Banana Trees for the Persistence in Time Series Experimentally

2024-05-28

arXiv.org

AI Summary

To address the computational inefficiency of updating persistent homology—a multiscale topological descriptor—for dynamic time series, this paper introduces the Banana Tree, a novel data structure specifically designed for maintaining persistent homology under streaming updates. The Banana Tree supports efficient incremental insertion and deletion operations, drastically reducing topological update overhead. Theoretical analysis and empirical evaluation demonstrate that, on unbiased random walk time series, our method achieves over an order-of-magnitude speedup compared to state-of-the-art static persistent homology algorithms. Further experiments on real-world time series—including financial and physiological signals—reveal highly similar persistent homology distributions, confirming their topological equivalence to the synthetic benchmark. This work substantially extends the practical applicability of topological data analysis to streaming time-series settings and establishes a new paradigm for dynamic topological modeling.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Data Mining & Knowledge Management: Data Stream Mining
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain persistent homology -- a multi-scale topological descriptor -- for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.

Problem

• Efficiently maintain persistent homology for dynamic time series.
• Assess banana tree data structure properties and runtime.
• Compare performance with state-of-the-art static algorithms.

Innovation

• Banana tree structure for dynamic time series
• Efficient persistent homology maintenance
• Outperforms static algorithms in random data