Probabilistic Fréchet means for time varying persistence diagrams

Munch, Elizabeth; Turner, Katharine; Bendich, Paul; Mukherjee, Sayan; Mattingly, Jonathan C.; Harer, John

doi:10.1214/15-ejs1030

Cited by 51 publications

(37 citation statements)

References 29 publications

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“…For example, Gamble and Heo (2010) found interesting structure using multidimensional scaling with a W p -dissimilarity matrix computed from a set of persistence diagrams, each one associated with a set of landmarks on a single tooth. One could go further, using methods such as the Fréchet mean approach of Mileyko, Mukherjee and Harer (2011) or Munch et al (2015) to find the center of the data followed by multidimensional scaling to analyze variation about the mean. We opted not to go that route because computation of the W p -metric is generally expensive.…”

Section: Detailed Analysis Of Brain Artery Datamentioning

confidence: 99%

Persistent homology analysis of brain artery trees

Bendich¹,

Marron²,

Miller³

et al. 2016

Ann. Appl. Stat.

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232

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New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.

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Section: Detailed Analysis Of Brain Artery Datamentioning

confidence: 99%

Persistent homology analysis of brain artery trees

Bendich¹,

Marron²,

Miller³

et al. 2016

Ann. Appl. Stat.

Self Cite

232

227

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show abstract

“…This is despite the fact that the measure-theoretic issues involved in defining probability spaces for objects related to persistent homology have indeed been solved; e.g. [32,47].…”

Section: Introductionmentioning

confidence: 99%

Modeling and replicating statistical topology and evidence for CMB nonhomogeneity

Adler

Agami

Pranav

2017

Proc. Natl. Acad. Sci. U.S.A.

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Under the banner of "big data," the detection and classification of structure in extremely large, high-dimensional, data sets are two of the central statistical challenges of our times. Among the most intriguing new approaches to this challenge is "TDA," or "topological data analysis," one of the primary aims of which is providing nonmetric, but topologically informative, preanalyses of data which make later, more quantitative, analyses feasible. While TDA rests on strong mathematical foundations from topology, in applications, it has faced challenges due to difficulties in handling issues of statistical reliability and robustness, often leading to an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis-the typical case for big data applications-the replications permit conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical framework for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the parametric approach to TDA modeling in an analysis of cosmic microwave background (CMB) nonhomogeneity.

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“…Many of the standard statistics techniques do not immediately apply, however, because unlike the standard literature, the statistic used to represent the data is not a real number, but a much more complicated object: the persistence diagram. Several methods have been proposed for looking at the mean of a collection of diagrams, including the Fréchet mean (Turner, Mileyko, Mukherjee, & Harer, 2014;Munch et al, 2015) and the persistence landscape (Bubenik, 2015). There are confidence intervals for persistence diagrams (Fasy et al, 2014b), as well as intersections with the machine learning framework (Niyogi, Smale, & Weinberger, 2011).…”

Section: Further Readingmentioning

confidence: 99%

A User’s Guide to Topological Data Analysis

Munch

2017

Learning Analytics

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116

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ABSTRACT. Topological data analysis (TDA) is a collection of powerful tools that can quantify shape and structure in data in order to answer questions from the data's domain. This is done by representing some aspect of the structure of the data in a simplified topological signature. In this article, we introduce two of the most commonly used topological signatures. First, the persistence diagram represents loops and holes in the space by considering connectivity of the data points for a continuum of values rather than a single fixed value. The second topological signature, the mapper graph, returns a 1-dimensional structure representing the shape of the data, and is particularly good for exploration and visualization of the data. While these techniques are based on very sophisticated mathematics, the current ubiquity of available software means that these tools are more accessible than ever to be applied to data by researchers in education and learning, as well as all domain scientists.

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Probabilistic Fréchet means for time varying persistence diagrams

Cited by 51 publications

References 29 publications

Persistent homology analysis of brain artery trees

Persistent homology analysis of brain artery trees

Modeling and replicating statistical topology and evidence for CMB nonhomogeneity

A User’s Guide to Topological Data Analysis

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