Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 22

2

Ey z ðÞy z ðÞ : ð 1 :

42 Þ

½

Kriging computes the best linear unbiased predictor (BLUP), y (z 0 ), based on a
stochastic model of the spatial dependence defined by the expectation, ʼ (z), and

covariance function, C (h), of the random field.
Traditional methods are simple kriging, ordinary kriging, and universal kriging.
Simple kriging assumes that the data have known mean and variance throughout the
study area. These assumptions are very restrictive for most problems, and hence this
method is rarely used.
If E ( y (z)) is unknown but constant across locations, the BLUP using the squared-
error loss function in Eq. ( 1.42 ) is known as ordinary kriging. This is the most
widely implemented of these procedures in GIS packages.
Now, suppose that

the general linear model holds for both the data and

unobserved variables such that

y ðÞ ¼X ðÞʲ þ ε ðÞ

ð 1 : 43 Þ

y z ðÞ ¼x t z ð ʲ þ ʵ z ð ;

where X(z) is the ( n x q ) matrix of covariates measured at locations z, and x(z 0 )is
the ( q x 1) vector of explanatory variables associated with location z 0 . Furthermore,
assume a general structure for the variance-covariance matrix , and that the data

and unobserved variables are spatially correlated. In this case, the optimal predictor
is given by universal kriging.
For more information regarding the form of the different predictors of the
simple, ordinary, and universal kriging methods, an interested reader can see
Cressie ( 1993 ) and Schabenberger and Gotway ( 2005 ).
Using the geoR library, an R code for ordinary kriging can be implemented as
follows. The contour plot is reported in Fig. 1.2 .

ʣ

Fig. 1.2 Contour plot