Function for fitting multivariate Bayesian spatial regression models

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, tuning, priors, cov.model,
      modified.pp = TRUE, amcmc, n.samples, 
      verbose=TRUE, n.report=100, ...)

Arguments

formula

a list of \(q\) symbolic regression model descriptions 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 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 knot grid. The third, optional, element sets the offset of the outermost knots from the extent of the coords.

starting

a list with tags corresponding to beta, A, phi, and nu. Depending on the specification of the non-spatial residual, tags are L or Psi for a block diagonal or diagonal covariance matrix, respectively.

The value portion of each tag is a vector that holds the parameter's starting values and are of length \(p\) for beta (where \(p\) is the total number of regression coefficients in the multivariate model), \(\frac{q(q+1)}{2}\) for A and L, and \(q\) for Psi, phi, and nu. Here, A and L hold the lower-triangle elements in column major ordering of the Cholesky square root of the spatial and non-spatial cross-covariance matrices, respectively.

tuning

a list with tags A, phi, and nu. Depending on the specification of the non-spatial residual, tags are L or Psi for a block diagonal or diagonal covariance matrix, respectively. The value portion of each tag defines the variance of the Metropolis sampler Normal proposal distribution. For A and L the vectors are of length \(\frac{q(q+1)}{2}\) and \(q\) for Psi, phi, and nu.

priors

a list with tags beta.flat, beta.norm, K.iw, Psi.iw, Psi.ig, phi.unif, and nu.unif. If the regression coefficients, i.e., beta vector, are assumed to follow a multivariate Normal distribution then pass the hyperparameters as a list of length two with the first and second elements corresponding to the mean vector and positive definite covariance matrix, respectively. If beta is assumed flat then no arguments are passed. The default is a flat prior. Use Psi.iw if the non-spatial residual covariance matrix is assumed block diagonal. Otherwise if the non-spatial residual covariance matrix is assumed diagonal then each of the \(q\) diagonal element are assumed to follow an inverse-Gamma in which case use Psi.ig. The hyperparameters of the inverse-Wishart, i.e., for cross-covariance matrices \(AA'\) K.iw and \(LL'\) Psi.iw, are passed as a list of length two, with the first and second elements corresponding to the \(df\) and \(q\times q\) scale matrix, respectively. If Psi.ig is specified, the inverse-Gamma hyperparameters of the diagonal variance elements are pass using a list of length two with the first and second list elements consisting of vectors of the \(q\) shape and scale hyperparameters, respectively. The hyperparameters of the Uniform phi.unif, and nu.unif are also passed as a list of vectors with the first and second list elements corresponding to the lower and upper support, respectively.

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", and "gaussian". See below for details.

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 acceptance and MCMC progress.

...

currently no additional arguments.

Value

An object of class spMvLM, 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 percent. Reported at batch.length or n.report intervals for amcmc specified and non-specified, respectively

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 Psi and L 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, Fla.

Finley, A.O., S. Banerjee, and A.E. Gelfand. (2015) spBayes for large univariate and multivariate point-referenced spatio-temporal data models. Journal of Statistical Software, 63:1–28. https://www.jstatsoft.org/article/view/v063i13.

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.

Finley, A.O., S. Banerjee, A.R. Ek, and R.E. McRoberts. (2008) Bayesian multivariate process modeling for prediction of forest attributes. Journal of Agricultural, Biological, and Environmental Statistics, 13:60–83.

Author

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

Examples

if (FALSE) { # \dontrun{
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)

##Generate some data
n <- 25 ##number of locations
q <- 2 ##number of outcomes at each location
nltr <- q*(q+1)/2 ##number of triangular elements in the cross-covariance matrix

coords <- cbind(runif(n,0,1), runif(n,0,1))

##Parameters for the bivariate spatial random effects
theta <- rep(3/0.5,q)

A <- matrix(0,q,q)
A[lower.tri(A,TRUE)] <- c(1,-1,0.25)
K <- A%*%t(A)

Psi <- diag(0,q)

C <- mkSpCov(coords, K, Psi, theta, cov.model="exponential")

w <- rmvn(1, rep(0,nrow(C)), C)

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

##Covariate portion of the mean
x.1 <- cbind(1, rnorm(n))
x.2 <- cbind(1, rnorm(n))
x <- mkMvX(list(x.1, x.2))

B.1 <- c(1,-1)
B.2 <- c(-1,1)
B <- c(B.1, B.2)

Psi <- diag(c(0.1, 0.5))

y <- rnorm(n*q, x%*%B+w, diag(n)%x%Psi)

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

##Call spMvLM
A.starting <- diag(1,q)[lower.tri(diag(1,q), TRUE)]
n.samples <- 1000

starting <- list("phi"=rep(3/0.5,q), "A"=A.starting, "Psi"=rep(1,q))
tuning <- list("phi"=rep(1,q), "A"=rep(0.01,length(A.starting)), "Psi"=rep(0.01,q))
priors <- list("beta.Flat", "phi.Unif"=list(rep(3/0.75,q), rep(3/0.25,q)),
               "K.IW"=list(q+1, diag(0.1,q)), "Psi.ig"=list(c(2,2), c(0.1,0.1)))

m.1 <- spMvLM(list(y.1~x.1-1, y.2~x.2-1), 
               coords=coords, starting=starting, tuning=tuning, priors=priors,
               n.samples=n.samples, cov.model="exponential", n.report=100)

burn.in <- 0.75*n.samples

m.1 <- spRecover(m.1, start=burn.in)

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

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

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

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