A User’s Guide to Topological Data Analysis

Munch, Elizabeth

doi:10.18608/jla.2017.42.6

Cited by 122 publications

(92 citation statements)

References 43 publications

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“…It was for this reason that topological data analysis (TDA) [25][26][27][28][29] has proven to be quite useful for time series analysis. TDA is a collection of methods arising from the mathematical field of algebraic topology [30,31] which provide concise, quantifiable, comparable, and robust summaries of the shape of data.…”

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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“…He favoured Riemann surfaces, and recommended that we examine the work of topologists such as Buser at Lausanne, Carlsson at Stanford, and Harer at Duke. In this vein, we find near-neighbour ideas proposed by fellow mathematicians Buser and Semmler (2017) and Munch (2017).…”

Section: Mika Seppälä and The Shape Of Datamentioning

confidence: 76%

Is Learning Data in the Right Shape?

Kelly

2017

Learning Analytics

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ABSTRACT:In this short thought-piece, I attempt to capture the type of freewheeling discussions I had with our late colleague, Mika Seppälä, a research mathematician from Helsinki. Mika, not being a psychometrician or learning scientist, was blissfully free from the design constraints that experts sometimes ingest, unwittingly. I also draw on delightful conversations with the German research mathematician, Heinz-Otto Peitgen, a polyglot whose work includes advances in medical imaging and explorations in fractal geometry for K-12 students. Together, they taught me to reconsider foundational assumptions about learning, how to describe it, and how to grow it. Accordingly, I use this set of papers as a prompt for examining assumptions that numerical precision ensures scientific insight, that linear models best capture growth in learning, and that relaxing a fixation with time (exemplified by the reification of pre-and post-testing) might open up new topologies for describing, predicting, and promoting learning in its myriad manifestations.

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“…Features are then extracted from the persistence diagrams and used for machine learning. This section briefly describes the main concepts of 1-D persistent homology, but we defer a more thorough treatment to references in the literature such as [30][31][32][33][34].…”

Section: Topological Data Analysismentioning

confidence: 99%

Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector

Yesilli

Tymochko

Khasawneh

et al. 2019

2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA)

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Chatter detection has become a prominent subject of interest due to its effect on cutting tool life, surface finish and spindle of machine tool. Most of the existing methods in chatter detection literature are based on signal processing and signal decomposition. In this study, we use topological features of data simulating cutting tool vibrations, combined with four supervised machine learning algorithms to diagnose chatter in the milling process. Persistence diagrams, a method of representing topological features, are not easily used in the context of machine learning, so they must be transformed into a form that is more amenable. Specifically, we will focus on two different methods for featurizing persistence diagrams, Carlsson coordinates and template functions. In this paper, we provide classification results for simulated data from various cutting configurations, including upmilling and downmilling, in addition to the same data with some added noise. Our results show that Carlsson Coordinates and Template Functions yield accuracies as high as 96% and 95%, respectively. We also provide evidence that these topological methods are noise robust descriptors for chatter detection.

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A User’s Guide to Topological Data Analysis

Cited by 122 publications

References 43 publications

Persistent homology of complex networks for dynamic state detection

Persistent homology of complex networks for dynamic state detection

Is Learning Data in the Right Shape?

Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector

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