High Dimensional Models for Multivariate Time Series Analysis

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This fellowship will focus on developing methods for high dimensional time series analysis. Methodology for high dimensional data is one of the most important current research topics in statistics and signal processing, where massive data sets have inspired the development of a new statistical paradigm based on sparsity. Such developments have mainly concerned deterministic structure immersed in noise, while this program will model the signal of interest as stochastic. The advantage of modeling an observed signal stochastically as a time series is that one can deduce properties of a population of series, important for the correct understanding of uncertainty or variability in structure.Traditional time series methods are restricted to stationary processes, whose structure is homogeneous in time. The project will instead develop theory and methodology for classes of nonstationary processes, that can experience changes in their generating mechanism over the time course of observation. Such processes are important as they allow us to model the evolution of an observable quantity, and also enable us to quantify this evolution explicitly. Nonstationary processes are observed in a number of applications such as geoscience (remote sensing and satellite observations), oceanography (drifter and float measurements), neuroscience (functional MRI and EEG) and ecology (species abundance) to mention but a few areas. In such applications single processes are rarely of interest, and so we shall develop methods for the analysis of multiple (or equivalently multivariate) signals, to quantify the evolving interdependencies of observed processes.The difficulty in analyzing nonstationary signals is their high degree of overparameterization, that is much exacerbated if inferences are to be made of multiple series. At first glance reliable estimation in such problems seems impossible, as a consequence of the extreme overparameterization. Assumptions on sparsity have recently been used to enable estimation in related overparameterized problems. Such methods need careful extension and substantial innovation to cover the case of multivariate and stochastic signals, that we propose to address via this project. Key to developing such methods is introducing new sparse classes of nonstationary processes, building on recent developments in statistics for high dimensional data. Sparse models despite a nominal degree of high complexity are described by some unknown but simpler structure of smaller complexity. Sparse models will be constructed to contain previously incompatible nonstationary processes, thus enabling us to treat series that lacked a natural analysis framework.This proposal therefore aims to a) introduce new classes of nonstationary processes for single signals using sparsity, b) extend these classes to rich families of multivariate processes for scenarios where either the group structure of the processes is known or has to be learned, c) develop a theoretical understanding of the estimability of such classes of processes and d) develop general estimation methods as well as application specific methodology.We expect this work to impact statistics much beyond time series. New forms of sparsity and methods will also be relevant to related problems in mathematics, machine learning and signal processing, especially in terms of defining new forms of signal group sparsity. The work will also have more than a methodological impact as the development of these methods will allow us to analyze multiple series that previously could not be analyzed, and we intend to develop application specific methods with our collaborators.