sp.lm {spBayes}R Documentation

Function for fitting univariate Bayesian spatial regression models

Description

The function sp.lm fits Gaussian univariate stationary Bayesian spatial regression models. Given a set of knots, sp.lm fits a predictive process model (see references below).

Usage

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

Arguments

formula for a univariate model, this is 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 sp.lm 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 recoding 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.
fixed a list with each tag corresponding to a parameter name. Valid list tags are beta, sigma.sq, tau.sq, phi, and nu. The value portion of each of each tag is value at which the parameter will be fixed.
starting a list with each tag corresponding to a parameter name (not already present in the fixed list). Valid list tags are beta, sigma.sq, tau.sq, phi, and nu. The value portion of each of each tag is the parameter's starting value.
sp.tuning a list with each tag corresponding to a variance, spatial range, or smoothness parameter names (not already present in the fixed list). Valid list tags are sigma.sq, tau.sq, phi, and nu. The value portion of each of each tag defines the step size of the proposal used in the Metropolis-Hastings sampling.
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 (not already present in the fixed list). Valid list tags are sigma.sq.ig, tau.sq.ig, phi.unif, and nu.unif (Beta priors are assumed flat). Variance parameters, simga.sq and tau.sq, are assumed to follow an inverse-Gamma distribution, whereas the spatial range phi and smoothness nu parameters is assumed to follow a Uniform distribution. 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 hyperpriors 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.
sp.effects a logical value indicating if spatial random effects should be recovered.
n.samples the number of MCMC iterations.
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 MH acceptance and MCMC progress.
... currently no additional arguments.

Value

An object of class sp.lm, 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-Hastings sampling acceptance rate.
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 sudiptob@biostat.umn.edu

References

Banerjee, S., Gelfand, A.E., Finley, A.O., and Sang, H. (In press). Gaussian predictive process models for large spatial datasets. Journal of the Royal Statistical Society Series B.

Finley, A.O., S. Banerjee, P. Waldmann, and T. Ericsson. (2007) Hierarchical spatial modeling of additive and dominance genetic variance for large spatial trial datasets. Research Report, Division of Biostatistics, University of Minnesota, 2007. Biometrics, In press.

Finley, A.O,. H. Sang, S. Banerjee, and A.E. Gelfand. (2008) Predictive process models for large multivariate spatial datasets. Research Report, Division of Biostatistics, University of Minnesota, 2008. Computational Statistics and Data Analysis, In review.

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.

See Also

ggt.sp

Examples

## Not run: 
data(rf.n200.dat)

Y <- rf.n200.dat$Y
coords <- as.matrix(rf.n200.dat[,c("x.coords","y.coords")])
w <- rf.n200.dat$w

##############################
##Estimating all parameters
##############################
m.1 <- sp.lm(Y~1, coords=coords,
             starting=list("phi"=0.6,"sigma.sq"=1, "tau.sq"=1),
             sp.tuning=list("phi"=0.01, "sigma.sq"=0.05, "tau.sq"=0.05),
             priors=list("phi.Unif"=c(0.3, 3), "sigma.sq.IG"=c(2, 1),
               "tau.sq.IG"=c(2, 1)),
             cov.model="exponential",
             n.samples=1000, verbose=TRUE, n.report=100, sp.effects=TRUE)

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

##Requires MBA package to
##make surfaces
library(MBA)
par(mfrow=c(1,2))
obs.surf <-
  mba.surf(cbind(coords, Y), no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(obs.surf, xaxs = "r", yaxs = "r", main="Observed response")
points(coords)
contour(obs.surf, add=T)

w.hat <- rowMeans(m.1$sp.effects)
w.surf <-
  mba.surf(cbind(coords, w.hat), no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(w.surf, xaxs = "r", yaxs = "r", main="Estimated random effects")
points(coords)
points(m.1$knot.coords, pch=19, cex=1)
contour(w.surf, add=T)

##############################
##Fixing spatial and variance
##parameters
##############################
m.2 <- sp.lm(Y~1, coords=coords,
             fixed=list("phi"=0.6,"sigma.sq"=10, "tau.sq"=1),
             cov.model="exponential",
             n.samples=1000, verbose=TRUE, n.report=100, sp.effects=TRUE)

print(summary(m.2$p.samples))
plot(m.2$p.samples)
 
##############################
##Modified predictive process
##############################
##Use some more observations
data(rf.n500.dat)

Y <- rf.n500.dat$Y
coords <- as.matrix(rf.n500.dat[,c("x.coords","y.coords")])
w <- rf.n500.dat$w

##############################
##Estimating all parameters
##############################
m.3 <- sp.lm(Y~1, coords=coords, knots=c(6,6,0),
             starting=list("phi"=0.6,"sigma.sq"=1, "tau.sq"=1),
             sp.tuning=list("phi"=0.01, "sigma.sq"=0.01, "tau.sq"=0.01),
             priors=list("phi.Unif"=c(0.3, 3), "sigma.sq.IG"=c(2, 1),
               "tau.sq.IG"=c(2, 1)),
             cov.model="exponential",
             n.samples=2000, verbose=TRUE, n.report=100, sp.effects=TRUE)

print(summary(m.3$p.samples))
plot(m.3$p.samples)

##Requires MBA package to
##make surfaces
library(MBA)
par(mfrow=c(1,2))
obs.surf <-
  mba.surf(cbind(coords, w), no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(obs.surf, xaxs = "r", yaxs = "r", main="Observed response")
points(coords)
contour(obs.surf, add=T)

w.hat <- rowMeans(m.3$sp.effects)
w.surf <-
  mba.surf(cbind(coords, w.hat), no.X=100, no.Y=100, extend=TRUE)$xyz.est
image(w.surf, xaxs = "r", yaxs = "r", main="Estimated random effects")
contour(w.surf, add=T)
points(coords, pch=1, cex=1)
points(m.3$knot.coords, pch=19, cex=1)
legend(1.5,2.5, legend=c("Obs.", "Knots"), pch=c(1,19), cex=c(1,1), bg="white")
## End(Not run)

[Package spBayes version 0.1-0 Index]