# Variational Bayesian Based Adaptive Shifted Rayleigh Filter for Bearings-Only Tracking in Clutters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Shifted Rayleigh Filter Algorithm

#### 2.1. The Bearing Model

#### 2.2. The Treatment of Clutter

## 3. Variational Bayesian Filtering

#### 3.1. Conjugate Exponential Model

#### 3.2. VB Approximation Method

- (1)
- The VB expectation step yields:$${Q}_{x}\left({\mathbf{x}}_{k}\right)\propto f({\mathbf{x}}_{k},{\mathbf{z}}_{k}){e}^{{\langle \varphi \left({\mathbf{r}}_{k}\right)\rangle}_{{\mathbf{r}}_{k}}^{T}u({\mathbf{x}}_{k},{\mathbf{z}}_{k})}=p\left({\mathbf{x}}_{k}\right|{\mathbf{z}}_{k},{\langle \varphi \left({\mathbf{r}}_{k}\right)\rangle}_{{\mathbf{r}}_{k}})$$
- (2)
- The VB maximization step yields that ${Q}_{r}\left({\mathbf{r}}_{k}\right)$ is conjugate and of the form$${Q}_{r}\left({\mathbf{r}}_{k}\right)=h({\alpha}_{k},{\beta}_{k}^{-}g{\left({\mathbf{r}}_{k}\right)}^{{\beta}_{k}}{e}^{\varphi {\left({\mathbf{r}}_{k}\right)}^{T}{\alpha}_{k}}$$$$\begin{array}{}(28)& \hfill {\alpha}_{k}& ={\alpha}_{k}^{-}+{\langle u({\mathbf{x}}_{k},{\mathbf{z}}_{k})\rangle}_{{\mathbf{x}}_{k}}\hfill (29)& \hfill {\beta}_{k}& ={\beta}_{k}^{-}+n\hfill \end{array}$$

## 4. VB Based Adaptive Shifted Rayleigh Filter with Unknown Clutter Probability

- (1)
- Optimization of ${Q}_{x}\left({\mathbf{x}}_{k}\right)$ for fixed $Q({r}_{k},\xi )$.

- (2)
- Optimization of $Q({r}_{k},\xi )$ for fixed ${Q}_{x}\left({\mathbf{x}}_{k}\right)$.

