A Stable Measure of Similarity for Time Series using Persistent Homology
利用持久同调构造稳定的时间序列相似度量AI 解读 ↓
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利用持久同调构造稳定的时间序列相似度量
- 研究问题
- 如何度量两条时间序列的相似性，使结果对数据与频率的小扰动保持稳定，并兼顾嵌入维数和降维影响？
- 方法与思路
- 作者基于持久同调提出双条件周期性评分，证明扰动稳定性、收敛所需嵌入维数及主成分降维下的稳定性，并给出算法。
- 材料陈述的贡献
- 摘要称，所提评分在合成序列和气候数据中比百分比确定性指标更稳定，计算只需一个参数，并提供算法复杂度分析。
- 阅读关注点
- 应核对稳定性界使用的距离、最小嵌入维数的含义、主成分方差条件，以及参数减少是否改变比较公平性。
- 阅读边界
- 所据为 arXiv 预印本，不能据此认定已同行评审或获期刊接收；摘要未给出稳定性界常数及实际计算规模，稳定性也不直接保证任务准确率。
原文摘要
Persistent homology, the study of holes that appear in data as one thickens balls centered around its points over time, has theoretically guaranteed stability. That is, small data perturbations guarantee small changes in the lifetimes of these holes. This stability has been used to construct a measure of periodicity for a single univariate time series, denoted score(f1). One popular measure of similarity between two time series is percent determinism (%DET), which measures the correlation between two time-series embeddings. We introduce a novel persistent-homology based measure of time-series similarity which we denote the bi-conditional periodicity score, score(f1,f2). We prove the stability of our measure under small time series and frequency perturbations, as well as the existence of a minimum embedding dimension for the convergence of our score. Our latter result implies that larger embedding dimensions may be necessary to reach desired levels of convergence. Since pairwise distances between points in these larger dimensions may start to concentrate, we also prove the stability of our measure under dimension reduction which guarantees that as long as the first K principal components capture a majority of the variance under orthogonal projection, the score will undergo small changes. We next introduce an algorithm for computing the bi-conditional periodicity score and deduce its computational complexity as O(N log N + PK^2 + P^6) for N the number of time series points, P the number of embedding points, and K the number of principal components. We experimentally verify the greater stability of our measure in comparison with %DET on both synthetic time series as well as real climate data. As well, score(f1,f2) requires only one parameter for its computation while %DET requires four.