Abstract
Abstract Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore, researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition, both through experiment and through simulations, provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at short time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Here, we study sufficient statistics computed from time averages, an approach that we demonstrate to lead to sufficient statistics on a variety of problems and that has the secondary benefit of obviating the need to match trajectories. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion. Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refinable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz ’63 model, and then in other applications, including dimension reduction in deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.
Highlights
The goal of this paper is to describe a straightforward ensemble based methodology that facilitates parameter estimation in ergodic stochastic differential equations (SDEs), using statistics derived from time-series data
The data are initially presented in two figures, one showing the ability of the ensemble Kalman inversion (EKI) method to fit the data, and a second showing how well the fitted SDE performs in terms of reproducing the invariant measure
We show other figures which differ between the different cases and are designed to illustrate the nature of the fitted SDE model
Summary
1.1 Overview and Literature ReviewThe goal of this paper is to describe a straightforward ensemble based methodology that facilitates parameter estimation in ergodic stochastic differential equations (SDEs), using statistics derived from time-series data. SDEs arise naturally as models in many disciplines, and the wish to describe complex phenomena without explicitly representing all of the interacting components within the system make them of widespread interest. Stochastic models are widespread in applications, including in biology (see, e.g., Goel & Richter-Dyn, 2016; Wilkinson, 2018), chemistry (see, e.g., Leimkuhler & Reich, 2004; Tuckerman, 2010; Boninsegna et al, 2018), engineering (see, e.g., Maybeck, 1982), the geophysical sciences (see, e.g., Majda & Kramer, 1999; Palmer, 2001; Arnold et al, 2013) and the social sciences (see, e.g., Diekmann & Mitter, 2014); the text (see, e.g., Gardiner, 2009) provides a methodological overview aimed at applications in the physical and social sciences
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