REVEALING BAND AND CIRCUMPLEX PATTERNS IN REORDERABLE MATRICES USING POLAR SORT AND FAST MULTIDIMENSIONAL PROJECTIONS

Abstract

Analysts may use matrix-based visualizations (such as heatmaps) to reveal patterns of a dataset with the help of reordering algorithms that permute matrix rows and columns properly. One of these algorithms is Polar Sort, a pattern-focused reordering method that uses a multidimensional projection technique – Classical MDS – to reveal Band and Circumplex patterns in reorderable matrices. Despite its good reordering results regarding the mentioned patterns, Polar sort is not scalable due to Classical MDS’ asymptotic time complexity (O(n3) for an input matrix with size n × n). In this paper, we propose a new version of this algorithm, in which we replace Classical MDS with FastMap, a method with asymptotic time complexity O(n). The new algorithm (Polar Sort with Fastmap, or PSF for short) permutes rows and columns according to their bidimensional projections and uses a barycenter-based ordering identical to Polar Sort’s approach. The results of an experiment indicate that PSF maintained the output quality of Polar Sort regarding minimal span loss function, Moore stress, and circular correlation when reordering synthetic matrices. Besides, PSF’s asymptotic time complexity is O(n log n). This complexity is coherent with our experiment results, which point out that PSF had lower execution time than other compared methods. We also show some examples in which real-world matrices reordered by PSF revealed patterns similar to Band and Circumplex.