With widespread technological developments, it has become commonplace to collect high-dimensional time-series data, that is, intensive repeated measurement data on many variables and subjects simultaneously during daily life. This includes sensor-based physiological measurements (e.g., heart rate, skin conductance) and health and movement data (e.g., calorie tracking, Fitbit, GPS) across people, various macro-level indicators recorded over time for different economies, as well as data from many other noisy and complex systems evolving over time across subjects. Although technological advances including machine learning have decreased the burden associated with collecting such high-dimensional time-series data, these developments have also brought a newfound appreciation of the rich heterogeneity inherent to many biological, behavioral and other systems. For example, in neuroscience, extensive between-person heterogeneity is observed in the anatomical organization of brain regions and dynamic network activation profiles. The heterogeneity of firms, countries and other subjects has been well recognized and studied in economics and finance. As such, how best to model processes that exhibit meaningful heterogeneity across subjects is a critical open question in many disciplines, including precision medicine, computational psychiatry and economics, machine learning, and artificial intelligence. This project aims to develop the theoretical foundation, methodological approaches and computational tools needed to model time-dependent systems characterized by unknown heterogeneity. Broader impacts activities will also involve education and training of undergraduate and graduate students. The project advances a unified statistical and machine learning framework for the analysis of complex multivariate time series arising from multiple heterogeneous subjects. It brings together two directions in modern time series methodology: high-dimensional multivariate modeling and joint