# Maximum Correntropy Criterion Kalman Filter for α-Jerk Tracking Model with Non-Gaussian Noise

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## Abstract

**:**

## 1. Introduction

- The first approach is to develop filters for the systems with non-Gaussian noises directly. Noise distributions such as heavy-tailed distributions and t-distributions are considered in these filters [18,19]. However, it is difficult to handle more than one dimension, which limits its applicability [20].
- Approximating the posteriori probability density is another practical approach to handle the non-Gaussian noises. The unscented Kalman filter (UKF) uses the unscented transformation (UT) technique to capture the mean and the covariance of the state estimation with sigma points [21]. The ensemble Kalman filter (EnKF) is a method to approximate the state estimation with a set of samples to handle non-Gaussian noises [22]. Gaussian sum filter (GSF) is an algorithm to obtain the filtering distribution and the predictive distribution recursively approximated as Gaussian mixtures [23,24,25].
- A new robust Kalman filter is proposed by Chang [31] in recent years. It handles the outliers based on the hypothesis testing theory, which defines a judging index as the square of the Mahalanobis distance from the observation to its prediction. It can effectively resist the heavy-tailed distribution of the observation noises and the outliers in the actual observations.
- Multi-sensor data fusion Kalman filter is a fuzzy logical method proposed by Rodger [32]. It can effectively improve the computational burden and the robustness of Kalman filter. Furthermore, it has been applied on the vehicle health maintenance system.
- Maximum correntropy criterion is the latest optimization criterion that is used for improving Kalman filter. Maximum correntropy Kalman filter (MCKF) is a newly proposed filter to process the non-Gaussian noises [33]. In addition, several improved MCKF algorithms have been proposed and applied on state estimation [34,35,36,37].

## 2. $\mathit{\alpha}$-Jerk Model

## 3. Design for Maximum Correntropy Criterion Kalman Filter with Non-Gaussian Noise

#### 3.1. Standard Kalman Filter

#### 3.2. Design of Maximum Correntropy Criterion Kalman Filter

#### 3.2.1. Correntropy

**Definition**

**1.**

_{▪},

_{▪}) is any continuous positive definite kernel function, $\lambda >0$ is the kernel size.

#### 3.2.2. Maximum Correntropy Criterion Kalman Filter Algorithm

- State Prediction$$\begin{array}{c}{\widehat{\mathit{X}}}_{k|k-1}={\mathbf{\Phi}}_{k|k-1}{\widehat{\mathit{X}}}_{k-1},\end{array}$$
- Covariance Prediction$$\begin{array}{c}{\mathit{P}}_{k|k-1}={\mathbf{\Phi}}_{k|k-1}{\mathit{P}}_{k-1|k-1}{\mathbf{\Phi}}_{k|k-1}^{\mathrm{T}}+{\mathit{Q}}_{k},\end{array}$$
- Filter Gain$$\begin{array}{c}{\mathit{K}}_{k}={\left(\right)}^{{\mathit{P}}_{k|k-1}^{-1}+{L}_{k}{\mathit{H}}_{k}^{\mathrm{T}}{\mathit{R}}_{k}^{-1}{\mathit{H}}_{k}}-1{L}_{k}{\mathit{H}}_{k}^{\mathrm{T}}{\mathit{R}}_{k}^{-1},\end{array}$$
- State Update$$\begin{array}{c}{\widehat{\mathit{X}}}_{k}={\widehat{\mathit{X}}}_{k|k-1}+{\mathit{K}}_{k}\left(\right)open="("\; close=")">{\mathit{Z}}_{k}-{\mathit{H}}_{k}{\widehat{\mathit{X}}}_{k|k-1},\end{array}$$
- Covariance Update$$\begin{array}{c}{\mathit{P}}_{k|k}=\left(\right)open="("\; close=")">\mathit{I}-{\mathit{K}}_{k}{\mathit{H}}_{k}{\mathit{P}}_{k|k-1}{\left(\right)}^{\mathit{I}-{\mathit{K}}_{k}{\mathit{H}}_{k}}\mathrm{T}& +{\mathit{K}}_{k}{\mathit{R}}_{k}{\mathit{K}}_{k}^{\mathrm{T}}.\end{array}$$

