In this paper we develop a novel Topological Data Analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram-a two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.
Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.
This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks’ shapes that are missed by traditional connectivity and centrality graph statistics.
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