spMvLM: Function for fitting multivariate Bayesian spatial regression models

Description

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

Usage

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

Arguments

formula

for a multivariate model with $q$ response variables, this is a list of univariate formulas. 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 spMvLM 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 list tags are beta, A, L, phi, and nu. The value portion of each tag is a vector of parameter's starting value. For

sp.tuning

a list with each tag corresponding to a parameter name. Valid list tags are A, L, phi, and nu. The value portion of each of each tag defines the step size of the proposal used in the Metr

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

priors

a list with each tag corresponding to a parameter name. Valid list tags are K.IW, Psi.IW, phi.unif, and nu.unif (Beta priors are assumed flat). Variance parameters, K<

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"

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 spMvLM, which is a list with the following tags:
coordsthe $n \times 2$ matrix specified by coords.
knot.coordsthe $m \times 2$ matrix as specified by knots.
p.samplesa coda object of posterior samples for the defined parameters.
acceptancethe Metropolis sampling acceptance rate.
sp.effectsa 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.

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.

Finley, A.O., S. Banerjee, P. Waldmann, and T. Ericsson. (2008). Hierarchical spatial modeling of additive and dominance genetic variance for large spatial trial datasets. Biometrics. DOI: 10.1111/j.1541-0420.2008.01115.x

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.

Examples

Run this code

###################################
##Multivariate spatial regression
###################################

##Generate some data
n <- 50 ##observed
q <- 3   
nltr <- q*(q-1)/2+q

coords <- cbind(runif(n),runif(n))
theta <- rep(3/0.5,q)

A <- matrix(0,q,q)
A[lower.tri(A,TRUE)] <- rnorm(nltr, 5, 1)
K <- A%*%t(A)

Psi <- diag(1,q)

c1 <- mvCovInvLogDet(coords=coords, knots=coords,
                     cov.model="exponential",
                     V=K, Psi=Psi, theta=theta,
                     modified.pp=FALSE, SWM=FALSE)

w <- mvrnorm(1,rep(0,nrow(c1$C)),c1$C)

w.1 <- w[seq(1,length(w),q)]
w.2 <- w[seq(2,length(w),q)]
w.3 <- w[seq(3,length(w),q)]

##Specify starting values and collect samples
K.starting <- diag(1,q)[lower.tri(diag(1,q), TRUE)]
Psi.starting <- diag(1,q)[lower.tri(diag(1,q), TRUE)]

n.samples <- 5000

m.1 <- spMvLM(list(w.1~1,w.2~1,w.3~1), coords=coords,
              starting=list("beta"=rep(1,q), "phi"=rep(3/0.5,q),
                "A"=K.starting, "L"=Psi.starting), 
              sp.tuning=list("phi"=rep(0.1,q),
                "A"=rep(0.1,nltr), "L"=rep(0.1,nltr)),
              priors=list("phi.Unif"=rep(c(3/1,3/0.1),q),
                "K.IW"=list(q+1, diag(1,q)), "Psi.IW"=list(q+1, diag(1,q))),
              modified.pp=TRUE, cov.model="exponential",
              n.samples=n.samples, sub.samples=c(2500,n.samples,5),
              verbose=TRUE, n.report=100)

m.1$p.samples[,paste("phi_",1:q,sep="")] <-
  3/m.1$p.samples[,paste("phi_",1:q,sep="")]

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

w.hat <- rowMeans(m.1$sp.effects)
w.hat.1 <- w.hat[seq(1,length(w.hat),q)]
w.hat.2 <- w.hat[seq(2,length(w.hat),q)]
w.hat.3 <- w.hat[seq(3,length(w.hat),q)]

##Take a look
par(mfrow=c(3,2))
surf <- mba.surf(cbind(coords,w.1),
                 no.X=100, no.Y=100, extend=T)$xyz.est
image(surf, main="Observed 1"); contour(surf, add=TRUE)

surf <- mba.surf(cbind(coords,w.hat.1),
                 no.X=100, no.Y=100, extend=T)$xyz.est
image(surf, main="Fitted 1"); contour(surf, add=TRUE)
points(m.1$knot.coords, pch=19, cex=1)

surf <- mba.surf(cbind(coords,w.2),
                 no.X=100, no.Y=100, extend=T)$xyz.est
image(surf, main="Observed 2"); contour(surf, add=TRUE)

surf <- mba.surf(cbind(coords,w.hat.2),
                 no.X=100, no.Y=100, extend=T)$xyz.est
image(surf, main="Fitted 2"); contour(surf, add=TRUE)
points(m.1$knot.coords, pch=19, cex=1)

surf <- mba.surf(cbind(coords,w.3),
                 no.X=100, no.Y=100, extend=T)$xyz.est
image(surf, main="Observed 3"); contour(surf, add=TRUE)

surf <- mba.surf(cbind(coords,w.hat.3),
                 no.X=100, no.Y=100, extend=T)$xyz.est
image(surf, main="Fitted 3"); contour(surf, add=TRUE)
points(m.1$knot.coords, pch=19, cex=1)