Quantifying Bayesian filter performance for turbulent dynamical systems through information theory

Abstract

Abstract. Incomplete knowledge of the true dynamics and its partial observability pose a notoriously difficult problem in many scientific applications which require predictions of high-dimensional dynamical systems with instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this high-dimensional setting are analyzed through appropriate information measures which naturally extend the common path-wise estimates of filter performance, like the mean-square error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. Particular emphasis is on the notion of practically achievable filter skill which requires trade-offs between different facets of filter performance; a new information criterion is introduced in this context. This information-theoretic framework for assessment of filter performance has natural generalizations to Kalman filtering with non-Gaussian statistically exactly solvable forecast models. Here, this approach is utilized to study the performance of imperfect, reduced-order filters involving Gaussian forecast models which use various spatio-temporal discretizations to approximate the dynamics of the stochastically forced advection-diffusion equation; important examples in this configuration include effects of biases due to model error in the filter estimates for the mean dynamics which are quantified through appropriate information measures.

Linear theory for filtering nonlinear multiscale systems with model error

“…However, this measure has the following useful property: a consistent filter which produces posterior mean estimates close to the true posterior mean estimates also has a covariance close to the true posterior covariance (see Appendix D in the electronic supplementary material for detail). We should point out that although this measure is much weaker than the pattern correlation measure advocated in [12], we shall see that many suboptimal filters are not even consistent in the sense of Definition 3.1. Table 1, as functions of time.…”

“…In the context of predictability, information theoretic criteria were advocated to ensure consistent covariance estimates [10]. While in the context of filtering, information theoretic criteria were also suggested for optimizing the filtering skill [12]. In the mathematical analysis below, we will enforce a different criteria which is based on orthogonal projection on Hilbert subspaces (see Theorem 6.1.2 in [49]) to find the unique set of reduced filter parameters that ensures not only consistent but also optimal filtering in the sense of least squares.…”

“…This metric is nothing else but the signal part of the relative entropy measure of two Gaussian distributions [12]. With this definition, it is obvious that the true filter is consistent.…”

Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems

“…Additionally, the interplay of sparsity and complex systems has been investigated with the goal of overcoming the curse of dimensionality associated with neuronal activity and neuro-sensory systems [20]. Compressive sensing may also play a role in similar statistical learning, library-based, and/or information theory methods [14,7] used in fluid dynamics [8,2], climate science [21,7] and oceanography [1]. Indeed, compressive sensing is already playing a critical role in model building and assessment in the physical sciences [31,41,47,44].…”

Mathematical Background

“…It is nonetheless included for completeness, and causes no harm. The Euclidean norm of the error is computed in line 16, and the results are plotted similarly to previous programs in lines [18][19][20][21][22][23][24][25]. This program is used to plot Figs.…”