# A New Variational Bayesian-Based Kalman Filter with Unknown Time-Varying Measurement Loss Probability and Non-Stationary Heavy-Tailed Measurement Noise

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## Abstract

**:**

## 1. Introduction

- (a)
- By employing a Bernoulli-distributed variable, the NSHTMN is modelled as a Gaussian-Student’s t-mixture distribution;
- (b)
- The measurement likelihood function is converted from the weight sum of two mixture distributions to an exponential product and a new hierarchical Gaussian state-space model is therefore derived;
- (c)
- The system state vector, UTVMLP, and the unknown variables are simultaneously estimated by utilizing the variational Bayesian technique;
- (d)
- Numerical simulation results indicate that the proposed filter has better performance than that of existing algorithms in the scenarios of NSHTMN and UTVMLP

## 2. Problem Formulation

## 3. Proposed Variational Bayesian-Based Kalman Filter

#### 3.1. Gaussian-Student’s t-Mixture Distribution

#### 3.2. New Hierarchical Gaussian State-Space Model (HGSSM)

**Remark**

**1.**

**Remark**

**2.**

#### 3.3. Variational Bayesian Approximation of the Joint Posterior PDFs

**Proposition**

**1.**

**Proof:**see Appendix A.

**Proposition**

**2.**

**Proof:**see Appendix B.

**Proposition**

**3.**

**Proof:**see Appendix C.

**Proposition**

**4.**

**Proof:**see Appendix D.

**Proposition**

**5.**

**Proof:**see Appendix E.

**Proposition**

**6.**

**Proof:**see Appendix F.

#### 3.4. Calculation of the Required Mathematical Expectations

## 4. Simulations

## 5. Conclusions

## 6. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

## Appendix B. Proof of Proposition 2

## Appendix C. Proof of Proposition 3

## Appendix D. Proof of Proposition 4

## Appendix E. Proof of Proposition 5

## Appendix F. Proof of Proposition 5

## References

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**Figure 5.**${\mathrm{RMSE}}_{x}$ of the proposed filter with forgetting factor $\rho =0.93,0.95,0.97,\mathrm{and}0.99$.

**Figure 6.**The estimated UTVMLPs of the proposed filter with $\rho =0.93,0.95,0.97,\mathrm{and}0.99$.

**Figure 7.**${\mathrm{AGRMSE}}_{\mathrm{x}}$ of the proposed filter when the iteration number ${N}_{I}=1,2,\cdots 10$.

**Table 1.**The proposed variational Bayesian-based Kalman filter with UTVMLP and NSHTMN (one-time step).

Inputs: ${\widehat{x}}_{t-1|t-1}$, ${P}_{t-1|t-1}$, ${Q}_{t-1|t-1}$, ${R}_{t-1|t-1}$, ${y}_{t}$, ${F}_{t-1}$, ${H}_{t}$, $n$, $m$, ${\mu}_{t}$, ${h}_{0}$, ${\widehat{\eta}}_{t-1}$, ${\widehat{\delta}}_{t-1}$, ${N}_{I}$, $\varsigma $ |

Time update: |

1. Obtain ${\widehat{x}}_{t|t-1}$ and ${P}_{t|t-1}$ utilizing Equations (18) and (19) (time update of typical Kalman filter). |

Variational measurement update: |

2. Initialization: ${\widehat{x}}_{t|t}^{\left(0\right)}={x}_{t|t-1}$, ${P}_{t|t}^{\left(0\right)}={P}_{t|t-1}$, ${\mathrm{E}}^{\left(0\right)}\left[{\beta}_{t}\right]=1$, ${\mathrm{E}}^{\left(0\right)}\left[\mathrm{log}\left({\beta}_{t}\right)\right]=0$, ${\mathrm{E}}^{\left(0\right)}\left[{b}_{t}\right]={\widehat{\eta}}_{t-1}/{\widehat{\delta}}_{t-1}$, ${\mathrm{E}}^{\left(0\right)}\left[1-{b}_{t}\right]=1-{\mathrm{E}}^{\left(0\right)}\left[{b}_{t}\right]$, ${\widehat{\eta}}_{t}^{\left(0\right)}={\widehat{\eta}}_{t-1}$, ${\widehat{\delta}}_{t}^{\left(0\right)}={\widehat{\delta}}_{t-1}$, ${\mathrm{E}}^{\left(0\right)}\left[{\zeta}_{t}\right]=1$, ${\mathrm{E}}^{\left(0\right)}\left[{\zeta}_{t}\right]=1-{\mathrm{E}}^{\left(0\right)}\left[{\zeta}_{t}\right]$, ${\mathrm{E}}^{\left(0\right)}\left[\mathrm{log}\left({\phi}_{t}\right)\right]=\Psi \left({h}_{0}\right)-\Psi \left(1\right)$, ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left(1-{\phi}_{t}\right)\right]=\Psi \left(1-{h}_{0}\right)-\Psi \left(1\right)$, ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left({\gamma}_{t}\right)\right]=\Psi \left({\widehat{\eta}}_{t}^{\left(0\right)}\right)-\Psi \left({\widehat{\eta}}_{t}^{\left(0\right)}+{\widehat{\delta}}_{t}^{\left(0\right)}\right)$, ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left(1-{\gamma}_{t}\right)\right]=\Psi \left({\widehat{\delta}}_{t}^{\left(0\right)}\right)-\Psi \left({\widehat{\eta}}_{t}^{\left(0\right)}+{\widehat{\delta}}_{t}^{\left(0\right)}\right)$ |

