Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 20

(z), and that the variance exists for
all z. The process is said to be strictly stationary if, for any given n 1, any set of
n sites (z 1 , z 2 ,

Suppose that our spatial process has a mean ʼ

... , z n ), and any h, the distribution of ( y (z 1 ), y (z 2 ), ... , y (z n )) is the

same as that of y z 1 þ h ð ð Þ . A process is weakly station-

ary if the mean is constant across the locations z, and the covariance relationship
between the values at any two locations can be described by a covariance function
C (h) that depends only on the distance between the spatial locations z and z + h.
Because C (h) does not depend on absolute coordinates, the variability of a
weakly stationary process is the same in every location. So, a weakly stationary
process has a constant mean, a constant variance, and a covariance function that
depends only on h. Note that because we have assumed that all variances exist,
strict stationarity implies weak stationarity. In general, the converse is not true, but
it does hold for Gaussian processes.
However, even if y (z) is not a weakly stationary process, the increments y (z) y
(z + h) may be. So, the process y (z) is said to be intrinsically stationary if

ð Þ , y z 2 þ h ð Þ , ... , y z n þ h

ʼ (z) ¼ ʼ

and

1
2 Var y ðÞy z þ h

½ ð Þ ¼ ʳ ðÞ: ð 1 : 38 Þ

The function (h) is denoted as the semivariogram of the spatial process. Con-

versely, the variogram can also be defined (Schabenberger and Gotway 2005 )as

ʳ

Var y ðÞy z þ h ð Þ ¼ 2 ʳ

ðÞ:

ð 1 :

39 Þ

½

Weak stationarity implies intrinsic stationarity, but the reverse is not implied.
If the covariance function C(h) can be written as a function of h ,where h
h 1 þ h 2 þ ...þ h d

1 = 2 (i.e. the Euclidean distance between the sites), we say that
the process is isotropic; if not, we say it is anisotropic. In the isotropic case, the
covariance depends on the size of the displacement, but not on its direction from z.
An isotropic process is thus invariant under coordinate shifts and rotations. Isotropic
processes are very popular, because of their simplicity and interpretability.
The basic R package for the analysis of geostatistical data is geoR. In the
exercises below, we have used the soya bean production data that are available in
geoR. The following code is useful for estimating the semivariogram (see Fig. 1.1 ):

> library(geoR)
> prod98 < - as.geodata(soja98, coords.col ¼ 1:2, data.col ¼ "PH")
> vario < - variog(prod98,max.dist ¼ 50)
> plot(vario,pch ¼ 19,cex ¼ 1)
> fitted < - variofit(vario)
> lines(fitted)