krige.bayes: Bayesian Analysis for Gaussian Geostatistical Models

Description

The function krige.bayes performs Bayesian analysis of geostatistical data allowing specifications of different levels of uncertainty in the model parameters.
It returns results on the posterior distributions for the model parameters and on the predictive distributions for prediction locations (if provided).

Usage

krige.bayes(geodata, coords = geodata$coords, data = geodata$data,
            locations = "no", borders, model, prior, output)
model.control(trend.d = "cte", trend.l = "cte", cov.model = "matern",
              kappa = 0.5, aniso.pars = NULL, lambda = 1)
prior.control(beta.prior = c("flat", "normal", "fixed"),
              beta = NULL, beta.var.std = NULL,
              sigmasq.prior = c("reciprocal", "uniform",
                                "sc.inv.chisq", "fixed"),
              sigmasq = NULL, df.sigmasq = NULL,
              phi.prior = c("uniform", "exponential","fixed",
                            "squared.reciprocal", "reciprocal"),
              phi = NULL, phi.discrete = NULL,
              tausq.rel.prior = c("fixed", "uniform", "reciprocal"),
              tausq.rel, tausq.rel.discrete = NULL)
post2prior(obj)

Value

An object with class

"krige.bayes" and

"kriging". The attribute prediction.locations containing the name of the object with the coordinates of the prediction locations (argument

locations) is assigned to the object. Returns a list with the following components:

posterior: results on on the posterior distribution of the model parameters. A list with the following possible components:

beta: summary information on the posterior distribution of the mean parameter \(\beta\).
sigmasq: summary information on the posterior distribution of the variance parameter \(\sigma^2\) (partial sill).
phi: summary information on the posterior distribution of the correlation parameter \(\phi\) (range parameter) .
tausq.rel: summary information on the posterior distribution of the relative nugget variance parameter \(\tau^2_{rel} \).
joint.phi.tausq.rel: information on discrete the joint distribution of these parameters.
sample: a data.frame with a sample from the posterior distribution. Each column corresponds to one of the basic model parameters.

predictive: results on the predictive distribution at the prediction locations, if provided. A list with the following possible components:

mean: expected values.
variance: expected variance.
distribution: type of posterior distribution.
mean.simulations: mean of the simulations at each locations.
variance.simulations: variance of the simulations at each locations.
quantiles.simulations: quantiles computed from the the simulations.
probabilities.simulations: probabilities computed from the simulations.
simulations: simulations from the predictive distribution.

prior: a list with information on the prior distribution and hyper-parameters of the model parameters (\(\beta, \sigma^2, \phi, \tau^2_{rel}\)).
model: model specification as defined by model.control.
.Random.seed: system random seed before running the function. Allows reproduction of results. If the .Random.seed is set to this value and the function is run again, it will produce exactly the same results.
max.dist: maximum distance found between two data locations.
call: the function call.

