exponential.semivariance {krige} | R Documentation |
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 class 'krige'
exponential.semivariance(object, ...)
## Default S3 method:
exponential.semivariance(nugget, decay, partial.sill, distance, power = 2, ...)
Arguments
... |
Additional arguments |
object |
A |
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: \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.
Value
A semivariance object. It will be a numeric vector with each bin's value of the semivariance.
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)))
## End(Not run)