geoBayes (version 0.3.0)

mcsglmm: MCMC samples from the Spatial GLMM

Description

Draw MCMC samples from the Spatial GLMM with known link function

Usage

mcsglmm(formula, family = c("gaussian", "binomial", "poisson", "Gamma",
  "GEV.binomial", "GEVD.binomial", "Wallace.binomial"), data, weights, subset,
  atsample, Nout, Nthin = 1, Nbi = 0, betm0, betQ0, ssqdf, ssqsc, phipars,
  omgpars, corrfcn = c("matern", "spherical", "powerexponential"), kappa,
  linkp, phisc, omgsc, zstart, phistart, omgstart, dispersion = 1,
  longlat = FALSE, test = FALSE)

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

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.

atsample

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

Nout

Number of MCMC samples to return.

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.

phipars

Parameters for the generalized inverse gamma prior for the spatial range parameter phi. A four dimensional vector with parameters scale, shape, exponent, location in that order. See Details.

omgpars

Parameters for the generalized inverse gamma prior for the relative nugget parameter omg. A four dimensional vector with parameters scale, shape, exponent, location in that order. See Details.

corrfcn

Spatial correlation function. See ebsglmm for details.

kappa

Spatial correlation parameter. Smoothness parameter for Matern, exponent for the power family.

linkp

Parameter of the link function. For binomial, a positive number for the degrees of freedom of the robit family or "logit" or "probit". For the other families any number for the exponent of the Box-Cox transformation.

phisc

Random walk parameter for phi. Smaller values increase the acceptance ratio. Set this to 0 for fixed phi. In this case the fixed value is given in the argument phistart.

omgsc

Random walk parameter for omg. Smaller values increase the acceptance ratio. Set this to 0 for fixed omg. In this case the fixed value is given in the argument omgstart.

zstart

Optional starting value for the MCMC for the GRF. This can be either a scalar, a vector of size n where n is the number of sampled locations.

phistart

Optional starting value for the MCMC for the spatial range parameter phi. Defaults to the mean of its prior. If phisc is 0, then this argument is required and it corresponds to the fixed value of phi.

omgstart

Optional starting value for the MCMC for the relative nugget parameter omg. Defaults to the mean of its prior. If omgsc is 0, then this argument is required and itcorresponds to the fixed value of omg.

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 phis

Value

• A list containing the MCMC samples and other variables as follows:
  • zA matrix containing the MCMC samples for the spatial random field. Each column is one sample.
  • muA matrix containing the MCMC samples for the mean response (a transformation of z). Each column is one sample.
  • betaA matrix containing the MCMC samples for the regressor coefficients. Each column is one sample.
  • ssqA vector with the MCMC samples for the partial sill parameter.
  • phiA vector with the MCMC samples for the spatial range parameter.
  • omgA vector with the MCMC samples for the relative nugget parameter.
  • nuThe link function parameter translated to numeric code used internally.
  • logLikA vector containing the value of the log-likelihood evaluated at each sample.
  • acc_ratioThe acceptance ratio for the joint update of the parametersphiandomg.
  • sys_timeThe total computing time for the MCMC sampling.
  • Nout,Nbi,NthinAs in input. Used internally in other functions.
  • responseThe value of the response variable at the observed locations. Used internally in other functions.
  • weightsThe response weights at the observed locations. Used internally in other functions.
  • modelmatrixThe model matrix at the observed locations. Used internally in other functions.
  • familyAs in input. Used internally in other functions.
  • betm0,betQ0,ssqdf,ssqsc,corrfcn,kappa,dispersionAs in input. Used internally in other functions.
  • locationsCoordinates of the observed locations. Used internally in other functions.
  • whichobsA logical vector indicated which rows in the data and in the MCMC samples for the spatial random field correspond to the observed locations.

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

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"
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
linkp <- "probit"

### MCMC sizes
Nout <- 100
Nthin <- 1
Nbi <- 0

### Trial run
emt <- mcsglmm(Infected ~ 1, 'binomial', rhizdata, weights = Total,
               atsample = ~ Xcoord + Ycoord,
               Nout = Nout, Nthin = Nthin, Nbi = Nbi,
               betm0 = betm0, betQ0 = betQ0, ssqdf = ssqdf, ssqsc = ssqsc,
               phipars = phiprior, omgpars = omgprior, linkp = linkp,
               corrfcn = corrf, kappa = kappa, phisc = phisc, omgsc = omgsc,
               dispersion = 1, test = 10)

### Full run
emc <- mcsglmm(Infected ~ 1, 'binomial', rhizdata, weights = Total,
               atsample = ~ Xcoord + Ycoord,
               Nout = Nout, Nthin = Nthin, Nbi = Nbi,
               betm0 = betm0, betQ0 = betQ0, ssqdf = ssqdf, ssqsc = ssqsc,
               phipars = phiprior, omgpars = omgprior, linkp = linkp,
               corrfcn = corrf, kappa = kappa, phisc = phisc, omgsc = omgsc,
               dispersion = 1, test = FALSE)


plot.ts(cbind(phi = emc$phi, omg = emc$omg, beta = c(emc$beta),
              ssq = emc$ssq), nc = 2)

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