Elliptical Wishart distributions: a new approach to classify covariance matrices

Frédéric Pascal1 Florent Bouchard1 and Imen Ayadi1
  • 1

    Université Paris Saclay, CNRS, CentraleSupélec, laboratoire des signaux et systèmes, France [frederic.pascal@centralesupelec.fr] 

Keywords: Elliptical Wishart distributions – covariance matrices – robust statistics – machine learning

Although the Elliptical Wishart distributions [Anderson, 1958], which generalize the Wishart distribution, have been defined a long time ago, they were not previously exploited in practice. In [Ayadi et al., 2023a, 2024], two algorithms are proposed to compute the maximum likelihood estimator (MLE): a fixed-point algorithm and a Riemannian optimization method based on the derived information geometry of Elliptical Wishart distributions. The existence and uniqueness of the MLE are characterized as well as the convergence of both estimation algorithms. Statistical properties of the MLE are also investigated, such as consistency, asymptotic normality, and an intrinsic version of Fisher’s efficiency. On the statistical learning side, novel classification and clustering methods are designed in [Ayadi et al., 2023b, 2024]. For the t-Wishart distribution, the performance of the MLE and statistical learning algorithms is evaluated on both simulated and real EEG and hyperspectral data, showcasing the interest of our proposed methods.

References

  • Anderson [1958] Theodore Wilbur Anderson. An introduction to multivariate statistical analysis. Wiley New York, 1958.
  • Ayadi et al. [2023a] Imen Ayadi, Florent Bouchard, and Frédéric Pascal. Elliptical Wishart distribution: maximum likelihood estimator from information geometry. ICASSP IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023.
  • Ayadi et al. [2023b] Imen Ayadi, Florent Bouchard, and Frédéric Pascal. t-WDA: A novel Discriminant Analysis applied to EEG classification. 31st European Signal Processing Conference (EUSIPCO), 2023.
  • Ayadi et al. [2024] Imen Ayadi, Florent Bouchard, and Frédéric Pascal. Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning. arXiv preprint arXiv:2411.02726, 2024.