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Functional Data Representation with Merge Trees

Matteo Pegoraro, Piercesare Secchi

Abstract

In this paper we face the problem of representation of functional data with the tools of algebraic topology. We represent functions by means of merge trees, which, like the more commonly used persistence diagrams, are invariant under homeomorphic reparametrizations of the functions they represent, thus allowing for a statistical analysis which is indifferent to functional misalignment. We consider a recently defined metric for merge trees and we prove some theoretical results related to its specific implementation when merge trees represent functions, establishing also a class of consistent estimators with convergence rates. To showcase the good properties of our topological approach to functional data analysis, we test it on the Aneurisk65 dataset replicating, from our different perspective, the supervised classification analysis which contributed to make this dataset a benchmark for methods dealing with misaligned functional data. In the Appendix we provide an extensive comparison between merge trees and persistence diagrams, highlighting similarities and differences, which can guide the analyst in choosing between the two representations.

Functional Data Representation with Merge Trees

Abstract

In this paper we face the problem of representation of functional data with the tools of algebraic topology. We represent functions by means of merge trees, which, like the more commonly used persistence diagrams, are invariant under homeomorphic reparametrizations of the functions they represent, thus allowing for a statistical analysis which is indifferent to functional misalignment. We consider a recently defined metric for merge trees and we prove some theoretical results related to its specific implementation when merge trees represent functions, establishing also a class of consistent estimators with convergence rates. To showcase the good properties of our topological approach to functional data analysis, we test it on the Aneurisk65 dataset replicating, from our different perspective, the supervised classification analysis which contributed to make this dataset a benchmark for methods dealing with misaligned functional data. In the Appendix we provide an extensive comparison between merge trees and persistence diagrams, highlighting similarities and differences, which can guide the analyst in choosing between the two representations.

Paper Structure

This paper contains 42 sections, 16 theorems, 41 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Merge Trees of Functions
  3. Sublevel Sets
  4. Path Connected Components
  5. Assumptions on $f:X\rightarrow \mathbb{R}$
  6. Tree Structures, Critical Values and Topological Changes
  7. Isomorphism classes
  8. Height and Weight Functions
  9. Properties
  10. The Metric for Merge Trees
  11. Persistence Diagrams
  12. Properties
  13. Metrics for Persistence Diagrams
  14. Stability, Estimators and Approximations
  15. Stability
  16. ...and 27 more sections

Key Result

Proposition 6

The isomorphism class of the merge tree of a function $f:X\rightarrow \mathbb{R}$ satisfying (A0), is invariant under homeomorphic re-parametrization of $f$.

Figures (10)

  • Figure 1: Sublevel sets of a function (a); the same function with its associated merge tree (b).
  • Figure 2: A graphical representation of the truncation process described in \ref{['sec:merge_and_weight']}.
  • Figure 3: (a)$\rightarrow$(e) form an edit path made by one deletion, one ghosting and a final shrinking.
  • Figure 4: A function (left) with its associated persistence diagram (centre) and merge tree (right). On the PD axes we see the birth and death coordinates of its points. The plot of the merge tree features the length of its branches (given by the weight function - \ref{['sec:merge_and_weight']}) on the horizontal axis, and the leaves (taxa) are displaced on the vertical axis. The vertical axis scale is only for visualization purposes. The merge tree is truncated at height $7$ - see \ref{['sec:merge_and_weight']}.
  • Figure 5: Plots to support the bandwidth choice described in \ref{['sec:band']}.
  • ...and 5 more figures

Theorems & Definitions (30)

  • Definition 1
  • Definition 2: merge_intrins, pegoraro2024finitelyfunc
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6: Invariance
  • Remark 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 20 more