# An Approach for Filter Divergence Suppression in a Sequential Data Assimilation System and Its Application in Short-Term Traffic Flow Forecasting

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## Abstract

**:**

_{1}-norm constraint for filter-divergence suppression in sequential data assimilation systems was proposed. The method adjusts the weights of the state-simulated values and measurements based on new measurements using an L

_{1}-norm constraint when filter divergence is about to occur. Results for simulation data and real-world traffic flow measurements collected from a sub-area of the highway between Leeds and Sheffield, England, showed that the proposed method produced a higher assimilation accuracy than the other filter-divergence suppression methods. This indicates the effectiveness of the proposed approach based on the L

_{1}-norm constraint for filter-divergence suppression.

## 1. Introduction

_{1}-norm constraint for filter-divergence suppression in the S-DA system was proposed and applied to short-term traffic flow forecasting to verify its effectiveness. The advantages of the filter-divergence suppression method based on the L

_{1}-norm constraint method are that it can guarantee high precision and it is easy to implement. Three critical items were investigated: (i) an inaccurate S-DA system that generates filter divergence for short-term traffic flow forecasting; (ii) an approach based on the L

_{1}-norm constraint for filter-divergence suppression; and (iii) the application of the S-DA system based on the L

_{1}-norm constraint method to short-term traffic flow forecasting to verify its effectiveness compared with other methods.

_{1}-norm constraint is proposed. Section 4 gives a numerical example. In Section 5, short-term traffic flow forecasting application experiments are described. Section 6 presents the results and Section 7 presents our conclusions.

## 2. Representation of Filter Divergence Phenomenon in Sequential Data Assimilation System for Short-Term Traffic Flow Forecasting

## 3. Methodology

_{1}-norm constraint in the KF assimilation method of the S-DA system.

_{1}-norm Constraint:

_{1}-norm criterion. The L

_{1}-norm and L

_{2}-norm are commonly used as objective functions to be minimized [51]. However, objective functions based on the L

_{1}-norm criterion have better resistance to noise. The L

_{1}-norm constraint function for calculating the Kalman gain matrix $\mathit{K}$ is as follows:

## 4. Numerical Study

_{1}-norm constraint, two commonly used and effective methods for suppressing filter divergence were employed, namely the C-W and A-KF methods. The original KF method [34], C-W method [41], A-KF method [44], and proposed method based on the L

_{1}-norm constraint were applied in the numerical example of filter-divergence suppression, as shown in Equation (4) and Figure 1. As selecting a weight or inflating factor in the C-W method is difficult, in following experiments, an adaptive inflation was used to acquire the weight or fading factor to inflate the model error covariance matrix [41]. The results were then compared and analyzed.

_{1}-norm constraint.

_{1}-norm constraint. The RMSE and MAPE values acquired from the proposed method were 0.0340 and 0.12%, respectively, which were the smallest values out of all methods. Compared with the C-W and A-KF methods, the RMSE values were reduced by 7.61% and 4.49%, respectively, and the MAPE values were reduced by 25% and 14.29%, respectively. This result indicates that the proposed method based on the L

_{1}-norm constraint could suppress the filter divergence problem efficiently with the highest assimilation accuracy. To verify the applicability of the proposed method, an empirical study is presented in the next section.

## 5. Empirical Study

#### 5.1. Study Area and Material Description

#### 5.2. Test Design

_{1}-norm constraint-based method was applied with the original KF method [34], the C-W method with an adaptive inflation [41], and the A-KF method [44] to acquire the traffic flow forecasting results of the paths shown in Figure 5a. The effectiveness of the proposed method was verified by comparing the results. To illustrate the analysis in detail, the forecasting performances of the six paths shown in Figure 5b are listed first.

## 6. Results and Discussion

_{1}-norm constraint-based method under the incorrect model shown in Equation (9) as an example to demonstrate the performance of the proposed method in detail and the results are shown in Figure 6 and Figure 7, respectively. The results presented showed that filter divergence occurred when only using the KF method under the incorrect model; however, the filter divergence was alleviated to different degrees after using the C-W, A-KF, and L

_{1}-norm constraint-based methods. Furthermore, the true values were added to verify the effectiveness of these three filter-divergence suppression methods. Results showed that the best performance for filter-divergence suppression was obtained using the L

_{1}-norm constraint-based method on both Monday and Sunday.

