NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 II.5-1 Tony E. Smith 5. Spatial Interpolation Models Given the above model of stationary random spatial effects {(): }ss R , our ultimate objective is to apply these concepts to spatial models involving global trends, ()s, i.e., to spatial stochastic models of the form, () () (),Ys s s s R. In continuous spatial data analysis, the most fully developed models of this type focus on spatial prediction, where values of spatial variables observed at certain locations are used to predict values at other locations. But it is important to emphasize here that many such models are in fact completely deterministic in nature [i.e., implicitly assume that ( ) 0s ]. Such models are typical referred to as spatial interpolation (or smoothing) models [so we reserve the term spatial prediction for stochastic models of this type, as discussed later]. Indeed the Inverse Distance Weighting (IDW) model used for the Sudan Rainfall example in Section 2.1 above is an interpolation model. Moreover, a variety of other such models are in common use, and indeed, are also available in ARCMAP. So before developing the spatial prediction models that are of central interest for our purposes, it is appropriate to begin with selected examples of these interpolation models. In Section 6 below, we shall then consider the simplest types of spatial prediction models in which the global trend is constant, i.e., with ( )s for all sR. This will be followed in Section 7 with a development of more general prediction models in which the global trend, ( )s, is allowed to vary over space, and takes on a more important role. 5.1 A Simple Example of Spatial Interpolation The basic idea of spatial interpolation is well illustrated by the “elevation” example shown in Figure 5.1 below (taken from the ESRI Desktop Help documentation) Figure 5.1. Interpolating Elevations 0 sNOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 II.5-2 Tony E. Smith Here it is assumed that elevations, ()ys, have been measured at a set of spatial locations { : 1,.., }isi n in some relevant region, R, as shown by the dots outlined in white. Given these measurements, one would like to estimate the elevation, 0()ys , at some new location 0sR , shown in the figure (outlined in black). Given the typical continuity properties of elevation, it is clear that those measurement locations closest to 0s are the most relevant ones for estimating 0()ys , as illustrated by the red dots lying in the neighborhood of 0s denoted by the yellow circle. While it is not obvious how large this neighborhood should be, let us suppose for the moment that it has somehow been determined (we return to this question in Section 6.4 below). Then the question is how to use this set of five elevations at locations, say 15,..,ss, to estimate 0()ys . These locations are displayed in more detail in Figure 5.2 below, where 00|| ||iidss denotes the distance from 0s to each point , 1,..,5isi. 5.2 Kernel Smoothing Models Intuitively, those points closer to 0s should have more influence in this estimate. For example, it is seen in the figure that point 3s is considerably closer to 0s than is point 4s . So it is reasonable to assume that 3()ys is more influential in the estimation of 0()ys than is 4()ys . Hence if we now designate the set of points used for estimation at 0s as the interpolation set, 0()Ss , [so that in the example above, 015( ) { ,.., }Ss s s] then it is natural to consider estimates, 0ˆ()ys , of the form, 0s 01d 05d 03d 02d 04d 1s 3s 2s 4s 5s Figure 5.2. Neighborhood of Point s 0NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 II.5-3 Tony E. Smith (5.2.1) 00000()00()00() ()()()()ˆ() ()() ()iijjiisSsiisSsjjsSs sSswd yswdysyswd wd where the weight function, ( )wd , is a positive decreasing function of distance, d . Interpolation models of this type are often referred to as kernel smoothers. The reason for the ratio form is that the effective weights on each ( )ys value, [as defined by the bracketed expression in (5.1.1)] must then sum to one, i.e., (5.2.2) 00000()0()00() ()()()1() ()iijjisSsisSsjjsSs sSswdwdwd wd and are thus interpretable as the “fractional contribution” of each ()ys to the estimate, 0ˆ()ys . Thus points closer to 0s in 0()Ss will have higher fractional contributions to 0ˆ()ys since for all 0,()ijss Ss (5.2.3) 00 0 0() ()ij i jd d wd wd  000000() ()()()() ()kkjikksSs sSswdwdwd wd We have already seen an example of a kernel smoother, namely the inverse distance weighting (IDW) smoother in Section 2.1 above. In this case, the weight function is a simple inverse power function of the form, (5.2.4) ( )awd d where  is a positive constant (typically 1 or 2). While this is the only kernel smoother available in ARCMAP, it worthwhile mentioning one other, namely the exponential smoother, in which the weights are given by a negative exponential function of the form (5.2.5) ( )dwd e for some positive constant, 0 . To compare these two smoothers, it is instructive to plot typical values of these weight functions. In Figure 5.3 below, an inverse power function with 2 (shown in blue) is compared to a negative exponential function with 1