Limit theorems for persistence diagrams

Hiraoka, Yasuaki; Shirai, Tomoyuki; Trinh, Khanh Duy

doi:10.1214/17-aap1371

Cited by 75 publications

(142 citation statements)

References 32 publications

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“…. , E k are pairwise disjoint and contain points above the diagonal, [25,Example 1.8] shows that under the homogeneous Poisson point process the event F ∩ F 2 has positive probability. Also F ∩ F 1 is of positive probability.…”

Section: Acmentioning

confidence: 99%

Testing goodness of fit for point processes via topological data analysis

Biscio¹,

Chenavier²,

Hirsch³

et al. 2020

Electron. J. Statist.

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We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

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Section: Acmentioning

confidence: 99%

Testing goodness of fit for point processes via topological data analysis

Biscio¹,

Chenavier²,

Hirsch³

et al. 2020

Electron. J. Statist.

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“…Point processes have gathered interest in TDA community, see for example [4,9,16]. We simulated six different classes of point processes on a unit square, see their descriptions below.…”

Section: Point Processesmentioning

confidence: 99%

Metrics and Stabilization in One Parameter Persistence

Chachólski¹,

Riihimäki²

2020

SIAM J. Appl. Algebra Geometry

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We propose the use of persistent homology in a supervised way. We believe homological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to a model. Fitting this model is what we believe supervised homological persistence is. We develop theory behind constructing such models and we give evidence of the usefulness of this approach in concrete data analysis tasks.

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“…These were [10,20,30,40,50,60,70,80,90,100,150,200,400] for the tree species and [10,20,30,40,50,60,70,80,90,100] for the point processes data. For each sampling rate in the point process data, 10 runs were conducted of the entire procedure, and the classi cation accuracy would be ultimately averaged for these runs.…”

Section: Methodsmentioning

confidence: 99%

“…The rst was a simulation of six di erent point processes on the unit square. Point processes have gathered interest in TDA community as case studies, see for example [9,10,11]. Let X ∼ PD(k) denote that random variable X follows probability disribution PD with parameter k. In particular, Poisson(λ) denotes the Poisson distribution with event rate λ.…”

Section: Data Descriptionmentioning

confidence: 99%

A topological data analysis based classification method for multiple measurements

Riihimäki

Chachólski

Theorell

et al. 2019

Preprint

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Background, Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classi er for repeated measurements which samples from the data space and builds a network graph based on the data topology. When applying this to two case studies, accuracy exceeds alternative models with additional bene ts such as reporting data subsets with high purity along with feature values. Results, For 300 examples of 3 tree species, the accuracy reached 80% after 30 datapoints, which was improved to 90% after increased sampling to 400 datapoints. Using data from 100 examples of each of 6 point processes, the classi er achieved 96.8% accuracy. In both datasets, the TDA classi er outperformed an alternative model. Conclusions, This algorithm and software can be bene cial for repeated measurement data common in biological sciences, as both an accurate classi er and a feature selection tool.

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Limit theorems for persistence diagrams

Cited by 75 publications

References 32 publications

Testing goodness of fit for point processes via topological data analysis

Testing goodness of fit for point processes via topological data analysis

Metrics and Stabilization in One Parameter Persistence

A topological data analysis based classification method for multiple measurements

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