Using Spatial Statistics to Select Model Complexity

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Abstract
Testing for nonindependence among the residuals from a regression or time series model is a common approach to evaluating the adequacy of a fitted model. This idea underlies the familiar Durbin–Watson statistic, and previous works illustrate how the spatial autocorrelation among residuals can be used to test a candidate linear model. We propose here that a version of Moran's I statistic for spatial autocorrelation, applied to residuals from a fitted model, is a practical general tool for selecting model complexity under the assumption of iid additive errors. The “space” is defined by the independent variables, and the presence of significant spatial autocorrelation in residuals is evidence that a more complex model is needed to capture all of the structure in the data. An advantage of this approach is its generality, which results from the fact that no properties of the fitted model are used other than consistency. The problem of smoothing parameter selection in nonparametric regression is used to illustrate the performance of model selection based on residual spatial autocorrelation (RSA). In simulation trials comparing RSA with established selection criteria based on minimizing mean square prediction error, smooths selected by RSA exhibit fewer spurious features such as minima and maxima. In some cases, at higher noise levels, RSA smooths achieved a lower average mean square error than smooths selected by GCV. We also briefly describe a possible modification of the method for non-iid errors having short-range correlations, for example, time-series errors or spatial data. Some other potential applications are suggested, including variable selection in regression models.

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