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The Fréchet distance between multivariate normal distributions

Author

Listed:
  • Dowson, D. C.
  • Landau, B. V.

Abstract

The Fréchet distance between two multivariate normal distributions having means [mu]X, [mu]Y and covariance matrices [Sigma]X, [Sigma]Y is shown to be given by d2 = [mu]X - [mu]Y2 + tr([Sigma]X + [Sigma]Y - 2([Sigma]X[Sigma]Y)1/2). The quantity d0 given by d02 = tr([Sigma]X + [Sigma]Y - 2([Sigma]X[Sigma]Y)1/2) is a natural metric on the space of real covariance matrices of given order.

Suggested Citation

  • Dowson, D. C. & Landau, B. V., 1982. "The Fréchet distance between multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 450-455, September.
    • Handle: RePEc:eee:jmvana:v:12:y:1982:i:3:p:450-455
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    Cited by:

    1. Xu, Ganggang & Zhu, Huirong & Lee, J. Jack, 2020. "Borrowing strength and borrowing index for Bayesian hierarchical models," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    2. Nabil Kahalé, 2019. "Efficient Simulation of High Dimensional Gaussian Vectors," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 58-73, February.
    3. Elham Yousefi & Luc Pronzato & Markus Hainy & Werner G. Müller & Henry P. Wynn, 2023. "Discrimination between Gaussian process models: active learning and static constructions," Statistical Papers, Springer, vol. 64(4), pages 1275-1304, August.
    4. Zhongzhi Lawrence He, 2018. "Comparing Asset Pricing Models: Distance-based Metrics and Bayesian Interpretations," Papers 1803.01389, arXiv.org.
    5. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    6. Olivier Ledoit & Michael Wolf, 2019. "Shrinkage estimation of large covariance matrices: keep it simple, statistician?," ECON - Working Papers 327, Department of Economics - University of Zurich, revised Jun 2021.
    7. Artur Karimov & Ekaterina Kopets & Tatiana Shpilevaya & Evgenii Katser & Sergey Leonov & Denis Butusov, 2023. "Comparing Neural Style Transfer and Gradient-Based Algorithms in Brushstroke Rendering Tasks," Mathematics, MDPI, vol. 11(10), pages 1-30, May.
    8. Mordant, Gilles & Segers, Johan, 2022. "Measuring dependence between random vectors via optimal transport," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    9. Abdulkabir Abdulraheem & Im Y. Jung, 2022. "A Comparative Study of Engraved-Digit Data Augmentation by Generative Adversarial Networks," Sustainability, MDPI, vol. 14(19), pages 1-14, September.
    10. Knott, Martin & Smith, Cyril, 2006. "Choosing joint distributions so that the variance of the sum is small," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1757-1765, September.
    11. Puccetti, Giovanni & Rüschendorf, Ludger & Vanduffel, Steven, 2020. "On the computation of Wasserstein barycenters," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    12. Rippl, Thomas & Munk, Axel & Sturm, Anja, 2016. "Limit laws of the empirical Wasserstein distance: Gaussian distributions," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 90-109.
    13. Zhongzhi Lawrence He, 2018. "Generalized Information Ratio," Papers 1803.01381, arXiv.org, revised Apr 2018.
    14. Whiteley, Nick, 2021. "Dimension-free Wasserstein contraction of nonlinear filters," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 31-50.

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