geoBayes (version 0.7.4)

mcsglmm: MCMC samples from the Spatial GLMM

Description

Draw MCMC samples from the Spatial GLMM with known link function

Usage

mcsglmm(
  formula,
  family = "gaussian",
  data,
  weights,
  subset,
  offset,
  atsample,
  corrfcn = "matern",
  linkp,
  phi,
  omg,
  kappa,
  Nout,
  Nthin = 1,
  Nbi = 0,
  betm0,
  betQ0,
  ssqdf,
  ssqsc,
  corrpriors,
  corrtuning,
  dispersion = 1,
  longlat = FALSE,
  test = FALSE
)

Value

A list containing the objects MODEL, DATA,

FIXED, MCMC and call. The MCMC samples are stored in the object MCMC as follows:

• z A matrix containing the MCMC samples for the spatial random field. Each column is one sample.

• mu A matrix containing the MCMC samples for the mean response (a transformation of z). Each column is one sample.

• beta A matrix containing the MCMC samples for the regressor coefficients. Each column is one sample.

• ssq A vector with the MCMC samples for the partial

• phi A vector with the MCMC samples for the spatial range parameter, if sampled.

• omg A vector with the MCMC samples for the relative nugget parameter, if sampled.

• logLik A vector containing the value of the log-likelihood evaluated at each sample.

• acc_ratio The acceptance ratio for the joint update of the parameters phi and omg, if sampled.

• sys_time The total computing time for the MCMC sampling.

• Nout, Nbi, Nthin As in input. Used internally in other functions.

The other objects contain input variables. The object call

contains the function call.

Arguments

formula

A representation of the model in the form response ~ terms. The response must be set to NA's at the prediction locations (see the examples on how to do this using the function stackdata). At the observed locations the response is assumed to be a total of replicated measurements. The number of replications is inputted using the argument weights.

family

The distribution of the data. The "GEVbinomial" family is the binomial family with link the GEV link (see Details).

data

An optional data frame containing the variables in the model.

weights

An optional vector of weights. Number of replicated samples for Gaussian and gamma, number of trials for binomial, time length for Poisson.

subset

An optional vector specifying a subset of observations to be used in the fitting process.

offset

See lm.

atsample

A formula in the form ~ x1 + x2 + ... + xd with the coordinates of the sampled locations.

corrfcn

Spatial correlation function. See geoBayes_correlation for details.

linkp

Parameter of the link function. A scalar value.

phi

Optional starting value for the MCMC for the spatial range parameter phi. Defaults to the mean of its prior. If corrtuning[["phi"]] is 0, then this argument is required and it corresponds to the fixed value of phi. This can be a vector of the same length as Nout.

omg

Optional starting value for the MCMC for the relative nugget parameter omg. Defaults to the mean of its prior. If corrtuning[["omg"]] is 0, then this argument is required and it corresponds to the fixed value of omg. This can be a vector of the same length as Nout.

kappa

Optional starting value for the MCMC for the spatial correlation parameter kappa (Matern smoothness or exponential power). Defaults to the mean of its prior. If corrtuning[["kappa"]] is 0 and it is needed for the chosen correlation function, then this argument is required and it corresponds to the fixed value of kappa. This can be a vector of the same length as Nout.

Nout

Number of MCMC samples to return. This can be a vector for running independent chains. 0 elements are dropped.

Nthin

The thinning of the MCMC algorithm.

Nbi

The burn-in of the MCMC algorithm.

betm0

Prior mean for beta (a vector or scalar).

betQ0

Prior standardised precision (inverse variance) matrix. Can be a scalar, vector or matrix. The first two imply a diagonal with those elements. Set this to 0 to indicate a flat improper prior.

ssqdf

Degrees of freedom for the scaled inverse chi-square prior for the partial sill parameter.

ssqsc

Scale for the scaled inverse chi-square prior for the partial sill parameter.

corrpriors

A list with the components phi, omg and kappa as needed. These correspond to the prior distribution parameters. For phi and omg it must be a vector of length 4. The generalized inverse gamma prior is assumed and the input corresponds to the parameters scale, shape, exponent, location in that order (see Details). For kappa it must be a vector of length 2. A uniform prior is assumed and the input corresponds to the lower and upper bounds in that order.

corrtuning

A vector or list with the components phi, omg and kappa as needed. These correspond to the random walk parameter for the Metropolis-Hastings step. Smaller values increase the acceptance ratio. Set this to 0 for fixed parameter value.

dispersion

The fixed dispersion parameter.

longlat

How to compute the distance between locations. If FALSE, Euclidean distance, if TRUE Great Circle distance. See spDists.

test

Whether this is a trial run to monitor the acceptance ratio of the random walk for phi and omg. If set to TRUE, the acceptance ratio will be printed on the screen every 100 iterations of the MCMC. Tune the phisc and omgsc parameters in order to achive 20 to 30% acceptance. Set this to a positive number to change the default 100. No thinning or burn-in are done when testing.

Details

The four-parameter prior for phi is defined by $$\propto (\phi - \theta_4)^{\theta_2 -1} \exp\{-(\frac{\phi - \theta_4}{\theta_1})^{\theta_3}\}$$ for \(\phi > \theta_4\). The prior for omg is similar. The prior parameters correspond to scale, shape, exponent, and location. See arXiv:1005.3274 for details of this distribution.

The GEV (Generalised Extreme Value) link is defined by $$\mu = 1 - \exp\{-\max(0, 1 + \nu x)^{\frac{1}{\nu}}\}$$ for any real \(\nu\). At \(\nu = 0\) it reduces to the complementary log-log link.

Examples

if (FALSE) {
data(rhizoctonia)

### Create prediction grid
predgrid <- mkpredgrid2d(rhizoctonia[c("Xcoord", "Ycoord")],
                         par.x = 100, chull = TRUE, exf = 1.2)

### Combine observed and prediction locations
rhizdata <- stackdata(rhizoctonia, predgrid$grid)
##'
### Define the model
corrf <- "spherical"
family <- "binomial.probit"
kappa <- 0
ssqdf <- 1
ssqsc <- 1
betm0 <- 0
betQ0 <- .01
phiprior <- c(100, 1, 1000, 100) # U(100, 200)
phisc <- 3
omgprior <- c(2, 1, 1, 0)        # Exp(mean = 2)
omgsc <- .1
##'
### MCMC sizes
Nout <- 100
Nthin <- 1
Nbi <- 0

### Trial run
emt <- mcsglmm(Infected ~ 1, family, rhizdata, weights = Total,
               atsample = ~ Xcoord + Ycoord,
               Nout = Nout, Nthin = Nthin, Nbi = Nbi,
               betm0 = betm0, betQ0 = betQ0, ssqdf = ssqdf, ssqsc = ssqsc,
               corrpriors = list(phi = phiprior, omg = omgprior), 
               corrfcn = corrf, kappa = kappa,
               corrtuning = list(phi = phisc, omg = omgsc, kappa = 0),
               dispersion = 1, test = 10)

### Full run
emc <- update(emt, test = FALSE)

emcmc <- mcmcmake(emc)
summary(emcmc[, c("phi", "omg", "beta", "ssq")])
plot(emcmc[, c("phi", "omg", "beta", "ssq")])
}