Algorithm 1 : VB-SRF. |

(1) Initialization: ${\overline{\mathbf{x}}}_{0|0}$, ${\overline{\mathbf{P}}}_{0|0}$, ${\mathbf{Q}}_{v}$, ${\mathbf{Q}}_{w}$, ${\eta}_{0}$, ${\alpha}_{1,0}$, ${\alpha}_{2,0}$(2) Prediction:$\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\widehat{\mathbf{x}}}_{k|k-1}={\mathbf{F}}_{k-1}{\overline{\mathbf{x}}}_{k-1|k-1}+{\mathbf{u}}_{k-1}^{s}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\mathbf{P}}_{k|k-1}={\mathbf{F}}_{k-1}{\overline{\mathbf{P}}}_{k-1|k-1}{\mathbf{F}}_{k-1}^{T}+{\mathbf{Q}}_{v}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\mathbf{S}}_{k}={\mathbf{H}}_{k}{\mathbf{P}}_{k|k-1}{\mathbf{H}}_{k}^{T}+{\mathbf{Q}}_{k}^{m}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\eta}_{k|k-1}=\rho {\eta}_{k-1},\phantom{\rule{1.em}{0ex}}{\alpha}_{1,k|k-1}=\rho {\alpha}_{1,k-1},\phantom{\rule{1.em}{0ex}}{\alpha}_{2,k|k-1}=\rho {\alpha}_{2,k-1}$ where $\rho $ is the scale factor and $0<\rho \le 1$. (3) Update: the update of VB-SRF utilizes iterate filtering framework.(3.a) First set: ${\overline{\mathbf{x}}}_{k|k}^{\left(0\right)}={\widehat{\mathbf{x}}}_{k|k-1}$, ${\overline{\mathbf{P}}}_{k|k}^{\left(0\right)}={\mathbf{P}}_{k|k-1}$, ${\eta}_{k}^{\left(0\right)}={\eta}_{k|k-1}$, ${\alpha}_{1,k}^{\left(0\right)}={\alpha}_{1,k|k-1}$, ${\alpha}_{2,k}^{\left(0\right)}={\alpha}_{2,k|k-1}$(3.b) Calculate state estimation and its covariance using SRF when the measurement is from the target:$\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\mathbf{K}}_{k}={\mathbf{P}}_{k|k-1}{\mathbf{H}}_{k}^{T}{\mathbf{S}}_{k}^{-1}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\epsilon}_{k}={\left({\mathbf{b}}_{k}^{T}{\mathbf{S}}_{k}^{-1}{\mathbf{b}}_{k}\right)}^{-1/2}{\mathbf{b}}_{k}^{T}{\mathbf{S}}_{k}^{-1}({\mathbf{H}}_{k}{\widehat{\mathbf{X}}}_{k|k-1}+{\mathbf{u}}_{k}^{m})$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\gamma}_{k}={\left({\mathbf{b}}_{k}^{T}{\mathbf{S}}_{k}^{-1}{\mathbf{b}}_{k}\right)}^{-1/2}{\rho}_{n}\left({\epsilon}_{k}\right)$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\delta}_{k}={\left({\mathbf{b}}_{k}^{T}{\mathbf{S}}_{k}^{-1}{\mathbf{b}}_{k}\right)}^{-1/2}[2+{\epsilon}_{k}{\rho}_{2}\left({\epsilon}_{k}\right)-{\rho}_{2}^{2}{\epsilon}_{k}]$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\rho}_{2}\left({\epsilon}_{k}\right)=\frac{{\epsilon}_{k}{e}^{-{\epsilon}_{k}^{2}/2}+\sqrt{2\pi}({\epsilon}_{k}^{2}+1){F}_{normal}\left({\epsilon}_{k}\right)}{{e}^{-{\epsilon}_{k}^{2}/2}+\sqrt{2\pi}\left({\epsilon}_{k}\right){F}_{normal}\left({\epsilon}_{k}\right)}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\widehat{\mathbf{x}}}_{k|k}=(\mathbf{I}-{\mathbf{K}}_{k}{\mathbf{H}}_{k}){\widehat{\mathbf{x}}}_{k|k-1}-{\mathbf{K}}_{k}{\mathbf{u}}_{k}^{m}+{\gamma}_{k}{\mathbf{K}}_{k}{\mathbf{b}}_{k}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\mathbf{P}}_{k|k}=(\mathbf{I}-{\mathbf{K}}_{k}{\mathbf{H}}_{k}){\mathbf{P}}_{k|k-1}+{\delta}_{k}{\mathbf{K}}_{k}{\mathbf{b}}_{k}{\mathbf{b}}_{k}^{T}{\mathbf{K}}_{k}^{T}$ (3.c) For $j=1:N$, iterate the following N (N denotes iterated times) steps:$\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\u2022$ Calculate the fused state estimation and its covariance:$\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)}=\frac{1}{2\pi c}{\eta}_{k}^{(j-1)}{\widehat{\mathbf{x}}}_{k|k-1}+\frac{1}{c}(1-{\eta}_{k}^{(j-1)})f\left({\theta}_{k}\right|{\mathbf{z}}_{1:k-1}){\widehat{\mathbf{x}}}_{k|k}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\overline{\mathbf{P}}}_{k|k}^{\left(j\right)}=\frac{1}{2\pi c}{\eta}_{k}^{(j-1)}({\mathbf{P}}_{k|k}+({\widehat{\mathbf{x}}}_{k|k}-{\overline{\mathbf{x}}}_{k|k}){({\widehat{\mathbf{x}}}_{k|k}-{\overline{\mathbf{x}}}_{k|k})}^{T})$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}+\frac{1}{c}(1-{\eta}_{k}^{(j-1)})f\left({\theta}_{k}\right|{\mathbf{z}}_{1:k-1})({\mathbf{P}}_{k|k-1}+({\widehat{\mathbf{x}}}_{k|k-1}-{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)}){({\widehat{\mathbf{x}}}_{k|k-1}-{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)})}^{T})$ where $c=\frac{1}{2\pi}{\eta}_{k}^{(j-1)}+f\left({\theta}_{k}\right)(1-{\eta}_{k}^{(j-1)})$ is a normalization term, and $f\left({\theta}_{k}\right|{\mathbf{z}}_{1:k-1})$ can be obtained using (A6). $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\u2022$ Update parameters:$\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}ln(\frac{{\eta}_{k}^{\left(j\right)}}{1-{\eta}_{k}^{(j)}})=\psi ({\alpha}_{1,k}^{(j-1)})-\psi ({\alpha}_{2,k}^{(j-1)})+ln(1/2\pi )-lnf({\theta}_{k}|{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)})$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\alpha}_{1,k}^{\left(j\right)}={\alpha}_{1,k}^{(j-1)}+{\eta}_{k}^{\left(j\right)}$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\alpha}_{2,k}^{\left(j\right)}={\alpha}_{2,k}^{(j-1)}-{\eta}_{k}^{\left(j\right)}+1$ $\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\u2022$ End for and set ${\overline{\mathbf{x}}}_{k|k}={\overline{\mathbf{x}}}_{k|k}^{\left(N\right)}$, ${\overline{\mathbf{P}}}_{k|k}={\overline{\mathbf{P}}}_{k|k}^{\left(N\right)}$, ${\eta}_{k}={\eta}_{k}^{\left(N\right)}$, ${\alpha}_{1,k}={\alpha}_{1,k}^{\left(N\right)}$, ${\alpha}_{2,k}={\alpha}_{2,k}^{\left(N\right)}$. |

