This paper considers the problem of robust iterative Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Iterative methods are known to improve smoothed estimates but are not guaranteed to converge, motivating the development of more robust versions of the algorithms. The aim of this article is to present Levenberg-Marquardt (LM) and line-search extensions of the classical iterated extended Kalman smoother (IEKS) as well as the iterated posterior linearisation smoother (IPLS). The IEKS has previously been shown to be equivalent to the Gauss-Newton (GN) method. We derive a similar GN interpretation for the IPLS. Furthermore, we show that an LM extension for both iterative methods can be achieved with a simple modification of the smoothing iterations, enabling algorithms with efficient implementations. Our numerical experiments show the importance of robust methods, in particular for the IEKS-based smoothers. The computationally expensive IPLS-based smoothers are naturally robust but can still benefit from further regularisation.
翻译:本文考虑了在非线性国家空间模型中使用高斯近似值添加噪音的添加性噪音使贝叶斯山平滑稳健的问题。已知迭代方法可以改进平滑的估算,但不能保证汇合,从而推动开发更稳健的算法。本篇文章的目的是介绍Levenberg-Marquardt(LM)和古典迭代扩展卡尔曼光滑(IEKS)的线性扩展以及迭代后线性平滑器(IPLS)的线性搜索扩展。IEKS以前被证明相当于高斯-牛顿(GN)方法。我们为IPLS得出了类似的GN解释。此外,我们展示了两种迭代方法的LM扩展,只要简单修改平滑的迭代法,使算法得以高效实施。我们的数字实验显示了稳健方法的重要性,特别是对以IEKS为基础的光滑器。计算成本高昂的IPLS光滑器是自然稳健的,但还可以从进一步的正规化中获益。