_{1}-norm constraint-based methods under the incorrect model and the KF method under the correct model (Equation (7)). The values are presented in Figure 8 and Figure 9, respectively. The RMSE and MAPE values calculated from the KF method using the correct model were set as a reference to verify the effectiveness of each filter-divergence suppression method. Figure 9 showed that the MAPE values of each path acquired using the KF method under the incorrect model reached almost 100%, which indicated that filter divergence occurred. The traffic flow forecasting performances improved to different degrees when using the C-W, A-KF, and L

_{1}-norm constraint-based methods. The RMSE and MAPE values of each path on Monday to Sunday acquired using the C-W, A-KF, and L

_{1}-norm constraint-based methods were smaller than those from the KF method under the incorrect model. Furthermore, compared with the results from the C-W and A-KF methods, the results obtained using the proposed L

_{1}-norm constraint-based method were much closer to those from the KF method under the correct model. This indicates the effectiveness of the proposed L

_{1}-norm constraint-based method.

_{1}-norm constraint-based methods are listed in Table 3 and Table 4, respectively. Compared with the KF method under the incorrect model, the average RMSE and MAPE values decreased significantly, especially those obtained using the proposed L

_{1}-norm constraint-based method. The best performance of the L

_{1}-norm constraint-based method reduced the average RMSE by 823.3 (from 901.86 to 78.56), and the relative accuracy improved by 91.29%. The corresponding average MAPE decreased by 92.51% (from 99.75% to 7.24%).

_{1}-norm constraint-based method were much closer to those from the KF method under the correct model than the values from the other two filter-divergence suppression methods. The smallest average RMSE and MAPE differences were 2.07% and 0.47%, respectively.

_{1}-norm constraint-based methods were calculated. Table 6 and Table 7 show the average RMSE and MAPE values, respectively, of each path from Monday to Sunday. The average MAPE values for all paths from Monday to Sunday obtained using the KF method under the incorrect model were all above 97%. This indicates that filter divergence occurred. The average MAPE values acquired from the C-W, A-KF, and L

_{1}-norm constraint-based methods decreased by different amounts. For the sake of analysis, the forecasting results for all paths on workday Monday and non-workday Saturday were taken as examples. The best filter-divergence suppression performances were obtained using the L

_{1}-norm constraint-based method. The average RMSE value from the L

_{1}-norm constraint-based method decreased by 524.82 (from 596.29 to 71.47), and the relative accuracy improved by 88.01% on Monday compared with the results from the KF method under the incorrect model. The corresponding relative accuracy improved by 90.87% on Saturday. Furthermore, the average MAPE value from the L

_{1}-norm constraint-based method decreased by 89.05% (from 98.39% to 9.34%) on Monday and 89.64% (from 98.50% to 8.86%) on Saturday compared with those obtained using the KF method under the incorrect model.

_{1}-norm constraint-based method were much closer to those obtained using the KF method under the correct model. On workday Monday, the difference in the average RMSE and MAPE values between the C-W and KF methods under the correct model was 177.69 and 24.44%, respectively. The corresponding differences between the A-KF and KF methods were 100.44 and 14.49%, respectively. The differences in the average RMSE and MAPE values between the proposed L

_{1}-norm constraint-based method and the KF method were 15.77 and 2.22% on Monday, respectively. Similar results were obtained for non-workday Saturday. The smallest average RMSE and MAPE differences were 11.77 and 2.44%, respectively, which were acquired using the L

_{1}-norm constraint-based method.

_{1}-norm constraint-based method outperformed the C-W and A-KF methods. This indicates that using the proposed L

_{1}-norm constraint-based method to suppress filter divergence is effective. Unlike suppressing the filter divergence phenomenon in the C-W method by adding an inflation factor to model the covariance matrix, in the proposed method, the gain matrix $\mathit{K}$ based on the actual conditions was directly and adaptively adjusted. The difficulty of selecting a weight or inflating factor and the propagation from the model or measurement error covariance to the $\mathit{K}$ matrix estimation in the C-W method were therefore also avoided. Moreover, compared with the A-KF method, this method was simpler and required less storage space. Furthermore, the proposed method adjusted the gain matrix directly without adjusting the model error covariance matrix. This prevented the uncertainties of the indirect operations in the A-KF method from affecting the assimilation results.

## 7. Conclusions

_{1}-norm criterion. The proposed L

_{1}-norm constraint-based method was compared with two other commonly used methods, the C-W and A-KF methods, to suppress filter divergence in short-term traffic flow forecasting. The empirical results confirmed the following.

