Nonparametric Probabilistic Forecasting Methods

Short Work Statement: The PI will develop analytical and computational tools for analyzing stochastic dynamical systems and their associated data assimilation.Objective: The PI's effort is to develop improved techniques to predict the behavior of dynamical systems that routinelymodel oceanic and atmospheric processes. The main goa lis to develop techniques that allow for data assimilation to improve model skill when some of the physical processes are not well known.Approach:The PI will develop an alternative parametric approach, which is physics-based, with nonparametric models. In contrast to the classical parametric modeling approach, this approach assumes no specific structural formto govern the underlying dynamics. In contrast, it relies on the historical data to avoid the two fundamental issues: (1) How to choose the appropriate parametric form? (2) How to estimate the parameters in such models especially when they are not directly observed?. While classical nonparametric models for estimating time-independent or static densities are not new, such as histograms or kernel density estimation methods, the proposed method is an extension to such classical notion of nonparametric modeling to approximate the evolution of time-dependent densities. Inparticular, the proposed projects below are to improve the recently introduced nonparametric forecasting method called the diffusion forecast and to utilize it in data assimilation applications: diffusion forecast is an operator estimation method that approximates the semigroup solutions of the Fokker-Planck PDE based on time series of the underlying dynamics at statistical steady state.Overall Merits: The PI will develop practical computational tools based on state-of-the-art mathematical analysis of stochastic dynamical systems.ONR Relevance:The proposed research, if successful, can have major impact for both long range forecasting and strategic planning studies of importance to ONR in particular and DoD in general, especially when the available modelsare imperfect and the missing dynamics are poorly understood.