NMMI: A Mass Compactness Measure for Spatial Pattern Analysis of Areal Features

Spatial pattern analysis plays an important role in geography for understanding geographical phenomena, identifying causes, and predicting future trends. Traditional pattern analysis tools assess cluster or dispersed patterns of geographical features based on the distribution of nonspatial attributes. These metrics ignore the shape of spatial objects—a critical consideration. The study of shape analysis, on the other hand, measures the compactness, elongation, or convexity of an areal feature based merely on geometry, without considering patterns of its attribute distribution. This article reports our efforts in developing a new pattern analysis method called the normalized mass moment of inertia (NMMI) that integrates both shape and nonspatial attributes into the analysis of compactness patterns. The NMMI is based on a well-known concept in physics—the mass moment of inertia—and is capable of detecting the degree of concentration or diffusion of some continuous attribute on an areal feature. We termed this the mass compactness. This measure can be reduced to a shape compactness measure when the attribute is evenly distributed on the feature. We first describe the theoretical model of the NMMI and its computation and then demonstrate its good performance through a series of experiments. We further discuss potentially broad applications of this approach in the contexts of urban expansion and political districting. In the political districting context, higher NMMI of a congressional district suggests a lower degree of gerrymander and vice versa. This work makes an original and unique contribution to spatial pattern and shape analysis by introducing this new, effective, and efficient measure of mass compactness that accounts for both geometric and spatial distribution.