tdaphantom 1.0.0

Statistical hypothesis testing for persistence diagrams and barcodes

TDA-PHANTOM

Topological data analysis - Persistent Homology Analysis via Null Testing On Manifolds (TDA-PHANTOM) is a tool for statistically analysing significance of persistence diagrams and barcodes.

This project implements hypothesis tests from:

• Confidence Sets for Persistence Diagrams Fasy et al. (2014) DOI: https://doi.org/10.1214/14-AOS1252

• A Universal Null-Distribution for Topological Data Analysis Bobrowski and Skraba (2023) DOI: https://doi.org/10.1038/s41598-023-37842-2

Installation

Via PyPI:

pip install tdaphantom

Or you can clone this repository and install it manually:

python setup.py install

Overview

This tool can build a Vietoris-Rips complex from either a point cloud or distance matrix.

It can then be used to visualise the persistence diagram for that complex, and run various hypothesis tests for it.

The results of these hypothesis tests can be analysed via a return results array, or visualised in a signifiance persistence diagram.

Example Usage

Generate data

def _make_circle(n=2000, noise=0.03, seed=1):
    rng   = np.random.default_rng(seed)
    theta = rng.uniform(0, 2 * np.pi, n)
    pts   = np.stack([np.cos(theta), np.sin(theta)], axis=1)
    return pts + rng.normal(0, noise, pts.shape)

pc_circle = _make_circle()

Init Phantom class

phantom_circle = Phantom(pc_circle)

Calculate persistence diagram

Here we go up to homological dimension 1

phantom_circle.calculate_dgms_from_point_cloud_ripser(k=1)

Display persistence diagram

phantom_circle.display_dgms()

Run hypothesis test

alpha      = 0.01
correction = None
methods    = ["universal_null"]

phantom_circle.hypothesis_test(alpha, correction_method=correction, methods=methods, k=1)

Display signifiance persistence diagram

phantom_circle.display_results()

Basic useage

Avaliable methods

Universal null median

Useage

Theory

Universal null mean

Useage

Theory

Bottleneck subsampling

Useage

Theory

TODO

• Add bottleneck shells
• Add bottleneck density
• Add bottleneck concentration
• Add more integeration tests
• Add more unit tests