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A Martingale-Free Introduction to Conditional Gaussian Nonlinear Systems

Marios Andreou, Nan Chen

Abstract

The conditional Gaussian nonlinear system (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise.

A Martingale-Free Introduction to Conditional Gaussian Nonlinear Systems

Abstract

The conditional Gaussian nonlinear system (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise.

Paper Structure

This paper contains 26 sections, 17 theorems, 224 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Conditional Gaussian Nonlinear System Modeling Framework
  3. Basic Concepts
  4. Conditionally Gaussian Nonlinear Systems
  5. Significance and Broad Applications of CGNSs
  6. Preliminaries
  7. Analytically Solvable Filter and Smoother Posterior Distributions
  8. Optimal Conditional Sampling Procedures
  9. Filter-Based Forward Sampling
  10. Smoother-Based Backward Sampling
  11. A Hierarchy of Uncertainty in the Sampling Formulae for the Unobserved Variables
  12. Pathwise Error Metrics for the Posterior Mean Time Series and the Sampled Trajectories
  13. The Temporal Autocorrelation Function
  14. Numerical Case Study: A Nonlinear Physics-Constrained Stochastic Reduced-Order Model
  15. The Model
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $\mathbf{x}(t)$ and $\mathbf{y}(t)$ satisfy eq:condgauss1--eq:condgauss2 and assume that the regularity conditions (1)--(8) hold. Additionally, assume that the initial conditional distribution $\mathbb{P}(\mathbf{y}(0)\leq \boldsymbol{\alpha}_0|\mathbf{x}(0))$The event $\{\mathbf{y}(s)\leq \bold Furthermore, assume $\mathbb{P}\left(\mathrm{tr}(\mathbf{R}_{\text{\normalfont{f}}}(0))<+\infty\rig

Figures (3)

  • Figure 4.1: Performance of filtering, smoothing, forward sampling, and backward sampling methods using the reduced-order climate model in \ref{['eq:study1']}--\ref{['eq:study3']}. Panels (a)–(b): Observed time series of $u_1$ and its corresponding PDF. Panels (c)–(d): State estimation of $u_2$ with filtering and smoothing, along with the corresponding PDFs. Panels (e)–(f): State estimation of $u_3$ with filtering and smoothing, and the corresponding PDFs. The PDFs correspond to the full-time-length signal, i.e., for $t\in[0,60]$.
  • Figure 4.2: Statistical comparison between the state estimation using different approaches. Panels (a)–(d): Comparison of the autocorrelation function (ACF). Panels (e)–(h): Comparison of the power spectral density (PSD). Panels (i)–(j): Correlation coefficient between the true time series and the recovered ones from different methods. All comparisons are based on a time series of 60 time units. The SRMSE leads to a similar conclusion as the Corr and is therefore omitted here.
  • Figure 4.3: Time evolution of the spectrum for the components of the sample-to-sample uncertainty with respect to the different procedures of simulating the unobservable variables. Panels (a)--(b): Comparison with regards to the expected real part of the maximal and minimal eigenvalues of the damping feedback in the unconditional forward run of \ref{['eq:study2']}--\ref{['eq:study3']} and the conditional forward and backward optimal nonlinear sampling of $(u_2,u_3)$. The x-axis is logarithmically scaled. Panels (c)--(d): Same as (a)--(b), but instead concerning the fluctuation part of the uncertainty, i.e., the noise feedback matrices. The time evolution of the expected maximal and minimal eigenvalues is also shown for the matrix difference between the unconditional and filter-based sampling noise coefficients.

Theorems & Definitions (36)

  • Theorem 2.1: Conditional Gaussianity
  • proof : Proof of Theorem \ref{['thm:condgaussianity']}
  • Theorem 2.2: Optimal Nonlinear Filter State Estimation Equations
  • Theorem 2.3: Optimal Nonlinear Smoother State Estimation Backward Equations
  • Theorem 3.1: Optimal Nonlinear Forward Sampling Formula
  • Theorem 3.2: Consistency Between Forward Sampling and Filter State Estimation
  • Theorem 3.3: Optimal Nonlinear Backward Sampling Formula
  • Theorem 3.4: Consistency Between Backward Sampling and Smoother State Estimation
  • Remark 3.5: Hierarchy in the Uncertainty of the Latent Dynamics
  • Theorem 3.6: Expected Sampling Squared Error
  • ...and 26 more