- The proposed approach based on the L
_{1}-norm constraint can suppress the filter-divergence phenomenon that occurs in the KF assimilation method of the S-DA system. - The proposed approach based on the L
_{1}-norm constraint had a higher assimilation accuracy for suppressing filter divergence than the other two methods.

_{1}-norm constraint is feasible and effective for filter-divergence suppression in an S-DA system for short-term traffic flow predictions. In future work, the proposed method is planned to be used in other fields to expand its range of applications.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Traffic flow forecasting results under the two assimilation models on (

**a**) Monday and (

**b**) Saturday.

**Figure 3.**Framework of the filter-divergence suppression method based on the L

_{1}-norm constraint. The main contents are shown in red.

**Figure 4.**Cumulative absolute errors average (Cum-AEA) values of the assimilation results acquired using four methods.

**Table 1.**Kalman filter method adapted from [34].

Forecast | ${x}_{k}^{f}={\mathit{M}}_{k,k-1}{x}_{k-1}^{a}$ |

${\mathit{P}}_{k}^{f}={\mathit{M}}_{k,k-1}{\mathit{P}}_{k-1}^{a}{\mathit{M}}_{k,k-1}{}^{T}+{\mathit{G}}_{k,k-1}{\mathit{Q}}_{k-1}{G}_{k,k-1}{}^{T}$ | |

Update | ${\mathit{K}}_{k}={\mathit{P}}_{k}^{f}{\mathit{H}}_{k}{}^{T}{\left({\mathit{H}}_{k}{\mathit{P}}_{k}^{f}{\mathit{H}}_{k}{}^{T}+{\mathit{R}}_{k}\right)}^{-1}$ |

${x}_{k}^{a}={x}_{k}^{f}+{\mathit{K}}_{k}\left({y}_{k}-{\mathit{H}}_{k}{x}_{k}^{f}\right)$ | |

${\mathit{P}}_{k}^{a}=\left(\mathit{I}-{\mathit{K}}_{k}{\mathit{H}}_{k}\right){\mathit{P}}_{k}^{f}$ |

**Table 2.**Root mean square error (RMSE) and the mean absolute percentage error (MAPE, %) values of the assimilation results acquired using the four methods.

RMSE | MAPE (%) | |
---|---|---|

KF method | 0.5647 | 31.81 |

C-W method | 0.0368 | 0.16 |

A-KF method | 0.0356 | 0.14 |

Proposed method | 0.0340 | 0.12 |

RMSE | |||||
---|---|---|---|---|---|

KF | C-W | A-KF | Proposed Method | KF (Correct Model) | |

Monday | 842.65 | 394.20 | 152.55 | 85.37 | 77.38 |

Tuesday | 829.18 | 340.98 | 178.67 | 85.69 | 74.59 |

Wednesday | 857.94 | 346.18 | 164.94 | 88.85 | 80.74 |

Thursday | 856.66 | 360.62 | 227.82 | 84.82 | 75.18 |

Friday | 901.86 | 412.18 | 211.86 | 78.56 | 69.80 |

Saturday | 584.92 | 210.12 | 134.08 | 41.29 | 39.22 |

Sunday | 579.88 | 145.76 | 104.04 | 35.77 | 30.16 |

MAPE (%) | |||||
---|---|---|---|---|---|

KF | C-W | A-KF | Proposed Method | KF (Correct Model) | |

Monday | 99.56 | 39.67 | 15.31 | 7.23 | 6.76 |

Tuesday | 99.71 | 35.72 | 17.99 | 7.91 | 6.82 |

Wednesday | 99.70 | 34.62 | 15.15 | 7.58 | 7.00 |

Thursday | 99.69 | 37.70 | 22.18 | 7.56 | 6.66 |

Friday | 99.75 | 40.51 | 20.83 | 7.24 | 6.26 |

Saturday | 99.65 | 35.94 | 22.05 | 6.70 | 6.15 |

Sunday | 99.48 | 28.73 | 21.58 | 6.60 | 5.74 |

**Table 5.**Differences in average RMSE and MAPE values between three filter-divergence suppression methods and KF method under the correct model.

