In this talk, I will describe the family of mean-mixtures of multivariate normal distributions and establish many of its properties, stochastic representations, moments, distributional shape characteristics, etc.

The multivariate Gaussian distribution is widely used in many statistical applications due to its appealing features. However, real-world data often violate its assumptions, showing skewness and/or tail-thickness.

Goodness-of-fit tests determine how well a set of observed data fits a particular probability distribution. They can also show if some categorical variable follows a hypothesized family of distributions.

It is widely acknowledged how the relentless surge of Volume, Velocity and Variety of data, as well as the simultaneous increase of computational resources have stimulated the development of data-driven methods with unprecedented flexibility and predictive power. However, not every environmental study entails a large data set: many applications ranging from astronomy or paleo-climatology have a high associated sampling cost and are instead constrained by physics-informed partial differential equations. Throughout the past few years, a new and powerful paradigm has emerged in the machine learning literature, merging data-driven and physics-informed problems, hence providing a unified framework for a whole spectrum of problems ranging from data-rich/context-poor to data-poor/context-rich. In this talk, I will present this new framework and discuss some of the most recent efforts to reformulate it as a stochastic model-based approach, thereby allowing calibrated uncertainty quantification.

In this talk, I will review, unify and extend recent advances in Bayesian inference and computation for such a class of models, proving that unified skew-normal (SUN) distributions (which include Gaussians as a special case) are conjugate to the general form of the likelihood induced by these formulations. This result opens new avenues for improved posterior inference, under a broad class of widely-implemented models, via novel closed-form expressions, tractable Monte Carlo methods based on independent and identically distributed samples from the exact SUN posterior, and more accurate and scalable approximations from variational Bayes and expectation-propagation. These results will be further extended, in asymptotic regimes, to the whole class of Bayesian generalized linear models via novel limiting approximations relying on skew-symmetric distributions.

As a branch of statistics, functional data analysis studies observations regarded as curves, surfaces, or other objects evolving over a continuum. Current methods in functional data analysis usually require data to be smoothed and analyzed marginally, which may hide some outlier information or take extra time on pretreating the data. After exploring model-based fitting for regularly observed multivariate functional data, we explore new visualization tools, clustering, and multivariate functional depths for irregularly observed (sparse) multivariate functional data.

We investigate privacy aspects of tests for symmetry equivalence null hypotheses. Specifically, we consider weighted L2 type tests as well as chi-squared type tests for multivariate symmetry based on the characteristic function, and their privacy properties are specifically quantified within the context of differential privacy. We consider both the case of known centre as well as tests for symmetry about an unknown centre.

In geostatistical analysis, we are faced with the formidable challenge of specifying a valid spatio-temporal covariance function, either directly or through the construction of processes. This task is difficult as these functions should yield positive definite covariance matrices. In recent years, we have seen a flourishing of methods and theories on constructing spatio-temporal covariance functions satisfying the positive definiteness requirement. The current state-of-the-art when modeling environmental processes are those that embed the associated physical laws of the system. The class of Lagrangian spatio-temporal covariance functions fulfills this requirement. Moreover, this class possesses the allure that they turn already established purely spatial covariance functions into spatio-temporal covariance functions by a direct application of the concept of Lagrangian reference frame. In this dissertation, several developments are proposed and new features are provided to this special class.

Due essentially to the difficulties associated with obtaining explicit forms of stationary marginal distributions of non-linear stationary processes, appropriate characterizations of such processes are worked upon little. After discussing an elaborate motivation behind this thesis and presenting preliminaries in Chapter 1, we characterize, in Chapter 2, the stationary marginal distributions of certain non-linear multivariate stationary processes. To do so, we show that the stationary marginal distributions of these processes belong to specific skew-distribution families, and for a given skew-distribution from the corresponding family, a process, with stationary marginal distribution identical to that given skew-distribution, can be found.

In this thesis, I firstly provide a comprehensive assessment of wind energy resources and associated spatio-temporal patterns over Saudi Arabia in both current and future climate conditions, based on a Regional Climate Model output. A high wind energy potential exists and is likely to persist at least until 2050 over a vast area of Western Saudi Arabia, particularly in the region between Medina and the Red Sea coast and during Summer months. Since an accurate assessment of wind extremes is crucial for risk management purposes, I then present the first high-resolution risk assessment of wind extremes over Saudi Arabia.

