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From Distance Correlation to Multiscale Graph Correlation

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  • Cencheng Shen
  • Carey E. Priebe
  • Joshua T. Vogelstein

Abstract

Understanding and developing a correlation measure that can detect general dependencies is not only imperative to statistics and machine learning, but also crucial to general scientific discovery in the big data age. In this paper, we establish a new framework that generalizes distance correlation (Dcorr)—a correlation measure that was recently proposed and shown to be universally consistent for dependence testing against all joint distributions of finite moments—to the multiscale graph correlation (MGC). By using the characteristic functions and incorporating the nearest neighbor machinery, we formalize the population version of local distance correlations, define the optimal scale in a given dependency, and name the optimal local correlation as MGC. The new theoretical framework motivates a theoretically sound sample MGC and allows a number of desirable properties to be proved, including the universal consistency, convergence, and almost unbiasedness of the sample version. The advantages of MGC are illustrated via a comprehensive set of simulations with linear, nonlinear, univariate, multivariate, and noisy dependencies, where it loses almost no power in monotone dependencies while achieving better performance in general dependencies, compared to Dcorr and other popular methods. Supplementary materials for this article are available online.

Suggested Citation

  • Cencheng Shen & Carey E. Priebe & Joshua T. Vogelstein, 2020. "From Distance Correlation to Multiscale Graph Correlation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 280-291, January.
  • Handle: RePEc:taf:jnlasa:v:115:y:2020:i:529:p:280-291
    DOI: 10.1080/01621459.2018.1543125
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    Cited by:

    1. Zdeněk Hlávka & Marie Hušková & Simos G. Meintanis, 2021. "Testing serial independence with functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 603-629, September.
    2. Lai, Tingyu & Zhang, Zhongzhan & Wang, Yafei, 2021. "A kernel-based measure for conditional mean dependence," Computational Statistics & Data Analysis, Elsevier, vol. 160(C).
    3. Apichit Maneengam, 2023. "Multi-Objective Optimization of the Multimodal Routing Problem Using the Adaptive ε-Constraint Method and Modified TOPSIS with the D-CRITIC Method," Sustainability, MDPI, vol. 15(15), pages 1-22, August.
    4. Lai, Tingyu & Zhang, Zhongzhan & Wang, Yafei & Kong, Linglong, 2021. "Testing independence of functional variables by angle covariance," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    5. Natalia Markovich & Marijus Vaičiulis, 2023. "Extreme Value Statistics for Evolving Random Networks," Mathematics, MDPI, vol. 11(9), pages 1-35, May.
    6. Meintanis, Simos G. & Hušková, Marie & Hlávka, Zdeněk, 2022. "Fourier-type tests of mutual independence between functional time series," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    7. Cencheng Shen & Joshua T. Vogelstein, 2021. "The exact equivalence of distance and kernel methods in hypothesis testing," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(3), pages 385-403, September.

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