Conference Agenda

This paper presents a novel copula-based autoregressive framework for multi-layer arrays of integer-valued time series with tensor structure. Our framework generalizes recent advances in tensor time series models for real-valued data to a context that accounts for the unique properties of integer-valued data, such as discreteness and non-negativity. The model incorporates feedback effects for the counts’ temporal dynamics and introduces identification constraints. An asymptotic theory is developed for a Two-Stage Maximum Likelihood Estimator (2SMLE) for the model’s parameters. The estimator balances the challenges of parameter dimensionality, interdependence of the different count series, and computational stability. Together, this substantially pushes the frontier for modeling multi-dimensional, structured tensor time series of counts. An application to tensor crime counts demonstrates the practical usefulness of the proposed methodology.

We consider the setting where the state dynamics at each node in a network depend on interactions with its neighbors. We model this using the general framework of Network Stochastic Differential Equations (N-SDEs). The evolution at each node arises from three components: intrinsic dynamics (a momentum term), feedback from adjacent nodes (a network term), and a stochastic volatility component driven by Brownian motion. Our goals are twofold: (i) parameter estimation for N-SDE systems and (ii) recovery of the underlying graph.
The main motivation is to handle very high-dimensional time series by exploiting sparsity in the network structure. We study two settings. i) Known network structure: the graph is given, and we provide identifiability conditions for the parameters, accounting for the fact that the parameter dimension grows with the number of edges. ii) Unknown network structure: the graph must be learned from data; for this case, we propose an iterative procedure based on adaptive Lasso, developed for a particular class of N-SDE models.
We focus on oriented graphs, which supports applications to causal inference by allowing the investigation of directed cause–effect relationships in dynamical systems. Using simulations and real data, we illustrate the performance of the proposed estimators across several graph topologies in high-dimensional regimes. We establish non-asymptotic bounds for parametric estimation when the system dimension is large, in two observation schemes: (1) high-frequency data from an ergodic diffusion, and (2) continuous observation in a small-diffusion, not necessarily ergodic, setting.

Based on joint works with S.M. Iacus and N. Yoshida.

Exact testing of model assumptions is often of limited relevance, especially in high-dimensional settings. Structural assumptions on large-dimensional covariance matrices, such as sphericity, are rarely expected to hold exactly for real data, and practitioners are often primarily interested in whether such model assumptions are approximately satisfied. In this work, we propose a test for approximate sphericity of high-dimensional covariance matrices, where the tolerated level of deviation from sphericity can be chosen by the user. Our test statistic is based on estimators of the largest and smallest eigenvalues of the population covariance matrix in a high-dimensional regime, where the corresponding sample eigenvalues are not consistent. We derive theoretical guarantees showing that the test keeps the prescribed asymptotic level under the null hypothesis and is power consistent under the alternative. Our key theoretical contribution is a joint central limit theorem for the estimators of the extreme eigenvalues of the population covariance matrix, provided the corresponding eigenvalues exceed the critical phase transition threshold.

Functional principal component analysis (FPCA) is a fundamental tool for exploring variation in samples of random curves or surfaces. We propose a new approach to FPCA for functional data observed irregularly and sparsely over their domains, based on smoothing directly at the level of the eigenfunctions. Our formulation leads to an efficient optimization-based procedure whose computational and storage costs are comparable to those of standard multivariate PCA for regularly observed data. The method is flexible with respect to domain geometry and model class, accommodates structural constraints and penalties, and facilitates uncertainty quantification via resampling and asymptotic theory.