spGLM {spBayes}R Documentation

Function for fitting univariate Bayesian generalized linear spatial regression models

Description

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

Usage

spGLM(formula, family="binomial", data = parent.frame(), coords, knots,
      amcmc, starting, tuning, priors, cov.model,
      n.samples, sub.samples, verbose=TRUE, n.report=100, ...)

Arguments

formula a symbolic description of the regression model to be fit. See example below.
family currently only supports binomial and poisson data using the logit and log link functions, respectively.
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 spGLM is called.
coords an n x 2 matrix of the observation coordinates in R^2 (e.g., easting and northing).
knots either a m x 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 knot grid. The third, optional, element sets the offset of the outermost knots from the extent of the coords extent.
amcmc a list with tags n.batch, batch.length, and accept.rate.
starting a list with each tag corresponding to a parameter name. Valid list tags are beta, sigma.sq, phi, nu, and w. The value portion of each tag is the parameter's starting value. If the predictive process is used then w must be of length m; otherwise, it must be of length n. Alternatively, w can be set as a scalar, in which case the value is repeated.
tuning a list with each tag corresponding to a parameter name. Valid list tags are beta, sigma.sq, phi, nu, and w. The value portion of each tag defines the variance of the Metropolis normal proposal distribution. The tuning value for beta can be a vector of length p or the lower-triangle of the pxp Cholesky square-root of the desired proposal variance matrix. If the predictive process is used then w must be of length m; otherwise, it must be of length n. Alternatively, w can be set as a scalar, in which the value is repeated.
priors a list with each tag corresponding to a parameter name. Valid list tags are beta.flat, beta.normal, sigma.sq.ig, phi.unif, and nu.unif. simga.sq is assumed to follow an inverse-Gamma distribution, whereas the spatial range phi and smoothness nu parameters are assumed to follow Uniform distributions. If beta.normal then covariate specific mean and variance hyperparameters are passed as the first and second list elements, respectively. The hyperparameters of the inverse-Gamma are passed as a vector of length two, with the first and second elements corresponding to the shape and scale, respectively. The hyperparameters of the Uniform are also passed as a vector of length two with the first and second elements corresponding to the lower and upper support, respectively.
cov.model a quoted key word that specifies the covariance function used to model the spatial dependence structure among the observations. Supported covariance model key words are: "exponential", "matern", "spherical", and "gaussian". See below for details.
n.samples the number of MCMC iterations.
sub.samples a vector of length 3 that specifies start, end, and thin, respectively, of the MCMC samples. The default is c(1, n.samples, 1) (i.e., all samples).
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 acceptance and MCMC progress.
... currently no additional arguments.

Value

An object of class spGLM, which is a list with the following tags:

coords the n x 2 matrix specified by coords.
knot.coords the m x 2 matrix as specified by knots.
p.samples a coda object of posterior samples for the defined parameters.
acceptance the Metropolis sampling acceptance rate. If amcmc is used then this will be a matrix of each parameter's acceptance rate at the end of each batch. Otherwise, the sampler is a Metropolis with a joint proposal of all parameters, and the acceptance rate is the average over all proposals.
acceptance.w If this is a non-predictive process model and amcmc is used then this will be a matrix of each random spatial effects acceptance rate at the end of each batch.
acceptance.w.str If this is a predictive process model and amcmc is used then this will be a matrix of each random spatial effects acceptance rate at the end of each batch.
sp.effects a matrix that holds samples from the posterior distribution of the spatial random effects. The rows of this matrix correspond to the n point observations and the columns are the posterior samples.

The return object might include additional data used for subsequent prediction and/or model fit evaluation.

Author(s)

Andrew O. Finley finleya@msu.edu,
Sudipto Banerjee baner009@umn.edu

References

Finley, A.O., S. Banerjee, and R.E. McRoberts. (2008) A Bayesian approach to quantifying uncertainty in multi-source forest area estimates. Environmental and Ecological Statistics, 15:241–258.

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.

Finley, A.O,. H. Sang, S. Banerjee, and A.E. Gelfand. (2008) Improving the performance of predictive process modeling for large datasets. Computational Statistics and Data Analysis, DOI: 10.1016/j.csda.2008.09.008

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

Roberts G.O. and Rosenthal J.S. (2006) Examples of Adaptive MCMC. http://probability.ca/jeff/ftpdir/adaptex.pdf Preprint.

