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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

Summary

This summary is machine-generated.

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 homology chaotic attractors fractal dimension topological 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.