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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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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Zhongzhi Lawrence He, 2018. "Generalized Information Ratio," Papers 1803.01381, arXiv.org, revised Apr 2018.
    4. 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).
    5. 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.
    6. Mordant, Gilles & Segers, Johan, 2022. "Measuring dependence between random vectors via optimal transport," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    7. Zhang, Kefei & Yang, Xiaolin & Xu, Liang & Thé, Jesse & Tan, Zhongchao & Yu, Hesheng, 2024. "Enhancing coal-gangue object detection using GAN-based data augmentation strategy with dual attention mechanism," Energy, Elsevier, vol. 287(C).
    8. Puccetti, Giovanni & Rüschendorf, Ludger & Vanduffel, Steven, 2020. "On the computation of Wasserstein barycenters," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    9. Nabil Kahalé, 2019. "Efficient Simulation of High Dimensional Gaussian Vectors," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 58-73, February.
    10. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    11. 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.
    12. 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.
    13. 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.
    14. Whiteley, Nick, 2021. "Dimension-free Wasserstein contraction of nonlinear filters," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 31-50.
    15. Zhongzhi Lawrence He, 2018. "Comparing Asset Pricing Models: Distance-based Metrics and Bayesian Interpretations," Papers 1803.01389, arXiv.org.

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