given careful consideration by the practitioner.

Kriging, as it is usually applied, is an exact inter-

polator. Questions may be raised, however, about

whether this is a desirable property if it is known

that the measurements are contaminated with a

considerable amount of measurement error. One

advantage of stochastic methods in general is that

existence of measurement error may be incorpo-

approaches to spatial prediction are discussed. At

rated objectively, and, in fact, some kriging soft-

the beginning of Chapter 2, the distinction between

ware packages (including STATPAC) have this

stochastic and nonstochastic techniques for spatial

feature, resulting in a surface that is not an exact

prediction was discussed. Kriging, the main sub-

interpolator. Several of the nonstochastic methods

ject of this ETL, is a stochastic technique because

discussed in this section depend on a parameter that

of the structure that is imposed in terms of an

controls the deviation from exact interpolation.

underlying random process (the regionalized

The ability to adjust such a parameter when using

variables) with joint probability distributions that

these techniques lends a degree of flexibility to the

obey certain assumptions. Kriging yields the

practitioner, but selecting the best value may not be

predictor that is statistically optimal in the sense

straightforward and may involve considerable

that it is the best linear unbiased predictor, given

subjectivity on the part of the practitioner.

certain assumptions that are detailed in Chapter 2.

There are other stochastic techniques that are less

well-known than kriging in applications, such as

predictor of the process at location *x*0 takes the

Markov-random-field prediction and Bayesian

form of a linear combination of the measurements

nonparametric smoothing (see Cressie (1991)), but

~

at locations *x*i, *i*=1, 2,..., *n*. Using *Z *(*x*0) to denote

these will not be discussed here.

an arbitrary predictor (the notation distinguishes

the predictors to be discussed in this section from

~

the kriging predictor, which is denoted by *Z (x*0),

in a nonstochastic setting will be discussed. Tech-

~

the definition of *Z (x*0) is

niques applied in such a setting are typically

applied strictly empirically and not evaluated with

respect to rigorous statistical criteria such as mean

~

(7-1)

squared prediction error, although, as discussed in

Chapter 2, such criteria may be applied in certain

of the techniques such as simple average and trend

Although this form is the same form that is taken

analysis. It has been shown in this ETL that there

by the kriging predictor, the difference is in the way

the coefficients *w*i are computed.

are some compelling advantages for assuming

some kind of stochastic setting. However, the sim-

plicity of not having to postulate and justify the

structure and assumptions inherent in stochastic

analyses might be considered one advantage of

nonstochastic techniques, and such an analysis

may be perfectly adequate for certain problems. In

location *x*0 is the simple average of the measure-

addition to statistical optimality and simplicity,

ments; that is, the weights *w*i are all equal and are

there are other considerations in selecting a spatial

prediction technique, such as ease of computation,

given by Cressie (1991)

sensitivity to data errors, and whether the predic-

tors are exact interpolators; that is, match the mea-

1

(7-2)

surements exactly at the measurement locations *x*1,

7-1