ETL 1110-1-175
30 Jun 97
Chapter 7
given careful consideration by the practitioner.
Kriging, as it is usually applied, is an exact inter-
Other Spatial Prediction Techniques
polator. Questions may be raised, however, about
whether this is a desirable property if it is known
that the measurements are contaminated with a
7-1. General
considerable amount of measurement error. One
advantage of stochastic methods in general is that
a. In this chapter, some alternative
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
underlying random process (the regionalized
The ability to adjust such a parameter when using
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
c. In most of the following techniques, the
well-known than kriging in applications, such as
predictor of the process at location x0 takes the
Markov-random-field prediction and Bayesian
form of a linear combination of the measurements
nonparametric smoothing (see Cressie (1991)), but
~
at locations xi, i=1, 2,..., n. Using Z (x0) to denote
these will not be discussed here.
an arbitrary predictor (the notation distinguishes
the predictors to be discussed in this section from
b. Several techniques that are often applied
~
the kriging predictor, which is denoted by Z (x0),
in a nonstochastic setting will be discussed. Tech-
~
the definition of Z (x0) is
niques applied in such a setting are typically
applied strictly empirically and not evaluated with
Z(x0) = j wiZ (xi)
n
respect to rigorous statistical criteria such as mean
~
(7-1)
squared prediction error, although, as discussed in
i=1
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 wi 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
7-2. Global Measure of Central
analyses might be considered one advantage of
Tendency (Simple Averaging)
nonstochastic techniques, and such an analysis
a. The predictor for the process at any
may be perfectly adequate for certain problems. In
location x0 is the simple average of the measure-
addition to statistical optimality and simplicity,
ments; that is, the weights wi 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
wi =
(7-2)
surements exactly at the measurement locations x1,
n
x2,..., xn. The last property is one that needs to be
7-1