It is seen that the weight assigned to a point is pro-

portional to the area of the triangle opposite the

point.

partitioned into what are referred to as Voronoi

locations closer to measurement location *x*i than to

than computation of those in sections 7-2, 7-3, and

any other measurement location. If any two poly-

7-4. The predictor is an exact interpolator, and the

gons, *V*i and *V*j, share a common boundary, *x*i and

surface produced is continuous, but not differen-

tiable at the edges of the triangulation.

all such lines defines what is known as the

Delaunay triangulation. There will be one such

triangle containing the prediction location *x*0; the

vertices of this triangle, which are measurement

locations, are labelled *x*j, *x*k, and *x*1. The spatial

prediction at *x*0 will be the planar interpolant

interpolated using combinations of certain so-called

basis functions. These basis functions are usually

(*x*1, z(*x*1)). Joining *x*0 and *x*j, *x*k, and *x*1, three sub-

taken to be piecewise polynomials of a certain

triangles are formed. The weights *w*i are (Cressie

degree, say *k*, which is determined by the user. The

1991)

coefficients of these polynomials are chosen so that

the function values and the first *k*-1 derivatives

agree at the locations where they join. The larger *k*

, *i *= *j*, *k*, or *l*

is, the smoother will be the prediction surface.

(7-5)

Spline techniques are often applied in a non-

stochastic framework; in such a context they

0,

otherwise

represent a way of fitting a surface with certain

where *A*i is the area of the subtriangle opposite

smoothness properties to measurements at a set of

vertex *x*i.

locations with no explicit consideration of statisti-

cal optimality. There is, however, a considerable

body of work in which this technique is applied in a

stochastic setting. Splines may be used, for

ure 7-1. In this figure, the dashed lines depict the

example, in nonparametric regression estimation

problems (Wegman and Wright 1983).

lation. Vertices of the triangle containing the pre-

diction point *x*0 are *x*1, *x*5, and *x*6, and dotted lines

problem is to pose it as an optimization problem.

show the subtriangles defining the associated area

In one special case, it is assumed that the first two

derivatives of the prediction surface exist, which is

general Equation 7-5 are 1, 5, and 6, so the

weights assigned to points *x*1, *x*5, and *x*6 are,

a way of imposing a certain degree of smoothness,

and that the spline function minimizes

respectively,

j * z *(*x*i) & *z *(*x*i) % 0 *Q*

1

~

,

2

(7-7)

where *Q *is a term that depends on the first two

, and

(7-6)

derivatives of the predictor surface. The parame-

ter 0 is a nonnegative number that needs to be

7-3

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