spaMM (version 4.5.0)

CauchyCorr: Cauchy correlation function and Cauchy formula term

Description

The Cauchy family of correlation functions is useful to describe spatial processes with power-law decrease of correlation at long distance. It is valid for Euclidean distances in spaces of any dimension, and for great-circle distances on spheres of any dimension. It has a scale parameter (rho, as in the Matérn correlation function), a shape (or “smoothness”, Gneiting 2013) parameter, and a long-memory dependence (or, more abstractly, “shape”; Gneiting 2013) parameter (Gneiting and Schlater 2004). The present implementation also accepts a Nugget parameter. The family can be invoked in two ways. First, the CauchyCorr function evaluates correlations, using distances as input. Second, a term of the form Cauchy(1|<...>) in a formula specifies a random effect with Cauchy correlation function, using coordinates found in a data frame as input. In the latter case, the correlations between realizations of the random effect for any two observations in the data will be the value of the Cauchy function at the scaled distance between coordinates specified in <...>, using “+” as separator (e.g., Cauchy(1|longitude+latitude)). A syntax of the form Cauchy(1|longitude+latitude %in% grp) can be used to specify a Cauchy random effect with independent realizations for each level of the grouping variable grp.

Usage

# S3 method for default
CauchyCorr(d, rho=1, shape, longdep, Nugget=NULL)
# Cauchy(1|...)

Value

Scalar/vector/matrix depending on input.

Arguments

d

Euclidean or great-circle distance

rho

The scaling factor for distance, a real >0.

shape

The shape (smoothness) parameter, a real 0<.<=2 for Euclidean distances and 0<.<=1 for great-circle distances. Smoothness increases, and fractal dimension decreases, with increasing shape (the fractal dimension of realizations in spaces of dimension \(d\) being \(d\)+1-shape/2).

longdep

The long-memory dependence parameter, a real >0. It gives the exponent of the asymptotic decrease of correlation with distance: the smaller longdep is, the longer the dependence.

Nugget

(Following the jargon of Kriging) a parameter describing a discontinuous decrease in correlation at zero distance. Correlation will always be 1 at \(d=0\), and from which it immediately drops to (1-Nugget). Defaults to zero.

...

Names of coordinates, using “+” as separator (e.g., Matern(1|longitude+latitude). The coordinates are numeric values found in the data data frame provided to the fitting function. No additional declaration of groups, factors, or other specific formatting is required.

Details

The correlation at distance \(d>0\) is $$(1-\textrm{Nugget}) (1+(\rho d)^\textrm{shape})^(-\textrm{longdep/shape})$$

References

Gneiting, T. and Schlater M. (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46: 269–282.

Gneiting T. (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli 19: 1327-1349.

Examples

data("blackcap")
fitme(migStatus ~ means+ Cauchy(1|longitude+latitude),
      data=blackcap,
      #  fixed=list(longdep=0.5,shape=0.5,rho=0.05)
     )
## The Cauchy family can be used in Euclidean spaces of any dimension:
set.seed(123)
randpts <- matrix(rnorm(20),nrow=5)
distMatrix <- as.matrix(proxy::dist(randpts))
CauchyCorr(distMatrix,rho=0.1,shape=1,longdep=10)

# See ?MaternCorr for examples of syntaxes for group-specific random effects,
#  also handled by Cauchy().