spBayes (version 0.2-9)

spLM: Function for fitting univariate Bayesian spatial regression models

Description

The function spLM fits Gaussian univariate Bayesian spatial regression models. Given a set of knots, spLM will also fit a predictive process model (see references below).

Usage

spLM(formula, data = parent.frame(), coords, knots,
      starting, tuning, priors, cov.model,
      modified.pp = TRUE, amcmc, n.samples, 
      verbose=TRUE, n.report=100, ...)

Arguments

formula

a symbolic description of the regression model to be fit. See example below.

data

an optional data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which spLM is called.

coords

an $n \times 2$ matrix of the observation coordinates in $R^2$ (e.g., easting and northing).

knots

either a $m \times 2$ matrix of the predictive process knot coordinates in $R^2$ (e.g., easting and northing) or a vector of length two or three with the first and second elements recording the number of columns and rows in the desired

starting

a list with each tag corresponding to a parameter name. Valid tags are beta, sigma.sq, tau.sq, phi, and nu. The value portion of each tag is the parameter's starting value.

tuning

a list with each tag corresponding to a parameter name. Valid tags are sigma.sq, tau.sq, phi, and nu. The value portion of each tag defines the variance of the Metropolis sampler Normal proposal

priors

a list with each tag corresponding to a parameter name. Valid tags are sigma.sq.ig, tau.sq.ig, phi.unif, nu.unif, beta.norm, and beta.flat. Variance parameters,

cov.model

a quoted keyword that specifies the covariance function used to model the spatial dependence structure among the observations. Supported covariance model key words are: "exponential", "matern", "spherical"

modified.pp

a logical value indicating if the modified predictive process should be used (see references below for details). Note, if a predictive process model is not used (i.e., knots is not specified) then this argument is ignored

amcmc

a list with tags n.batch, batch.length, and accept.rate. Specifying this argument invokes an adaptive MCMC sampler, see Roberts and Rosenthal (2007) for an explanation.

n.samples

the number of MCMC iterations. This argument is ignored if amcmc is specified.

verbose

if TRUE, model specification and progress of the sampler is printed to the screen. Otherwise, nothing is printed to the screen.

n.report

the interval to report Metropolis sampler acceptance and MCMC progress.

...

currently no additional arguments.

Value

• An object of class spLM, which is a list with the following tags:
• coords
  the $n \times 2$ matrix specified by coords.
• knot.coords
  the $m \times 2$ matrix as specified by knots.
• p.theta.samples
  a coda object of posterior samples for the defined parameters.
• acceptance
  the Metropolis sampling acceptance rate.
• The return object might include additional data used for subsequent prediction and/or model fit evaluation.

Details

Model parameters can be fixed at their starting values by setting their tuning values to zero.

The no nugget model is specified by removing tau.sq from the starting list.

References

Banerjee, S., A.E. Gelfand, A.O. Finley, and H. Sang. (2008) Gaussian Predictive Process Models for Large Spatial Datasets. Journal of the Royal Statistical Society Series B, 70:825--848.

Banerjee, S., Carlin, B.P., and Gelfand, A.E. (2004). Hierarchical modeling and analysis for spatial data. Chapman and Hall/CRC Press, Boca Raton, FL.

Finley, A.O., H. Sang, S. Banerjee, and A.E. Gelfand. (2009) Improving the performance of predictive process modeling for large datasets. Computational Statistics and Data Analysis, 53:2873--2884. Roberts G.O. and Rosenthal J.S. (2006). Examples of Adaptive MCMC. http://probability.ca/jeff/ftpdir/adaptex.pdf.

