OLM and ICAR model probabilities for areal data

Description

Performs simultaneous selection of covariates and spatial model structure for areal data.

Usage

probs.icar(
  Y,
  X,
  H,
  H.spectral = NULL,
  Sig_phi = NULL,
  b = 0.05,
  verbose = FALSE
)

Arguments

Value

A list containing a data frame with all posterior model probabilities and other selection information.

Author(s)

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

References

Porter EM, Franck CT, Ferreira MAR (2024). “Objective Bayesian model selection for spatial hierarchical models with intrinsic conditional autoregressive priors.” Bayesian Analysis, 19(4), 985–1011. doi:10.1214/23-BA1375.

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

Value

A list containing MCMC chains and parameter summaries.

Author(s)

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

References

Keefe MJ, Ferreira MAR, Franck CT (2019). “Objective Bayesian analysis for Gaussian hierarchical models with intrinsic conditional autoregressive priors.” Bayesian Analysis, 14(1), 181 – 209. doi:10.1214/18-BA1107.

Keefe MJ, Ferreira MAR, Franck CT (2018). “On the formal specification of sum-zero constrained intrinsic conditional autoregressive models.” Spatial Statistics, 24, 54–65. doi:10.1016/j.spasta.2018.03.007.

Ferreira MAR, Porter EM, Franck CT (2021). “Fast and scalable computations for Gaussian hierarchical models with intrinsic conditional autoregressive spatial random effects.” Computational Statistics and Data Analysis, 162, 107264. ISSN 0167-9473, doi:10.1016/j.csda.2021.107264.

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)

MCMC Analysis and Summaries for Reference Prior on an Intrinsic Autoregressive Model for Areal Data

Description

Performs analysis on a geographical areal data set using the objective prior for intrinsic conditional autoregressive (ICAR) random effects (Keefe et al. 2019). It takes a shapefile, data, and region names to construct a neighborhood matrix and perform Markov chain Monte Carlo sampling on the unstructured and spatial random effects. Finally, the function obtains regional estimates and performs posterior inference on the model parameters.

Usage

ref.analysis(
  shape.file,
  X,
  y,
  x.reg.names,
  y.reg.names,
  shp.reg.names = NULL,
  iters = 10000,
  burnin = 5000,
  verbose = TRUE,
  tauc.start = 1,
  beta.start = 1,
  sigma2.start = 1,
  step.tauc = 0.5,
  step.sigma2 = 0.5
)

Arguments

Value

A list containing H, MCMC chains, parameter summaries, fitted regional values, and regional summaries.

Author(s)

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

Examples

## Refer to the vignette attached to the package.

Trace Plots for Parameters in ICAR Model

Description

This function creates trace plots for the parameters in the ICAR reference prior model (Keefe et al. 2019).

Usage

ref.plot(MCMCchain, X, burnin, num.reg)

Arguments

Value

Trace plots for the fixed effect regression coefficients, the precision parameter, and the unstructured random effects variance.

Author(s)

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

Examples

## Refer to the vignette attached to the package.

Parameter Summaries for MCMC Analysis

Description

Takes a matrix of MCMC chains, iterations, and acceptance values to return posterior summaries of the parameters, including posterior medians, intervals, and acceptance rates.

Usage

ref.summary(
  MCMCchain,
  tauc.MCMC,
  sigma2.MCMC,
  beta.MCMC,
  phi.MCMC,
  accept.phi,
  accept.sigma2,
  accept.tauc,
  iters = 10000,
  burnin = 5000
)

Arguments

Value

Parameter summaries

Author(s)

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

Examples

## Refer to the vignette attached to the package.

Regional Summaries for Areal Data Modeled by ICAR Reference Prior Model

Description

This function takes data and sampled MCMC chains for an areal data set and gives fitted posterior values and summaries by region using the model by (Keefe et al. 2019).

Usage

reg.summary(MCMCchain, X, Y, burnin)

Arguments

Value

A list of the fitted distributions by region, and medians and credible intervals by region.

Author(s)

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

Examples

## Refer to the vignette attached to the package.

Creating a Neighborhood Matrix for Areal Data from a Shapefile

Description

Takes a path to a shape file and creates a neighborhood matrix. This neighborhood matrix can be used with the objective ICAR model (Keefe et al. 2018).

Usage

shape.H(shape.file)

Arguments

Value

A list containing a neighborhood matrix and the SpatialPolygonsDataFrame object corresponding to the shape file.

Author(s)

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

Examples

#### Load extra libraries
library(sp)
library(sf)

### Read in a shapefile of the contiguous U.S. from package data
system.path <- system.file("extdata", "us.shape48.shp", package = "ref.ICAR", mustWork = TRUE)
shp.layer <- gsub('.shp','',basename(system.path))
shp.path <- dirname(system.path)
us.shape48 <- st_read(dsn = path.expand(shp.path), layer = shp.layer)

shp.data <- shape.H(system.path)
names(shp.data)