State and Measurement Noise in Positioning and Tracking: Covariance Matrices Estimation and Gaussianity Assessment
Jindrich Dunik, Honeywell International, Advanced Technology Europe, Czech Republic; Oliver Kost, Ondrej Straka, University of West Bohemia, Czech Republic; Erik Blasch, Air Force Research Laboratory, USA
Location: Big Sur
Knowledge of an appropriate system state-space model is a key prerequisite for an optimal design of many signal processing algorithms for applications such as global navigation satellite system (GNSS) based positioning or radar based object tracking. The state-space system model includes the parameterised deterministic part and the distribution-based stochastic part. While the deterministic model often arises from the first principles based on physical, kinematical, and mathematical laws governing the system behaviour, the description of the stochastic part is often difficult to assess by modelling and has to be identified from the measured data.
A tremendous research interest has been, therefore, focused on a design of the methods estimating the properties of the stochastic part of the model, namely on the estimation of the covariance matrices (CMs) of the state and measurement noise appearing in the state-space model (NOTE 1) . Over the past five decades, various noise CMs estimation methods have been proposed. The methods can be divided into four groups according to their underlying idea, namely
- Correlation methods, where the innovation sequence of a linear estimator, which is not optimal in the mean square error sense, is statistically analysed , ,
- Maximum likelihood methods, which are based on a batch processing of the joint estimation of the state and noise CMs elements by the maximisation of a likelihood function often utilising the expectation-maximisation algorithm ,
- Covariance matching methods, where the estimate error CMs computed by a statistical filter are made consistent with the actual state and measurement estimation error statistics ,
- Bayesian methods, which are based on a recursive joint estimation of the state together with the noise CMs elements by a nonlinear state estimator .
The state-of-the-art noise CMs estimation methods, surveyed in , have been designed for wide range of the models (linear/non-linear, time-invariant/time-varying), may provide unbiased and consistent estimates, and offer a trade-off between estimation accuracy and computational complexity. Unfortunately, none of these methods provides information whether the state and measurement noises are Gaussian or not.
Such information is, however, essential for an optimal design of positioning and tracking algorithms. Indeed, these algorithms are often based on the local (or Gaussian) Bayesian state estimation methods (NOTE 2) for nonlinear stochastic dynamic systems, such as the extended Kalman filter, unscented Kalman filter, or cubature Kalman filter , , , which are designed under assumption of the Gaussian state and measurement noise . Violation of the Gaussianity assumption of the noises may result in inconsistent state estimates, which in turn results in inconsistent navigation information with unreliable integrity assessment. Therefore, without any information regarding the Gaussianity of the state and measurement noise, a consistent and reliable positioning and tracking algorithm with integrity assured output cannot be proposed.
The goals of the paper are, therefore, the following:
1. To develop a computationally efficient noise CMs estimation method for linear time-varying (LTV) state-space models with explicit assessment of the Gaussianity of the state and measurement noise. The method is further denoted as the noise CMs estimation method with Gaussianity assessment (NEA),
2. To extend the NEA method for nonlinear models typically appearing in the area of positioning and tracking.
3. To analyse and illustrate the performance and properties of the proposed NEA method using a set of simulation studies.
Considering the first goal, the developed NEA method extends the concept of the recently introduced correlation measurement difference autocovariance (MDA) method for the noise CMs estimation . The extension, resulting in the noises Gaussianity assessment, is based on a statistical analysis and hypothesis testing of the measurement prediction error, which is a by-product of the MDA method. The developed method, thus, provides an unbiased and consistent estimate of the state and measurement noise CMs together with a decision whether the noises are normally distributed or not. The decision regarding the noises’ Gaussianity is provided with a significance level typically specified by the user in terms of probability.
Considering the second goal, the NEA method is modified for nonlinear models. The modification lies in the utilisation of a linearized model used by the positioning and tracking algorithms. As a consequence, the method modified for nonlinear models provides not only the noise CMs estimate with evaluation of the noise PDF Gaussianity, but it also implicitly assesses, whether the linearized model is still a valid approximation of the nonlinear system dynamics. Such ability takes advantage of the fact that the linearization error, which can be understood as a contribution to the state or measurement noise, cannot be typically described by a Gaussian PDF.
Considering the third goal, the performance of the proposed NEA method is verified using synthetic and realistic examples. Whereas the former offer further insight into the properties and expected performance of the proposed method, the latter illustrate the expected behaviour of the proposed noise CMs estimation method with test on Gaussian distribution using models typically appearing in the positioning and tracking.
 J. Dunik, O. Straka, O. Kost, and J. Havlik, “Noise covariance matrices in state-space models: A survey and comparison - part I,” Accepted for International Journal of Adaptive Control and Signal Processing, DOI: 10.1002/acs.2783, 2017.
 R. K. Mehra, “On the identification of variances and adaptive filtering,” IEEE Transactions on Automatic Control, vol. 15, no. 2, pp. 175–184, 1970.
 B. J. Odelson, M. R. Rajamani, and J. B. Rawlings, “A new autocovariance least-squares method for estimating noise covariances,” Automatica, vol. 42, no. 2, pp. 303–308, 2006.
 R. H. Shumway and D. S. Stoffer, Time Series Analysis and its Applications. Springer-Verlag, 2000.
 K. A. Myers and B. D. Tapley, “Adaptive sequential estimation with unknown noise statistics,” IEEE Transactions on Automatic Control, vol. 21, no. 8, pp. 520–523, 1976.
 E. Ozkan, V. Smidl, S. Saha, C. Lundquist, and F. Gustafsson, “Marginalized adaptive particle filtering for nonlinear models with unknown time-varying noise parameters,” Automatica, vol. 49, no. 6, pp. 1566–1575, 2013.
 S. Sarkka, Bayesian Filtering and Smoothing. Cambridge University Press, 2013.
 I. Arasaratnam and S. Haykin, “Cubature Kalman filters,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1254–1269, 2009.
 J. Dunik, O. Straka, M. Simandl, and E. Blasch, “Random-point-based filters: Analysis and comparison in target tracking,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 2, pp. 303–308, 2015.
 J. Dunik, O. Straka, and O. Kost, “Measurement difference autocovariance method for noise covariance matrices estimation,” in Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, USA, Dec. 2016.
Note 1: In general, for a complete description of the noise the probability density function (PDF) is required. Nevertheless, for many signal processing and decision-making methods knowledge of the first two moments of the noise, i.e., the mean and the covariance matrix, is sufficient. Often, the noises are assumed zero-mean and the problem of finding a description of the noises reduces to an estimation of the noise covariance matrices.
Note 2: Within the Bayesian approach, local state estimation methods extend applicability of the Kalman filter, as the optimal estimator for the linear Gaussian systems, for nonlinear systems using various assumptions and approximations of the system model or state estimate. The local methods provide estimates in the form of the (inherently approximate) conditional mean and covariance matrix.