Topological characterization and early detection of bifurcations and chaos in complex systems using persistent homology

Mittal, Khushboo; Gupta, Savita

doi:10.1063/1.4983840

Cited by 37 publications

(15 citation statements)

References 17 publications

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“…Therefore, we summarize the information from topological dimension 1 (loops) in five statistically inferred features: number of loops, the average diameter of birth, the average diameter of duration, the standard deviation of birth diameters, and the standard deviation of duration diameters. This is similar to what Mittal and Gupta suggested in [34] to summarize persistence diagram-that is using six features from the persistence diagram including the number of holes, the average lifetime of holes, the maximum diameter of holes and the maximum distance between holes in each dimension. Here we utilize some similar features.…”

Section: Topological Features From Term Frequency Spacementioning

confidence: 57%

Topological Data Analysis in Text Classification: Extracting Features with Additive Information

Gholizadeh,

Savle,

Seyeditabari

et al. 2020

Preprint

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While the strength of Topological Data Analysis has been explored in many studies on high dimensional numeric data, it is still a challenging task to apply it to text. As the primary goal in topological data analysis is to define and quantify the shapes in numeric data, defining shapes in the text is much more challenging, even though the geometries of vector spaces and conceptual spaces are clearly relevant for information retrieval and semantics. In this paper, we examine two different methods of extraction of topological features from text, using as the underlying representations of words the two most popular methods, namely word embeddings and TF-IDF vectors. To extract topological features from the word embedding space, we interpret the embedding of a text document as high dimensional time series, and we analyze the topology of the underlying graph where the vertices correspond to different embedding dimensions. For topological data analysis with the TF-IDF representations, we analyze the topology of the graph whose vertices come from the TF-IDF vectors of different blocks in the textual document. In both cases, we apply homological persistence to reveal the geometric structures under different distance resolutions. Our results show that these topological features carry some exclusive information that is not captured by conventional text mining methods.In our experiments we observe adding topological features to the conventional features in ensemble models improves the classification results (up to 5%). On the other hand, as expected, topological features by themselves may be not sufficient for effective classification. It is an open problem to see whether TDA features from word embeddings might be sufficient, as they seem to perform within a range of few points from top results obtained with a linear support vector classifier.Keywords topological data analysis • natural language processing • persistent homology • text analytics • computational topology

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Section: Topological Features From Term Frequency Spacementioning

confidence: 57%

Topological Data Analysis in Text Classification: Extracting Features with Additive Information

Gholizadeh,

Savle,

Seyeditabari

et al. 2020

Preprint

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“…This idea is considered one of the most straightforward way to extract features from persistence diagrams (Pun et al, 2018). Despite its simplicity, it has been used in several studies, such as skin lesions classification (Chung et al, 2018), bifurcations analysis in dynamical systems (Mittal and Gupta, 2017), and protein classification (Cang et al, 2015).…”

Section: Persistence Statisticsmentioning

confidence: 99%

A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification

Chung

et al. 2021

Front. Physiol.

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Persistent homology is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply persistent homology to study time series, particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects—the persistent homologies and hence persistence diagrams of its sub-level set and Taken's lag map. Second, we propose a systematic and computationally efficient approach to summarize persistence diagrams, which we coined persistence statistics. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets.

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“…Recently, Mittal and Gupta have explored persistence homology as a tool for detecting early bifurcations and chaos in dynamic systems [23]. In [23], persistent homology is used to identify the presence of holes in the phase portraits of deterministic dynamical systems, thus an indicator for bifurcations. The method has shown promising results in [23] even with a signal to noise ratio of 30 dB.…”

Section: A Topological Approachmentioning

confidence: 99%

A look into chaos detection through topological data analysis

Tempelman

Khasawneh

2020

Physica D: Nonlinear Phenomena

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Traditionally, computation of Lyapunov exponents has been the marque method for identifying chaos in a time series. Recently, new methods have emerged for systems with both known and unknown models to produce a definitive 0-1 diagnostic. However, there still lacks a method which can reliably perform an evaluation for noisy time series with no known model. In this paper, we present a new chaos detection method which utilizes tools from topological data analysis. Bi-variate density estimates of the randomly projected time series in the p-q plane described in Gottwald and Melbourne's approach for 0-1 detection are used to generate a gray-scale image. We show that simple statistical summaries of the 0D sub-level set persistence of the images can elucidate whether or not the underlying time series is chaotic. Case studies on the Lorenz and Rossler attractors as well as the Logistic Map are used to validate this claim. We demonstrate that our test is comparable to the 0-1 correlation test for clean time series and that it is able to distinguish between periodic and chaotic dynamics even at high noise-levels. However, we show that neither our persistence based test nor the 0-1 test converge for trajectories with partially predicable chaos, i.e. trajectories with a cross-distance scaling exponent of zero and a non-zero cross correlation.

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Topological characterization and early detection of bifurcations and chaos in complex systems using persistent homology

Cited by 37 publications

References 17 publications

Topological Data Analysis in Text Classification: Extracting Features with Additive Information

Topological Data Analysis in Text Classification: Extracting Features with Additive Information

A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification

A look into chaos detection through topological data analysis

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