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On distributional autoregression and iterated transportation

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  • Laya Ghodrati
  • Victor M. Panaretos

Abstract

We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of ℝ. An order‐1 autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one‐step conditional Fréchet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalizes or extends certain existing models of transportation‐based regression and autoregression, and in doing so also provides some additional insights on existing models.

Suggested Citation

  • Laya Ghodrati & Victor M. Panaretos, 2024. "On distributional autoregression and iterated transportation," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(5), pages 739-770, September.
    • Handle: RePEc:bla:jtsera:v:45:y:2024:i:5:p:739-770
    DOI: 10.1111/jtsa.12736
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    References listed on IDEAS

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    1. Chao Zhang & Piotr Kokoszka & Alexander Petersen, 2022. "Wasserstein autoregressive models for density time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 30-52, January.
    2. Lajos Horváth & Piotr Kokoszka & Ron Reeder, 2013. "Estimation of the mean of functional time series and a two-sample problem," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(1), pages 103-122, January.
    3. Newey, Whitney K, 1991. "Uniform Convergence in Probability and Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 59(4), pages 1161-1167, July.
    4. Delicado, P., 2011. "Dimensionality reduction when data are density functions," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 401-420, January.
    5. Kneip A. & Utikal K. J, 2001. "Inference for Density Families Using Functional Principal Component Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 519-542, June.
    6. Laya Ghodrati & Victor M Panaretos, 2022. "Distribution-on-distribution regression via optimal transport maps [Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line]," Biometrika, Biometrika Trust, vol. 109(4), pages 957-974.
    7. Petersen, Alexander & Zhang, Chao & Kokoszka, Piotr, 2022. "Modeling Probability Density Functions as Data Objects," Econometrics and Statistics, Elsevier, vol. 21(C), pages 159-178.
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