Extremal Event Graphs: A (Stable) Tool for Analyzing Noisy Time Series Data

TL;DR

This paper introduces extremal event DAGs, a noise-robust graph-based method for analyzing noisy time series data, with applications in genomics and biology, supported by theoretical stability guarantees and practical algorithms.

Contribution

It presents a novel extremal event DAG construction using persistent homology, along with a stable distance measure and publicly available algorithms for analysis.

Findings

Extremal event DAGs are stable under measurement noise.

The method effectively captures extremal features in noisy data.

Algorithms and software are provided for practical use.

Abstract

Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise $L_{\infty}$ distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.