#### 3.3. Robustness of MCCKF

_{▪}) denotes the mixture distribution, ${F}_{0}$ (

_{▪}) is a known normal distribution, ${F}_{c}$ (

_{▪}) is a contaminated distribution that is unknown, $\upsilon $ denotes the contamination ratio [49], which is usually much smaller than 1. For Equation (43), it is impractical to solve the maximum likelihood estimation by Equation (42).

#### 3.3.1. Influence Function of MCCKF

**Definition**

**2.**

_{▪}) denotes the probability distribution. Similar to Equation (41), Equation (61) is equal to the following form:

_{▪}) is the cost or penalty function. Then, a necessary condition for theminimizer in Equation (62) is

**Remark**

**1.**

**Proof.**

- Fix $\epsilon >0$, and ${\varphi}_{MCC}\left(r\right)$ is bounded in the interval $\left(\right)$.
- In the interval $\left(\right)$, according to the L’Hospital’s rule [53], Equation (72) can be transformed as:$$\begin{array}{c}\underset{r\to \infty}{lim}{\varphi}_{MCC}\left(r\right)\hfill \\ =\underset{r\to \infty}{lim}\frac{1}{\sqrt{2\pi}{\lambda}^{3}}\left./\phantom{\frac{1}{\sqrt{2\pi}{\lambda}^{3}}\sqrt{2\pi}\lambda rexp\left(\right)open="("\; close=")">\frac{{r}^{2}}{2{\lambda}^{2}}}\right)\phantom{\rule{0.0pt}{0ex}}\sqrt{2\pi}\lambda rexp\left(\right)open="("\; close=")">\frac{{r}^{2}}{2{\lambda}^{2}}\hfill \end{array}=0.\hfill $$

#### 3.3.2. Comparison with Huber Filter

_{▪}) is the designed robust function that is bounded and convex. In other words, a Huber-based filter can resist the non-Gaussian noises because the robust function does not increase rapidly as the estimate residual grows. In addition, the M-estimation function with the expression shown in Equation (74) has several forms. Huber M-estimator [46] is a common one that can be shown as:

_{▪}) function in which the kernel size $\lambda $ plays a role similar to $\delta $.

#### 3.4. Kernel Size Selection

## 4. Simulation

#### 4.1. Simulation Conditions

#### 4.2. The Kernel Size Adaptive Method

#### 4.3. The Presence of Gaussian Noise

#### 4.4. The Presence of Large Outliers

#### 4.5. The Presence of Gaussian Mixture Noises

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**$\psi $ with different kernel size selection methods (where ${\lambda}_{0}$ is the novel kernel size selection method).

**Figure 4.**Target trajectories of two kernel size adaptive methods on $\alpha $-jerk model where ${\lambda}_{0}$ is the novel kernel size adaptive method.

**Figure 5.**Residual of estimated position with two kernel size adaptive methods where ${\lambda}_{0}$ is the novel kernel size adaptive method.

**Figure 6.**Target trajectories of different filter algorithms on the $\alpha $-jerk model in the presence of Gaussian noises.

**Figure 7.**Residual of estimated position with different filter algorithms in the presence of Gaussian noises.

**Figure 8.**Target trajectories of different filter algorithms on $\alpha $-jerk model in the presence of large outliers.

**Figure 9.**Residual of estimated position with different filter algorithms in the presence of large outliers.

**Figure 10.**Target trajectories of different filter algorithms on the $\alpha $-jerk model in the presence of Gaussian mixture noises.

**Figure 11.**Residual of estimated position with different filter algorithms in the presence of Gaussian mixture noises.