## 5. Simulation Results

#### 5.1. Scenario 1

#### 5.2. Scenario 2

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

VB | Variational Bayesian |

SRF | Shifted Rayleigh Filter |

PDA | Probability Data Association |

EKF | Extended Kalman Filter |

MPEKF | Polar Coordinate EKF |

PLE | Pseudo-Linear Estimator |

UKF | Unscented Kalman Filter |

CKF | Cubature Kalman Filter |

PF | Particle Filter |

MEFPDA | Maximum Entropy Fuzzy Probabilistic Data Association |

SCKF | Square-root Cubature Kalman Filter |

CE | Conjugate Exponential |

KL | Kullback- Leibler |

EM | Expectation-Maximum |

RMS | Root Mean Square |

## Appendix A. Derivation of f(θ_{k}|x_{k})

## Appendix B. Derivation of f(θ_{k}|z_{1:k−1})

## References

- Leong, P.H.; Arulampalam, S.; Lamahewa, T.A.; Abhayapala, T.D. A Gaussian-sum based cubature Kalman filter for bearings-only tracking. IEEE Trans. Aerosp. Electron. Syst.
**2013**, 49, 1161–1176. [Google Scholar] [CrossRef] - Aidala, V.J. Kalman filter behavior in bearings-only tracking applications. IEEE Trans. Aerosp. Electron. Syst.
**1979**, 15, 29–39. [Google Scholar] [CrossRef] - Aidala, V.J.; Hammel, S. Utilization of modified polar coordinates for bearings-only tracking. IEEE Trans. Autom. Control
**1983**, 28, 283–294. [Google Scholar] [CrossRef] [Green Version] - Aidala, V.J.; Nardone, S.C. Biased estimation properties of the pseudo linear tracking filter. IEEE Trans. Aerosp. Electron. Syst.
**1982**, 18, 432–441. [Google Scholar] [CrossRef] - Doğançay, K. On the efficiency of a bearings-only instrumental variable estimator for target motion analysis. Signal Process.
**2005**, 85, 481–490. [Google Scholar] [CrossRef] - Doğançay, K. Bias compensation for the bearings-only pseudolinear target track estimator. IEEE Trans. Signal Process.
**2006**, 54, 59–68. [Google Scholar] [CrossRef] - Nguyen, N.H.; Doğançay, K. Improved pseudolinear Kalman filter algorithms for bearings-only target tracking. IEEE Trans. Signal Process.
**2017**, 65, 6119–6134. [Google Scholar] [CrossRef] - Wang, W.P.; Liao, S.; Xing, T.W. The unscented Kalman filter for state estimation of 3-dimension bearing-only tracking. In Proceedings of the 2009 International Conference on Information Engineering and Computer Science, Wuhan, China, 19–20 December 2009; pp. 1–5. [Google Scholar]
- Yang, R.; Ng, G.W.; Bar-Shalom, Y. Bearings-only tracking with fusion from heterogenous passive sensors: ESM/EO and acoustic. In Proceedings of the 2015 18th International Conference on Information Fusion (Fusion), Washington, DC, USA, 6–9 July 2015; pp. 1810–1816. [Google Scholar]
- Hong, S.H.; Shi, Z.G.; Chen, K.S. Novel roughening algorithm and hardware architecture for bearings-only tracking using particle filter. J. Electromagn. Waves Appl.
**2008**, 22, 411–422. [Google Scholar] [CrossRef] - Chang, D.C.; Fang, M.W. Bearing-only maneuvering mobile tracking with nonlinear filtering algorithms in wireless sensor networks. IEEE Syst. J.
**2014**, 8, 160–170. [Google Scholar] [CrossRef] - Clark, J.M.C.; Vinter, R.B.; Yaqoob, M.M. The shifted Rayleigh filter for bearings only tracking. In Proceedings of the 2005 7th International Conference on Information Fusion, Philadelphia, PA, USA, 25–28 July 2005; pp. 93–100. [Google Scholar]
- Clark, J.M.C.; Vinter, R.B.; Yaqoob, M.M. Shifted Rayleigh filter: A new algorithm for bearings-only tracking. IEEE Trans. Aerosp. Electron. Syst.
**2007**, 43, 1373–1384. [Google Scholar] [CrossRef] - Arulampalam, S.; Clark, M.; Vinter, R. Performance of the shifted Rayleigh filter in single-sensor bearings-only tracking. In Proceedings of the 2007 10th International Conference on Information Fusion, Quebec, QC, Canada, 9–12 July 2007; pp. 1–6. [Google Scholar]
- Mei, D.; Liu, K.; Wang, Y. MEFPDA-SCKF for underwater single observer bearings-only target tracking in clutter. In Proceedings of the 2013 OCEANS—San Diego, San Diego, CA, USA, 23–27 September 2013; pp. 1–6. [Google Scholar]
- Sarkka, S.; Nummenmaa, A. Recursive noise adaptive Kalman filtering by variational Bayesian approximations. IEEE Trans. Autom. Control
**2009**, 54, 596–600. [Google Scholar] [CrossRef] - Li, K.; Chang, L.; Hu, B. A variational Bayesian-based unscented Kalman filter with both adaptivity and robustness. IEEE Sens. J.
**2016**, 16, 6966–6976. [Google Scholar] [CrossRef] - Sun, J.; Zhou, J.; Li, X.R. State estimation for systems with unknown inputs based on variational Bayes method. In Proceedings of the 2012 15th International Conference on Information Fusion, Singapore, 9–12 July 2012; pp. 983–990. [Google Scholar]
- Beal, M.J. Variational Algorithms for Approximate Bayesian Inference. Ph.D. Dissertation, Gatsby Computational Neuroscience Unit, University College London, London, UK, 2003. [Google Scholar]
- Hou, J. jaylin254/VB-SRF1. Available online: https://github.com/jaylin254/VB-SRF1 (accessed on 23 March 2019).

