2] <- mcmcsample$xi[j] * mcmcsample$sigma2[j]
vgfit$range[2] <- mcmcsample$phi[j]
predictions <- krige(
dummy ~ 1,
mygrid,
newdata = mysample,
model = vgfit,
nmax = 100
)
MKV[i, j] <- mean(predictions$var1.var)
}
}
save(MKV, file = "MOKV_Bayesian_Ethiopia.RData")
(MMKV <- apply(MKV, MARGIN = 1, FUN = mean))
# Set target for MKV
t.MKV <- 0.8
# Compute for each variogram parameter vector the spacing for which the MKV equals the target variance
spacing.tol <- numeric(length = ncol(MKV))
for (i in 1:ncol(MKV)) {
spacing.tol[i] <- approx(x = MKV[, i], y = spacing, xout = t.MKV)$y
}
summary(spacing.tol)
pdf("Histogram_TolerableGridSpacing_Ethiopia.pdf", width = 5, height = 5)
ggplot()+
geom_histogram(mapping = aes(spacing.tol), binwidth = 1, colour = "orange")+
scale_x_continuous(name = "Tolerable grid spacing", breaks = seq(2:12))+
scale_y_continuous(name = "# MCMC samples")
dev.off()
# Compute for each grid spacing the porportion of MCMC samples with MKV smaller or equal to target MKV
F <- numeric(length = length(spacing))
for (i in 1:length(spacing)) {
F[i] <- sum(MKV[i, ] < t.MKV)
}
F <- F/ncol(MKV)
df <- data.frame(spacing, F)
pdf("ProportionvsGridspacing_Ethiopia.pdf", width = 5, height = 5)
ggplot(df)+
geom_line(mapping = aes(x = spacing, y = F), se = FALSE, colour = "red")+
scale_x_continuous(breaks = spacing, name = "Grid spacing")+
scale_y_continuous(limits = c(0, 1), breaks = seq(from = 0, to = 1, by = 0.2), name = "Proportion of MCMC samples")
dev.off()