Measuring Diversity and Segregation with Convex Functions
This study addresses the challenge of quantifying population segregation by proposing a general measurement framework based on convex functions over the probability simplex. Methodologically, it leverages Jensen’s inequality to define segregation as the discrepancy between local and global diversity, thereby constructing Jensen-information-based measures. To incorporate spatial structure, three strategies are introduced: spatial smoothing, aggregation, and local Jensen information. By integrating convex optimization with spatial statistics theory, this work establishes a comprehensive mathematical system for measuring segregation. Computational examples validate the effectiveness of various spatial modeling approaches. Ultimately, this research provides a unified theoretical foundation for quantifying the interplay between population attributes and spatial separation.