Scenario 1 | Scenario 2 | |||||
---|---|---|---|---|---|---|

${\mathit{p}}_{\mathit{c}}\mathbf{=}\mathbf{0.7}$ | ${\mathit{p}}_{\mathit{c}}\mathbf{=}\mathbf{0.5}$ | ${\mathit{p}}_{\mathit{c}}\mathbf{=}\mathbf{0.3}$ | ${\mathit{p}}_{\mathit{c}}\mathbf{=}\mathbf{0.667}$ | ${\mathit{p}}_{\mathit{c}}\mathbf{=}\mathbf{0.5}$ | ${\mathit{p}}_{\mathit{c}}\mathbf{=}\mathbf{0.3}$ | |

VB-SRF | 0 | 0 | 0.1% | 0 | 0 | 0 |

SRF | 0.9% | 1.6% | 2.9% | 0 | 0 | 2.7% |

MEFPDA-SCKF | 0 | 0 | 0 | 13.5% | 13.3% | 14.2% |

PDA-SCKF | 0 | 0 | 0 | 20.1% | 20.8% | 22.4% |

Scenario 1 | Scenario 2 | |
---|---|---|

VB-SRF | 0.7406 s | 1.0236 s |

SRF | 0.3690 s | 0.5779 s |

MEFPDA-SCKF | 0.2066 s | 0.3314 s |

MEFPDA-SCKF | 0.2092 s | 0.3128 s |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hou, J.; Yang, Y.; Gao, T.
Variational Bayesian Based Adaptive Shifted Rayleigh Filter for Bearings-Only Tracking in Clutters. *Sensors* **2019**, *19*, 1512.
https://doi.org/10.3390/s19071512

**AMA Style**

Hou J, Yang Y, Gao T.
Variational Bayesian Based Adaptive Shifted Rayleigh Filter for Bearings-Only Tracking in Clutters. *Sensors*. 2019; 19(7):1512.
https://doi.org/10.3390/s19071512

**Chicago/Turabian Style**

Hou, Jing, Yan Yang, and Tian Gao.
2019. "Variational Bayesian Based Adaptive Shifted Rayleigh Filter for Bearings-Only Tracking in Clutters" *Sensors* 19, no. 7: 1512.
https://doi.org/10.3390/s19071512