Dynamic equations parameterized by differential equations are used to represent a variety of real-world processes. The equations used to describe these processes are generally derived based on physical principles and a scientific understanding of the process. Statisticians have embedded these physically-inspired differential equations into a probabilistic framework, providing uncertainty quantification to parameter estimates and model specification. These statistical models typically rely on a predefined differential equation or class of models to represent the dynamics of the system. Recently, methods have been developed to discover the governing equation of complex systems. However, these approaches rarely account for uncertainty in the discovered equations, and when uncertainty is accounted for, it is not for the complete system. This dissertation begins with a statistical model for the seasonal temperature cycle over North America, where the dynamics of the system are parameterized by a specified functional form. The model highlights how the seasonal cycle is changing in space and time, motivating the need to better understand the driving mechanisms of such systems. Then, a statistical approach to data-driven discovery is proposed, where uncertainty is incorporated throughout the complete modeling process. The novelty of the approach is the dynamics are treated as a random process, which has not be considered previously in the data-driven discovery literature. The proposed approach sits at the junction between the statistical approach of incorporating dynamic equations in a probabilistic framework and the data-driven discovery methods proposed in computer science, physics, and applied mathematics. The proposed method is put into context within the broader literature, highlighting its contribution to the field of data-driven discovery.Includes bibliographical references
