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Tian-cheng Li

http://orcid.org/0000-0002-0499-5135

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Frontiers of Information Technology & Electronic Engineering  2017 Vol.18 No.12 P.1913-1939

10.1631/FITEE.1700379


Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond


Author(s):  Tian-cheng Li, Jin-ya Su, Wei Liu, Juan M. Corchado

Affiliation(s):  School of Sciences, University of Salamanca, Salamanca 37007, Spain; more

Corresponding email(s):   t.c.li@usal.es, J.Su2@lboro.ac.uk, w.liu@sheffield.ac.uk, corchado@usal.es

Key Words:  Kalman filter, Gaussian filter, Time series estimation, Bayesian filtering, Nonlinear filtering, Constrained filtering, Gaussian mixture, Maneuver, Unknown inputs


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Tian-cheng Li, Jin-ya Su, Wei Liu, Juan M. Corchado. Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond[J]. Frontiers of Information Technology & Electronic Engineering, 2017, 18(12): 1913-1939.

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Abstract: 
Since the landmark work of R.~E.~Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov&x2013;Bayes recursion, e.g., recursion from a Gaussian or gaussian mixture (GM) prior to a Gaussian/GM posterior (termed &x2018;Gaussian conjugacy&x2019; in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity.

近似高斯共轭:非线性、多模态、不确定以及约束下的参数递归滤波等

概要:自上世纪60年代作为现代估计开山之作的卡尔曼滤波器(Kalman filter)的诞生,时间序列状态空间模型应用于各类动态估计问题吸引了大量的研究关注。特别是,寻求实现闭环马尔科夫?贝叶斯递归(比如,从一个高斯先验到一个高斯后验,本文称之为高斯共轭)的解析解成为一般时间序列滤波器设计的主流思路。其面临的主要挑战包括:系统的非线性、多模态(包括机动模型)、复杂不确定性(比如未知的系统输入,非高斯噪声等)和系统约束(包括循环随机变量)等。这些挑战不断触生新的理论、算法与滤波技术,以实现所期望的参数共轭递归。本文对最新研究进行分类、系统回顾,强调了一些容易被忽略的要点。着重介绍了高精观测非线性系统、高斯后验和机动多模态、以及复杂未知系统输入与约束,以弥补当前文献介绍的不足。同时,本文提出一些新的思考:一是一阶马尔科夫转移模型的替代模型,二是有关计算复杂度的滤波器评价。

关键词:卡尔曼滤波;高斯滤波;时间序列估计;贝叶斯滤波;非线性滤波;约束滤波;高斯混合;机动;未知输入

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