Stationarity of Manifold Time Series

TL;DR

This paper introduces formal definitions of stationarity for manifold time series and develops statistical tests to assess stationarity, accounting for manifold geometry, with applications in biology and finance.

Contribution

It provides the first formal definitions of stationarity for manifold time series and develops novel testing procedures with proven asymptotic properties.

Findings

First-order stationarity test aligns with biological findings.

Second-order stationarity test supports key assumptions in biological data.

Methods demonstrate effectiveness through simulations and real data analysis.

Abstract

In modern interdisciplinary research, manifold time series data have been garnering more attention. A critical question in analyzing such data is ``stationarity'', which reflects the underlying dynamic behavior and is crucial across various fields like cell biology, neuroscience and empirical finance. Yet, there has been an absence of a formal definition of stationarity that is tailored to manifold time series. This work bridges this gap by proposing the first definitions of first-order and second-order stationarity for manifold time series. Additionally, we develop novel statistical procedures to test the stationarity of manifold time series and study their asymptotic properties. Our methods account for the curved nature of manifolds, leading to a more intricate analysis than that in Euclidean space. The effectiveness of our methods is evaluated through numerical simulations and their practical merits are demonstrated through analyzing a cell-type proportion time series dataset from a paper recently published in Cell. The first-order stationarity test result aligns with the biological findings of this paper, while the second-order stationarity test provides numerical support for a critical assumption made therein.