Measuring the circularity of congressional districts

Shape analysis has special importance in the detection of manipulated redistricting, which is called gerrymandering. In most of the US states, this process is made by non-independent actors and often causes debates about partisan manipulation. The somewhat ambiguous concept of compactness is a standard criterion for legislative districts. In the literature, circularity is widely used as a measure of compactness, since it is a natural requirement for a district to be as circular as possible. In this paper, we introduce a novel and parameter-free circularity measure that is based on Hu moment invariants. This new measure provides a powerful tool to detect districts with abnormal shapes. We examined some districts of Arkansas, Iowa, Kansas, and Utah over several consecutive periods and redistricting plans, and also compared the results with classical circularity indexes. We found that the fall of the average circularity value of the new measure indicates potential gerrymandering.

A Note on Cluster Validity Indices SV and OS

Krista Rizman Zalik and Borut Zalik proposed indices SV and OS associating with separation and compactness or overlap. Compactness and overlap were calculated by a few data points of a cluster, which makes the indices able to identify the number of clusters underlying the data set of different sizes and densities. However, the measure of overlap depends on two undefined variants and the measure of compactness are build on randomly selected ten percent data points of a cluster, which makes them difficult to compute and unpractical. This paper supposes to measure the compactness using ten percent of data points of a cluster that are farthest away from the center of the cluster, and revises the measure of overlap so that two undefined variants are eliminated from the measure of overlap. Experiments show that the modified index SV can identify the optional number of clusters underlying the data set of different size and prefers to the indexOS.

On the Measure of Compactness of Fuzzy Clustering

This paper tested the measures of compactness of fuzzy partitions. Over the same labeled data, Fuzzy k-Means clustering algorithm generates the first partition, then the proposed revision function in (7) revises it several times to generate various fuzzy partitions with different pattern recognition rates computed by (6), finally the measures of compactness measure the compactness of each fuzzy partition. Experimental results on real data show that the measures of compactness in literatures fail to measure the compactness of a fuzzy clustering in some cases, for they argue that the fuzzy clustering with higher pattern recognition rate is less compact and worse than that with lower pattern recognition rate.