# Persistent Homology Analysis of Brain Artery Trees.

@article{Bendich2016PersistentHA, title={Persistent Homology Analysis of Brain Artery Trees.}, author={Paul Bendich and J. S. Marron and Ezra Miller and Alex Pieloch and Sean Skwerer}, journal={The annals of applied statistics}, year={2016}, volume={10 1}, pages={ 198-218 } }

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to…

## Figures from this paper

## 201 Citations

PERSISTENT HOMOLOGY ANALYSIS OF BRAIN

- Environmental Science
- 2016

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of…

Can neural networks learn persistent homology features?

- MathematicsArXiv
- 2020

This work explores the possibility of learning several types of features extracted from persistence diagrams using neural networks, including connected components or holes across the one-parameter family.

Topological Data Analysis for Multivariate Time Series Data

- Computer Science
- 2022

The aim in this article is to introduce TDA concepts to a statistical audience and provide an approach to analyze multivariate time series data and focus on multivariate brain signals and brain connectivity networks.

Topological data analysis of task-based fMRI data from experiments on schizophrenia

- Computer ScienceJournal of Physics: Complexity
- 2021

This paper uses persistent homology to analyze networks that are constructed from task-based fMRI data from schizophrenia patients, healthy controls, and healthy siblings of schizophrenia patients and finds that the members of the sibling cohort have topological features that are distinct from the other two cohorts.

Statistical shape analysis of brain arterial networks (BAN)

- MathematicsArXiv
- 2020

A mathematical representation, a Riemannian metric and other geometrical tools, such as computations of geodesics, means and covariances, and PCA for analyzing elastic graphs and BANs are developed and found an increased variance in BAN shapes as age increases.

An introduction to persistent homology for time series

- Computer ScienceWIREs Computational Statistics
- 2021

This paper discusses persistent homology, a part of computational (algorithmic) topology that converts data into simplicial complexes and elicits information about the persistence of homology classes in the data, and computes and outputs the birth and death of such topologies via a persistence diagram.

The Accumulated Persistence Function, a New Useful Functional Summary Statistic for Topological Data Analysis, With a View to Brain Artery Trees and Spatial Point Process Applications

- MathematicsJournal of Computational and Graphical Statistics
- 2019

Abstract We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the…

Promises and pitfalls of Topological Data Analysis for brain connectivity analysis

- BiologybioRxiv
- 2021

This work estimates connectivity from functional magnetic resonance imaging and electrophysiology data, employs Persistent Homology and quantifies its ability to distinguish healthy from diseased brain states by applying a support vector machine to features quantifying persistent homology structure.

Metrics for comparing neuronal tree shapes based on persistent homology

- Computer SciencePloS one
- 2017

This work aims to develop a flexible yet powerful framework to support comparison and classification of large collection of neuron structures efficiently and shows that using a certain descriptor function will give a persistence-based signature containing strictly more information than the classical Sholl analysis.

The importance of the whole: Topological data analysis for the network neuroscientist

- Biology, Computer ScienceNetwork Neuroscience
- 2019

An introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as the authors move through a combinatorial object such as a weighted network.

## References

SHOWING 1-10 OF 43 REFERENCES

Tree-Oriented Analysis of Brain Artery Structure

- BiologyJournal of Mathematical Imaging and Vision
- 2013

This analysis of MRA data shows a statistically significant effect of age and sex on brain artery structure, and variation in the proximity of brain arteries to the cortical surface results in strong statistical difference between sexes and statistically significant age effect.

Functional Data Analysis of Tree Data Objects

- Computer ScienceJournal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America
- 2014

This article establishment of a connection between tree-data spaces and the well-developed area of functional data analysis (FDA), where the data objects are curves, through two tree representation approaches, the Dyck path representation and the branch length representation.

A Principal Component Analysis for Trees

- Computer Science
- 2008

An analog of Principal Component Analysis for trees is developed, based on the notion of tree-lines, and numerically fast algorithms are proposed to solve the resulting optimization problems.

Homological scaffolds of brain functional networks

- Computer ScienceJournal of The Royal Society Interface
- 2014

The results show that the homological structure of the brain's functional patterns undergoes a dramatic change post-psilocybin, characterized by the appearance of many transient structures of low stability and of a small number of persistent ones that are not observed in the case of placebo.

A Nonparametric Regression Model With Tree-Structured Response

- Computer Science, Mathematics
- 2012

The notion of nonparametric regression is generalized to the case of a tree-structured response variable and the proposed method to analyze a dataset of human brain artery trees is implemented.

Exploring uses of persistent homology for statistical analysis of landmark-based shape data

- PhysicsJ. Multivar. Anal.
- 2010

Object oriented data analysis: Sets of trees

- Mathematics
- 2007

Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean…

Statistical topological data analysis using persistence landscapes

- MathematicsJ. Mach. Learn. Res.
- 2015

A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined.

Digital reconstruction and morphometric analysis of human brain arterial vasculature from magnetic resonance angiography

- BiologyNeuroImage
- 2013

Principal components analysis in the space of phylogenetic trees

- Biology
- 2011

A novel geometrical approach to PCA in tree-space that constructs the first principal path in an analogous way to standard linear Euclidean PCA is described, illustrated by application to simulated sets of trees and to a set of gene trees from metazoan (animal) species.