Comparing Circular Histograms by Using Modulo Similarity and Maximum Pair-Assignment Compatibility Measure

Abstract

Histograms are an intuitively understandable tool for graphically presenting frequency data that is available for and useful in modern data-analysis, this also makes comparing histograms an interesting field of research. The concept of similarity and similarity measures have been gaining in importance, because similarity and similarity measures can be used to replace the simpler distance measures in many data-analysis applications. In this paper we concentrate on circular histograms that are well-suited for time or direction-stamped frequency data and especially on the comparison of circular histograms by way of similarity. We focus on Łukasiewicz many-valued logic based similarities and introduce a new similarity measure, the “modulo similarity” for circular problems. We prove that modulo similarity is a similarity measure in the strict sense. We also present a new compatibility measure, the “maximum pair assignment compatibility” that can be used in lossless sample-based comparison of histograms. We demonstrate the usefulness of these two new concepts by numerically applying them to a comparison of circular histograms and comparatively analyze the results with results from a comparison with a bin-based Łukasiewicz many-valued logic based method for the comparison of histograms.

Keywords

• Circular histograms
• similarity
• Łukasiewicz logic
• Similarity
• Modulo similarity
• Compatibility