for$s=0:{N}_{I}-1$. |

3. Update ${q}_{a}^{\left(s+1\right)}\left({x}_{t}\right)$ by Equation (30). |

4. Obtain ${\widehat{x}}_{t|t}^{\left(s+1\right)}$, ${P}_{t|t}^{\left(s+1\right)}$, and ${\tilde{R}}_{t}^{\left(s+1\right)}$ by utilizing Equations (31)–(34) (typical Kalman filter). |

5. Update the Gamma-distributed ${q}_{a}^{\left(s+1\right)}\left({\beta}_{t}\right)$ by Equation (35). |

6. Obtain ${\pi}_{t}^{\left(s+1\right)}$, ${\nu}_{t}^{\left(s+1\right)},$ and ${G}_{t}^{\left(s+1\right)}$ by utilizing Equations (36)–(38). |

7. Update the Bernoulli-distributed ${q}_{a}^{\left(s+1\right)}\left({b}_{t}\right)$. |

8. Obtain ${p}^{\left(s+1\right)}\left({b}_{t}=1\right)$, ${p}^{\left(s+1\right)}\left({b}_{t}=0\right)$, ${C}_{t}^{\left(s+1\right)}$, and ${D}_{t}^{\left(s+1\right)}$ by utilizing Equations (39)–(42). |

9. Obtain ${\mathrm{E}}^{\left(s+1\right)}\left[{b}_{t}\right]$ and ${\mathrm{E}}^{\left(s+1\right)}\left[1-{b}_{t}\right]$ by utilizing Equations (55) and (56). |

10. Update the Bernoulli-distributed ${q}_{a}^{\left(s+1\right)}\left({\zeta}_{t}\right)$. |

11. Obtain ${p}^{\left(s+1\right)}\left({\zeta}_{t}=1\right)$, ${p}^{\left(s+1\right)}\left({\zeta}_{t}=0\right)$, ${V}_{t}^{\left(s+1\right)},$ and ${W}_{t}^{\left(s+1\right)}$ by utilizing Equations (43)–(46). |

12. Obtain ${\mathrm{E}}^{\left(s+1\right)}\left[{\zeta}_{t}\right]$ and ${\mathrm{E}}^{\left(s+1\right)}\left[1-{\zeta}_{t}\right]$ by utilizing Equations (57) and (58). |

13. Update the Beta-distributed ${q}_{a}^{\left(s+1\right)}\left({\phi}_{t}\right)$ by Equation (47). |

14. Obtain ${h}_{t}^{\left(s+1\right)}$ and ${d}_{t}^{\left(s+1\right)}$ by utilizing Equations (48) and (49). |

15. Obtain ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left({\phi}_{t}\right)\right]$ and ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left(1-{\phi}_{t}\right)\right]$ by utilizing Equations (59) and (60). |

16. Update the Beta-distributed ${q}_{a}^{\left(s+1\right)}\left({\gamma}_{t}\right)$ by Equation (50). |

17. Obtain ${\widehat{\eta}}_{t}^{\left(s+1\right)}$ and ${\widehat{\delta}}_{t}^{\left(s+1\right)}$ by utilizing Equations (51) and (52). |

18. Obtain ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left({\gamma}_{t}\right)\right]$ and ${\mathrm{E}}^{\left(s+1\right)}\left[\mathrm{log}\left(1-{\gamma}_{t}\right)\right]$ by utilizing Equations (61) and (62). |

19. If $\left(\parallel {\widehat{x}}_{t|t}^{\left(s+1\right)}-{\widehat{x}}_{t|t}^{\left(s\right)}\parallel /\parallel {\widehat{x}}_{t|t}^{\left(s\right)}\parallel \right)\le \varsigma $, the iteration stopped. |

