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A statistical framework for analyzing shape in a time series of random geometric objects

Anne van Delft, Andrew J. Blumberg

TL;DR

A new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds, and derives complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relates these invariants to so-called ball volume processes.

Abstract

We introduce a new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds. At the core of our approach is the point of view that the data arises as sampled recordings from a metric space-valued stochastic process, possibly of nonstationary nature, thereby integrating geometric data analysis into the realm of functional time series analysis. Our framework allows for natural incorporation of spatial-temporal dynamics, heterogeneous sampling, and the study of convergence rates. Further, we derive complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relate these invariants to so-called ball volume processes. Under mild dependence conditions, a weak invariance principle in $D([0,1]\times [0,\mathscr{R}])$ is established for sequential empirical versions of the latter, assuming the probabilistic structure possibly changes over time. Finally, we use this result to introduce novel test statistics for topological change, which are distribution-free in the limit under the hypothesis of stationarity. We explore these test statistics on time series of single-cell mRNA expression data, using shape descriptors coming from topological data analysis.

A statistical framework for analyzing shape in a time series of random geometric objects

TL;DR

A new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds, and derives complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relates these invariants to so-called ball volume processes.

Abstract

We introduce a new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds. At the core of our approach is the point of view that the data arises as sampled recordings from a metric space-valued stochastic process, possibly of nonstationary nature, thereby integrating geometric data analysis into the realm of functional time series analysis. Our framework allows for natural incorporation of spatial-temporal dynamics, heterogeneous sampling, and the study of convergence rates. Further, we derive complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relate these invariants to so-called ball volume processes. Under mild dependence conditions, a weak invariance principle in is established for sequential empirical versions of the latter, assuming the probabilistic structure possibly changes over time. Finally, we use this result to introduce novel test statistics for topological change, which are distribution-free in the limit under the hypothesis of stationarity. We explore these test statistics on time series of single-cell mRNA expression data, using shape descriptors coming from topological data analysis.
Paper Structure (22 sections, 27 theorems, 181 equations, 7 figures, 2 tables)

This paper contains 22 sections, 27 theorems, 181 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Framework
  3. Stable shape descriptors
  4. Locally stationary metric space-valued processes
  5. Inference on the latent process via point clouds
  6. Sampling regime
  7. Characterizing $(S,\partial_S)$-valued processes
  8. A reconstruction theorem for metric space-valued stochastic processes
  9. Characterizing measures via ball volumes
  10. Application: Detecting changes in geometric features
  11. Methodology: a weak invariance principle in $D([0,1]\times [0,\mathscr{R}])$
  12. Self-normalized tests
  13. Data-driven weight function
  14. Simulation study
  15. Application in the context of topological data analysis (TDA)
  16. ...and 7 more sections

Key Result

Lemma 1

For all $X, Y \in S=(C( M,M^\prime),\rho)$ and $\tilde{X}^n = e_{m_1,\ldots,m_n} \circ {X}, \tilde{Y}^n = e_{m_1,\ldots,m_n} \circ {Y}$,

Figures (7)

  • Figure 1.1: Differentiated cell types emerging over time.
  • Figure 1.2: Hierarchical clustering dendrograms reveal the change in shape of the developmental process. Each panel corresponds to the spherical Gaussians from \ref{['fig:toy-example']}; as they separate, cluster structure emerges.
  • Figure 1.3: Each slice of the dendrogram corresponds to a clustering of the data.
  • Figure 2.1: Each panel corresponds to $PH_0$ of the example in \ref{['fig:toy-example']}. The $x$-axis represents the scale at which the cluster appears and the $y$-axis the scale at which it disappears. The two points appearing far away from the diagonal represent the emergence of two distinct cell types.
  • Figure 5.1: The Vietoris-Rips complexes and persistence diagram from a noisy circle.
  • ...and 2 more figures

Theorems & Definitions (64)

  • Definition 1
  • Example 1: Dendograms
  • Example 2: Persistent homology
  • Definition 2
  • Lemma 1
  • proof : Proof of \ref{['lem:dghds']}
  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2
  • ...and 54 more