**Table 1.**RMSE of every state variable in the x-direction of two different kernel size selection methods.

Position (m) | Velocity (m/s) | Accelaration (m/s${}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

${\mathit{\lambda}}_{\mathbf{0}}$ | 1.2331 | 1.3251 | 0.7766 | 0.3512 | 0.2637 |

$\lambda $ | 1.2821 | 1.3590 | 0.7961 | 0.3531 | 0.2764 |

**Table 2.**RMSE of every state variable in the y-direction of two different kernel size selection methods.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

${\mathit{\lambda}}_{\mathbf{0}}$ | 1.2325 | 1.2203 | 0.6863 | 0.3248 | 0.3232 |

$\lambda $ | 1.2538 | 1.2451 | 0.6905 | 0.32 | 0.3232 |

**Table 3.**RMSE of every state variable in the z-direction of two different kernel size selection methods.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

${\mathit{\lambda}}_{\mathbf{0}}$ | 1.3477 | 1.3768 | 0.8037 | 0.3581 | 0.7747 |

$\lambda $ | 1.3747 | 1.3968 | 0.8256 | 0.3682 | 0.7936 |

**Table 4.**RMSE of every state variable in the x-direction of EnKF, UKF, GSF, HF and MCCKF algorithms.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 7.0350 | 4.7926 | 2.3029 | 0.5543 | 1.7291 |

UKF | 1.2892 | 1.3825 | 0.8940 | 0.4424 | 1.6125 |

GSF | 1.5990 | 1.2220 | 0.8072 | 0.4202 | 1.6125 |

HF | 1.2843 | 1.2220 | 0.8072 | 0.4202 | 1.6125 |

MCCKF | 1.2901 | 1.2252 | 0.8076 | 0.4201 | 1.6125 |

**Table 5.**RMSE of every state variable in the y-direction of EnKF, UKF, GSF, HF and MCCKF algorithms.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/\mathbf{s}{}^{2}$) | Jerk ($\mathbf{m}/\mathbf{s}{}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 3.2812 | 3.9866 | 2.1125 | 0.5019 | 3.2963 |

UKF | 1.2856 | 1.5343 | 0.9418 | 0.4299 | 1.1582 |

GSF | 1.5749 | 1.3181 | 0.8666 | 0.4245 | 1.1582 |

HF | 1.2695 | 1.3181 | 0.8666 | 0.4245 | 1.1582 |

MCCKF | 1.2695 | 1.3195 | 0.8685 | 0.4245 | 1.1582 |

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 4.5702 | 3.5353 | 1.9189 | 0.5649 | 1.9978 |

UKF | 1.2178 | 1.4724 | 0.9319 | 0.4700 | 1.1227 |

GSF | 1.5931 | 1.2563 | 0.8887 | 0.4738 | 1.1227 |

HF | 1.2071 | 1.2562 | 0.8887 | 0.4738 | 1.1227 |

MCCKF | 1.2125 | 1.2617 | 0.8907 | 0.4741 | 1.1227 |

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 6.9390 | 6.4846 | 3.7753 | 0.6530 | 1.3750 |

UKF | 3.3104 | 1.7170 | 0.9083 | 0.4238 | 0.9707 |

GSF | 3.0822 | 1.5156 | 0.8306 | 0.4258 | 0.9707 |

HF | 2.3903 | 1.4708 | 1.7734 | 1.3180 | 1.0783 |

MCCKF | 1.4252 | 1.3336 | 0.8153 | 0.4245 | 0.9707 |

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 3.7977 | 2.8730 | 1.7031 | 0.5630 | 5.5318 |

