# Draws independent samples from joint posterior in Bayesian CAR models

### Description

This package fits Bayesian conditional autoregressive models (also called intrinsic Gaussian Markov random fields or IGMRFs) for spatial and spatiotemporal data on a lattice. It uses graphical processing units (GPUs) to perform rejection sampling to obtain independent samples from the joint posterior distribution of model parameters, including spatial(temporal) random effects, precision parameters, and regression coefficients for some polynomial trend surfaces.

The CARrampsOcl package can handle models with up to three structure matrices to accommodate different patterns and strengths of spatial association in different dimensions. Such models were used by Besag and Higdon (1999) and by Reich, Hodges, and Carlin (2007) are discussed in detail in Kunsch (1994) and He, Hodges, and Carlin (2007).

Rue and Held (2005), Chapter 3, describe (polynomial) intrinsic GMRFs of first, second, and higher orders. If all of the structure matrices representing association in a model are random walk 1 or 2, then the CARrampsOcl package can estimate the coefficients of the polynomial trend surface, the design matrix of which spans the null space of the Kronecker sum of the structure matrices.

### Details

Package: | CARrampsOcl |

Type: | Package |

Version: | 0.1.3 |

Date: | 2013-04-04 |

License: | GPL>=3 |

LazyLoad: | yes |

### Author(s)

Kate Cowles and Michael Seedorff and Alex Sawyer Maintainer: Kate Cowles <kate-cowles@uiowa.edu>

### References

Besag, J. E. and Higdon, D. M. (1993), "Bayesian inference for agricultural Field experiments." Bull. Int. Statist. Inst., 55, no.1, 121-136.

Cowles, M.K. (2011) "Back to the future: New hardware + an old algorithm equals fast sampling for Bayesian spatial models." Department of Statistics and Actuarial Science, The University of Iowa.

Cowles M.K., Yan, J., Smith, B. (2009), "Reparameterized and Marginalized Posterior and Predictive Sampling for Complex Bayesian Geostatistical Models," Journal of Computational and Graphical Statistics, 18(2), 262-282.

He, Y., Hodges, J.S., and Carlin, B.P. (2007), "Reconsidering the variance parameterization in multiple precision models," Bayesian Analysis, 2, 529-556.

Kunsch, H.R. (1994), "Robust priors for smoothing and image restoration," Annals of the Institute of Statistical Mathematics, 55, no. 1, 1-19.

Reich, B.J., Hodges, J.S., and Carlin, B.P. (2007), "Spatial analyses of periodontal data using conditionally autoregressive priors having two classes of neighbor relations," J. Amer. Statist. Assoc., 102, 44-55.

Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications, volume 104 of Monographs on Statistics and Applied Probability. Chapman & Hall, London.

### Examples

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# load data
data(iowaSW06)
# construct structure matrices
Q1<- makeRW2Q(33) # RW2 within rows (east-west)
Q2<- makeRW2Q(24) # RW2 within columns (north-south)
iowaQ <- list( list( type="Gen", content=Q1 ), list( type="Gen", content=Q2))
# dimenstions of Q1, Q2, in that order
na<- nrow(Q1)
nb<- nrow(Q2)
# construct the design matrix with with as many columns as there are
# in null space of kronecker prod of Q's
X2 <- cbind( rep(1,nb), 1:nb)
X1 <- cbind( rep(1,na), 1:na)
X <- kronecker( X2, X1)
# parameters of gamma prior densities on tausqy, tausqphi[1], tausqphi[2]
alpha2 = beta2 <- c(.1, .1, .1)
# number of samples
nsamp = 100
#random seed
myseed = 314
output <- CARrampsOcl.fit(alpha=alpha2, beta=beta2,
Q=iowaQ, y=iowaSW06, nsamp=nsamp,
seed=myseed,
fixed = FALSE, randeffs=TRUE, coefs=TRUE,designMat=X,
mult= 50)
# summarize marginal posterior densities of precision parameters
library(coda)
summary(as.mcmc( output$params ))
# summarize marginal posterior densities of regression coefficients
# intercept, slope within rows (west-to-east linear trend),
# slope for columns (north to south linear trend),
# interaction between rows and columns
summary(as.mcmc(output$regcoefs))
# summary statistics for site-specific random effects at first 10 sites
print( cbind( output$phi$phimean, output$phi$phisd)[1:10,] )
# plot the raw data and the posterior means of the site-specific random effects
plot2Q( output, numcols=16)
``` |