Andreou Marios, Chen Nan
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA.
Entropy (Basel). 2024 Dec 24;27(1):2. doi: 10.3390/e27010002.
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.
条件高斯非线性系统（CGNS）是一类广泛的非线性随机动力系统。给定状态变量子集的轨迹，其余部分服从高斯分布。尽管具有条件线性结构，但CGNS表现出很强的非线性，从而通过其联合分布和边缘分布捕捉到自然界中观察到的许多非高斯特征。理想的是，它具有关于其条件高斯统计量时间演化的封闭解析公式，这有助于研究数据同化和其他相关主题。在本文中，我们开发了一种无鞅方法来增进对CGNS的理解。这种方法提供了一种易于处理的方法，通过时间离散化方案推导结果来证明条件统计量的时间演化，当离散化时间步长消失时，通过形式上的极限过程获得连续时间状态。这种离散化方法进一步允许为具有相关噪声的未观测状态变量的最优后验采样开发解析公式。这些工具对于研究极端事件和间歇性特别有价值，并适用于高维系统。此外，该方法增进了对不同采样方法在表征不确定性方面的理解。通过一个具有立方非线性和状态依赖交叉相互作用噪声的物理约束三元相互作用气候模型，证明了该框架的有效性。
