covariog: Empirical Covariogram for a Model with log-link and an...
In geoRglm: A Package for Generalised Linear Spatial Models

Description Usage Arguments Details Value Author(s) References See Also Examples

Computes the sample empirical (sample) covariogram described in Christensen, Moller and Waagepetersen (2000). Output is returned as a binned covariogram. The function is NOT a general function for computing the covariogram, and it is in fact of very limited use.

covariog(geodata, coords = geodata$coords, data = geodata$data,
         units.m = "default", uvec = "default", bins.lim = "default",
         estimator.type = c("poisson", "not-poisson"),
         max.dist = NULL, pairs.min = 2)

`geodata`	a list containing elements `data` and `coords` as described next. Typically an object of the class `"geodata"` - a geoR data set. If not provided the arguments `data` and `coords` must be provided instead. The list may also contain an argument `units.m` as described below.
`coords`	an n x 2 matrix containing coordinates of the n data locations in each row. Default is `geodata$coords`, if provided.
`data`	a vector or matrix with data values. If a matrix is provided, each column is regarded as one variable or realization. Default is `geodata$data`, if provided.
`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.
`uvec`	a vector with values defining the covariogram binning. The values of `uvec` defines the midpoints of the bins. If uvec[1] > 0 the first bin is: 0 < u <= uvec[2] - 0.5(uvec[2] - uvec[1]). If uvec[1] = 0* first bin is: 0 < u <= 0.5uvec[2], and uvec[1]* is replaced by the midpoint of this interval. The default (`uvec = "default"`) is that uvec[i]=max.dist(i-1)/14* for i=1,...,15.
`bins.lim`	separating values for the binning. By default these values are defined via the argument of `uvec`.
`estimator.type`	`"poisson"` estimates the value \hat{C}(0) using the Poisson assumption. `"not-poisson"` doesn't compute \hat{C}(0).
`max.dist`	a number defining the maximal distance for the covariogram. Pairs of locations separated by a larger distance than this value are ignored in the covariogram calculation. Default is the maximum distance between pairs of data locations.
`pairs.min`	An integer number defining the minimum number of pairs for the bins. Bins with number of pairs smaller than this value are ignored.

Covariograms can be used in geostatistical analysis for exploratory purposes, to estimate covariance parameters and/or to compare theoretical and fitted models against the empirical covariogram.

The covariogram computed by this function assumes a specific model, a spatial GLMM, and furthermore it assumes that the link-function is the logarithm (i.e. it should not be used for the binomial-logistic model !).

Assume that the conditional distribution of Y_i given S_i has mean t_i*exp(S_i), where the values of t_i are given in units.m. The estimator implemented is

hat{C}(u) = log(frac{1/|W_u^{Δ}|∑_{(i,j) in W_u^{Δ}} Y(x_i)*Y(x_j) /(t_i*t_j)}{(1/n sum_i Y(x_i)/t_i)^2}), u > 0

When a Poisson distribution is assumed, then

hat{C}(0) = log(frac{1/n sum_i Y(x_i)*(Y(x_i)-1)/t_i^2}{(1/n sum_i Y(x_i)/t_i)^2}).

An object of the class covariogram which is a list with the following components:

`u`	a vector with distances.
`v`	a vector with estimated covariogram values at distances given in `u`. When `estimator.type = "poisson"`, the first value in `v` is the estimate of sigma^2, \hat{C}(0).
`n`	number of pairs in each bin. When `estimator.type = "poisson"`, the first value in `n` is `v0`.
`v0`	the estimate of sigma^2, \hat{C}(0).
`bins.lim`	Separating values for the binning provided in the function call.
`estimator.type`	echoes the type of estimator used.
`call`	The function call.

Ole F. Christensen OleF.Christensen@agrsci.dk,
Paulo J. Ribeiro Jr. Paulo.Ribeiro@est.ufpr.br.

Christensen, O. F., Moller, J. and Waagepetersen R. (2000). Analysis of spatial data using generalized linear mixed models and Langevin-type Markov chain Monte Carlo. Research report R-00-2009, Aalborg University.

Further information about geoRglm can be found at:
http://gbi.agrsci.dk/~ofch/geoRglm.

covariog.model.env for covariogram envelopes and plot.covariogram for graphical output.

data(p50)
covar <- covariog(p50, uvec=c(1:10))
plot(covar)
## Now excluding the bin at zero (only assuming log-link).
covar2 <- covariog(p50,uvec=c(1:10), estimator.type="no")
plot(covar2)