Spherically Embedded Time Series with Unknown Trend and Periodic Components

2026-04-03

AI Summary

This study addresses the challenge of modeling nonstationary spherical time series that exhibit intrinsic spherical geometry and contain unknown trend and periodic components, which conventional Euclidean methods fail to capture effectively. The work proposes the first unified geometric framework that introduces a novel nonparametric trend–cycle decomposition based on optimal transport. This approach sequentially extracts smooth trend and periodic components while preserving the spherical topology, followed by fitting a spherical autoregressive model to the residual stationary component. The method achieves both interpretability and predictive accuracy; theoretical analysis establishes its consistency, and extensive simulations alongside real-world applications—such as electricity generation mix and bike-sharing flow data—demonstrate its significant superiority over existing approaches in uncovering structural dynamics and enhancing forecasting performance.

Abstract

Spherically embedded time series are time series with values naturally residing on or can be equivalently mapped to the sphere. Despite their ubiquity in diverse scientific fields, these data frequently exhibit complex non-stationarity driven by latent trend and periodic components. Traditional Euclidean time series methods fail to account for the intrinsic non-Euclidean geometry of the sphere, leaving a critical gap in rigorous methodologies for modelling and forecasting nonstationary spherically embedded time series. To address this methodological gap, we propose a unified geometric framework to analyse nonstationary spherically embedded time series. Central to our approach is a novel nonparametric spherical trend-periodicity decomposition model that uses an optimal-transport-based removal operation to sequentially extract the smooth trend and periodic components while preserving spherical topology. The resulting de-trended and de-seasonalised stationary residuals can be further modelled using a spherical autoregressive model, formalising a novel trend-periodic spherical autoregressive model. Theoretical foundations for the modelling procedure are established on the consistency under temporal dependence. Extensive simulations corroborate these theoretical guarantees and demonstrate the superior finite-sample predictive performance of the trend-periodic spherical autoregressive model. Finally, we validate the practical utility of our methodology through applications to electricity generation compositions and bike trip volume profiles, yielding significantly enhanced forecasting accuracy while providing interpretable insights into the underlying structural dynamics.

Problem

• spherically embedded time series
• nonstationarity
• trend and periodic components
• non-Euclidean geometry
• time series forecasting

Innovation

• spherical time series
• optimal transport
• trend-periodicity decomposition
• non-Euclidean geometry
• spherical autoregressive model