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摘要: 为实时动态评估交通态势，结合低秩张量补全理论，提出了一种基于自回归正则项与拉普拉斯正则项的交通速度预测模型；为提高模型在全局空间维度的表达能力，构建基于低秩张量补全框架的拉普拉斯卷积正则项表示路段间的关联关系；为提高模型在局部空间维度的表达能力，利用自回归模型的时间序列趋势捕获能力提高模型在时间维度的短时与长时表达能力，更精确地捕获交通数据的时空信息；为提高算法效率，通过时域与频域信号的转换降低了矩阵运算量，并采用截断核范数作为低秩张量逼近模型；使用交替方向乘子法实现高效的低秩拉普拉斯自回归张量补全(LLATC)预测方法；基于出租车行驶速度数据集和高速公路交通速度数据集，分析了LLATC算法在不同缺失率情况下的补全效果，对比了LLATC算法与其他基线预测算法的预测精度。研究结果表明：在交通数据随机缺失模式下，缺失率为20%~70%时，相对于传统的低秩张量补全模型，LLATC算法补全平均绝对误差降低了2%~6%，相比于传统的预测方法，LLATC算法预测平均绝对误差降低了4%~22%；在交通数据非随机缺失模式下，相对于传统的低秩张量补全模型，LLATC算法的平均绝对误差降低了2%~6%，相比于传统的预测方法，LLATC算法的预测平均绝对误差降低了13%~25%。可见，在2种交通数据缺失模式下，改进低秩张量补全方法降低了交通量数据的补全误差，能有效提高交通数据的预测精度，简化了数据处理流程。Abstract: To dynamically evaluate traffic condition in real time, a traffic speed prediction model based on autoregressive regularization terms and Laplacian regularization terms was proposed. To improve the model's expression capability in global dimensions, a Laplace convolutional regularization term based on a low-rank tensor completion framework was introduced to represent the correlations of road segments. To improve the model's expression capability in local spatial dimensions, the time series trend-capturing capabilities of autoregressive models were utilized, and the short- and long-term expression capabilities of the models in the time dimension were improved to capture the spatio-temporal information of traffic data more effectively. The implementation of the truncated kernel norm as the low-rank tensor approximation model and the conversion of time- and frequency-domain signals leaded to improve the computation efficiency. An efficient low-rank Laplacian autoregressive tensor completion (LLATC) prediction method was developed by using the alternating direction multiplier method. Based on taxi speed data set and expressway traffic speed data set, the completion performances of the LLATC algorithm under different missing rates were systematically analyzed, and the prediction accuracy of the LLATC algorithm was compared with other baseline prediction algorithms. Research results show that under the random missing pattern with a missing rate of 20% to 70%, the mean absolute error (MAE) of the LLATC algorithm reduces by 2% to 6% compared to the traditional low-rank tensor completion models, and the MAE reduces by 4% to 22% compared to the traditional prediction methods. Under the non-random missing pattern, the MAE of the LLATC algorithm reduces by 2% to 6% compared to the traditional low-rank tensor completion models, and the MAE reduces by 13% to 25% compared to the traditional prediction methods. The finding indicates that the LLATC algorithm effectively reduces the completion error of traffic volume data, significantly enhances the prediction accuracy of traffic volume data under two kinds of missing data patterns, and simplifies the data processing workflow.