RMSE | MAPE (%) | |||||
---|---|---|---|---|---|---|

C-W | A-KF | Proposed Method | C-W | A-KF | Proposed Method | |

Monday | 316.82 | 75.17 | 7.99 | 32.91 | 8.55 | 0.47 |

Tuesday | 266.39 | 104.08 | 11.10 | 28.90 | 11.17 | 1.09 |

Wednesday | 265.44 | 84.20 | 8.11 | 27.62 | 8.15 | 0.58 |

Thursday | 285.44 | 152.64 | 9.64 | 31.04 | 15.52 | 0.90 |

Friday | 342.38 | 142.06 | 8.76 | 34.25 | 14.57 | 0.98 |

Saturday | 170.90 | 94.86 | 2.07 | 29.79 | 15.90 | 0.55 |

Sunday | 115.60 | 73.88 | 5.61 | 22.99 | 15.84 | 0.86 |

RMSE | |||||
---|---|---|---|---|---|

KF | C-W | A-KF | Proposed Method | KF (Correct Model) | |

Monday | 596.29 | 233.39 | 156.14 | 71.47 | 55.70 |

Tuesday | 591.97 | 204.79 | 145.25 | 70.72 | 53.58 |

Wednesday | 607.66 | 208.28 | 143.06 | 72.54 | 54.74 |

Thursday | 606.60 | 213.69 | 161.17 | 95.87 | 52.45 |

Friday | 649.17 | 252.80 | 222.20 | 68.37 | 50.28 |

Saturday | 419.31 | 127.81 | 111.40 | 38.30 | 26.53 |

Sunday | 414.71 | 102.03 | 96.47 | 36.77 | 23.62 |

MAPE (%) | |||||
---|---|---|---|---|---|

KF | C-W | A-KF | Proposed Method | KF (Correct Model) | |

Monday | 98.39 | 31.56 | 21.61 | 9.34 | 7.12 |

Tuesday | 98.88 | 28.13 | 19.98 | 9.63 | 7.19 |

Wednesday | 98.43 | 27.77 | 19.08 | 9.57 | 7.07 |

Thursday | 97.99 | 28.91 | 21.59 | 10.06 | 6.87 |

Friday | 98.15 | 32.42 | 25.47 | 8.90 | 6.55 |

Saturday | 98.50 | 26.12 | 22.56 | 8.86 | 6.42 |

Sunday | 97.94 | 25.80 | 24.11 | 9.23 | 6.71 |

**Table 8.**Differences in the average RMSE and MAPE values between three filter-divergence suppression methods and the KF method under the correct model.

RMSE | MAPE (%) | |||||
---|---|---|---|---|---|---|

C-W | A-KF | Proposed Method | C-W | A-KF | Proposed Method | |

Monday | 177.69 | 100.44 | 15.77 | 24.44 | 14.49 | 2.22 |

Tuesday | 151.21 | 91.67 | 17.14 | 20.94 | 12.79 | 2.44 |

Wednesday | 153.54 | 88.32 | 17.80 | 20.70 | 12.01 | 2.50 |

Thursday | 161.24 | 108.72 | 43.42 | 22.04 | 14.72 | 3.19 |

Friday | 202.52 | 171.92 | 18.09 | 25.87 | 18.92 | 2.35 |

Saturday | 101.28 | 84.87 | 11.77 | 19.70 | 16.14 | 2.44 |

Sunday | 78.41 | 72.85 | 13.15 | 19.09 | 17.40 | 2.52 |

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**MDPI and ACS Style**

Tong, X.; Wang, R.; Shi, W.; Li, Z.
An Approach for Filter Divergence Suppression in a Sequential Data Assimilation System and Its Application in Short-Term Traffic Flow Forecasting. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 340.
https://doi.org/10.3390/ijgi9060340

**AMA Style**

Tong X, Wang R, Shi W, Li Z.
An Approach for Filter Divergence Suppression in a Sequential Data Assimilation System and Its Application in Short-Term Traffic Flow Forecasting. *ISPRS International Journal of Geo-Information*. 2020; 9(6):340.
https://doi.org/10.3390/ijgi9060340

**Chicago/Turabian Style**

Tong, Xiaohua, Runjie Wang, Wenzhong Shi, and Zhiyuan Li.
2020. "An Approach for Filter Divergence Suppression in a Sequential Data Assimilation System and Its Application in Short-Term Traffic Flow Forecasting" *ISPRS International Journal of Geo-Information* 9, no. 6: 340.
https://doi.org/10.3390/ijgi9060340