Detection of Irregularities and Rips by Finding Critical Points of Morse Theory - Preliminary Results on Analysis on Sales Data

• Main Authors: Sato, Hiroyuki, Seino, Yoshihiro, Ogata, Takanori, Shirakawa, Tatsuya
• Format: Conference Proceeding
• Language: English
• Subjects: anomaly detection; critical points; Data analysis; Data mining; Manifolds; Morse theory; Robustness; Statistical analysis; Three-dimensional displays; Time series analysis; topological data analysis
• ISSN: 0730-3157
• DOI: 10.1109/COMPSAC.2017.106

As data mining grows its significance in real world information extraction, we need a robust methodology without using rich assumptions. In detection of irregularities, statistical methods are very successful when we assume some distribution. However, even when we encounter data with unknown distribution, we must first assume some distribution, which may cause incorrect inference. This paper applies topological data analysis (TDA) methodologies to detection of irregularities and rips. We also apply Morse theory together with persistent homology to the analysis of dynamical system of big data set such as the surveillance video data and the sales dynamics. The idea is that data tendency changes can be observed as a change of data connectivity. In this paper, we explore the theory of critical points by using Morse theory. Furthermore, we compute the persistent homology groups over daily sales data in a store, a typical dynamical system that we know little about. We show the computed critical points of dimension 1 through 3 in the experiment. The growth of dimensions shows that some irregular event may have happened on the corresponding date point. Although we need refinement on the interpretation of the result, anyway we can collect the candidates of irregularities.