Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
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DOI：
10.1007/s10208-014-9206-z
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发表时间：
2015-06-01
影响因子：
3
通讯作者：
Harer, John
中科院分区：
文献类型：
--
作者：
Perea, Jose A.;Harer, John
本文提出了一个时间序列数据拓扑研究的理论框架。从广义上讲，我们描述了滑动窗口嵌入的几何和拓扑性质，就像通过持续同源的透镜看到的那样。特别是，我们证明了点云级别的最大持久性可以用于量化信号级别的周期性，证明了所得到的持久性图的结构定理和收敛定理，并推导了它们对窗口大小和嵌入维数的依赖估计。我们将这种方法应用于量化合成数据集的周期性，并将结果与使用最先进的基因表达分析方法获得的结果进行比较。我们称这种新方法为SW1PerS，它代表滑动窗口和一维持久性评分。
We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring.