The Codispersion Coefficient: An Application in the Evaluation of the Performance of Different Spatial Interpolators

We examine the quantitative assessment of the association between two spatial sequences to evaluate the performance between different spatial interpolators. In the context of natural resources this problem is relevant in several different fields. For example, in precision farming and forest productivity it is of interest to study the performance of interpolators in the generation of digital elevation models. In this work the codispersion coefficient is considered for intrinsic stationary spatial processes. Recent asymptotic results allow the construction of hypothesis testing and confidence intervals for the coefficient. To illustrate the applicability of the coefficient, a real data example was considered in the context of digital elevation models, in a study area located in a micro-basin in the south of Chile. Three surfaces generated by kriging, spline and inverse distance weighted (IDW) were evaluated using the conventional approach based on the root mean square error. Since this approach does not consider the spatial dependence present in the data, the codispersion coefficient was computed to compare all possible pairs of interpolations. The results show that the codispersion coefficient captures spatial association that is not possible to obtain using the conventional methods. 1 NOTATION AND DEFINITIONS The association between two spatial processes has been extensively studied in spatial statistics. Several coefficients of association have been proposed to summarized in a single number the association characteristics of two spatial or temporal sequences. In a nonparametric context Tjøtheim [15] introduced a coefficient for spatial variables. Subsequently Clifford et. al, [5], [7] proposed a coefficient that is a corrected version of the correlation coefficient. That measure was generalized later by Richardson and Clifford [12]. As a result, tests of spatial association based on the correlation coefficient have been implemented (see also [10]). The codispersion coefficient was first introduced by Matheron [11] to study the spatial association between two spatial sequences. This coefficient is defined as a normalized version of the cross-variogram between two spatial variables at distance lag h. Thus it can be interpreted as a measure of the dependence of the spatial structure of different variables. The codispersion coefficient can also be interpreted as a linear correlation coefficient between spatial increments of both atributes (Goovaerts, [8]). This coefficient has been applied in several different fields such as soil sciences [8], time series [4], hydrology [1], biology [2] among others. Definition 1. Consider two weakly stationary processes, X and Y , defined on D ⊂ Z. The cross-variogram between X and Y is defined as γ(h) = E[X(s + h)−X(s)][Y (s + h)− Y (s)], (1) such that s, s + h ∈ D. Definition 2. The codispersion coefficient is a normalized version of (1) given by ρ(h) = γ(h) √ VX(h)VY (h) , (2) where VX(h) = E[(X(s + h)−X(s))]. This coefficient has been derived for several different types of parametric models, for example, for MA(∞), AR(1) and first order spatial AR models explicit expressions for ρ(h) can be found in Rukhin and Vallejos [13]. In a one-dimensional case, this coefficient can be interpreted as a comovement coefficient for two time series [16]. It is related to the mean squared successive difference ∑n−1 i=1 (xi+1 − xi)/(n − 1), that was studied by Von Neumann in the forties. Its exact distribution has been derived [9] and its variance in the i.i. d. normal case is 4σ(3n − 4)/(n − 1). The ratio of the mean squared successive difference statistic to the sample variance ∑n−1 i=1 (xi+1 − xi)/ ∑n i=1(xi − x) is known variously as the Von Neumann ratio, or the Durbin-Watson statistic in the context of regression. It is used as a test statistic for the independence of Gaussian observations, or as a test for independence versus the alternative of nonzero first order autocorrelation. 2 ASYMPTOTIC RESULTS For Z(s) = (X(s), Y (s)) , assume that Z(s1 + h1, s2 + h2)− Z(s1, s2) = ∞ ∑