Statistical Learning of Nonlinear Stochastic Differential Equations from Nonstationary Time Series using Variational Clustering

Multiscale Modeling and Simulation, vol. 20, pp. 1251–1283

Abstract

Abstract. 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 flexible 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.

Authors 3

1. Vyacheslav Boyko

   Freie Universität Berlin

   Affiliation as printed

   Department of Mathematics and Computer Sciences, Freie Universität Berlin, 14195 Berlin, Germany ()

2. Sebastian Krumscheid Aachen Faculty of Mathematics, Computer Science and Natural Sciences

   RWTH Aachen University

   Affiliation as printed

   Faculty of Mathematics, Computer Science and Natural Sciences, RWTH Aachen University, 52062 Aachen, Germany ()

3. Nikki Vercauteren

   University of Oslo

   Affiliation as printed

   Department of Geosciences, University of Oslo, Blindern, 0316 Oslo, Norway ()