See Also

spGGT, spMvLM, spMvGLM

Examples

## Not run: 
###########################
##Spatial poisson
###########################
##Generate count data
set.seed(1)

n <- 100

coords <- cbind(runif(n,1,100),runif(n,1,100))

phi <- 3/50
sigma.sq <- 2

R <- sigma.sq*exp(-phi*as.matrix(dist(coords)))
w <- mvrnorm(1, rep(0,n), R)

x <- as.matrix(rep(1,n))
beta <- 0.1
y <- rpois(n, exp(x%*%beta+w))

##Collect samples
beta.starting <- coefficients(glm(y~x-1, family="poisson"))
beta.tuning <- t(chol(vcov(glm(y~x-1, family="poisson"))))

n.batch <- 500
batch.length <- 50
n.samples <- n.batch*batch.length

##Note tuning list is now optional

m.1 <- spGLM(y~1, family="poisson", coords=coords,
             starting=
             list("beta"=beta.starting, "phi"=0.06,"sigma.sq"=1, "w"=0),
             tuning=
             list("beta"=0.1, "phi"=0.5, "sigma.sq"=0.1, "w"=0.1),
             priors=
             list("beta.Flat", "phi.Unif"=c(0.03, 0.3), "sigma.sq.IG"=c(2, 1)),
             amcmc=
             list("n.batch"=n.batch,"batch.length"=batch.length, "accept.rate"=0.43),
             cov.model="exponential",
             n.samples=n.samples, sub.samples=c(1000,n.samples,10),
             verbose=TRUE, n.report=10)

##Just for fun check out the progression of the acceptance
##as it moves to 43% (same can be seen for the random spatial effects).
plot(mcmc(t(m.1$acceptance)), density=FALSE, smooth=FALSE)

##Now parameter summaries, etc.
m.1$p.samples[,"phi"] <- 3/m.1$p.samples[,"phi"]

plot(mcmc(m.1$p.samples))
print(summary(mcmc(m.1$p.samples)))

beta.hat <- mean(m.1$p.samples[,"(Intercept)"])
w.hat <- rowMeans(m.1$sp.effects)

y.hat <-exp(x%*%beta.hat+w.hat)

##Take a look
par(mfrow=c(1,2))
surf <- mba.surf(cbind(coords,y),no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(surf, main="obs")
contour(surf, drawlabels=FALSE, add=TRUE)
text(coords, labels=y, cex=1, col="blue")

surf <- mba.surf(cbind(coords,y.hat),no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(surf, main="Fitted counts")
contour(surf, drawlabels=FALSE, add=TRUE)
text(coords, labels=round(y.hat,0), cex=1, col="blue")

###########################
##Spatial logistic
###########################

##Generate binary data
n <- 100

coords <- cbind(runif(n,1,100),runif(n,1,100))

phi <- 3/50
sigma.sq <- 2

R <- sigma.sq*exp(-phi*as.matrix(dist(coords)))
w <- mvrnorm(1, rep(0,n), R)

x <- as.matrix(rep(1,n))
beta <- 0.1
p <- 1/(1+exp(-(x%*%beta+w)))
y <- rbinom(n, 1, prob=p)

##Collect samples
beta.starting <- coefficients(glm(y~x-1, family="binomial"))
beta.tuning <- t(chol(vcov(glm(y~x-1, family="binomial"))))
            
n.batch <- 500
batch.length <- 50
n.samples <- n.batch*batch.length

m.1 <- spGLM(y~1, family="binomial", coords=coords, 
             starting=
             list("beta"=beta.starting, "phi"=0.06,"sigma.sq"=1, "w"=0),
             tuning=
             list("beta"=beta.tuning, "phi"=0.5, "sigma.sq"=0.1, "w"=0.01),
             priors=
             list("beta.Normal"=list(0,10), "phi.Unif"=c(0.03, 0.3),
                  "sigma.sq.IG"=c(2, 1)),
             amcmc=
             list("n.batch"=n.batch,"batch.length"=batch.length, "accept.rate"=0.43),
             cov.model="exponential",
             n.samples=n.samples, sub.samples=c(1000,n.samples,10),
             verbose=TRUE, n.report=10)

m.1$p.samples[,"phi"] <- 3/m.1$p.samples[,"phi"]

print(summary(mcmc(m.1$p.samples)))

beta.hat <- mean(m.1$p.samples[,"(Intercept)"])
w.hat <- rowMeans(m.1$sp.effects)

y.hat <- 1/(1+exp(-(x%*%beta.hat+w.hat)))

##Take a look
par(mfrow=c(1,2))
surf <- mba.surf(cbind(coords,y),no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(surf, main="Observed")
contour(surf, add=TRUE)
points(coords[y==1,], pch=19, cex=1)
points(coords[y==0,], cex=1)

surf <- mba.surf(cbind(coords,y.hat),no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(surf, main="Fitted probabilities")
contour(surf, add=TRUE)
points(coords[y==1,], pch=19, cex=1)
points(coords[y==0,], cex=1)

## End(Not run)

[Package spBayes version 0.1-5 Index]