Persistent topological features of dynamical systems

Maletić, Slobodan; Zhao, Yi; Rajković, Milan

doi:10.1063/1.4949472

Cited by 78 publications

(40 citation statements)

References 47 publications

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“…One particularly useful tool for this analysis is 1-dimensional persistent homology [32,33], which encodes how circular structures persist over the course of a filtration in a topological signature called a persistence diagram. This and its variants have been quite successful in applications, particularly for the analysis of periodicity [34][35][36][37][38][39][40][41], including for parameter selection [42,43], data clustering [44], machining dynamics [45][46][47][48][49], gene regulatory systems [50,51], financial data [52][53][54], wheeze detection [55], sonar classification [56], video analysis [57][58][59], and annotation of song structure [60,61].…”

Section: Introductionmentioning

confidence: 99%

Persistent homology of complex networks for dynamic state detection

2019

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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.

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Section: Introductionmentioning

confidence: 99%

Persistent homology of complex networks for dynamic state detection

2019

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“…Exploration of stable topological structures (or 'shapes') in nosy multidimensional datasets has led to new insights, including the discovery of a subgroup of breast cancers [8], is actively used in image processing [9], in signal and time-series analysis [10,11,12,13,14,15]. The latter has primarily been applied to detect and quantify periodic patterns in data [16,17], to understand the nature of chaotic attractors in the phase space of complex dynamical systems [18], to analyze turbulent flows [19], and stock correlation networks [20].…”

Section: Introductionmentioning

confidence: 99%

Topological data analysis of financial time series: Landscapes of crashes

Gidea

Katz²

2018

Physica A: Statistical Mechanics and its Applications

183

123

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We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional time series. Using a sliding window, we extract time-dependent point cloud data sets, to which we associate a topological space. We detect transient loops that appear in this space, and we measure their persistence. This is encoded in real-valued functions referred to as a 'persistence landscapes'. We quantify the temporal changes in persistence landscapes via their L p -norms. We test this procedure on multidimensional time series generated by various non-linear and non-equilibrium models. We find that, in the vicinity of financial meltdowns, the L p -norms exhibit strong growth prior to the primary peak, which ascends during a crash. Remarkably, the average spectral density at low frequencies of the time series of L p -norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03/10/2000, or to the Lehman bankruptcy on 09/15/2008. Our study suggests that TDA provides a new type of econometric analysis, which complements the standard statistical measures. The method can be used to detect early warning signals of imminent market crashes. We believe that this approach can be used beyond the analysis of financial time series presented here.

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“…For the detection of persistent structures in trajectories, we have implemented methods using persistent topology for the analysis of time series (Maletić et al, 2016), which have become a promising way to detect patterns in data different to entropy based methods. Despite the implementation of persistent homology is not straightforward, it shows the possibility to better select the data for training and points to the possibility to introduce clever bias in the models to reduce the amount of training data.…”

Section: Resultsmentioning

confidence: 99%

Relative distances between homology groups to assess persistent defects in time series

Ochoa¹

2019

Preprint

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It is common to consider a data-intensive strategy to be an appropriate way to develop systemic analyses in biology and physiology. Therefore, options for data collection, sampling, standardization, visualization, and interpretation determine how causes are identified in time series to build mathematical models. However, there are often biases in the collected data that can affect the validity of the model: while collecting enough large datasets seems to be a good strategy for reducing the bias of the collected data, persistent and dynamical anomalies in the data structure can affect the overall validity of the model. In this work we present a methodology based on the definition of homological groups to evaluate persistent anomalies in the structure of the sampled time series. In this evaluation relevant patterns in the combination of different time series are clustered and grouped to customize the identification of causal relationships between parameters. We test this methodology on data collected from patients using mobile sensors to test the response to physical exercise in real-world conditions and outside the lab. With this methodology we plan to obtain a patient stratification of the time series to customize models in medicine.

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Persistent topological features of dynamical systems

Cited by 78 publications

References 47 publications

Persistent homology of complex networks for dynamic state detection

Persistent homology of complex networks for dynamic state detection

Topological data analysis of financial time series: Landscapes of crashes

Relative distances between homology groups to assess persistent defects in time series

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