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.
coords: an \(n \times 2\) matrix where each row has the 2-D coordinates of the \(n\) data locations. By default it takes the component coords of the argument geodata, if provided.
data: a vector with n data values. By default it takes the component data of the argument geodata, if provided.
locations: an \(N \times 2\) matrix or data-frame with the 2-D coordinates of the \(N\) prediction locations, or a list for which the first two components are used. Input is internally checked by the function check.locations. Defaults to "no" in which case the function returns only results on the posterior distributions of the model parameters.
borders: optional. If missing, by default reads the element borders from the geodata object, if present. Setting to NULL preents this behavior. If a two column matrix defining a polygon is provided the prediction is performed only at locations inside this polygon.
model: a list defining the fixed components of the model. It can take an output to a call to model.control or a list with elements as for the arguments in model.control. Default values are assumed for arguments not provided. See section DETAILS below.
prior: a list with the specification of priors for the model parameters. It can take an output to a call to prior.control or a list with elements as for the arguments in prior.control. Default values are assumed for arguments not provided. See section DETAILS below.
output: a list specifying output options. It can take an output to a call to output.control or a list with elements as for the arguments in output.control. Default values are assumed for arguments not provided. See documentation for output.control for further details.
trend.d: specifies the trend (covariates) values at the data locations. See documentation of trend.spatial for further details. Defaults to "cte".
trend.l: specifies the trend (covariates) at the prediction locations. Must be of the same type as defined for trend.d. Only used if prediction locations are provided in the argument locations.
cov.model: string indicating the name of the model for the correlation function. Further details in the documentation for cov.spatial.
kappa: additional smoothness parameter. Only used if the correlation function is one of: "matern", "powered.exponential", "cauchy" or "gneiting.matern". In the current implementation this parameter is always regarded as fixed during the Bayesian analysis.
aniso.pars: fixed parameters for geometric anisotropy correction. If aniso.pars = FALSE no correction is made, otherwise a two elements vector with values for the anisotropy parameters must be provided. Anisotropy correction consists of a transformation of the data and prediction coordinates performed by the function coords.aniso.
lambda: numerical value of the Box-Cox transformation parameter. The value \(\lambda = 1\) corresponds to no transformation. The Box-Cox parameter \(\lambda\) is always regarded as fixed and data transformation is performed before the analysis. Prediction results are back-transformed and returned is the same scale as for the original data. For \(\lambda = 0\) the log-transformation is performed. If \(\lambda < 0\) the mean predictor doesn't make sense (the resulting distribution has no expectation).
beta.prior: prior distribution for the mean (vector) parameter \(\beta\). The options are "flat" (default), "normal" or "fixed" (known mean).
beta: mean hyperparameter for the distribution of the mean (vector) parameter \(\beta\). Only used if beta.prior = "normal" or beta.prior = "fixed". For the later beta defines the value of the known mean.
beta.var.std: standardised (co)variance hyperparameter(s) for the prior for the mean (vector) parameter \(\beta\). The (co)variance matrix for\(\beta\) is given by the multiplication of this matrix by \(\sigma^2\). Only used if beta.prior = "normal".
sigmasq.prior: specifies the prior for the parameter \(\sigma^2\). If "reciprocal" (the default), the prior \(\frac{1}{\sigma^2}\) is used. Otherwise the parameter is regarded as fixed.
sigmasq: fixed value of the sill parameter \(\sigma^2\). Only used if sigmasq.prior = FALSE.
df.sigmasq: numerical. Number of degrees of freedom for the prior for the parameter \(\sigma^2\). Only used if sigmasq.prior = "sc.inv.chisq".
phi.prior: prior distribution for the range parameter \(\phi\). Options are: "uniform", "exponential", "reciprocal" , "squared.reciprocal" and "fixed". Alternativelly, a user defined discrete distribution can be specified. In this case the argument takes a vector of numerical values of probabilities with corresponding support points provided in the argument phi.discrete.
If "fixed" the argument \(\phi\) must be provided and is regarded as fixed when performing predictions.
For the exponential prior the argument phi must provide the value for of hyperparameter \(\nu\) which corresponds to the expected value for this distribution.
phi: fixed value of the range parameter \(\phi\). Only needed if phi.prior = "fixed" or if phi.prior = "exponential".
phi.discrete: support points of the discrete prior for the range parameter \(\phi\). The default is a sequence of 51 values between 0 and 2 times the maximum distance between the data locations.
tausq.rel.prior: specifies a prior distribution for the relative nugget parameter \(\frac{\tau^2}{\sigma^2}\). If tausq.rel.prior = "fixed" the relative nugget is considered known (fixed) with value given by the argument tausq.rel. If tausq.rel.prior = "uniform" a discrete uniform prior is used with support points given by the argument tausq.rel.discrete. Alternativelly, a user defined discrete distribution can be specified. In this case the argument takes the a vector of probabilities of a discrete distribution and the support points should be provided in the argument tausq.rel.discrete.
tausq.rel: fixed value for the relative nugget parameter. Only used if tausq.rel.prior = "fixed".
tausq.rel.discrete: support points of the discrete prior for the relative nugget parameter \(\frac{\tau^2}{\sigma^2}\).
obj: an object of the class krige.bayes or posterior.krige.bayes with the output of a call to krige.bayes. The function post2prior takes the posterior distribution computed by one call to krige.bayes and prepares it to be used a a prior in a subsequent call. Notice that in this case the function post2prior is used instead of prior.control.

Author

Paulo J. Ribeiro Jr. paulojus@leg.ufpr.br,
Peter J. Diggle p.diggle@lancaster.ac.uk.

Details

krige.bayes is a generic function for Bayesian geostatistical analysis of (transformed) Gaussian where predictions take into account the parameter uncertainty.

It can be set to run conventional kriging methods which use known parameters or plug-in estimates. However, the functions krige.conv and ksline are preferable for prediction with fixed parameters.

PRIOR SPECIFICATION

The basis of the Bayesian algorithm is the discretisation of the prior distribution for the parameters \(\phi\) and \(\tau^2_{rel} =\frac{\tau^2}{\sigma^2}\). The Tech. Report (see References below) provides details on the results used in the current implementation.
The expressions of the implemented priors for the parameter \(\phi\) are:

"uniform":: \(p(\phi) \propto 1\).
"exponential":: \(p(\phi) = \frac{1}{\nu} \exp(-\frac{1}{\nu} * \phi)\).
"reciprocal":: \(p(\phi) \propto \frac{1}{\phi}\).
"squared.reciprocal":: \(p(\phi) \propto \frac{1}{\phi^2}\).
"fixed":: fixed known or estimated value of \(\phi\).

