Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes
- 74 Downloads
Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen–Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality of these operators. In this paper, we describe the manifold-like geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric (Pigoli et al. Biometrika 101, 409–422, 2014). We identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a detailed description of those aspects of this manifold-like geometry that are important in terms of statistical inference; to establish key properties of the Fréchet mean of a random sample of covariances; and to define generative models that are canonical for such metrics and link with the problem of registration of warped functional data.
Keywords and phrases.Functional data analysis Fréchet mean Manifold statistics Optimal coupling Tangent space PCA Trace-class operator.
AMS (2000) subject classification.Primary 60G15 Gaussian processes 60D05 Geometric probability and stochastic geometry Secondary 60H25 Random operators and equations 62M99 None of the above but in this section.
Unable to display preview. Download preview PDF.
We wish to warmly thank a reviewer for providing constructive and insightful comments that led to genuine improvements in our presentation. This research is supported in part by a Swiss National Science Foundation grant to V. M. Panaretos.
- Ambrosio, L. and Gigli, N. (2013). A User’s Guide to Optimal Transport. In Modelling and Optimisation of Flows on Networks. Springer, pp. 1–155.Google Scholar
- Bhatia, R., Jain, T. and Lim, Y. (2018). On the Bures-Wasserstein distance between positive definite matrices. Expo. Math. https://doi.org/10.1016/j.exmath.2018.01.002.
- Bigot, J. and Klein, T. (2012). Characterization of barycenters in the Wasserstein space by averaging optimal transport maps. arXiv:1212.2562.
- Descary, M.-H. and Panaretos, V.M. (2016). Functional data analysis by matrix completion. Ann. Stat. arXiv:1609.00834.
- Panaretos, V.M. and Zemel, Y. (2018). Introduction to statistics in the Wasserstein space. Springer Briefs in Probability and Mathematical Statistics. To appear.Google Scholar
- Paparoditis, E. and Sapatinas, T. (2014). Bootstrap-based testing for functional data. arXiv:1409.4317.
- Ramsay, J. and Silverman, B. (2005). Springer Series in Statistics.Google Scholar
- Schwartzman, A. (2006). Random Ellipsoids and False Discovery Rates: Statistics for Diffusion Tensor Imaging Data. PhD thesis, Stanford University.Google Scholar
- Zemel, Y. (2017). Fréchet Means in Wasserstein Space: Theory and Algorithms. PhD thesis, École Polytechnique Fédérale de Lausanne.Google Scholar
- Zemel, Y. and Panaretos, V.M. (2017). Fréchet means and Procrustes analysis in Wasserstein space. Bernoulli (to appear), available on arXiv:1701.06876.
- Zhang, J. (2013). Analysis of Variance for Functional Data. Monographs on statistics and applied probability. Chapman & Hall, London.Google Scholar
- Ziezold, H. (1977). On Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the 1974 European Meeting of Statisticians. Springer, pp. 591–602.Google Scholar