Elsevier

Signal Processing

Volume 169, April 2020, 107413
Signal Processing

Black box variational inference to adaptive kalman filter with unknown process noise covariance matrix

https://doi.org/10.1016/j.sigpro.2019.107413Get rights and content

Highlights

  • We prove that the probabilistic model for the joint posterior distribution of the state and the PNCM is non-conjugate.

  • A novel adaptive KF based on the black box variational inference method is proposed to approximate the posterior distributions efficiently.

  • The BBVI inference for the posterior of PNCM is derived and a special case when the structure of PNCM is known is discussed.

Abstract

Adaptive Kalman filter (AKF) is concerned with jointly estimating the system state and the unknown parameters of the state-space models. In this paper, we treat the model uncertainty of the process noise covariance matrix (PNCM) from black box variational inference (BBVI) perspective. In order to lay the foundation for research, we prove that the probabilistic model for online Bayesian inference of the system state and PNCM is non-conjugate, so the traditional coordinate-ascent variational inference (CAVI) cannot deal with this problem. To fill this gap, we propose an AKF in the presence of unknown PNCM based on the BBVI method (which is recently introduced to conduct the approximate Bayesian inference for the non-conjugate probabilistic model). Firstly, we introduce a structured posterior model of the system state and PNCM, by which the posterior distributions of the system state and the PNCM can be calculated efficiently. Then, the BBVI online inference for the posterior distribution of the PNCM is derived. In what follows, we use the intrinsically Bayesian robust KF (IBR-KF) to calculate the state posterior distribution. In addition, a special case, when the structure of the PNCM is known, is explored. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed filters.

Introduction

State estimation is essential for a wide range of engineering applications from time series analysis, navigation guidance, and target tracking, to name but a few. The Kalman filter (KF), as an optimal state estimator under the linear and Gaussian assumptions, is one of the most widely used state estimation methods in practice. Despite its wide applicability, KF is highly sensitive to the accuracy of the noise statistics [1]. In practical applications, as is often the case, the statistical information about the system models may be unknown or inaccurate. For example, the covariance matrixes of the process noise and the measurement noise may be unknown [2], [3]. In the case of unknown covariance matrixes, the state estimation performance of KF will be decreased and it may even diverge [4]. Due to this fact, in the KF community, special attention has been paid to develop adaptive filters with model uncertainty. In this paper, we focus on the model uncertainty of the process noise; specifically, the process noise covariance matrix (PNCM) is unknown.

In recent decades, a variety of filters have been proposed to address the state estimation problem with unknown noise covariance matrixes, such as robust KF [5], [6], adaptive KF [3], [7], [8], [9], [10], [11], [12], [13], and others [14], [15]. In the robust KF, the estimation performance is optimized for the worst case to ensure a lower bound of the performance. In the adaptive KF, the system state along with the noise statistics need to be estimated simultaneously to adjust the system models online. Generally speaking, the adaptive KF methods can be divided into covariance matching, maximum likelihood (ML), Bayesian methods, and so on. Among the above methods, the Bayesian methods have been intensively studied due to their ability to calculate the joint posterior distribution of the system state and noise covariance matrix, which integrates all available information contained in the measurement data. Unfortunately, direct Bayesian inference for the joint posterior distribution is usually analytically intractable and it is necessary to resort to approximate inference.

More recently, the variational Bayesian (VB) [16], [17], [18] method has been introduced into the filtering framework to achieve approximate Bayesian inference. In [11], a VB adaptive Kalman filter (VB-AKF) is proposed to jointly estimate the state and the variances of the measurement noise where the prior distribution of the variance is governed by an inverse-Gamma distribution. In [19], the VB-AKF is further extended to estimate the measurement noise covariance matrix (MNCM) with an inverse-Wishart distribution as its prior distribution. In addition, the VB methods have been extended to the jump Markov linear systems [20], nonlinear adaptive filters [21], [22], and others [23], [24]. However, the VB-based adaptive filters mentioned above are only suitable for online state estimation in the presence of the unknown MNCM. To the best of our knowledge, far too little attention has been paid to the online state estimation problem in the presence of the unknown PNCM based on the Bayesian inference method. In [10], the online state estimation with inaccurate PNCM is treated, in which the predicted error covariance matrix (PECM) rather than the PNCM is approximated by the VB method. However, in [10], the initial value of PECM is determined by the inaccurate PNCM through the time prediction procedure of KF, resulting in a limited performance improvement compared with KF due to the fact that it does not make use of the historical estimation information. Besides, the proposed adaptive filter in [10] cannot give the estimated value of the unknown PNCM. In [25], the authors experimentally show that the state estimation performance of the proposed adaptive filter in [10] may be worse than that of the traditional KF in some cases, which indicates that there is still much room for improvement.

