Linear theory for filtering nonlinear multiscale systems with model error
Linear theory for filtering nonlinear multiscale systems with model error
复制标题
用于过滤具有模型误差的非线性多尺度系统的线性理论
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
J. Harlim
中科院分区：
文献类型：
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作者：
Tyrus Berry;J. Harlim
在本文中，我们研究了滤波的多尺度动力系统的模型误差所产生的限制，在解决较小的尺度过程。特别是，分析假设的连续时间噪声观测的慢变量的所有组件的可用性。在数学上，本文给出了条件测度前两阶矩的高阶渐近展开式的新结果。特别是，我们感兴趣的过滤多尺度问题，其中的条件分布是定义在慢变量的应用程序，单独的慢变量的噪声观测。从数学分析中，我们得知，对于带高斯噪声的连续时间线性模型，当只观测到慢变量时，对于慢变量的线性约简模型存在唯一的参数选择，从而给出最优滤波。此外，这些参数同时给出了底层系统的最佳平衡统计估计，因此它们可以根据真实信号的平衡统计进行离线估计。通过研究一个非线性测试模型，我们表明，线性理论扩展在这个非高斯，非线性配置，只要我们知道最佳的随机参数化和正确的观测模型。然而，当随机参数化模型不合适时，为良好的过滤器性能选择的参数可能会给出较差的平衡统计估计，反之亦然;这一发现是基于我们的非线性测试模型和两层Lorenz-96模型的分析和数值结果。最后，即使当正确的随机估计给出，它是必要的估计参数同时考虑到非线性反馈的随机参数到减少过滤器估计。在两层Lorenz-96模型的数值实验中，我们发现在线估计的参数，作为过滤过程的一部分，同时产生准确的过滤和平衡统计预测。相比之下，基于线性回归的离线估计技术，其将参数拟合到训练数据集而不使用滤波器，产生比观察更差的滤波器估计，或者当未完全观察到慢变量时甚至发散。这一发现并不意味着所有的离线方法本质上是劣于在线方法的非线性估计问题，它只是表明，一个理想的估计技术应该同时估计所有的参数，无论是在线或离线。
In this paper, we study filtering of multiscale dynamical systems with model error arising from limitations in resolving the smaller scale processes. In particular, the analysis assumes the availability of continuous-time noisy observations of all components of the slow variables. Mathematically, this paper presents new results on higher order asymptotic expansion of the first two moments of a conditional measure. In particular, we are interested in the application of filtering multiscale problems in which the conditional distribution is defined over the slow variables, given noisy observation of the slow variables alone. From the mathematical analysis, we learn that for a continuous time linear model with Gaussian noise, there exists a unique choice of parameters in a linear reduced model for the slow variables which gives the optimal filtering when only the slow variables are observed. Moreover, these parameters simultaneously give the optimal equilibrium statistical estimates of the underlying system, and as a consequence they can be estimated offline from the equilibrium statistics of the true signal. By examining a nonlinear test model, we show that the linear theory extends in this non-Gaussian, nonlinear configuration as long as we know the optimal stochastic parametrization and the correct observation model. However, when the stochastic parametrization model is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa; this finding is based on analytical and numerical results on our nonlinear test model and the two-layer Lorenz-96 model. Finally, even when the correct stochastic ansatz is given, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that the parameters estimated online, as part of a filtering procedure, simultaneously produce accurate filtering and equilibrium statistical prediction. In contrast, an offline estimation technique based on a linear regression, which fits the parameters to a training dataset without using the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed. This finding does not imply that all offline methods are inherently inferior to the online method for nonlinear estimation problems, it only suggests that an ideal estimation technique should estimate all parameters simultaneously whether it is online or offline.