Berry Tyrus, Harlim John
Department of Mathematics , The Pennsylvania State University , University Park, PA 16802, USA.
Department of Mathematics , The Pennsylvania State University , University Park, PA 16802, USA ; Department of Meteorology , The Pennsylvania State University , University Park, PA 16802, USA.
Proc Math Phys Eng Sci. 2014 Jul 8;470(2167):20140168. doi: 10.1098/rspa.2014.0168.
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 , 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.
在本文中，我们研究了具有模型误差的多尺度动力系统的滤波问题，该模型误差源于解析较小尺度过程时的局限性。具体而言，分析假设可获得慢变量所有分量的连续时间噪声观测值。在数学上，本文给出了关于条件测度前两阶矩的高阶渐近展开的新结果。特别地，我们感兴趣的是滤波多尺度问题的应用，其中条件分布是在仅给定慢变量噪声观测值的情况下，针对慢变量定义的。从数学分析中我们了解到，对于具有高斯噪声的连续时间线性模型，在慢变量的线性简化模型中存在唯一的参数选择，当仅观测慢变量时，该选择能给出最优滤波。此外，这些参数同时给出基础系统的最优平衡统计估计，因此可以根据真实信号的平衡统计量离线估计它们。通过研究一个非线性测试模型，我们表明只要我们知道最优随机参数化和正确的观测模型，线性理论就能在这种非高斯、非线性配置中扩展。然而，当随机参数化模型不恰当时，为获得良好滤波性能而选择的参数可能会给出较差的平衡统计估计，反之亦然；这一发现基于我们对非线性测试模型和两层Lorenz - 96模型的解析和数值结果。最后，即使给出了正确的随机假设，同时估计参数并考虑随机参数对简化滤波估计的非线性反馈也是至关重要的。在两层Lorenz - 96模型的数值实验中，我们发现作为滤波过程一部分估计的参数同时能产生准确的滤波和平衡统计预测。相比之下，基于线性回归的离线估计技术，即在不使用滤波器的情况下将参数拟合到训练数据集，当慢变量未被充分观测时，会产生比观测值更差甚至发散的滤波估计。这一发现并不意味着所有离线方法在非线性估计问题上本质上都不如在线方法，它只是表明理想的估计技术应该同时估计所有参数，无论其是在线还是离线。
