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Example2_SpatialData.R
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Example2_SpatialData.R
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rm(list=ls())
library(INLA) # R-INLA stable version 23.09.09
library(RColorBrewer)
library(sf)
library(splines)
library(tmap)
setwd(dirname(rstudioapi::getSourceEditorContext()$path))
#########################
## Auxiliary functions ##
#########################
DIC <- function(x){
data.frame(mean.deviance=x$dic$mean.deviance,
p.eff=x$dic$p.eff,
DIC=x$dic$dic,
WAIC=x$waic$waic)
}
CV <- function(x,O,E){
LS <- -mean(log(x$cv))
# Computation of E[Y_i | y_{I_i}] using trapezoid numerical integration #
expectation = numeric(n)
for (i in 1:n){
mu = x$mean[i]
sd = x$sd[i]
xx = seq(mu-6*sd,mu+6*sd,length.out = 100)
yy = E[i]*exp(xx)*dnorm(xx, mean=mu, sd=sd)
expectation[i] = pracma::trapz(xx,yy)
}
MSPE <- mean((expectation-O)^2)
data.frame(LS=LS, MSPE=MSPE)
}
marginal.basis <- function(x, p=3, q=10){
x <- (x-min(x))/(max(x)-min(x))
dist <- (max(x)-min(x))/q
xl <- min(x)-dist*0.05
xr <- max(x)+dist*0.05
dx <- (xr-xl)/q
knots <- seq(xl-p*dx, xr+p*dx, by=dx)
B <- spline.des(knots,x,p+1)$design
return(B)
}
Rten <- function(X1,X2){ ## Row-wise Kronecker product
one1 <- matrix(1,1,ncol(X1))
one2 <- matrix(1,1,ncol(X2))
kronecker(X1,one2)*kronecker(one1,X2)
}
sdunif="expression:
logdens=-log_precision/2;
return(logdens)"
lunif = "expression:
a = 1;
b = 1;
beta = exp(theta)/(1+exp(theta));
logdens = lgamma(a+b)-lgamma(a)-lgamma(b)+(a-1)*log(beta)+(b-1)*log(1-beta);
log_jacobian = log(beta*(1-beta));
return(logdens+log_jacobian)"
###########################################
## Load dowry mortality data (year 2011) ##
###########################################
load("data_UP.Rdata")
data_UP$X1 <- as.numeric(scale(data_UP$x1)) # sex ratio
data_UP$X2 <- as.numeric(scale(data_UP$x2)) # per capita income
data_UP$X3 <- as.numeric(scale(data_UP$x3)) # number of murders per 100000 inhabitants
paleta <- brewer.pal(6,"Blues")
values <- c(-Inf,seq(-2,2),Inf)
tmap_mode("plot")
Mapa <- tm_shape(data_UP) +
tm_polygons(col=c("SMR",c("X1","X2","X3")), palette=paleta, title="", legend.show=T,
legend.reverse=T, style="fixed", breaks=values, interval.closure="left") +
tm_layout(main.title="", main.title.position=0.2, legend.text.size=1, panel.label.size=1.5,
panel.labels=c("SMR","Sex ratio","Per Capita Income","Murder rate"),
panel.label.bg.color="lightskyblue", legend.outside=T, legend.outside.position="right", legend.frame=F) +
tm_facets(nrow=1, ncol=4)
tmap_save(Mapa, file="SpatialData_MapCovariates.pdf")
##############################
## Fit the models with INLA ##
##############################
n <- nrow(W)
Rs <- Diagonal(n,colSums(W))-W
data.INLA <- data.frame(O=data_UP$O, E=data_UP$E, ID.area=seq(1,n),
X1=data_UP$X1, X2=data_UP$X2, X3=data_UP$X3)
## M1: Intercept + fixed effects
#################################
formula1 <- O ~ 1 + X1 + X2 + X3
M1 <- inla(formula1, family="poisson", data=data.INLA, E=E,
control.predictor=list(compute=TRUE, link=1, cdf=c(log(1))),
control.compute=list(cpo=TRUE, dic=TRUE, waic=TRUE, config=TRUE))
summary(M1)
## M2: CAR model
#################
formula2 = O ~ 1 +
f(ID.area, model='bym2', graph=Rs, constr=TRUE,
hyper=list(prec=list(prior=sdunif), phi=list(prior=lunif)))
# formula2 = O ~ 1 +
# f(ID.area, model='bym2', graph=Rs, constr=TRUE, scale.model=TRUE,
# hyper=list(prec=list(prior='pc.prec', param=c(1,0.01)),
# phi=list(prior='pc', param=c(0.5,0.5))))
M2 <- inla(formula2, family="poisson", data=data.INLA, E=E,
control.predictor=list(compute=TRUE, link=1, cdf=c(log(1))),
control.compute=list(cpo=TRUE, dic=TRUE, waic=TRUE, config=TRUE))
summary(M2)
## M3: CAR + covariates
########################
formula3 = O ~ 1 + X1 + X2 + X3 +
f(ID.area, model='bym2', graph=Rs, constr=TRUE,
hyper=list(prec=list(prior=sdunif), phi=list(prior=lunif)))