The expressions of the implemented priors for the parameter \(\tau^2_{rel}\) are:

"fixed":: fixed known or estimated value of \(\tau^2_{rel}\). Defaults to zero.
"uniform":: \(p(\tau^2_{rel}) \propto 1\).
"reciprocal":: \(p(\tau^2_{rel}) \propto \frac{1}{\tau^2_{rel}}\).

Apart from those a user defined prior can be specifyed by entering a vector of probabilities for a discrete distribution with suport points given by the argument phi.discrete and/or tausq.rel.discrete.

CONTROL FUNCTIONS

The function call includes auxiliary control functions which allows the user to specify and/or change the specification of model components (using model.control), prior distributions (using prior.control) and output options (using output.control). Default options are available in most of the cases.

References

Diggle, P.J. & Ribeiro Jr, P.J. (2002) Bayesian inference in Gaussian model-based geostatistics. Geographical and Environmental Modelling, Vol. 6, No. 2, 129-146.

The technical details about the implementation of krige.bayes can be found at:
Ribeiro, P.J. Jr. and Diggle, P.J. (1999) Bayesian inference in Gaussian model-based geostatistics. Tech. Report ST-99-08, Dept Maths and Stats, Lancaster University.
Available at: http://www.leg.ufpr.br/geoR/geoRdoc/bayeskrige.pdf

Further information about geoR can be found at:
http://www.leg.ufpr.br/geoR/.

For a extended list of examples of the usage see http://www.leg.ufpr.br/geoR/tutorials/examples.krige.bayes.R and/or the geoR tutorials page at http://www.leg.ufpr.br/geoR/tutorials/.

Examples

Run this code

if (FALSE) {
# generating a simulated data-set
ex.data <- grf(70, cov.pars=c(10, .15), cov.model="matern", kappa=2)
#
# defining the grid of prediction locations:
ex.grid <- as.matrix(expand.grid(seq(0,1,l=21), seq(0,1,l=21)))
#
# computing posterior and predictive distributions
# (warning: the next command can be time demanding)
ex.bayes <- krige.bayes(ex.data, loc=ex.grid,
                 model = model.control(cov.m="matern", kappa=2),
                 prior = prior.control(phi.discrete=seq(0, 0.7, l=51),
                             phi.prior="reciprocal"))
#
# Prior and posterior for the parameter phi
plot(ex.bayes, type="h", tausq.rel = FALSE, col=c("red", "blue"))
#
# Plot histograms with samples from the posterior
par(mfrow=c(3,1))
hist(ex.bayes)
par(mfrow=c(1,1))

# Plotting empirical variograms and some Bayesian estimates:
# Empirical variogram
plot(variog(ex.data, max.dist = 1), ylim=c(0, 15))
# Since ex.data is a simulated data we can plot the line with the "true" model 
lines.variomodel(ex.data, lwd=2)
# adding lines with summaries of the posterior of the binned variogram
lines(ex.bayes, summ = mean, lwd=1, lty=2)
lines(ex.bayes, summ = median, lwd=2, lty=2)
# adding line with summary of the posterior of the parameters
lines(ex.bayes, summary = "mode", post = "parameters")

# Plotting again the empirical variogram
plot(variog(ex.data, max.dist=1), ylim=c(0, 15))
# and adding lines with median and quantiles estimates
my.summary <- function(x){quantile(x, prob = c(0.05, 0.5, 0.95))}
lines(ex.bayes, summ = my.summary, ty="l", lty=c(2,1,2), col=1)

# Plotting some prediction results
op <- par(no.readonly = TRUE)
par(mfrow=c(2,2), mar=c(4,4,2.5,0.5), mgp = c(2,1,0))
image(ex.bayes, main="predicted values")
image(ex.bayes, val="variance", main="prediction variance")
image(ex.bayes, val= "simulation", number.col=1,
      main="a simulation from the \npredictive distribution")
image(ex.bayes, val= "simulation", number.col=2,
      main="another simulation from \nthe predictive distribution")
#
par(op)
}
##
## For a extended list of exemples of the usage of krige.bayes()
## see http://www.leg.ufpr.br/geoR/tutorials/examples.krige.bayes.R
##

Run the code above in your browser using DataLab