Computation of persistent homology on streaming data using topological data summaries

Abstract

Persistent homology is a computationally intensive and yet extremely powerful tool for Topological Data Analysis. Applying the tool on potentially infinite sequence of data objects is a challenging task. For this reason, persistent homology and data stream mining have long been two important but disjoint areas of data science. The first computational model, that was recently introduced to bridge the gap between the two areas, is useful for detecting steady or gradual changes in data streams, such as certain genomic modifications during the evolution of species. However, that model is not suitable for applications that encounter abrupt changes of extremely short duration. This paper presents another model for computing persistent homology on streaming data that addresses the shortcoming of the previous work. The model is validated on the important real-world application of network anomaly detection. It is shown that in addition to detecting the occurrence of anomalies or attacks in computer networks, the proposed model is able to visually identify several types of traffic. Moreover, the model can accurately detect abrupt changes of extremely short as well as longer duration in the network traffic. These capabilities are not achievable by the previous model or by traditional data mining techniques.