Academic journal article Geographical Analysis

The Saddlepoint Approximation of Moran's I's and Local Moran's [I.Sub.I]'s Reference Distributions and Their Numerical Evaluation

Academic journal article Geographical Analysis

The Saddlepoint Approximation of Moran's I's and Local Moran's [I.Sub.I]'s Reference Distributions and Their Numerical Evaluation

Article excerpt

Global Moran's I and local Moran's [I.sub.i] are the most commonly used test statistics for spatial autocorrelation in univariate map patterns or in regression residuals. They belong to the general class of ratios of quadratic forms for whom a whole array of approximation techniques has been proposed in the statistical literature, such as the prominent saddlepoint approximation by Offer Lieberman (1994). The saddlepoint approximation outperforms other approximation methods with respect to its accuracy and computational costs. In addition, only the saddle point approximation is capable of handling, in analytical terms, reference distributions of Moran's I that are subject to significant underlying spatial processes.

The accuracy and computational benefits of the saddlepoint approximation are demonstrated for a set of local Moran's [I.sub.i] statistics under either the assumption of global spatial independence or subject to an underlying global spatial process. Local Moran's [I.sub.i] is known to have an excessive kurtosis and thus void the use of the simple approximation methods of its reference distribution. The results demonstrate how well the saddlepoint approximation fits the reference distribution of local Moran's [I.sub.i]. Furthermore, for local Moran's [I.sub.i] under the assumption of global spatial independence several algebraic simplifications lead to substantial gains in numerical efficiency. This makes it possible to evaluate local Moran's [I.sub.i]'s significance in large spatial tessellations.

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Moran's I and several related spatial statistics, such as Geary's c (see Cliff and Ord 1981, P. 167), which can be expressed as quadratic forms, are frequently encountered in the spatial statistical literature and implemented in several spatial software packages to test for spatial autocorrelation in regression residuals. So far, the assessment of the significances of the observed values of these statistics is performed either under the assumption that the statistics follow approximately a normal distribution for which the expectation and variance can be calculated, or by extensive simulation experiments that either randomize the locations of the observed residuals or assume a specific error structure of the underlying regression disturbances. In addition, we can give the exact reference distribution of these ratios of quadratic forms for normally distributed regression residuals. These exact distributions allow us to evaluate the performance of any approximation. This paper focuses on an investigation of the performance of the saddlepoint approximation method as a substitute for the exact reference distribution of Moran's I. The results demonstrate that saddlepoint approximation provides a substantial improvement over simpler approximation methods, and that it can be applied in a wider range of conditions than the conventional methods.

While, in most instances, under the assumption of spatial independence a correctly specified normal approximation is quite feasible for empirical tessellations with more than 100 spatial objects, there are numerous situations in spatial statistics where the normal approximation leads to a misjudgment of the significance of an observed test statistics. Most exceptions are related to unusual forms of the spatial link matrix or to peculiar sets of exogenous variables in the regression model. For example, the reference distributions of local Moran's [I.sub.i] and also global Moran's I, which is defined by higher-order neighbor link matrices (Boots and Tiefelsdorf 2000), as well as the general spatial cross-product statistic (see Costanzo, Hubert, and Golledge 1983) do not necessarily converge asymptotically toward the normal distribution as the number of spatial observations increases. Another often overlooked issue in practical applications of the Moran's I test is the specification of its expectation and varia nce. …

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