The thesis focuses on the computation of high-dimensional multivariate normal (MVN) and multivariate Student-t (MVT) probabilities. Firstly, a generalization of the conditioning method for MVN probabilities is proposed and combined with the hierarchical matrix representation. Next, I revisit the Quasi-Monte Carlo (QMC) method and improve the state-of-the-art QMC method for MVN probabilities with block reordering, resulting in a ten-time-speed improvement. The thesis proceeds to discuss a novel matrix compression scheme using Kronecker products. This novel matrix compression method has a memory footprint smaller than the hierarchical matrices by more than one order of magnitude. A Cholesky factorization algorithm is correspondingly designed and shown to accomplish the factorization in 1 million dimensions within 600 seconds. To make the computational methods for MVN probabilities more accessible, I introduce an R package that implements the methods developed in this thesis and show that the package is currently the most scalable package for computing MVN probabilities in R. Finally, as an application, I derive the posterior properties of the probit Gaussian random field and show that the R package I introduce makes the model selection and posterior prediction feasible in high dimensions.

Functional data analysis is a very active research area due to the overwhelming existence of functional data. In the first part of this talk, I will introduce how functional data depth is used to carry out exploratory data analysis and explain recently-developed depth techniques. In the second part, I will discuss spatio-temporal statistical modeling. It is challenging to build realistic space-time models and assess the validity of the model, especially when datasets are large. I will present a set of visualization tools we developed using functional data analysis techniques for visualizing covariance structures of univariate and multivariate spatio-temporal processes. I will illustrate the performance of the proposed methods in the exploratory data analysis of spatio-temporal data.

Abstract

There are several Bayesian models where the posterior density

2019 Statistics and Data Science Workshop confirmed speakers include Prof. Alexander Aue, University of California Davis, USA, Prof. Francois Bachoc, University Toulouse 3, France, Prof. Rosa M. Crujeiras Casais, University of Santiago de Compostela, Spain, Prof. Emanuele Giorgi, Lancaster University, UK, Prof. Jeremy Heng, ESSEC Asia-Pacific, Singapore, Prof. Birgir Hrafnkelsson, University of Iceland, Iceland, Prof. Ajay Jasra, KAUST, Saudi Arabia, Prof. Emtiyaz Khan, RIKEN Center for Advanced Intelligence Project, Japan, Prof. Robert Krafty, University of Pittsburgh, USA, Prof. Guido Kuersteiner, University of Maryland, USA, Prof. Paula Moraga, University of Bath, UK, Prof. Tadeusz Patzek, KAUST, Saudi Arabia, Prof. Brian Reich, North Carolina State University, USA, Prof. Dag Tjostheim, University Bergen, Norway, Prof. Xiangliang Zhang, KAUST, Saudi Arabia, Sylvia Rose Esterby, University of British Colombia, Canada, Prof. Abdel El-Shaarawi, Retired Professor at the National Water Research Institute, Canada. View Workshop schedule and abstracts here.

We consider the problem of recovering a low-rank signal matrix in the presence of a general, unknown additive noise; more specifically, noise where the eigenvalues of the sample covariance matrix have a general bulk distribution. We assume given an upper bound for the rank of the assumed orthogonally invariant signal, and develop a selector for hard thresholding of singular values, which adapts to the unknown correlation structure of the noise.

A variety of intriguing patterns in eigenvalues were observed and speculated about in ML conference papers. We describe the work of Vardan Papyan showing that the traditional subdisciplines, properly deployed, can offer insights about these objects that ML researchers had.

Dynamic treatment regimes are a set of decision rules and each treatment decision is tailored over time according to patients’ responses to previous treatments as well as covariate history. There is a growing interest in development of correct statistical inference for optimal dynamic treatment regimes to handle the challenges of nonregularity problems in the presence of nonrespondents who have zero-treatment effects, especially when the dimension of the tailoring variables is high.

Workshop on Computational Space-Time Statistics