Gaussian process

Summary

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions

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Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

Covariance operators play a fundamental role 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 the space of such operators, as well as with their intrinsic infinite-dimensionality. We will show that we can identify the space of infinite-dimensional covariance operators equipped with the Procrustes size-and-shape metric from shape theory, with that of centred Gaussian processes, equipped with the Wasserstein metric of optimal transportation. We then describe key geometrical and topological aspects of the space of covariance operators endowed with the Procrustes metric. Through the notion of multicoupling of Gaussian measures, we establish existence, uniqueness and stability for the Fréchet mean of covariance operators with respect to the Procrustes metric. Furthermore, we will provide generative models that are canonical for such metric. We then turn to the problem of comparing several samples of stochastic processes with respect to their second-order structure, and we subsequently describe the main modes of variation in this second order structure. These two tasks are carried out via an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators respectively. In order to perform ANOVA, we introduce a novel approach based on optimal (multi)transport and identify each covariance with an optimal transport map. These maps are then contrasted with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, outperforms the state-of-the-art in the functional case. If the null hypothesis postulating equality of the operators is rejected, thanks to a geometric interpretation of the transport maps we can construct a PCA on the tangent space with the aim of understanding the sample variability. Finally, we provide a further example of use of the optimal transport framework, by applying it to the problem of clustering of operators. Two different clustering algorithms are presented, one of which is innovative. The transportation ANOVA, PCA and clustering are validated both on simulated scenarios and real dataset.

EPFL2019

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalize the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(nlogn) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and realworld settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper.

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Global Optimization with Sparse and Local Gaussian Process Models

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Procrustes Metrics and Optimal Transport for Covariance Operators

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