spatialDecorrelation {gmGeostats}

Compute diagonalisation measures

Description

Compute one or more diagonalisation measures out of an empirical multivariate variogram.

Usage

spatialDecorrelation(vgemp, ...)

## S3 method for class 'gstatVariogram'
spatialDecorrelation(
  vgemp,
  vgemp0 = NULL,
  method = "add",
  quadratic = method[1] != "rdd",
  ...
)

## S3 method for class 'logratioVariogram'
spatialDecorrelation(vgemp, vgemp0 = NULL, method = "add", ...)

## S3 method for class 'gmEVario'
spatialDecorrelation(vgemp, vgemp0 = NULL, method = "add", ...)

Arguments

vgemp

the empirical variogram to qualify

...

ignored

vgemp0

optionally, a reference variogram (see below; necessary for method="sde")

method

which quantities are desired? one or more of c("rdd", "add", "sde")

quadratic

should the quantities be computed for a variogram or for its square? see below

Details

The three measures provided are

absolute deviation from diagonality ("add")

defined as the sum of all off-diagonal elements of the variogram, possibly squared ($p=2$ if quadratic=TRUE the default; otherwise $p=1$)

\zeta(h)=\sum_{k=1}^n\sum_{j\neq k}^n \gamma_{k,j}^p(h)

relative deviation from diagonality ("rdd")

comparing the absolute sum of off-diagonal elements with the sum of the diagonal elements of the variogram, each possibly squared ($p=2$ if quadratic=TRUE; otherwise $p=1$ the default)

\tau(h)=\frac{\sum_{k=1}^n\sum_{j \neq k}^n |\gamma_{k,j}(h)|^p}{\sum_{k=1}^n|\gamma_{k,k}(h)|^p}

spatial diagonalisation efficiency ("sde")

is the only one requiring vgemp0, because it compares an initial state with a diagonalised state of the variogram system

\kappa(h)=1- \frac{\sum_{k=1}^n\sum_{j \neq k}^n |\gamma_{k,j}(h)|^p}{\sum_{k=1}^n\sum_{j \neq k}^n |\gamma_{(0)k,j}(h)|^p }

The value of $p$ is controlled by the first value of method. That is, the results with method=c("rdd", "add") are not the same as those obtained with method=c("add", "rdd"), as in the first case $p=1$ and in the second case $p=2$.

Value

an object of a similar nature to vgemp, but where the desired quantities are reported for each lag. This can then be plotted or averages be computed.

Methods (by class)

• gstatVariogram: Compute diagonalisation measures

• logratioVariogram: Compute diagonalisation measures

• gmEVario: Compute diagonalisation measures

Examples

data("jura", package="gstat")
X = jura.pred[, 1:2]
Z = jura.pred[,-(1:6)]
gm1 = make.gmCompositionalGaussianSpatialModel(data=Z, coords=X, V="alr")
vg1 = variogram(as.gstat(gm1)) 
(r1 = spatialDecorrelation(vg1, method=c("add", "rdd")))
plot(r1)
mean(r1)
require("compositions")
pc = princomp(acomp(Z))
v = pc$loadings
colnames(v)=paste("pc", 1:ncol(v), sep="")
gm2 = make.gmCompositionalGaussianSpatialModel(data=Z, coords=X, V=v, prefix="pc")
vg2 = variogram(as.gstat(gm2)) 
(r2 = spatialDecorrelation(vg2, method=c("add", "rdd")))
plot(r2)
mean(r2)
(r21 = spatialDecorrelation(vg2, vg1, method="sde") )
plot(r21)
mean(r21)