corExp: Exponential Correlation Structure

Description

This function is a constructor for the "corExp" class, representing an exponential spatial correlation structure. Letting $d$ denote the range and $n$ denote the nugget effect, the correlation between two observations a distance $r$ apart is $\exp(-r/d)$ when no nugget effect is present and $(1-n) \exp(-r/d)$ when a nugget effect is assumed. Objects created using this constructor must later be initialized using the appropriate Initialize method.

Usage

corExp(value, form, nugget, metric, fixed)

Arguments

value

an optional vector with the parameter values in constrained form. If nugget is FALSE, value can have only one element, corresponding to the "range" of the exponential correlation structure, which must be gre

form

a one sided formula of the form ~ S1+...+Sp, or ~ S1+...+Sp | g, specifying spatial covariates S1 through Sp and, optionally, a grouping factor g. When a grouping factor is presen

nugget

an optional logical value indicating whether a nugget effect is present. Defaults to FALSE.

metric

an optional character string specifying the distance metric to be used. The currently available options are "euclidean" for the root sum-of-squares of distances; "maximum" for the maximum difference; and "manhattan

fixed

an optional logical value indicating whether the coefficients should be allowed to vary in the optimization, or kept fixed at their initial value. Defaults to FALSE, in which case the coefficients are allowed to vary.

Value

an object of class "corExp", also inheriting from class "corSpatial", representing an exponential spatial correlation structure.

References

Cressie, N.A.C. (1993), "Statistics for Spatial Data", J. Wiley & Sons. Venables, W.N. and Ripley, B.D. (2002) "Modern Applied Statistics with S", 4th Edition, Springer-Verlag. Littel, Milliken, Stroup, and Wolfinger (1996) "SAS Systems for Mixed Models", SAS Institute.

Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models in S and S-PLUS", Springer, esp. p. 238.

Examples

Run this code

sp1 <- corExp(form = ~ x + y + z)

# Pinheiro and Bates, p. 238
spatDat <- data.frame(x = (0:4)/4, y = (0:4)/4)

cs1Exp <- corExp(1, form = ~ x + y)
cs1Exp <- Initialize(cs1Exp, spatDat)
corMatrix(cs1Exp)

cs2Exp <- corExp(1, form = ~ x + y, metric = "man")
cs2Exp <- Initialize(cs2Exp, spatDat)
corMatrix(cs2Exp)

cs3Exp <- corExp(c(1, 0.2), form = ~ x + y,
                 nugget = TRUE)
cs3Exp <- Initialize(cs3Exp, spatDat)
corMatrix(cs3Exp)

# example lme(..., corExp ...)
# Pinheiro and Bates, pp. 222-247
# p. 222
options(contrasts = c("contr.treatment", "contr.poly"))
fm1BW.lme <- lme(weight ~ Time * Diet, BodyWeight,
                   random = ~ Time)
# p. 223
fm2BW.lme <- update(fm1BW.lme, weights = varPower())
# p. 246
fm3BW.lme <- update(fm2BW.lme,
           correlation = corExp(form = ~ Time))
# p. 247
fm4BW.lme <-
      update(fm3BW.lme, correlation = corExp(form =  ~ Time,
                        nugget = TRUE))
anova(fm3BW.lme, fm4BW.lme)

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