UKF | 3.3085 | 1.6968 | 0.9586 | 0.4464 | 0.3219 |

GSF | 3.5637 | 1.6479 | 0.8553 | 0.4259 | 0.3219 |

HF | 2.2420 | 3.2366 | 1.6525 | 1.3194 | 0.8478 |

MCCKF | 1.3119 | 1.2571 | 0.8020 | 0.4244 | 0.3219 |

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 11.6641 | 9.4672 | 4.7836 | 0.8071 | 2.8136 |

UKF | 3.1840 | 1.6303 | 0.9613 | 0.4324 | 0.6726 |

GSF | 2.8279 | 1.3632 | 0.8259 | 0.4192 | 0.6726 |

HF | 2.2469 | 3.2751 | 1.6901 | 1.3161 | 0.9380 |

MCCKF | 1.3696 | 1.2799 | 0.8209 | 0.4188 | 0.6726 |

**Table 10.**RMSE of every state variable in the x-direction of EnKF, UKF, GSF, HF and MCCKF algorithms.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 6.4009 | 6.4574 | 3.3219 | 0.5756 | 2.0893 |

UKF | 1.3171 | 1.5839 | 1.0426 | 0.4578 | 0.9669 |

GSF | 1.6669 | 1.4316 | 0.9840 | 0.4498 | 0.9669 |

HF | 1.5411 | 1.4907 | 0.9834 | 0.4703 | 0.9669 |

MCCKF | 1.3080 | 1.4194 | 0.9714 | 0.4497 | 0.9669 |

**Table 11.**RMSE of every state variable in the y-direction of EnKF, UKF, GSF, HF and MCCKF algorithms.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 5.6661 | 6.9973 | 3.8323 | 0.6798 | 1.6025 |

UKF | 1.2481 | 1.4203 | 0.9216 | 0.4233 | 1. 5972 |

GSF | 1.6537 | 1.2715 | 0.8655 | 0.412 | 1. 5972 |

HF | 1.2348 | 1.2715 | 0.8655 | 0.4122 | 1. 5972 |

MCCKF | 1.0214 | 1.2607 | 0.8226 | 0.4117 | 1.5972 |

**Table 12.**RMSE of every state variable in the z-direction of EnKF, UKF, GS, HF and MCCKF algorithms.

Position (m) | Velocity (m/s) | Accelaration ($\mathbf{m}/{\mathbf{s}}^{2}$) | Jerk ($\mathbf{m}/{\mathbf{s}}^{3}$) | $\mathit{\alpha}$ | |
---|---|---|---|---|---|

EnKF | 2.4590 | 2.6990 | 1.5505 | 0.5314 | 3.0559 |

UKF | 1.2806 | 1.5766 | 1.0119 | 0.4717 | 0.4726 |

GSF | 1.5664 | 1.3625 | 0.9533 | 0.4678 | 0.4726 |

HF | 1.2877 | 1.3625 | 0.9529 | 0.4678 | 0.4726 |

MCCKF | 1.2548 | 1.3602 | 0.9134 | 0.4678 | 0.4726 |

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**MDPI and ACS Style**

Hou, B.; He, Z.; Zhou, X.; Zhou, H.; Li, D.; Wang, J.
Maximum Correntropy Criterion Kalman Filter for *α*-Jerk Tracking Model with Non-Gaussian Noise. *Entropy* **2017**, *19*, 648.
https://doi.org/10.3390/e19120648

**AMA Style**

Hou B, He Z, Zhou X, Zhou H, Li D, Wang J.
Maximum Correntropy Criterion Kalman Filter for *α*-Jerk Tracking Model with Non-Gaussian Noise. *Entropy*. 2017; 19(12):648.
https://doi.org/10.3390/e19120648

**Chicago/Turabian Style**

Hou, Bowen, Zhangming He, Xuanying Zhou, Haiyin Zhou, Dong Li, and Jiongqi Wang.
2017. "Maximum Correntropy Criterion Kalman Filter for *α*-Jerk Tracking Model with Non-Gaussian Noise" *Entropy* 19, no. 12: 648.
https://doi.org/10.3390/e19120648