Impacts of Spatial Autocorrelation in Georeferenced Beta and Multinomial Random Variables

The literature is replete with acknowledgments that spatial autocorrelation (SA) inflates the variance of a random variable (RV), and that it also may alter other RV distributional properties. In most studies, impacts of SA have been examined only for the three most commonly used distributions: the normal, Poisson (and its negative binomial counterpart), and binomial distributions; much less is known about its effects on two other RVs that are utilized in GIScience research: the beta and the multinomial. The beta distribution—which is considered to be very flexible because it can mimic a uniform, exponential, sinusoidal, and normal RV—can be utilized to analyze the radiance of a remotely sensed image, for example. The multinomial distribution, a generalization of the binomial distribution, has been widely used for land use classification, and to describe land use change. The literature also suggests that RV impacts of negative SA, a neglected topic in spatial analysis, may differ from those of positive SA, at least for some RVs (e.g., the Poisson RV). The purpose of this article is to extend the investigation of effects of SA to beta and multinomial RVs, with both positive SA and negative SA assessed and contrasted with each other, using simulation experiments. The simulation experiments are designed to support this assessment. One of the major discoveries is that impacts of positive SA and negative SA behave similarly when a RV conforms to a normal distribution; however, maximum negative SA is unable to materialize for asymmetric RV, whereas positive SA always converges upon its maximum.