# formula3 = O ~ 1 + X1 + X2 + X3 +
# f(ID.area, model='bym2', graph=Rs, constr=TRUE, scale.model=TRUE,
# hyper=list(prec=list(prior='pc.prec', param=c(1,0.01)),
# phi=list(prior='pc', param=c(0.5,0.5))))
M3 <- inla(formula3, family="poisson", data=data.INLA, E=E,
control.predictor=list(compute=TRUE, link=1, cdf=c(log(1))),
control.compute=list(cpo=TRUE, dic=TRUE, waic=TRUE, config=TRUE))
summary(M3)
## M4: P-spline model
######################
B1 <- marginal.basis(x=coordinates(as(data_UP,"Spatial"))[,1], p=3, q=10)
k1 <- ncol(B1)
B2 <- marginal.basis(x=coordinates(as(data_UP,"Spatial"))[,2], p=3, q=10)
k2 <- ncol(B2)
Bs <- Rten(B2,B1)
ks <- ncol(Bs)
D1 <- diff(diag(k1),differences=1)
D2 <- diff(diag(k2),differences=1)
P1 <- t(D1)%*%D1
P2 <- t(D2)%*%D2
data.INLA <- list(O=data_UP$O, E=data_UP$E,
Intercept=c(1,rep(NA,ks)),
idx=c(NA,1:ks))
Ps <- list(as(kronecker(diag(k2),P1),"dgCMatrix"),
as(kronecker(P2,diag(k1)),"dgCMatrix"))
formula4 <- O ~ -1 + Intercept +
f(idx, model="generic3", Cmatrix=Ps, constr=TRUE, diagonal=1e-6,
hyper=list(prec1=list(prior=sdunif), prec2=list(prior=sdunif)))
# formula4 <- O ~ -1 + Intercept +
# f(idx, model="generic0", Cmatrix=Ps, rankdef=1, constr=TRUE, hyper=list(prec=list(prior=sdunif)))
#
# Ps <- kronecker(diag(k2),P1) + kronecker(P2,diag(k1))
M4 <- inla(formula4, family="poisson", data=data.INLA, E=E,
control.predictor=list(compute=TRUE, A=cbind(rep(1,n), Bs), link=1, cdf=c(log(1))),
control.compute=list(cpo=TRUE, dic=TRUE, waic=TRUE, config=TRUE))
summary(M4)
## M5: P-spline + covariates
#############################
data.INLA <- list(O=data_UP$O, E=data_UP$E, X1=data_UP$X1, X2=data_UP$X2, X3=data_UP$X3,
Intercept=c(1,rep(NA,3+ks)),
beta1=c(rep(NA,1),1,rep(NA,2+ks)),
beta2=c(rep(NA,2),1,rep(NA,1+ks)),
beta3=c(rep(NA,3),1,rep(NA,ks)),
idx=c(rep(NA,4),1:ks))
formula5 <- O ~ -1 + Intercept + beta1 + beta2 + beta3 +
f(idx, model="generic3", Cmatrix=Ps, constr=TRUE, diagonal=1e-6,
hyper=list(prec1=list(prior=sdunif), prec2=list(prior=sdunif)))
M5 <- inla(formula5, family="poisson", data=data.INLA, E=E,
control.predictor=list(compute=TRUE, A=cbind(rep(1,n), X1, X2, X3, Bs), link=1, cdf=c(log(1))),
control.compute=list(cpo=TRUE, dic=TRUE, waic=TRUE, config=TRUE))
summary(M5)
###############################################
## Table: Model comparison for temporal data ##
###############################################
MODELS <- list(CAR=M2, `CAR+cov`=M3, Pspline=M4, `Pspline+cov`=M5)
Table.DIC <- round(do.call(rbind,lapply(MODELS, DIC)),1)
loocv <- lapply(MODELS, function(x) inla.group.cv(x, num.level.sets=-1))
Table.LOOCV <- round(do.call(rbind,lapply(loocv, function(x) CV(x, O=data_UP$O, E=data_UP$E))),3)
lgocv <- lapply(MODELS, function(x) inla.group.cv(x, num.level.sets=3, groups=NULL))
Table.LGOCV <- round(do.call(rbind,lapply(lgocv, function(x) CV(x, O=data_UP$O, E=data_UP$E))),3)
Table <- cbind(Table.DIC,Table.LOOCV,Table.LGOCV)
print(Table)
#######################
## Plot some results ##
#######################
data_UP$risks.M1 <- M1$summary.fitted.values$`0.5quant`
data_UP$risks.M2 <- M2$summary.fitted.values$`0.5quant`
data_UP$risks.M3 <- M3$summary.fitted.values$`0.5quant`
data_UP$risks.M4 <- M4$summary.fitted.values$`0.5quant`[1:n]
data_UP$risks.M5 <- M5$summary.fitted.values$`0.5quant`[1:n]
paleta <- brewer.pal(9,"YlOrRd")
values <- c(-Inf,0.25,0.5,0.75,1,1.25,1.5,1.75,2,Inf)
tmap_mode("plot")
Mapa.risks <- tm_shape(data_UP) +
tm_polygons(col=c("risks.M2","risks.M3","risks.M4","risks.M5"), palette=paleta, title="Relative risk", legend.show=T,
legend.reverse=T, style="fixed", breaks=values, interval.closure="left") +
tm_layout(main.title="", main.title.position=0.2, legend.text.size=1, panel.label.size=1.5,
panel.labels=c("CAR model","CAR + covariates","Pspline model","Pspline + covariates"),
panel.label.bg.color="lightskyblue", legend.outside=T, legend.outside.position="right", legend.frame=F) +
tm_facets(nrow=1, ncol=4)
tmap_save(Mapa.risks, file="SpatialData_MapRisks.pdf")