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图 5 非随机和随机缺失下各算法的均方根误差
Figure 5. Root-mean-square errors of each algorithm under non-random and random missing
图 6 基于SZ数据集，非随机缺失率为20%时，路段3、28、32、44的1 d预测值与原始值对比
Figure 6. Based on SZ data set, comparison of predicted values and original values for road sections 3, 28, 32 and 44 at non-random missing rate of 20% in 1 d
图 7 基于SZ数据集，随机缺失率为20%时，路段3、28、32、44的1 d预测值与原始值对比
Figure 7. Based on SZ data set, comparison of predicted values and original values for road sections 3, 28, 32 and 44 at random missing rate of 20% in 1 d
图 8 基于SZ数据集，非随机缺失率为20%时，路段3、28、32、44的1 h预测值与原始值对比
Figure 8. Based on SZ data set, comparison of predicted values and original values for road sections 3, 28, 32 and 44 at non-random missing rate of 20% in 1 h
图 9 基于SZ数据集，随机缺失率为20%时，路段3、28、32、44的1 h预测值与原始值对比
Figure 9. Based on SZ data set, comparison of predicted values and original values for road sections 3, 28, 32 and 44 at random missing rate of 20% in 1 h
图 10 基于LP数据集，非随机缺失率为20%时，传感器5、14、19、31的1 d预测值与原始值对比
Figure 10. Based on LP data set, comparison of predicted values and original values for sensors 5, 14, 19 and 31 at non-random missing rate of 20% in 1 d
图 11 基于LP数据集，随机缺失率为20%时，传感器5、14、19、31的1 d预测值与原始值对比
Figure 11. Based on LP data set, comparison of predicted values and original values for sensors 5, 14, 19 and 31 at random missing rate of 20% in 1 d
图 12 基于LP数据集，非随机缺失率为20%时，传感器5、14、19、31的1 h预测值与原始值对比
Figure 12. Based on LP data set, comparison of predicted values and original values for sensors 5, 14, 19 and 31 at non-random missing rate of 20% in 1 h
图 13 基于LP数据集，随机缺失率为20%时，传感器5、14、19、31的1 h预测值与原始值对比
Figure 13. Based on LP data set, comparison of predicted values and original values for sensors 5, 14, 19 and 31 at random missing rate of 20% in 1 h
表 1 随机缺失下各方法数据补全精度(E1/E2)
Table 1. Data completion accuracy of each method for random missing (E1/E2)
数据集 缺失率/% 不同算法的数据补全精度/(km·h-1) CP-ALS HaLRTC BGCP LRTC_TNN BTTF LLATC SZ 20 3.78/5.44 3.73/5.41 3.76/5.42 3.60/5.26 3.78/5.44 3.54/5.14 30 3.76/5.43 3.77/5.45 3.76/5.41 3.63/5.28 3.78/5.44 3.57/5.20 40 3.79/5.46 3.81/5.51 3.76/5.42 3.65/5.32 3.80/5.46 3.62/5.27 50 3.80/5.48 3.87/5.57 3.78/5.45 3.68/5.35 3.79/5.45 3.64/5.29 60 3.82/5.51 3.95/5.66 3.80/5.46 3.72/5.37 3.80/5.46 3.68/5.33 70 3.87/5.58 4.05/5.79 3.82/5.48 3.76/5.43 3.85/5.52 3.72/5.38 80 3.97/5.76 4.20/6.00 3.85/5.53 3.81/5.49 3.87/5.55 3.80/5.45 LP 20 3.55/5.73 3.32/8.13 3.97/6.53 2.66/4.31 3.55/5.73 2.65/4.26 30 3.57/5.75 3.47/8.48 3.99/6.55 2.74/4.46 3.56/5.75 2.74/4.43 40 3.58/5.76 3.61/8.86 4.00/6.57 2.83/4.61 3.57/5.77 2.84/4.58 50 3.59/5.79 3.77/9.29 4.00/6.56 2.94/4.81 3.58/5.79 2.96/4.76 60 3.61/5.83 3.96/9.83 4.01/6.59 3.09/5.05 3.60/5.83 3.09/5.01 70 3.65/5.89 4.18/10.46 4.04/6.62 3.29/5.39 3.63/5.88 3.28/5.35 80 3.73/6.00 4.48/11.33 4.06/6.66 3.63/5.90 3.70/5.97 3.60/5.91 
