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Autoregressive optimal transport models.

Changbo Zhu1, Hans-Georg Müller2

  • 1Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA.

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|July 31, 2023
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Summary
This summary is machine-generated.

We introduce autoregressive transport models to analyze distributional time series. These models extend classical autoregressive methods by operating in optimal transport map spaces, offering new insights into complex data patterns.

Keywords:
Wasserstein spacedistributional data analysisdistributional regressiondistributional time seriesiterated random functionoptimal transport

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Area of Science:

  • Distributional Data Analysis
  • Optimal Transport Theory
  • Time Series Analysis

Background:

  • Distributional time series are common in various applications.
  • Analyzing these complex datasets presents significant challenges.
  • Existing methods may not fully capture the dynamics of distributional data.

Purpose of the Study:

  • To propose novel intrinsic autoregressive models for distributional time series.
  • To quantify and analyze temporal dependencies within sequences of distributions.
  • To extend classical autoregressive models to the space of optimal transport maps.

Main Methods:

  • Developed autoregressive transport models based on regressing optimal transport maps.
  • Utilized geodesics in Wasserstein space to link predictor and response transport maps.
  • Established conditions for the existence of unique stationary solutions using iterated random functions.

Main Results:

  • Demonstrated the existence of unique stationary solutions under specific contraction conditions.
  • Showcased the models' applicability through simulations.
  • Illustrated the models with real-world data, including house price distributions and temperature data.

Conclusions:

  • Autoregressive transport models provide a powerful new framework for distributional time series analysis.
  • These models naturally extend Euclidean autoregressive approaches to optimal transport spaces.
  • The proposed methods offer robust tools for understanding complex temporal distributional dynamics.