Machine Learning and Inference for Topological Data Analysis.

TDApplied

TDApplied is an R package for analyzing persistence diagrams using machine learning and statistical inference, and is designed to interface with persistent (co)homology calculations from the R packages TDA and TDAstats. Please note that during the development of TDApplied, TDA was available on CRAN and therefore included in package examples and tests, however since that is presently not the case the dependency on TDA has been removed (and therefore some examples and tests have been modified) but TDApplied will still work with TDA computed persistence diagrams and TDA functions if a user already has a working version installed.

R package TDA:

Fasy, Brittany T., Jisu Kim, Fabrizio Lecci, Clement Maria, David L. Millman, and Vincent Rouvreau. 2021. TDA: Statistical Tools for Topological Data Analysis. https://CRAN.R-project.org/package=TDA.

R package TDAstats:

Wadhwa, Raoul R., Drew R. K. Williamson, Andrew Dhawan, and Jacob G. Scott. 2018. TDAstats: R pipeline for computing persistent homology in topological data analysis. https://CRAN.R-project.org/package=TDAstats.

Installation

To install the latest version of this R package directly from github:

install.packages("devtools")
library(devtools)
devtools::install_github("shaelebrown/TDApplied")
library(TDApplied)

To install from Github you might need:

• Windows: Rtools (https://cran.r-project.org/bin/windows/Rtools/)
• OS X: xcode (from the app store)
• Linux: apt-get install r-base-dev (or similar).

To install the stable version of this R package from CRAN:

install.packages("TDApplied")

Citation

To cite this package in publication please use the BibTex entry:

@Manual{TDApplied, title = {TDApplied: Machine Learning and Inference for Topological Data Analysis}, author = {Shael Brown and Dr. Reza Farivar}, note = {R package version 3.0.0}, url = {https://github.com/shaelebrown/TDApplied}, }

If you wish to cite a particular method used in TDApplied see the REFERENCES.bib file in the vignette directory.

Functionality

TDApplied has three major modules:

1. Computing and interpreting persistence diagrams. The PyH function connects with python creating a fast persistent (co)homology engine compared to alternatives. The plot_diagram function can be used to plot diagrams computed from PyH or the TDA and TDAstats packages. The rips_graphs and plot_rips_graphs functions can be used to visualize dataset structure at the scale of particular topological features. The bootstrap_persistence_thresholds function can be used to identify statistically significant topological features in a dataset.
2. Machine learning. The functions diagram_mds, diagram_kpca and predict_diagram_kpca can be used to project a group of diagrams into a low dimensional space (i.e. dimension reduction). The functions diagram_kkmeans and predict_diagram_kkmeans can be used to cluster a group of diagrams. The functions diagram_ksvm and predict_diagram_ksvm can be used to link, through a prediction function, persistence diagrams and an outcome (i.e. dependent) variable.
3. Statistics. The permutation_test function acts like an ANOVA test for identifying group differences of persistence diagrams. The independence_test function can determine if two groups of paired persistence diagrams are likely independent or not.

Not only does TDApplied provide methods for the applied analysis of persistence diagrams which were previously unavailable, but an emphasis on speed and scalability through parallelization, C code, avoiding redundant slow computations, etc., makes TDApplied a powerful tool for carrying out applied analyses of persistence diagrams.

Example Code

This example creates six persistence diagrams, plots one and projects all six into 2D space using multidimensional scaling (MDS) to demonstrate TDApplied functionalities.

library(TDApplied)

# create 6 persistence diagrams
# 3 from circles and 3 from spheres
circ1 <- TDAstats::calculate_homology(TDAstats::circle2d[sample(1:100,size = 50),],dim = 1,threshold = 2)
circ2 <- TDAstats::calculate_homology(TDAstats::circle2d[sample(1:100,size = 50),],dim = 1,threshold = 2)
circ3 <- TDAstats::calculate_homology(TDAstats::circle2d[sample(1:100,size = 50),],dim = 1,threshold = 2)
sphere1 <- TDAstats::calculate_homology(TDAstats::sphere3d[sample(1:100,size = 50),],dim = 1,threshold = 2)
sphere2 <- TDAstats::calculate_homology(TDAstats::sphere3d[sample(1:100,size = 50),],dim = 1,threshold = 2)
sphere3 <- TDAstats::calculate_homology(TDAstats::sphere3d[sample(1:100,size = 50),],dim = 1,threshold = 2)

# plot a diagram
plot_diagram(circ1,title = "Circle 1")

# project into 2D and plot
proj_2D <- diagram_mds(list(circ1,circ2,circ3,sphere1,sphere2,sphere3),dim = 1,k = 2)
plot(x = proj_2D[,1],y = proj_2D[,2])

Documentation

TDApplied has five major vignettes:

1. “TDApplied Theory and Practice”, which documents the background theory and practical usage of all functions (on simple simulated data).
2. “Human Connectome Project Analysis”, which provides a sample analysis of real neurological data using TDApplied.
3. “Benchmarking and Speedups”, which describes all implemented optimizations of TDApplied functions and compares the runtime of TDApplied functions with functions from other packages.
4. “Personalized Analyses with TDApplied”, which demonstrates how machine learning (or statistical) models and pipelines, other than those implemented in TDApplied, can be fit to persistence diagrams.
5. “Comparing Distance Calculations”, which accounts for differences in distance functions of persistence diagrams across R packages.

Contribute

To contribute to TDApplied you can create issues for any bugs/suggestions on the issues page. You can also fork the TDApplied repository and create pull requests to add features you think will be useful for users.