一种改进的高维非线性系统资料同化方案
An Improved Data Assimilation Scheme for High Dimensional Nonlinear Systems
摘要：
非线性/非高斯滤波在生命科学的许多领域有着广泛的应用，这些领域要么是动态的非线性，要么是不确定状态的概率密度函数是非高斯的。在这样的问题中，估计量的准确性高度依赖于如何准确地近似其后验概率密度函数。在低维状态空间中，基于序贯重要性抽样（SIS）的方法可以很好地逼近后验概率密度函数。然而，对于更高维的问题，这些技术通常是不合适的，因为所需的粒子数量，以达到满意的估计增长指数与状态空间的维度。另一方面，集合卡尔曼滤波（EnKF）及其变体更适合于大规模的问题，由于在贝叶斯更新步骤中的粒子的变换。它已被证明，后一类的方法可能会导致次优解的强非线性问题，由于高斯假设的更新步骤。在本文中，我们介绍了一种新的技术的基础上的高斯和扩展，捕捉非高斯特征更准确，而所需的计算工作量仍然在合理的高维问题。我们通过几个例子，包括强非线性洛伦兹模型的非高斯过程的方法的性能。结果表明，显着改善的均方误差相比，EnKF，和一个理想的收敛行为的粒子数的增加。
英文摘要：
Nonlinear/non-Gaussian filtering has broad applications in many areas of life sciences where either the dynamic is nonlinear and/or the probability density function of uncertain state is non-Gaussian. In such problems, the accuracy of the estimated quantities depends highly upon how accurately their posterior pdf can be approximated. In low dimensional state spaces, methods based on Sequential Importance Sampling (SIS) can suitably approximate the posterior pdf. For higher dimensional problems, however, these techniques are usually inappropriate since the required number of particles to achieve satisfactory estimates grows exponentially with the dimension of state space. On the other hand, ensemble Kalman filter (EnKF) and its variants are more suitable for large-scale problems due to transformation of particles in the Bayesian update step. It has been shown that the latter class of methods may lead to suboptimal solutions for strongly nonlinear problems due to the Gaussian assumption in the update step. In this paper, we introduce a new technique based on the Gaussian sum expansion which captures the non-Gaussian features more accurately while the required computational effort remains within reason for high dimensional problems. We demonstrate the performance of the method for non-Gaussian processes through several examples including the strongly nonlinear Lorenz models. Results show a remarkable improvement in the mean square error compared to EnKF, and a desirable convergence behavior as the number of particles increases.
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