Persistent Homology of Geospatial Data: A Case Study with Voting

Feng, Michelle; Porter, Mason A.

doi:10.48550/arxiv.1902.05911

Cited by 8 publications

(15 citation statements)

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“…Generators with longer associated half-open intervals [b x , d x ) are more persistent. It is traditional to construe more-persistent intervals as better indicators of a signal and construe less-persistent intervals as noise, although recent work (including our own [21,44]) indicates that it is not always possible to interpret persistence in this way.…”

Section: A Computing Persistent Homologymentioning

confidence: 96%

“…In this paper, we use the software package Gudhi [45,46] to compute PH from the filtered simplicial complex {X i }. We construct the filtered simplicial complex {X i } from X using two different constructions, which we developed recently in a paper on voting data [21].…”

Section: A Computing Persistent Homologymentioning

confidence: 99%

“…We then proceed with the above adjacency construction, but we substitute f for f . We recently introduced our main adjacency construction in [21], and we introduce this adaptation of it to edge-based data in the present work.…”

Section: B Adjacency Construction Of Phmentioning

confidence: 99%

“…By reframing spatial networks into the language of topological spaces, we can leverage existing topological tools to better understand their structures. For a case study with voting data, see our recent paper [21].…”

Section: Introductionmentioning

confidence: 99%

“…(For approaches other than PH for analyzing maps while accounting for density variation, see [40,41].) In particular, in a recent paper [21], we examined the shape of voting patterns in the state of California and found that traditional methods for computing PH are more likely to capture disparities in population than to detect the presence of interesting voting patterns. To address this issue, we developed two novel PH constructions -one based on network adjacency and one based on the physical geometry of a map -that were more successful at capturing these voting patterns.…”

Section: Introductionmentioning

confidence: 99%

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Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence

Feng,

Porter

2020

Preprint

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Spatial networks are ubiquitous in social, geographic, physical, and biological applications. To understand their large-scale structure, it is important to develop methods that allow one to directly probe the effects of space on structure and dynamics. Historically, algebraic topology has provided one framework for rigorously and quantitatively describing the global structure of a space, and recent advances in topological data analysis (TDA) have given scholars a new lens for analyzing network data. In this paper, we study a variety of spatial networks -including both synthetic and natural ones -using novel topological methods that we recently developed specifically for analyzing spatial networks. We demonstrate that our methods are able to capture meaningful quantities, with specifics that depend on context, in spatial networks and thereby provide useful insights into the structure of those networks, including a novel approach for characterizing them based on their topological structures. We illustrate these ideas with examples of synthetic networks and dynamics on them, street networks in cities, snowflakes, and webs spun by spiders under the influence of various psychotropic substances.

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Section: A Computing Persistent Homologymentioning

confidence: 96%

Section: A Computing Persistent Homologymentioning

confidence: 99%

Section: B Adjacency Construction Of Phmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence

Feng,

Porter

2020

Preprint

Self Cite

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Inferring symbolic dynamics of chaotic flows from persistence

Gökhan

Budanur

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We introduce state space persistence analysis for deducing the symbolic dynamics of time-series data obtained from high-dimensional chaotic attractors. To this end, we adapt a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits. By comparing the shapes along a chaotic trajectory to those of the periodic orbits, state space persistence analysis quantifies the geometric similarities of chaotic trajectory segments and the periodic orbits. We demonstrate the method by applying it to the three-dimensional Rössler system and a thirty-dimensional discretization of the Kuramoto-Sivashinsky partial differential equation in (1 + 1) dimensions.

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Persistent homology detects curvature

et al. 2020

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In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space. In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning.

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Persistent Homology of Geospatial Data: A Case Study with Voting

Cited by 8 publications

References 0 publications

Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence

Spatial Applications of Topological Data Analysis: Cities, Snowflakes, Random Structures, and Spiders Spinning Under the Influence

Inferring symbolic dynamics of chaotic flows from persistence

Persistent homology detects curvature

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