Missing Data Prediction Based on Kronecker Compressing Sensing in Multivariable Time Series
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摘要:
针对现有算法在预测多变量时间序列中的缺失数据时不适用或只适用于缺失数据较少的情况，该文提出一种基于克罗内克压缩感知的缺失数据预测算法。首先，利用多变量时间序列的时域平滑特性和序列之间的潜在相关性从时空两个方面设计了稀疏表示基，从而将缺失数据预测问题建模成稀疏向量恢复问题。模型求解部分，根据缺失数据的位置特点设计了适合当前应用场景且与稀疏表示基相关性低的观测矩阵。接着，从稀疏表示向量是否足够稀疏和感知矩阵是否满足有限等距特性两个方面验证了模型的性能。最后，仿真结果表明，所提算法在数据缺失严重的情况下具有良好的性能。
Abstract:In view of the problem that the existing methods are not applicable or are only feasible to the case where only a low ratio of data are missing in multivariable time series, a missing data prediction algorithm is proposed based on Kronecker Compressed Sensing (KCS) theory. Firstly, the sparse representation basis is designed to largely utilize both the temporal smoothness characteristic of time series and potential correlation between multiple time series. In this way, the missing data prediction problem is modeled into the problem of sparse vector recovery. In the solution part of the model, according to the location of missing data, the measurement matrix is designed suitable for the current application scenario and low correlation with the sparse representation basis. Then, the validity of the model is verified from two aspects: Whether the sparse representation vector is sufficiently sparse and the sensing matrix satisfies the restricted isometry property. Simulation results show that the proposed algorithm has good performance in the case where a high ratio of data are missing.
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表 1 多变量时间序列
数据源 t1 t2 t3 t4 $\cdots$ tK s1 0.2 ? ? 0.3 1.9 s2 ? 0.4 0.5 ? ? s3 ? 0.7 ? 0.6 ? s4 0.8 ? ? ? 2.1 $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ sN ? 0.1 1.2 1.3 ? 
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表 2 稀疏表示基稀疏性能比较
n 100 200 300 500 MOTES_1 0.9897 0.9952 0.9973 0.9992 MOTES_2 0.7428 0.8731 0.9301 0.9786 GSA_1 0.9687 0.9995 0.9998 0.9999 GSA_2 0.5227 0.6841 0.9018 0.9721 SST_1 0.8759 0.9429 0.9722 0.9935 SST_2 0.7390 0.8785 0.9393 0.9849
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表 3 稀疏表示基和观测矩阵非相关性的大小
NK 2000 3000 4000 6000 8000 $I({ {{ψ}}_R},{ {φ}_{{\rm{KCS}}1}})$ 1998 2995 3997 5996 7993 $I({ {{ψ}}_R},{ {φ}_{{\rm{KCS}}2}})$ 1998 2999 4000 5999 8000 $I({ {{ψ}}_E},{ {φ}_{{\rm{KCS}}1}})$ 1999 2996 3997 5999 7995 $I({ {{ψ}}_E},{ {φ}_{{\rm{KCS}}2}})$ 1999 2999 3999 6000 8000 $I({ {{ψ}}_C},{ {φ}_{{\rm{KCS}}1}})$ 1997 2996 3997 5995 7995 $I({ {{ψ}}_C},{ {φ}_{{\rm{KCS}}2}})$ 1998 2999 3998 5999 7998
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表 4 各方法在不同数据缺失率下的性能比较
数据缺失率(%) 20 50 80 90 95 MOTES KCS-GAMP-SBL 0.0266 0.0356 0.0840 0.1101 0.1509 TDMF 0.0287 0.0871 0.1962 0.2604 0.3684 SI 0.0862 0.1604 1.0172 1.7662 3.5204 RNN 0.0392 0.0951 0.8875 0.9174 1.0529 KPPCA 0.0278 0.0608 0.1826 0.2769 0.4516 GSA KCS-GAMP-SBL 0.0357 0.0460 0.1985 0.2769 0.3304 TDMF 0.0548 0.1278 0.3557 0.4764 0.5688 SI 0.0935 0.1455 0.8348 2.9442 4.6315 RNN 0.0526 0.0945 1.0858 1.1639 1.7984 KPPCA 0.0498 0.1012 0.3892 0.4879 0.5872 SST KCS-GAMP-SBL 0.0181 0.0327 0.0956 0.1225 0.1767 TDMF 0.0242 0.0544 0.1788 0.2374 0.2961 SI 0.0485 0.0851 0.7451 1.3802 2.0596 RNN 0.0262 0.0488 0.3417 0.5896 1.0421 KPPCA 0.0226 0.0644 0.1736 0.2705 0.2909
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表 5 不同算法平均运行时间(ART)的比较(s)
SI RNN KPPCA TDMF KCS-GAMP-SBL MOTES 0.7350 3.1832 11.5781 8.4362 3.0802 GSA 1.5361 7.0320 24.8685 20.4093 15.8633 SST 0.7216 2.5216 10.8683 8.9381 2.9032
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表 6 稀疏表示基的选择对算法均方根误差(RMSE)的影响
数据缺失率(%) 20 50 80 90 95 MOTES_1 0.0266 0.0356 0.0930 0.1201 0.1509 MOTES_2 0.0312 0.0422 0.1151 0.1387 0.1954 GSA_1 0.0357 0.0460 0.1985 0.2769 0.3304 GSA_2 0.0412 0.0681 0.2313 0.3343 0.4068 SST_1 0.0181 0.0327 0.0996 0.1275 0.1767 SST_2 0.0199 0.0340 0.1265 0.1534 0.2463
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