This Ph.D. research focuses on proposing new statistical methods for two types of time series data: integer-valued data and multivariate nonstationary extreme data. For the former, the researcher proposes a novel approach to building an integer-valued autoregressive (INAR) model that offers the flexibility to specify both marginal and innovation distributions, leading to several new INAR processes. For the latter, the researcher proposes new extreme value theory methods for analyzing multivariate nonstationary extreme data, specifically EEG recordings from patients with epilepsy. Two extreme-value methods, Conex-Connect and Club Exco, are proposed to study alterations in the brain network during extreme events such as epileptic seizures.

Risk assessment for natural hazards and financial extreme events requires the statistical analysis of extreme events, often beyond observed levels. The characterization and extrapolation of the probability of rare events rely on assumptions about the extremal dependence type and about the specific structure of statistical models. In this thesis, we develop models with flexible tail dependence structures, in order to provide a reliable estimation of tail characteristics and risk measures. Our novel methodologies are illustrated by a range of applications to financial, climatic, and health data.

The Mat\'ern family of covariance functions is currently the most commonly used for the analysis of geostatistical data due to its ability to describe different smoothness behaviors. Yet, in many applications the smoothness parameter is set at an arbitrary value.

We study theoretical problems of fault diagnosis in circuits and switching networks, which are among the most fundamental models for computing Boolean functions. We investigate two main cases: when the scheme (circuit or switching network) has the same mode of operation for both calculation and diagnostics, and when the scheme has two modes of operation -normal for calculation and special for diagnostics.

The impressive success of Deep Learning (DL) in predictive performance tasks has fueled the hopes that this can help addressing societal challenges by supporting sound decision making. However, many open questions remain about their suitability to hold up to this promise. In this talk, I will discuss some of the current limitations of DL, which directly affect their wide adoption. I will focus in particular on the poor ability of DL models to quantify uncertainty in predictions, and I will present Bayesian DL as an attractive approach combining the flexibility of DL with probabilistic reasoning. I will then discuss the challenges associated with carrying out inference and specifying sensible priors for DL models. After presenting a few of my contributions to address these problems, I will conclude by presenting some interesting emerging research trends and open problems which define my current research agenda.

Refined characterizations of the probabilistic behavior of a stationary time-series by focusing on re normalized Markov processes that are conditioned to attain an extreme event, subject to the level of the extremity tending to the upper end point of the marginal distribution

Propagation of acoustic waves in time-varying and/or moving media has attracted a lot of attentions and is expected to lead to many intriguing applications. In this talk, I will discuss our recent work on acoustic wave propagation in spinning media (air or water). I will start with a review of the theoretical foundation built upon the Mie scattering framework, in which both the wave equation and the boundary conditions will be specifically discussed. The study is limited in the linear regime and exhibit the peculiar scattering features.

Theoretical and practical aspects associated with the limiting distributions of block-maxima and peaks-over-threshold events in the case of stationary time-series data. Special emphasis will be placed on the extremal index, a key measure of extremal dependence that allows us to quantify the degree of clustering at the tail of a time-series.

In this talk, we present theoretical asymptotic results for the nonparametric estimation of the conditional density of a scalar response variable Y given the explanatory X taking values in a finite-dimensional space when the sample of observations is considered as a sequence of dependent random variables.

In this talk we first provide an introduction to point processes, which are stochastic models for the occurrence of events in space and time. We then discuss the application of point processes to investigate the relationship between law enforcement drug seizures and accidental overdoses in Indianapolis. We will also discuss results from a field-experiment in Indianapolis where point process based harm indices were used to inform the distribution of addiction treatment information. 

The standard approach to analyzing brain electrical activity is to examine the spectral density function (SDF) and identify frequency bands, defined apriori, that have the most substantial relative contributions to the overall variance of the signal. However, a limitation of this approach is that the precise frequency and bandwidth of oscillations are not uniform across cognitive demands. Thus, these bands should not be arbitrarily set in any analysis. To overcome this limitation, we propose three Bayesian Non-parametric models for time series decomposition, which are data-driven approaches that identify (i) the number of prominent spectral peaks, (ii) the frequency peak locations, and (iii) their corresponding bandwidths (or spread of power around the peaks).

Biological systems are distinguished by their enormous complexity and variability. That is why mathematical modelling and computational simulation of those systems is very difficult, in particular thinking of detailed models which are based on first principles. The difficulties start with geometric modelling which needs to extract basic structures from highly complex and variable phenotypes, on the other hand also has to take the statistic variability into account.

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. Explicit examples show that the singular set could be, in general, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has codimension 3 inside the free boundary, solving a conjecture of Schaeffer in dimension n ≤ 4. The aim of this talk is to give an overview of these results.

In this paper, we propose a new methodological framework for performing extreme quantile regression using artificial neural networks, which are able to capture complex non-linear relationships and scale well to high-dimensional data.

Infinity-harmonic functions have recently found application in Semi-Supervised Learning, in the context of the so-called Lipschitz Learning. With this application in mind, we will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions and its several drawbacks, leading to the notion of AMLE (Absolutely Minimising Lipschitz Extension).

Statistical analysis for the purpose of prediction is preferably accompanied by uncertainty quantification, often in the form of prediction intervals. Deep learning approaches have been extensively shown to provide accurate point predictions in many applications.

The talk will give an overview of recent results for models of collective behavior governed by functional differential equations. It will focus on models of interacting agents with applications in biology (flocking, swarming), social sciences (opinion formation) and engineering (swarm robotics), where latency (delay) plays a significant role.

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.

Surface water waves are a physically important phenomenon with which we all have some experience. They are also surprisingly complex and interesting from a mathematical perspective. I will discuss two recent projects in water wave modeling. The first deals with ocean waves, such as tsunamis, passing over the continental slope. It has long been known that the amplification of such waves is greater than what the traditional transmission coefficient would predict.

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.

The workshop will cover current trends in statistics and the statistical modeling of big and complex data.

This dissertation consists of four major contributions to subasymptotic modeling of multivariate and spatial extremes. The dissertation proposes a multivariate skew-elliptical link model for correlated highly-imbalanced (extreme) binary responses, and shows that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior.

Models for flows on networks arise in the study of traffic and pedestrian crowds. These models encode congestion effects, the behavior and preferences of agents, such as aversion to crowds and their attempts to minimize travel time. We will present the Wardrop equilibrium model on networks with flow-dependent costs and its connection with stationary mean-field game.

In this work, we proved statistical inference for the common Z-estimator based on adaptively sampled data. Adaptive sampling methods, such as reinforcement learning (RL) and bandit algorithms, are increasingly used for the real-time personalization of interventions in digital applications like mobile health and education. As a result, there is a need to be able to use the resulting adaptively collected user data to address a variety of inferential questions, including questions about time-varying causal effects.

Reinforcement Learning provides an attractive suite of online learning methods for personalizing interventions in Digital Behavioral Health. However, after a reinforcement learning algorithm has been run in a clinical study, how do we assess whether personalization occurred? We might find users for whom it appears that the algorithm has indeed learned in which contexts the user is more responsive to a particular intervention. But could this have happened completely by chance? We discuss some first approaches to addressing these questions.