spBayes (version 0.2-4)

spGLM: 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", weights, data = parent.frame(),
      coords, knots, amcmc, starting, tuning, priors, cov.model,
      modified.pp = TRUE, 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.

weights

an optional vector of weights to be used in the fitting process when family is binomial. Unless otherwise specified a vector of ones is assumed i.e., one trial at each location.

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 $n \times 2$ 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

amcmc

a list with tags n.batch, batch.length, and accept.rate. Specifying this argument invokes an adaptive MCMC sampler (see Roberts and Rosenthal, 2006 and adaptMetropGibbs function).

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 p

tuning

a list with each tag corresponding to a parameter name. Valid list tags are beta, sigma.sq, phi, nu, w, and e. The value portion of each tag defines the variance of the

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

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"

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 ignor

n.samples

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

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 $n \times 2$ matrix specified by coords.
• knot.coords
  the $m \times 2$ 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.

Details

If a binomial model is specified the response vector is the number of successful trials at each location. Note, this is different from the glm argument which is passed as a rate.

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

library(MASS)
library(MBA)
library(fields)

################################
##Spatial binomial
################################

##Generate binary data
coords <- as.matrix(expand.grid(seq(0,100,length.out=8), seq(0,100,length.out=8)))
n <- nrow(coords)

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)))

weights <- rep(1, n)
weights[coords[,1]>mean(coords[,1])] <- 10

y <- rbinom(n, weights, prob=p)

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

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

m.1 <- spGLM(y~1, family="binomial", coords=coords, weights=weights, 
             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)

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

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

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

y.hat.mu <- apply(y.hat,1,mean)
y.hat.var <- apply(y.hat,1,var)

##Take a look
par(mfrow=c(1,2))
surf <- mba.surf(cbind(coords,y.hat.mu),no.X=100, no.Y=100, extend=TRUE)$xyz.est
image.plot(surf, main="Interpolated mean of posterior rate\n(observed rate)")
text(coords, label=paste("(",y/weights,")",sep=""))

surf <- mba.surf(cbind(coords,y.hat.var),no.X=100, no.Y=100, extend=TRUE)$xyz.est
image.plot(surf, main="Interpolated variance of posterior rate\n(observed # of trials)")
text(coords, label=paste("(",weights,")",sep=""))

###########################
##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)