See Also

spMvLM

Examples

rmvn <- function(n, mu=0, V = matrix(1)){
  p <- length(mu)
  if(any(is.na(match(dim(V),p))))
    stop("Dimension problem!")
  D <- chol(V)
  t(matrix(rnorm(n*p), ncol=p)%*%D + rep(mu,rep(n,p)))
}

set.seed(1)

n <- 100
coords <- cbind(runif(n,0,1), runif(n,0,1))
X <- as.matrix(cbind(1, rnorm(n)))

B <- as.matrix(c(1,5))
p <- length(B)

sigma.sq <- 2
tau.sq <- 0.1
phi <- 3/0.5

D <- as.matrix(dist(coords))
R <- exp(-phi*D)
w <- rmvn(1, rep(0,n), sigma.sq*R)
y <- rnorm(n, X%*%B + w, sqrt(tau.sq))

n.samples <- 2000

starting <- list("phi"=3/0.5, "sigma.sq"=50, "tau.sq"=1)

tuning <- list("phi"=0.1, "sigma.sq"=0.1, "tau.sq"=0.1)

priors.1 <- list("beta.Norm"=list(rep(0,p), diag(1000,p)),
                 "phi.Unif"=c(3/1, 3/0.1), "sigma.sq.IG"=c(2, 2),
                 "tau.sq.IG"=c(2, 0.1))

priors.2 <- list("beta.Flat", "phi.Unif"=c(3/1, 3/0.1),
                 "sigma.sq.IG"=c(2, 2), "tau.sq.IG"=c(2, 0.1))

cov.model <- "exponential"

n.report <- 500
verbose <- TRUE

m.1 <- spLM(y~X-1, coords=coords, starting=starting,
            tuning=tuning, priors=priors.1, cov.model=cov.model,
            n.samples=n.samples, verbose=verbose, n.report=n.report)

m.2 <- spLM(y~X-1, coords=coords, starting=starting,
            tuning=tuning, priors=priors.2, cov.model=cov.model,
            n.samples=n.samples, verbose=verbose, n.report=n.report)

burn.in <- 0.5*n.samples

##recover beta and spatial random effects
m.1 <- spRecover(m.1, start=burn.in, verbose=FALSE)
m.2 <- spRecover(m.2, start=burn.in, verbose=FALSE)

round(summary(m.1$p.theta.recover.samples)$quantiles[,c(3,1,5)],2)
round(summary(m.2$p.theta.recover.samples)$quantiles[,c(3,1,5)],2)

round(summary(m.1$p.beta.recover.samples)$quantiles[,c(3,1,5)],2)
round(summary(m.2$p.beta.recover.samples)$quantiles[,c(3,1,5)],2)

m.1.w.summary <- summary(mcmc(t(m.1$p.w.recover.samples)))$quantiles[,c(3,1,5)]
m.2.w.summary <- summary(mcmc(t(m.2$p.w.recover.samples)))$quantiles[,c(3,1,5)]

plot(w, m.1.w.summary[,1], xlab="Observed w", ylab="Fitted w",
     xlim=range(w), ylim=range(m.1.w.summary), main="Spatial random effects")
arrows(w, m.1.w.summary[,1], w, m.1.w.summary[,2], length=0.02, angle=90)
arrows(w, m.1.w.summary[,1], w, m.1.w.summary[,3], length=0.02, angle=90)
lines(range(w), range(w))

points(w, m.2.w.summary[,1], col="blue", pch=19, cex=0.5)
arrows(w, m.2.w.summary[,1], w, col="blue", m.2.w.summary[,2], length=0.02, angle=90)
arrows(w, m.2.w.summary[,1], w, col="blue", m.2.w.summary[,3], length=0.02, angle=90)

###########################
##Predictive process model
###########################
m.1 <- spLM(y~X-1, coords=coords, knots=c(6,6,0.1), starting=starting,
            tuning=tuning, priors=priors.1, cov.model=cov.model,
            n.samples=n.samples, verbose=verbose, n.report=n.report)

m.2 <- spLM(y~X-1, coords=coords, knots=c(6,6,0.1), starting=starting,
            tuning=tuning, priors=priors.2, cov.model=cov.model,
            n.samples=n.samples, verbose=verbose, n.report=n.report)

burn.in <- 0.5*n.samples

round(summary(window(m.1$p.beta.samples, start=burn.in))$quantiles[,c(3,1,5)],2)
round(summary(window(m.2$p.beta.samples, start=burn.in))$quantiles[,c(3,1,5)],2)

round(summary(window(m.1$p.theta.samples, start=burn.in))$quantiles[,c(3,1,5)],2)
round(summary(window(m.2$p.theta.samples, start=burn.in))$quantiles[,c(3,1,5)],2)