exponential.semivariogram: Parametric Exponential Semivariogram

In lbaole17/krige.ext: Geospatial Kriging with Metropolis Sampling (Experimental Version)

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.semivariogram(nugget,decay,partial.sill,distance,power=2)
     

Arguments

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.

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: γ(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 γ(d)=τ^2+σ^2(1-\exp(-|φ 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.

Value

Returns an object of class numeric with the parametric value of the semivariance corresponding to each input value of distance.

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.

Examples

#Examine Data
summary(ContrivedData)

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

#Examine the Parametric Semivariogram Graphically with Population Parameter Values
raw.semivar<-semivariogram(x=ContrivedData$y,east=ContrivedData$s.1,
     north=ContrivedData$s.2)
resid.semivar<-semivariogram(x=contrived.ols$residuals,east=ContrivedData$s.1,
     north=ContrivedData$s.2,draw.plot=FALSE)
points(resid.semivar,pch=3,col='blue')
lines(exponential.semivariogram(nugget=0.5,decay=2.5,partial.sill=0.5,
     distance=as.numeric(names(resid.semivar))),col='red')

lbaole17/krige.ext documentation built on Sept. 29, 2020, 1:52 a.m.