近似滤波方法的统计准确性
Statistical Accuracy of Approximate Filtering Methods
AI总结：
本文研究了集合卡尔曼滤波在非线性和非自治动态系统中的统计准确性，填补了线性高斯设定之外的严格分析空白，并以易于理解的方式呈现了相关技术细节。
摘要：
从部分和含噪观测中估计动态系统状态的统计特性，不仅在数学上具有挑战性，而且有着广泛的应用。此外，这些应用具有重大的社会意义，包括概率天气预报和流行病预测等问题。粒子滤波为该问题提供了一种有充分理论基础的方法，能够得到可证明准确的统计近似。然而，这些方法在高维情况下表现不佳。1994年，Evensen提出了集合卡尔曼滤波（ensemble Kalman filtering）的思想，由此产生的方法在地球物理科学中被广泛采用，并应用于相当一般的反问题。然而，除了线性高斯设定外，集合卡尔曼滤波的统计准确性一直难以进行严格分析。在本文中，我们描述了最近的工作，这些工作迈出了在线性高斯设定之外分析集合卡尔曼滤波统计准确性的第一步。该主题本质上是技术性的，因为它涉及根据非线性和非自治动态系统演化概率测度，以及对这种演化的近似。尽管如此，它仍可以以一种相当易于理解的方式呈现，只需具备动态系统、数值分析和概率论的基础知识即可理解。
英文摘要：
Estimating the statistics of the state of a dynamical system, from partial and noisy observations, is both mathematically challenging and finds wide application. Furthermore, the applications are of great societal importance, including problems such as probabilistic weather forecasting and prediction of epidemics. Particle filters provide a well-founded approach to the problem, leading to provably accurate approximations of the statistics. However these methods perform poorly in high dimensions. In 1994 the idea of ensemble Kalman filtering was introduced by Evensen, leading to a methodology that has been widely adopted in the geophysical sciences and also finds application to quite general inverse problems. However, ensemble Kalman filters have defied rigorous analysis of their statistical accuracy, except in the linear Gaussian setting. In this article we describe recent work which takes first steps to analyze the statistical accuracy of ensemble Kalman filters beyond the linear Gaussian setting. The subject is inherently technical, as it involves the evolution of probability measures according to a nonlinear and nonautonomous dynamical system; and the approximation of this evolution. It can nonetheless be presented in a fairly accessible fashion, understandable with basic knowledge of dynamical systems, numerical analysis and probability.
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