tutorials

In this one-hour tutorial, I will demonstrate how to use the PLab package (download page with demo) for extracting topological features in signals. I will also show how to perform statistical inference on these features in applications to electroencephalography (EEG).

Part 1 - Topological feature extraction from signals in the temporal domain using PLab (25 min)

• Persistent homology on a signal: Pairing algorithm

• Persistence features: Barcode, persistence landscape

Part 2 - Statistical inference on persistence landscapes (15 min)

• Permutation test

• Exact topological inference

Part 3 - Applications to EEG (15 min)

• Single-trial EEG

• Event-related potential

Q&A (5 min)

Instructors

• James Steve Marron, UNC Chapel Hill

• Moo K. Chung, University of Wisconsin - Madison

• Yuan Wang, University of South Carolina

Abstract

The era of big data in biology and medicine brings exciting opportunities for new scientific discoveries and new challenges for biostatistics. Yet, valuable information in the sheer amount of complex data may be hidden in patterns that cannot be decoded easily with standard statistical tools. The emerging area of topological data analysis (TDA) is a promising avenue of research to answer the challenge. TDA characterizes topological changes of multivariate representations of data in multidimensional scales. Not only does TDA reveal topological features in data only visible on a multi-scale level, the fact that overall topological changes hold more significance in TDA summary statistics over fleeting structures also makes the approach particularly robust at the presence of noise and artifacts. The unique powers of TDA are demonstrated by a decade worth of theory development and applications in computer vision, engineering, and neuroimaging. TDA has also provided tools for solving deep challenges in object-oriented data analysis (OODA), where the focus is in analyzing complex heterogeneous data. This short course is aimed at popularizing the state-of-the-art in TDA computation and methodology with detailed illustration using real datasets coming from medical imaging studies and genetics.

Prerequisites

Participants will benefit the most from the course if they have prior knowledge in statistics (including estimation and inference, linear regression), linear algebra and calculus. The course will be self- contained as much as possible.

Learning Objectives

Participants are expected to achieve the following learning outcomes at the end of the course:

1. Understand the basic concepts and principles of OODA and TDA.

2. Understand appropriate statistical procedures and approaches for OODA and TDA.

3. How to apply the basic concepts and procedures to complex real-world data including brain imaging time series and genetic data

4. How to use the distributed R/Matlab tools codes to perform OODA and TDA in practice.

Course Syllabus

Three leading experts in Object Oriented Data Analysis (OODA) and Topological Data Analysis (TDA) will give the full day course. Each expert will give a 2-hour lecture split into two 1 hour sessions. The first secession will be on theory and the second session will be on applications and R/MATLAB demonstration.

Session 1-1. Introduction to Object Oriented Data Analysis (OODA). Marron will introduce the basic concepts and theory of OODA.

Session 1-2. Application of OODA. Marron will explain how the OODA principles can be used in analyzing complex real world data, e.g., analysis of the variation in populations of tree-structured objects, such as brain arteries, forming a non-Euclidean space. R/MATLAB demonstration will be given.

Session 2-1. Introduction to TDA. Wang will introduce the key TDA concepts and technique persistent homology (PH) through point cloud data and simplicial complex, and demonstrate the standard algorithm for computing filtration and standard PH features (barcodes, persistence diagram, persistence landscape).

Session 2-2. Topological Signal Processing. Wang will give a tutorial on persistent homology (PH) applied to various time series data such as electroencephalographic (EEG) signals, functional magnetic resonance imaging (fMRI) signals, and audial signals, and designing appropriate statistical test for PH features in these settings. R/MATLAB demonstration will be given.

Session 3-1. TDA and OODA on graph and network data. Chung will give a tutorial on analyzing graphs and network data using real world data coming from various network data including social networks and brain networks. R/MATLAB demonstration will be given.

Session 3-2. TDA and OODA on genetics. Chung will give a hand on tutorial on how to analyze SNP data using TDA/OODA techniques. R/MATLAB demonstration will be given.

Textbook

A self-contained textbook in a form of PDF will be distributed before the course.

For OODA, a 141-page book draft by Marron and Dryden is recommended: http://stor881fall2017.web.unc.edu/files/2017/08/OODAbookV4FE.pdf.

For TDA, the following textbook is recommended:

Edelsbrunner and Harer (2010), Computational Topology, American Mathematical Society.

Software

To maximize the benefit from this course, the participants will be encouraged to bring their own laptops so that they can implement the codes during the hands-on demonstration. The instructors will make R and Matlab codes available for downloads under the Software page. We will schedule a 2-hour zoom meeting a few days before the workshop to assist those with difficulty in downloading the software.