This predictor represents the smoothest possible

predictor surface. In using this predictor, a certain

constant on the Voronoi polygons (see section 7-5)

degree of spatial homogeneity is assumed. No

induced by the measurement locations.

attempt is made to incorporate any detectable

patterns (or trends) in the mean or variance of the

data as a function of location, and the fact that

dictor. In one such variation, a distance r may be

fixed (rather than fixing *k*) and averages over loca-

measurements made at points that are close to each

tions that are within distance r of *x*0 taken. Addi-

other may be related is disregarded. Such a pre-

dictor has the advantage of being very simple to

tionally, a moving-median may be used rather than

compute; it needs no estimation of a variogram or

a moving average. Sorting and testing distances

other model parameters. The disadvantage is that

can slow computations relative to obtaining the

representing the spatial field by a single value

simple average, and use of medians rather than

ignores much of the relevant and interesting struc-

means leads to a more resistant (to outliers)

ture that may be very helpful in improving

predictor.

predictions.

stochastic setting, this predictor would be optimal

(best linear unbiased) if there is no drift and if

residuals are uncorrelated and have a common

variance.

1978)

1

0

(7-4)

j

1

be the ordered (from smallest to largest) distances,

2

and fix 1 # *k *# *n*. Then the weights *w*i are

(Cressie 1991)

where again *h*i0 is the distance of *x*0 from *x*i.

1

, *h *# *h*[*k *0]

(7-3)

the same, provided measurement locations are

0, *h*i0 > *h*[*k *0]

sufficiently close to the prediction location and are

zero otherwise. For the inverse-distance squared

method, weights are forced to decrease in a

Thus, this predictor is the average of the measure-

smoother manner as distance from the prediction

ments at the k nearest locations from *x*0.

location increases. This predictor again has the

advantage of being easy to compute. Another

feature of this predictor is that it is an exact inter-

to the simple average, with weights as given in

polator. In addition, the exponent 2 of *h*i0 may be

Equation 7-2. A choice of *k *smaller than *n *reflects

changed to any positive number, giving the user

an assumption that the predictor needs to incor-

some flexibility in determining the rate of decrease

porate more of the local fluctuation observed in the

of weights as a function of distance from *x*0. Isaaks

data, or, equivalently, that measurements at loca-

and Srivastava (1989, pp. 257-259) present an

tions near *x*0 should be more informative than

example illustrating the effects on weights of

measurements at other locations in predicting *z*(*x*0);

changing the exponent.

the smaller *k *is, the more variable the predictor. If

7-2