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Hi-C: A Method to Study the Three-dimensional Architecture of Genomes.
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FRACTAL DIMENSION ESTIMATION WITH PERSISTENT HOMOLOGY: A COMPARATIVE STUDY.

Jonathan Jaquette1, Benjamin Schweinhart2

  • 1Department of Mathematics, Brandeis University Waltham, MA 02453.

Communications in Nonlinear Science & Numerical Simulation
|April 8, 2020
PubMed
Summary

Persistent homology dimensions offer a new, practical method for estimating fractal dimensions from point data. This approach performs comparably to correlation dimensions and surpasses box-counting for fractal analysis.

Keywords:
Persistent homologychaotic attractorsfractal dimensiontopological data analysis

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Area of Science:

  • Topology
  • Data Analysis
  • Fractal Geometry

Background:

  • Estimating fractal dimensions is crucial for characterizing complex datasets.
  • Classical methods like box-counting and correlation dimensions have limitations.

Purpose of the Study:

  • To evaluate persistent homology dimensions as a practical tool for fractal dimension estimation.
  • To compare the performance of persistent homology dimension estimation against classical methods.

Main Methods:

  • Implementation of an algorithm to estimate persistent homology dimensions.
  • Comparison with correlation and box-counting dimension estimation methods.
  • Application to self-similar fractals, chaotic attractors, and empirical datasets.

Main Results:

  • The 0-dimensional persistent homology dimension shows performance comparable to the correlation dimension.
  • Persistent homology dimension estimation outperformed the box-counting method.
  • Demonstrated practical utility across diverse fractal datasets.

Conclusions:

  • Persistent homology dimensions provide a robust and effective method for fractal dimension estimation.
  • This topological approach offers advantages over traditional geometric methods for point cloud data.