Noise covariance matrices in state‐space models: A survey and comparison of estimation methods—Part I

Abstract

This paper deals with the estimation of the noise covariance matrices of systems described by state-space models. Stress is laid on the systematic survey and classification of both the recursive and batch processing methods proposed in the literature with a special focus on the correlation methods. Besides the correlation methods, representatives of other groups are introduced also with respect to their basic idea, estimate properties, assumptions and possible extensions, and user-defined parameters. Common and dual properties of the methods are highlighted, and a simulation comparison using exemplary MATLAB implementations of the methods is provided. KEYWORDSadaptive systems, noise covariance matrix, state estimation, state-space model, system identification INTRODUCTIONKnowledge of a system model is a key prerequisite for many state estimation, signal processing, fault detection, and optimal control problems. The model is often designed to be consistent with random behaviour of the system quantities and properties of the measurements. While the deterministic part of the model often arises from mathematical modelling on the basis of physical, chemical, or biological laws governing the behaviour of the system, the statistics of the stochastic part are often difficult to find by the modelling and have to be identified using the measured data. Incorrect description of the noise statistics may result in significant worsening of estimation, signal processing, detection, or control quality or even in a failure of the underlying algorithms.In the last 5 decades, therefore, a significant research interest has been focused on a design of the methods for the estimation of the properties of the stochastic part of the model. The attention has been devoted to both the input-output models 1-5 and the state-space (SS) models 6-14 and both recursive and batch processing methods. This paper focuses on the methods estimating the properties of the stochastic part of the system described by an SS model discrete in time. In particular, the methods estimating the covariance matrices (CMs) † of noises in the state and measurement equation from a sequence of measured data are of interest. The methods are further denoted as the noise CM estimation methods.

Survey on signal processing for GNSS under ionospheric scintillation: Detection, monitoring, and mitigation

“…Notice that the correct estimation of both noise covariance matrices is not possible due to identifiability issues (Vilà‐Valls, Closas, & Fernández‐Prades, 2015a), then typically only the measurement noise is adjusted. Some of the standard noise estimation techniques based on covariance matching and the autocorrelation of the innovation function (Duník, Straka, Kost, & Havlík, 2017) were evaluated in Vilà‐Valls, Fernández‐Prades, Closas, and Arribas (2017) and a new approach based on Bayesian covariance estimation in LaMountain, Vilà‐Valls, and Closas (2018); the latter being a very promising approach.…”

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

“…There is a large body of literature on noise covariance estimation in both state and observation equations. Interested readers can refer to a cuttingedge, comprehensive survey offered by Duník et al (2017b). A remarkable result which appeared recently (Ristic et al, 2017) states that: Highlight 7…”

“…There have been many excellent tutorials, surveys, and textbooks, primarily in the context of nonlinearity (Nørgaard et al, 2000;Wu et al, 2006;Crassidis et al, 2007;Hendeby, 2008;Šimandl and Duník, 2009;Li and Jilkov, 2012;Patwardhan et al, 2012;Morelande and García-Fernández, 2013;Stano et al, 2013;Duník et al, 2015;García-Fernández and Svensson, 2015;Huber, 2015;Roth et al, 2016;Särkkä et al, 2016;Afshari et al, 2017) or on some sub-topics such as noise covariance metrics estimation (Duník et al, 2017b) and circular Bayes filtering (Kurz et al, 2016). However, some important issues have not been addressed or only addressed briefly, including: (1) a unifying framework to analyze the common essences of different filters; (2) very informative observation systems (i.e., observation noise is insignificant); (3) the classification of multimodal systems, intractable uncertainties, and constraints.…”

Adaptive Kalman Filter for Detectable Linear Time-Invariant Systems

“…A recent survey of covariance estimation results for LTI systems using correlation methods can be found in [23].…”

“…Standing out from among the various methods surveyed in [23] is the measurement averaging correlation method (MACM) which forms a stacked measurement model wherein the observability matrix is referred to as the observation matrix. This paper shares some of the algebraic features of the MACM approach wherein we too construct a measurement stack as part of our overall formulation.…”