End for: |

20. ${\widehat{x}}_{t|t}={\widehat{x}}_{t|t}^{\left(s\right)}$, ${P}_{t|t}={P}_{t|t}^{\left(s\right)}$, ${h}_{t}={h}_{t}^{\left(s\right)}$, ${d}_{t}={d}_{t}^{\left(s\right)}$, ${\eta}_{t}={\eta}_{t}^{\left(s\right)}$, ${\delta}_{t}={\delta}_{t}^{\left(s\right)}$ |

Outputs: ${\widehat{x}}_{t|t}$, ${P}_{t|t}$, ${h}_{t}$, ${d}_{t}$, ${\eta}_{t}$, ${\delta}_{t}$, ${h}_{t}/\left({h}_{t}+{d}_{t}\right)$, ${\eta}_{t}/\left({\eta}_{t}+{\delta}_{t}\right)$ |

Measurement Stage | Measurement Noise | UTVMLP |
---|---|---|

Stage 1, time 1 s~100s | ${g}_{t}~\mathrm{N}\left(0,{R}_{t}\right)$ (Gaussian) | 0.1 (slight loss) |

Stage 2, time 101 s~200 s | ${g}_{t}~\{\begin{array}{c}\mathrm{N}\left(0,{R}_{t}\right)\mathrm{w}.\mathrm{p}.=0.98\\ \mathrm{N}\left(0,500{R}_{t}\right)\mathrm{w}.\mathrm{p}.=0.02\end{array}$ (slightly heavy-tailed) | 0.15 (slight loss) |

Stage 3, time 201 s~300 s | ${g}_{t}~\{\begin{array}{c}\mathrm{N}\left(0,{R}_{t}\right)\mathrm{w}.\mathrm{p}.=0.95\\ \mathrm{N}\left(0,500{R}_{t}\right)\mathrm{w}.\mathrm{p}.=0.05\end{array}$ (moderately heavy-tailed) | 0.3 (moderate loss) |

Stage 4, time 301 s~400 s | ${g}_{t}~\mathrm{N}\left(0,{R}_{t}\right)$ (Gaussian) | 0.1 (slight loss) |

**Table 3.**${\mathrm{AGRMSE}}_{\mathrm{x}}$s and single-step running times (SSRT) of different filters.

Filters | KF | IKF | VBAKF | GSTKF | The Proposed Filter |
---|---|---|---|---|---|

${\mathrm{AGRMSE}}_{\mathrm{x}}$ in Stage 1 | 10.1336 | 4.4326 | 4.7705 | 8.7683 | 4.7704 |

${\mathrm{AGRMSE}}_{\mathrm{x}}$ in Stage 2 | 25.4009 | 11.1049 | 18.7603 | 14.4343 | 5.0660 |

${\mathrm{AGRMSE}}_{\mathrm{x}}$ in Stage 3 | 58.0561 | 17.5274 | 35.8194 | 28.0354 | 6.3044 |

${\mathrm{AGRMSE}}_{\mathrm{x}}$ in Stage 4 | 24.9129 | 4.4448 | 27.0252 | 9.5892 | 4.5081 |

${\mathrm{AGRMSE}}_{\mathrm{x}}$ in all stages | 29.6259 | 9.3774 | 21.5944 | 15.2068 | 5.1623 |

SSRT (ms) | 0.0276 | 0.0925 | 0.1166 | 0.1491 | 0.2989 |

Iteration Number${\mathit{N}}_{\mathit{I}}$ | 1 | 2 | 3 | 4 | 5 |

SSRT (ms) | 0.0327 | 0.0550 | 0.0894 | 0.1307 | 0.1570 |

Iteration Number${\mathit{N}}_{\mathit{I}}$ | 6 | 7 | 8 | 9 | 10 |

SSRT (ms) | 0.1883 | 0.2243 | 0.2451 | 0.2835 | 0.2989 |

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**MDPI and ACS Style**

Shan, C.; Zhou, W.; Yang, Y.; Shan, H.
A New Variational Bayesian-Based Kalman Filter with Unknown Time-Varying Measurement Loss Probability and Non-Stationary Heavy-Tailed Measurement Noise. *Entropy* **2021**, *23*, 1351.
https://doi.org/10.3390/e23101351

**AMA Style**

Shan C, Zhou W, Yang Y, Shan H.
A New Variational Bayesian-Based Kalman Filter with Unknown Time-Varying Measurement Loss Probability and Non-Stationary Heavy-Tailed Measurement Noise. *Entropy*. 2021; 23(10):1351.
https://doi.org/10.3390/e23101351

**Chicago/Turabian Style**

Shan, Chenghao, Weidong Zhou, Yefeng Yang, and Hanyu Shan.
2021. "A New Variational Bayesian-Based Kalman Filter with Unknown Time-Varying Measurement Loss Probability and Non-Stationary Heavy-Tailed Measurement Noise" *Entropy* 23, no. 10: 1351.
https://doi.org/10.3390/e23101351