Multiscale Statistical Models for Hierarchical Spatial Aggregation

Scale dependency is an inherent property of geographic phenomena since most geographic patterns under observation vary with scale. Across numerous disciplines, including geography, various types of so‐called “multiscale” models have been used for the task of modeling and understanding the effects of scale. However, most of these models are descriptive—as opposed to inferential—in nature, and few of them (particularly outside geography) are well adapted to the wide variety of data structures typically encountered in geography. In this paper, we introduce a new, general framework for multiscale statistical modeling and inference that is explicitly designed for a broad class of geographic data. The key structural assumption underlying these models is that of a set of hierarchically defined partitions, corresponding to successive aggregations of an initial data space. Within our framework the effects of scale associated with such aggregation are captured through a fundamental decomposition of the data likelihood, directly induced by the hierarchical nature of the partitions, into individual components of local information at all possible spatial resolutions. Upon combining these multiscale likelihoods with an appropriately defined Bayesian prior probability structure, a powerful inferential framework results. We describe in detail how this framework may be used for the tasks of statistical estimation and classification, and illustrate its usage with an analysis of data from census geography.