gmGeostats (version 0.11.3)

spatialDecorrelation: Compute diagonalisation measures

Description

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

Usage

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

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.

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

Methods (by class)

• gstatVariogram: Compute diagonalisation measures

• logratioVariogram: Compute diagonalisation measures

• gmEVario: Compute diagonalisation measures

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 $p=2$.

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)