Topological data analysis for true step detection in periodic piecewise constant signals

Khasawneh, Firas A.; Munch, Elizabeth

doi:10.1098/rspa.2018.0027

Cited by 12 publications

(8 citation statements)

References 70 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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“…Using TDA to recognize different regimes of behavior of an underlining complex system has received a lot of attention in the last few years. Hence, there is a well-grounded theoretical foundation, as well as numerous experimental results; see, e.g., [2,3,38,39,28,29,43,34,30]. However, the application of TDA to financial markets is in the early stages of development; see, e.g., [21,22,47].…”

Section: Introductionmentioning

confidence: 99%

Topological Recognition of Critical Transitions in Time Series of Cryptocurrencies

Gidea

Goldsmith

Katz³

et al. 2018

SSRN Journal

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We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital market crash at the end of 2017 -beginning 2018. We introduce a methodology that combines topological data analysis with a machine learning technique -k-means clustering -in order to automatically recognize the emerging chaotic regime in a complex system approaching a critical transition. We first test our methodology on the complex system dynamics of a Lorenz-type attractor, and then we apply it to the four major cryptocurrencies. We find early warning signals for critical transitions in the cryptocurrency markets, even though the relevant time series exhibit a highly erratic behavior.

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“…With C α = b − a and the values of a and b from Equation (21) and Equation ( 22), respectively, we can solve for our general cutoff expression as…”

Section: Cutoff Backgroundmentioning

confidence: 99%

ANAPT: Additive noise analysis for persistence thresholding

Myers

Khasawneh

Fasy

2022

FoDS

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<p style='text-indent:20px;'>We introduce a novel method for Additive Noise Analysis for Persistence Thresholding (ANAPT) which separates significant features in the sublevel set persistence diagram of a time series based on a statistics analysis of the persistence of a noise distribution. Specifically, we consider an additive noise model and leverage the statistical analysis to provide a noise cutoff or confidence interval in the persistence diagram for the observed time series. This analysis is done for several common noise models including Gaussian, uniform, exponential, and Rayleigh distributions. ANAPT is computationally efficient, does not require any signal pre-filtering, is widely applicable, and has open-source software available. We demonstrate the functionality of ANAPT with both numerically simulated examples and an experimental data set. Additionally, we provide an efficient <inline-formula><tex-math id="M1">\begin{document}$ \Theta(n\log(n)) $\end{document}</tex-math></inline-formula> algorithm for calculating the zero-dimensional sublevel set persistence homology.</p>

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Topological data analysis for true step detection in periodic piecewise constant signals

Cited by 12 publications

References 70 publications

Persistent homology of complex networks for dynamic state detection

Persistent homology of complex networks for dynamic state detection

Topological Recognition of Critical Transitions in Time Series of Cryptocurrencies

ANAPT: Additive noise analysis for persistence thresholding

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