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Time Series, Persistent Homology and Chirality

Yuliy Baryshnikov

Abstract

We investigate the point process of persistent diagram for Brownian motions with drift, obtaining some of its basic characteristics. Further we introduce and study the refinement of the persistent homology, assigning to each bar its chirality.

Time Series, Persistent Homology and Chirality

Abstract

We investigate the point process of persistent diagram for Brownian motions with drift, obtaining some of its basic characteristics. Further we introduce and study the refinement of the persistent homology, assigning to each bar its chirality.

Paper Structure

This paper contains 40 sections, 18 theorems, 41 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Persistent Homology and Merge Trees
  3. Persistent Homology Diagrams
  4. Merge trees
  5. From merge trees to $\mathbf{PH}_0$
  6. Univariate Functions and Trees
  7. From $\mathbf{PH}_0$ to merge trees
  8. Chirality
  9. Bars and Windings
  10. Winding of a function around an interval
  11. Persistence Diagrams for Brownian Motions with Drift
  12. Persistence Diagrams as Point Processes
  13. Process decomposition
  14. Invariance properties
  15. Other Brownian motions
  16. ...and 25 more sections

Key Result

Lemma 2.2

The contour walk corresponding to the merge tree $\mathtt T_f$ of a continuous function on an interval is right-equivalent (i.e. equal up to a reparametrization of the argument) to $f$.

Figures (6)

  • Figure 1: Left to right: graph of a function, its merge tree, the pile of stems (bars) resulting from pruning, and the persistence diagram.
  • Figure 2: Coupled critical points: N in red and N in green.
  • Figure 3: Left display: There are $5$ bars straddling $I$ in this picture. The permutations of maxima and minima are, correspondingly, $\sigma_+=(514326)$ and $\sigma_-=(0 43251)$. The walls for $r=\sigma_+(3)=4$ are $s=5$ and $t=6$, at the positions $1$ and $6$ respectively, the left well is $3$, the right well is $1$ and $r=4$ is coupled with the $\sigma_-(2)=3$, thus a N . Right display: swapping the segments.
  • Figure 4: The finite automaton justifies the term windings: the number times a function straddles an interval is equal to the number of times its lift cycles around. The symbolic trajectory corresponding to the function shown above is $\alpha\to\delta\to\beta\to\delta\to\beta\to\delta\to\omega$. There are two transitions $\delta\to\beta$, and two windings around the interval $[b,d]$.
  • Figure 5: Windings around a pair of intervals. We do not indicate the states corresponding to the exits from the interval 01 on the left, as they do not contribute to the final result.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3: C
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 28 more