Data modeling and dynamical systems

Over the last decade, the use of discrete-time or continuous-time dynamics has become pervasive in data science for the purpose of modeling high-dimensional data generation processes, both explicit (e.g., for sampling or training) and implicit (e.g., as latent variables in deep generaive models). The use of dynamical models has been behind tremendous recent empirical successes in deep learning, probabilistic inference, and generative adversarial networks.

High-dimensional dynamical systems are inherent in massive problems involving multivariate time series and functional data. New data acquisition technologies are recording data continuously during a time interval or intermittently at several discrete time points. Such functional data are ubiquitous in a variety of domains, including biological and physical sciences, finance, and engineering. Nevertheless, few tools have been developed for understanding the underlying statistical structure.

Currently, there is a need for a framework for “next generation” functional data that are multivariate and arise in modern applications, such as electroencephalogram (EEG) and functional magnetic resonance imaging (fMRI). A crucial step in this endeavor will be to abandon the “static” formalism of classical univariate time series analysis in favor of an explicitly dynamic point of view based on generative modeling of multivariate functional data by means of deterministic and stochastic dynamical systems.