krige (version 0.6.2)

exponential.semivariance: Parametric Exponential Semivariance

Description

This function returns the value of a parametric powered exponential semivariogram given the values of the parameters and the distance between observations.

Usage

exponential.semivariance(...)
# S3 method for krige
exponential.semivariance(object, ...)
# S3 method for default
exponential.semivariance(nugget, decay, partial.sill, distance, power = 2, ...)

Arguments

…

Additional arguments

object

A krige object of which the values of the estimates are used to calculate the exponential semivariance.

nugget

The value of the non-spatial variance, or nugget term.

decay

The value of the decay term that sets the level of correlation given distance.

partial.sill

The value of the spatial variance, or partial sill term.

distance

The distance among observations for which the semivariance value is desired.

power

The exponent specified in the powered exponential semivariogram. Defaults to 2, which corresponds to a Gaussian semivariance function.

Value

A semivariance object. It will be a numeric vector with each bin's value of the semivariance.

Details

The models estimated by the krige package assume a powered exponential covariance structure. Each parametric covariance function for kriging models corresponds to a related semivariance function, given that highly correlated values will have a small variance in differences while uncorrelated values will vary widely. More specifically, semivariance is equal to half of the variance of the difference in a variable's values at a given distance. That is, the semivariance is defined as: \(\gamma(h)=0.5*E[X(s+h)-X(s)]^2\), where \(X\) is the variable of interest, s is a location, and h is the distance from s to another location.

The powered exponential covariance structure implies that the semivariance follows the specific functional form of \(\gamma(d)=\tau^2+\sigma^2(1-\exp(-|\phi d|^p))\) (Banerjee, Carlin, and Gelfand 2015, 27). A perk of this structure is that the special case of p=1 implies the commonly-used exponential semivariogram, and the special case of p=2 implies the commonly-used Gaussian semivariogram. Upon estimating a model, it is advisable to graph the functional form of the implied parametric semivariance structure. By substituting estimated values of the nugget, decay, and partial.sill terms, as well as specifying the correct power argument, it is possible to compute the implied semivariance from the model. The distance argument easily can be a vector of observed distance values.

References

Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand. 2015. Hierarchical Modeling and Analysis for Spatial Data. 2nd ed. Boca Raton, FL: CRC Press.

See Also

semivariogram, plot.semivariance, exponential.semivariance

Examples

# NOT RUN {
# Summarize data
summary(ContrivedData)

# Set seed
set.seed(1241060320)

M <- 100

contrived.run <- metropolis.krige(y ~ x.1 + x.2, coords = c("s.1","s.2"), 
  data = ContrivedData, n.iter = M, range.tol = 0.05)
  
# Parametric powered exponential semivariogram
exponential.semivariance(contrived.run)

#OLS Model for Residuals
contrived.ols<-lm(y~x.1+x.2,data=ContrivedData)

# Residual semivariance
(resid.semivar <- semivariance(contrived.ols, coords = c("s.1", "s.2"), terms = "residual"))

# Parametric exponential semivariance
exponential.semivariance(nugget=0.5,decay=2.5,partial.sill=0.5, 
                         distance=as.numeric(names(resid.semivar)))
# }
# NOT RUN {
# }