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表 2 非随机缺失下各方法数据补全精度(E1/E2)
Table 2. Data completion accuracy of each method for non-random missing (E1/E2)
数据集 缺失率/% 不同算法的数据补全精度/(km·h-1) CP-ALS HaLRTC BGCP LRTC_TNN BTTF LLATC SZ 20 3.95/5.62 3.82/5.48 3.95/5.62 3.62/5.23 3.95/5.62 3.58/5.17 30 3.97/5.66 3.89/5.59 3.97/5.66 3.66/5.30 3.97/5.66 3.62/5.27 40 3.95/5.66 3.96/5.68 3.95/5.66 3.68/5.33 3.96/5.66 3.64/5.27 50 3.95/5.65 4.06/5.79 3.95/5.65 3.70/5.35 3.95/5.65 3.67/5.32 60 3.97/5.67 4.22/5.99 3.97/5.67 3.75/5.41 3.97/5.67 3.72/5.36 70 3.96/5.67 4.59/6.47 3.96/5.66 3.79/5.45 3.96/5.66 3.77/5.43 80 3.98/5.67 5.61/7.90 3.97/5.66 3.86/5.52 3.97/5.67 3.89/5.62 LP 20 3.51/5.73 3.42/5.04 3.91/6.52 2.82/4.70 3.50/5.74 2.79/4.57 30 3.51/5.75 3.81/5.41 3.93/6.51 2.94/4.89 3.51/5.73 2.89/4.75 40 3.48/5.70 4.33/5.81 3.90/6.46 3.00/4.99 3.48/5.69 2.99/4.96 50 18.22/115.26 5.84/7.21 3.95/6.51 3.21/5.33 3.52/5.74 3.17/5.27 60 108.12/292.34 11.16/13.86 4.00/6.61 4.26/9.19 3.59/6.42 4.18/7.30 70 229.93/609.04 21.60/26.37 4.34/6.94 8.93/19.69 6.42/11.30 5.60/8.97 80 155.33/521.23 37.14/40.70 17.48/30.72 16.13/29.04 22.04/36.81 6.98/10.71
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表 3 随机缺失下各方法数据预测精度(E1/E2)
Table 3. Prediction accuracy of each method for random missing(E1/E2)
数据集 缺失率/% 不同算法的数据预测精度/(km·h-1) HA ARIMA LSTM GRU T-GCN LLATC SZ 0 1.69/2.42 2.18/2.98 1.60/2.07 1.60/2.06 2.34/3.01 1.38/1.75 20 1.63/2.35 2.18/2.97 1.57/2.01 1.65/2.08 2.23/2.88 1.38/1.77 30 1.59/2.31 2.17/2.96 1.62/2.11 1.55/1.99 1.75/2.32 1.38/1.79 40 1.57/2.25 2.17/2.96 1.53/2.05 1.45/1.91 1.83/2.37 1.41/1.82 50 1.51/2.13 2.17/2.96 1.57/2.07 1.56/2.01 2.19/2.75 1.43/1.84 60 1.51/2.09 2.17/2.96 1.59/2.07 1.47/1.93 1.67/2.21 1.43/1.85 70 1.64/2.27 2.17/2.96 1.61/2.03 1.51/1.93 1.50/1.98 1.43/1.85 80 10.76/10.93 2.17/2.95 1.46/1.92 1.51/1.96 1.87/2.41 1.46/1.87
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表 4 非随机缺失下各方法数据预测精度(E1/E2)
Table 4. Prediction accuracy of each method for non-random missing(in E1/E2)
数据集 缺失率/% 不同算法的数据预测精度/(km·h-1) HA ARIMA LSTM GRU T-GCN LLATC SZ 0 1.69/2.42 2.18/2.98 1.60/2.07 1.60/2.06 2.34/3.01 1.35/1.75 20 1.56/2.18 2.17/2.97 1.59/2.06 1.57/2.02 2.10/2.69 1.33/1.73 30 1.52/2.13 2.18/2.97 1.59/2.02 1.54/1.97 2.20/2.84 1.34/1.73 40 1.52/2.13 2.17/2.97 1.59/2.03 1.53/1.99 2.32/2.95 1.33/1.75 50 1.60/2.23 2.17/2.96 1.58/2.03 1.54/2.01 2.18/2.84 1.35/1.81 60 1.59/2.22 2.17/2.96 1.60/2.05 1.58/2.04 2.25/2.86 1.38/1.83 70 1.56/2.15 2.17/2.95 1.54/1.96 1.52/1.95 2.52/3.13 1.39/1.81 80 1.53/2.15 2.17/2.95 1.51/1.94 1.51/1.93 1.82/2.44 1.93/2.44
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