In this paper, we address the online state estimation problem in the presence of the unknown PNCM. The main contributions of this paper are summarized as follows:

  • (1)

    We prove that the probabilistic model for the estimation of the joint posterior distribution of the system state and the PNCM is non-conjugate, which means that the estimation cannot be performed by the traditional coordinate-ascent variational inference (CAVI).

  • (2)

    A novel adaptive KF based on the black box variational inference (BBVI) method is proposed. By simply decomposing the joint posterior distribution into a structured formulation, the proposed filter can calculate the state and PNCM posterior distributions efficiently. a) The proposed filter uses the intrinsically Bayesian robust KF (IBR-KF) to compute the state posterior distribution, such that only the posterior distribution of the PNCM needs to be approximated; b) The BBVI online inference for the posterior distribution of the PNCM is derived, and a special case, when the structure of the PNCM is known, is discussed.

This paper is organized as follows: Section 2 presents the linear state-space models in the presence of unknown PNCM and reviews the variational inference. Section 3 presents the proposed BBVI based adaptive KF, which is the main contribution of the paper, and some important issues of the proposed filter are discussed. Simulation results are then illustrated in Section 4, and finally, the conclusions are given in Section 5.

Section snippets

The state-space models

Consider the following linear and Gaussian discrete time state-space models:xk=Fk1xk1+vk1,vk1iidN(0,Q)zk=Hkxk+nk,nkiidN(0,R)Here, kN is the discrete time index, xkRnx is the state vector, and zkRnz is the measurement vector. Fk1Rnx×nx and HkRnz×nx denote the state transition matrix and the measurement matrix, respectively. The process noise vector vk1 and measurement noise vector nk are uncorrelated sequence. In this paper, we assume that Q is unknown and time-invariant. As is well

Black box variational inference adaptive kalman filter

In this section, we propose a novel adaptive KF based on the BBVI method. It should be noted that the adaptive property of the proposed filter is attached by the BBVI method. Thus, we refer to the proposed filter as a black box variational inference adaptive Kalman filter (BBVI-AKF) for convenience. Section 3.1 proves that the probabilistic model for estimating the system state and PNCM is non-conjugate. In Section 3.2, we introduce a structured posterior formulation for efficient

Simulations

In this section, a target tracking scenario is carried out to evaluate the performance of the proposed BBVI-AKF. The target dynamic is modeled by a CV model where the state vector is xk=[xkx˙kyky˙k]T; The measurement vector zk consists of xk and yk. The state-space parameters Fk and Hk are given as follows:Fk=[1T000100001T0001],Hk=[10000010]where the time interval is T=1s, the sample length is NT=400. The PNCM Q is defined by Eq. (50) where σ2=0.1m2/s4. The MNCM is set to R=diag(σxx2,σyy2)

Conclusions

In this paper, we propose an adaptive Kalman filter in the presence of unknown PNCM based on BBVI method. In order to explain the difficulty in the estimation of the joint posterior distribution of the system state and the PNCM, we prove that the probabilistic model for the variational inference is non-conjugate, which means that the estimation cannot be performed by the CAVI method. To reduce the difficulty of the approximation, a structured posterior model has been introduced, by which the

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Black Box Variational Inference to Adaptive Kalman Filter with Unknown Process Noise Covariance Matrix”.

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    This work has been partially supported by the Natural Science Foundation of China (61871397, 61501505).

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