ref.MCMC: MCMC for Reference Prior on an Intrinsic Conditional...

In ref.ICAR: Objective Bayes Intrinsic Conditional Autoregressive Model for Areal Data

MCMC for Reference Prior on an Intrinsic Conditional Autoregressive Random Effects Model for Areal Data

Description

Implements the Metropolis-within-Gibbs sampling algorithm proposed by Ferreira et al. (2021), to perform posterior inference for the intrinsic conditional autoregressive model with spatial random effects. This algorithm uses the spectral domain for the hierarchical model to create the Spectral Gibbs Sampler (SGS), which provides notable speedups to the MCMC algorithm proposed by Keefe et al (2019).

Usage

ref.MCMC(
  y,
  X,
  H,
  iters = 10000,
  burnin = 5000,
  verbose = TRUE,
  tauc.start = 1,
  beta.start = 1,
  sigma2.start = 1,
  step.tauc = 0.5,
  step.sigma2 = 0.5
)

Arguments

y	Vector of responses.
X	Matrix of covariates. This should include a column of 1's for models with a non-zero intercept.
H	The neighborhood matrix.
iters	Number of MCMC iterations to perform. Defaults to 10,000.
burnin	Number of MCMC iterations to discard as burn-in. Defaults to 5,000.
verbose	If FALSE, MCMC progress is not printed.
tauc.start	Starting value for the spatial dependence parameter.
beta.start	Starting value for the vector of fixed effect regression coefficients.
sigma2.start	Starting value for the variance of the unstructured random effects.
step.tauc	Step size for the spatial dependence parameter
step.sigma2	Step size for the variance of the unstructured random effects.

Value

A list containing MCMC chains and parameter summaries.

MCMCchain	Matrix of MCMC chains.
tauc.MCMC	MCMC chains for the spatial dependence parameter.
sigma2.MCMC	MCMC chains for the variance of the unstructured random effects.
phi.MCMC	MCMC chains for the spatial random effects.
beta.MCMC	MCMC chains for the fixed effect regression coefficients.
accept.sigma2	Final acceptance number for variance of the unstructured random effects.
accept.tauc	Final acceptance number for spatial dependence parameter.
accept.phi	Final acceptance number for spatial random effects.

Author(s)

Erica M. Porter, Matthew J. Keefe, Christopher T. Franck, and Marco A.R. Ferreira

References

\insertRef

keefe2018ref.ICAR

\insertRef

Keefe_2018ref.ICAR

\insertRef

Ferreira_2021ref.ICAR

Examples

#### Fit the model for simulated areal data on a grid ####

### Load extra libraries
library(sp)
library(methods)
library(spdep)
library(mvtnorm)

### Generate areal data on a grid
rows=5; cols=5
tauc=1
sigma2=2; beta=c(1,5)

### Create grid
grid <- GridTopology(c(1,1), c(1,1), c(cols,rows))
polys <- as(grid, "SpatialPolygons")
spgrid <- SpatialPolygonsDataFrame(polys,data=data.frame(row.names=row.names(polys)))

### Create neighborhood matrix
grid.nb <- poly2nb(spgrid,queen=FALSE)
W <- nb2mat(grid.nb, style="B")

### Put spatially correlated data in grid
p <- length(beta)
num.reg <- (rows*cols)
if(p>1){x1<-rmvnorm(n=num.reg,mean=rep(0,p-1),sigma=diag(p-1))} else{x1<-NULL}
X <- cbind(rep(1,num.reg),x1)
Dmat <- diag(apply(W,1,sum))
H <- Dmat - W
row.names(H) <- NULL

### Obtain true response vector
theta_true <- rnorm(num.reg,mean=0,sd=sqrt(sigma2))
Q <- eigen(H,symmetric=TRUE)$vectors
eigH <- eigen(H,symmetric=TRUE)$values
D <- diag(eigH)
Qmat <- Q[,1:(num.reg-1)]
phimat <- diag(1/sqrt(eigH[1:(num.reg-1)]))
z <- t(rmvnorm(1,mean=rep(0,num.reg-1),sigma=diag(num.reg-1)))
phi_true <- sqrt((1/tauc)*sigma2)*(Qmat%*%phimat%*%z)
Y <- X%*%beta + theta_true + phi_true

### Fit the model
set.seed(5432)
model <- ref.MCMC(y=Y,X=X,H=H,iters=15000,burnin=5000,verbose=TRUE,tauc.start=.1,beta.start=-1,
sigma2.start=.1,step.tauc=0.5,
      step.sigma2=0.5)
      

#### Small example for checking
model <- ref.MCMC(y=Y,X=X,H=H,iters=1000,burnin=50,verbose=TRUE,tauc.start=.1,beta.start=-1,
sigma2.start=.1,step.tauc=0.5,
      step.sigma2=0.5)

ref.ICAR documentation built on Aug. 22, 2023, 9:12 a.m.