STATISTICAL LEARNING OF NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS FROM NONSTATIONARY TIME SERIES USING VARIATIONAL CLUSTERING.

Data-driven stochastic parameterization methods use observational data to support and improve existing prediction systems. Specifically in atmospheric sciences, uncertainty in observations is a challenge for modeling because several aspects of the physical processes are yet to be understood in some contexts. A key goal in constructing atmospheric models is to express unresolved processes using the resolved variables of a prediction system. Estimating parameterisations for highly nonlinear systems with non stationary data requires application-specific tools. Previous work on methods for analyzing nonstationary data presents a exible model-based nonparametric clustering methodology that can be exploited for parameterization development. This paper adapts the existing framework for stochastic parameterization by combining the continuous-time formulation of stochastic differential equations (SDE) with the clustering method. We use a closed-form likelihood function approach based on a suitable Hermite expansion to approximate the parameter values of the SDE with arbitrary nonlinear drift and nonlinear diffusion. The novel parameterization framework provides a smooth classification function that allows us to recover the underlying temporal parameter modulation of a nonstationary one-dimensional SDE. The numerical examples show that the clustering approach recovers a hidden functional relationship between the parameters of the SDE model and an additional auxiliary process. The study builds upon this functional relationship to develop closed-form, nonstationary, data-driven stochastic models for multiscale dynamical systems in real-world applications. [ABSTRACT FROM AUTHOR]