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Nonlinear sufficient dimension reduction for distribution-on-distribution regression

Author

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  • Zhang, Qi
  • Li, Bing
  • Xue, Lingzhou

Abstract

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space, while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

Suggested Citation

  • Zhang, Qi & Li, Bing & Xue, Lingzhou, 2024. "Nonlinear sufficient dimension reduction for distribution-on-distribution regression," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    • Handle: RePEc:eee:jmvana:v:202:y:2024:i:c:s0047259x24000095
    DOI: 10.1016/j.jmva.2024.105302
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    References listed on IDEAS

    as
    1. Ding, Shanshan & Cook, R. Dennis, 2015. "Tensor sliced inverse regression," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 216-231.
    2. Wei Luo & Lingzhou Xue & Jiawei Yao & Xiufan Yu, 2022. "Inverse moment methods for sufficient forecasting using high-dimensional predictors [Eigenvalue ratio test for the number of factors]," Biometrika, Biometrika Trust, vol. 109(2), pages 473-487.
    3. Chao Ying & Zhou Yu, 2022. "Fréchet sufficient dimension reduction for random objects [Some asymptotic theory for the bootstrap]," Biometrika, Biometrika Trust, vol. 109(4), pages 975-992.
    4. Dong, Yushen & Wu, Yichao, 2022. "Fréchet kernel sliced inverse regression," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    5. Fan, Jianqing & Xue, Lingzhou & Yao, Jiawei, 2017. "Sufficient forecasting using factor models," Journal of Econometrics, Elsevier, vol. 201(2), pages 292-306.
    6. Yaqing Chen & Zhenhua Lin & Hans-Georg Müller, 2023. "Wasserstein Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(542), pages 869-882, April.
    7. Wei Luo & Bing Li, 2016. "Combining eigenvalues and variation of eigenvectors for order determination," Biometrika, Biometrika Trust, vol. 103(4), pages 875-887.
    8. Elsa Cazelles & Vivien Seguy & Jérémie Bigot & Marco Cuturi & Nicolas Papadakis, 2017. "Log-PCA versus Geodesic PCA of histograms in the Wasserstein space," Working Papers 2017-85, Center for Research in Economics and Statistics.
    9. Xiufan Yu & Jiawei Yao & Lingzhou Xue, 2022. "Nonparametric Estimation and Conformal Inference of the Sufficient Forecasting With a Diverging Number of Factors," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 342-354, January.
    10. 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.
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