pois.krige: Conditional Simulation and Prediction for the Poisson Spatial model with a link function from the Box-Cox class

Description

This function performs conditional simulation (by MCMC) and spatial prediction in the Poisson normal model with link function from the Box-Cox class for fixed covariance parameters. Available types of prediction are: sk (simple kriging; fixed beta), ok (ordinary kriging; uniform prior on beta).

Usage

pois.krige(geodata, coords = geodata$coords, data = geodata$data,
               units.m = "default", locations = NULL, borders,
               mcmc.input, krige, output)

Arguments

geodata

a list containing elements coords and data as described next. Typically an object of the class "geodata" - a geoR data set. If not provided the arguments coords and data must be provided instead. The list may also contain an argument units.m as described below.

coords

an $n \times 2$ matrix, each row containing Euclidean coordinates of the n data locations. By default it takes the element coords of the argument geodata.

data

a vector with data values. By default it takes the element data of the argument geodata.

units.m

$n$-dimensional vector of observation times for the data. By default (units.m = "default"), it takes geodata$units.m in case this exist and else a vector of 1's.

locations

an $N \times 2$ matrix or data frame, or a list with the 2-D coordinates of the $N$ prediction locations.

borders

optional. If a two column matrix defining a polygon is provided the prediction is performed only at locations inside this polygon.

mcmc.input

input parameter for the MCMC algorithm. It can take an output from mcmc.control or a list with elements as for the arguments in mcmc.control. See documentation for mcmc.control. ATTENTION: the argument S.scale is necessary while all the others have default values.

krige

defines the model components and the type of kriging. It can take an output from krige.glm.control or a list with elements as for the arguments in krige.glm.control. See documentation for krige.glm.control.

output

parameters for controlling the output. It can take an output from output.glm.control or a list with elements as for the arguments in output.glm.control. See documentation for output.glm.control.

Value

A list with the following components:

predict

a vector with predicted values.

krige.var

a vector with predicted variances.

mcmc.error

estimated Monte Carlo errors on the predicted values.

beta.est

estimate of the $\beta$ parameter. Not included in the output if type.krige = "sk".

intensity

an $n \times n.sim$ matrix with $n.sim$ being the number of MCMC simulations. containing $g^{-1}(S_i)$. Each column corresponds to a conditional simulation of the conditional distribution of $g^{-1}(S_i)$ at the data locations. Only returned when no prediction locations are given.

acc.rate

matrix with acceptance rates from MCMC. Only returned when no prediction locations are given.

simulations

an $ni \times n.sim$ matrix where $ni$ is the number of prediction locations and $n.sim$ is the number of MCMC simulations. Each column corresponds to a conditional simulation of the predictive distribution $g^{-1}(S^{*})$. Only returned if sim.predict = TRUE.

call

the function call.

Details

For simulating the conditional distribution of $S$ given $y$, the Langevin-Hastings algorithm with the parametrisation in Papaspilliopoulus, Roberts and Skold (2003) is used. This algorithm is a Metropolis-Hastings algorithm, where the proposal distribution uses gradient information from the log-posterior distribution.

The proposal variance (called S.scale; see mcmc.control) for the algorithm needs to be scaled such that approximately 60 percent of the proposals are accepted. We also recommend that the user to studies plots of the autocorrelations.

The prediction part of the program consist of performing trans-Gaussian kriging on each of the simulated $g^{-1}(S)$-``datasets'' from the conditional distribution. Afterwards the predictor is obtained by taking the mean of prediction means, and the prediction variance is obtained by taking the mean of the prediction variances plus the variance of the prediction means. The trans-Gaussian kriging is done by calling an internal function which is an extension of krige.conv allowing for more than one ``data set'', and using a second order Taylor approximation of the inverse Box-Cox transformation function $g^{-1}$ when the transformation parameter $lambda>0$; for the exponential function, i.e. logarithmic link and $lambda=0$, an exact formula is used instead of the Taylor approximation.

References

O. Papaspiliopoulus and G. O. Roberts and M. Skold (2003). Non-centered parameterizations for hierarchical models and data augmentation. Bayesian statistics 7 (eds. J. M. Bernardo, S. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West), Oxford University Press, 307-326.

Examples

Run this code

# NOT RUN {
if(!exists(".Random.seed", envir=.GlobalEnv, inherits = FALSE)) set.seed(1234)
data(p50)
# First we scale the algorithm, and study how well the chain is mixing.
test <- pois.krige(p50, krige = list(cov.pars = c(1,1), beta = 1),
      mcmc.input = mcmc.control(S.scale = 0.2, thin = 1))
plot(log(test$intensity[45,]), type = "l")
acf(log(test$intensity[45,]), type = "correlation", plot = TRUE)
# }
# NOT RUN {
# Now we make prediction (we decide to thin to every 10, which is the default),
# where we now use S.scale = 0.55.
test2 <- pois.krige(p50, locations = cbind(c(0.5,0.5), c(1,0.4)), 
      krige = krige.glm.control(cov.pars = c(1,1), beta = 1), 
      mcmc.input = mcmc.control(S.scale = 0.55)) 
# }

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