Problem
Research questions and friction points this paper is trying to address.
Time-series Analysis
Probabilistic Data
Temporal Pattern Recognition
Wasserstein multivariate auto-regressive models for modeling distributional time series
🤖 AI Summary
This work addresses modeling distributional time series, aiming to characterize dynamic dependencies among multiple time-varying probability distributions. We propose the first multivariate autoregressive framework defined on the Wasserstein space with simplex constraints—rigorously guaranteeing existence, uniqueness, and stationarity of solutions from both geometric and stochastic process perspectives, while naturally inducing sparse coefficients for learning temporal dependency graphs. Our method integrates Wasserstein geometry, the theory of iterated random functions, and constrained optimization to construct a consistent estimator for model coefficients. Simulation studies confirm its statistical validity. Applied to age-distribution time series across multiple countries, the framework successfully uncovers cross-national dynamic interdependencies in demographic structural evolution.
📝 Abstract
This paper is focused on the statistical analysis of data consisting of a collection of multiple series of probability measures that are indexed by distinct time instants and supported over a bounded interval of the real line. By modeling these time-dependent probability measures as random objects in the Wasserstein space, we propose a new auto-regressive model for the statistical analysis of multivariate distributional time series. Using the theory of iterated random function systems, results on the existence, uniqueness and stationarity of the solution of such a model are provided. We also propose a consistent estimator for the auto-regressive coefficients of this model. Due to the simplex constraints that we impose on the model coefficients, the proposed estimator that is learned under these constraints, naturally has a sparse structure. The sparsity allows the application of the proposed model in learning a graph of temporal dependency from multivariate distributional time series. We explore the numerical performances of our estimation procedure using simulated data. To shed some light on the benefits of our approach for real data analysis, we also apply this methodology to a data set made of observations from age distribution in different countries.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Wasserstein Space
Time-varying Probability Data
Parameter Estimation Tool
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