An Improved Method Based on EEMD-LSTM to Predict Missing Measured Data of Structural Sensors
Abstract
:1. Introduction
2. Theoretical Background of the Models
2.1. Ensemble Empirical Mode Decomposition
- (1).
- Setting the original signals’ processing times to be equal, m;
- (2).
- Adding random white noise of different amplitudes to each of these m original signals to form a new series of signals;
- (3).
- Performing EMD decomposition on each of these new signals to obtain a series of IMF components;
- (4).
- The EEMD decomposition results are obtained by averaging the IMF components of the corresponding modes separately.
2.2. Long Short-Term Memory
2.3. The Procedure of the Proposed Method
- (1).
- The use of the algorithm to decompose the original measured signal data can enhance the robustness of prediction;
- (2).
- The LSTM in the framework of the proposed method can make full use of the long-term correlation of the time series data;
- (3).
- The underlying EEMD-LSTM relies on a data-driven approach with no assumption restrictions on the input data.
3. Experimental Design and Evaluation Criteria
3.1. Data Description
3.2. Acceleration Decomposition Based on EEMD
3.3. Parameters Selection
3.4. Evaluation Criteria
4. Comparative Analysis of the Results
4.1. Performance Evaluation of EEMD
4.2. Performance Evaluation of the Proposed Models
4.3. Performance Evaluation of the Multi-Steps
4.4. Generalization Ability of the Proposed Model
5. Conclusions
- (1).
- Compared with the traditional single prediction models, the proposed EEMD-based methods well captured the micro change features of the measured signal, thereby obtaining a better data complementary performance;
- (2).
- The results of the measured signal data recovery showed that compared with EEMD-BiGRU, EEMD-GRU, and EEMD-DNN, the proposed EEMD-LSTM method could make full use of the long-term correlation of time series data and, thus, performed optimally in the missing data complementation performance indicators. It was also demonstrated that EEMD-LSTM could effectively correlate the historical information with the current input features and achieve accurate missing data completion;
- (3).
- The data recovery capability of most of the algorithms may decrease with the increase in missing data. The results showed that the model’s accuracy decreased with the increasing in step size. It is worth mentioning that EEMD-LSTM had the best missing data complementation performance among all methods when the missing data increased.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Dataset | Point | Metrics | EEMD-BiGRU | EEMD-GRU | EEMD-LSTM | EEMD-DNN | BiGRU | GRU | LSTM | DNN |
---|---|---|---|---|---|---|---|---|---|---|
El-Centro | 1 | MSE | 0.006 | 0.0056 | 0.0052 | 0.0101 | 0.0247 | 0.0284 | 0.0289 | 0.0325 |
1 | RMSE | 0.0774 | 0.0746 | 0.0722 | 0.1007 | 0.1571 | 0.1685 | 0.1699 | 0.1801 | |
1 | MAE | 0.0585 | 0.0566 | 0.0543 | 0.0787 | 0.1243 | 0.1322 | 0.1338 | 0.1414 | |
1 | R2 | 97.92% | 98.06% | 98.19% | 96.47% | 91.42% | 90.13% | 89.96% | 88.72% | |
5 | MSE | 0.0014 | 0.0014 | 0.0013 | 0.0037 | 0.0077 | 0.0076 | 0.0075 | 0.0057 | |
5 | RMSE | 0.0371 | 0.0375 | 0.0365 | 0.0607 | 0.0878 | 0.0869 | 0.0867 | 0.0752 | |
5 | MAE | 0.0273 | 0.0272 | 0.027 | 0.0433 | 0.0621 | 0.0629 | 0.0622 | 0.0549 | |
5 | R2 | 94.58% | 94.46% | 94.76% | 85.52% | 69.72% | 70.31% | 70.42% | 77.75% | |
10 | MSE | 0.0034 | 0.005 | 0.0027 | 0.0065 | 0.0132 | 0.0131 | 0.0152 | 0.0118 | |
10 | RMSE | 0.0579 | 0.071 | 0.0522 | 0.0807 | 0.1151 | 0.1145 | 0.1235 | 0.1086 | |
10 | MAE | 0.0396 | 0.0408 | 0.0378 | 0.0562 | 0.0747 | 0.0793 | 0.0781 | 0.0708 | |
10 | R2 | 86.52% | 79.76% | 89.07% | 73.85% | 46.82% | 47.36% | 38.77% | 52.67% | |
15 | MSE | 0.0061 | 0.0071 | 0.0061 | 0.0076 | 0.0165 | 0.0167 | 0.1794 | 0.0124 | |
15 | RMSE | 0.0784 | 0.084 | 0.0779 | 0.0874 | 0.1285 | 0.1293 | 0.4236 | 0.1114 | |
15 | MAE | 0.0511 | 0.0564 | 0.0463 | 0.0586 | 0.0856 | 0.0861 | 0.3272 | 0.0734 | |
15 | R2 | 75.30% | 71.60% | 75.60% | 69.26% | 33.60% | 32.74% | 37.29% | 50.04% |
Dataset | Point | Metrics | EEMD-BiGRU | EEMD-GRU | EEMD-LSTM | EEMD-DNN | BiGRU | GRU | LSTM | DNN |
---|---|---|---|---|---|---|---|---|---|---|
RG | 1 | MSE | 0.0016 | 0.0011 | 0.001 | 0.0024 | 0.0072 | 0.0077 | 0.0072 | 0.0089 |
1 | RMSE | 0.0399 | 0.0333 | 0.031 | 0.0488 | 0.085 | 0.0877 | 0.0851 | 0.0942 | |
1 | MAE | 0.0286 | 0.0237 | 0.0226 | 0.0358 | 0.0628 | 0.0662 | 0.0638 | 0.0698 | |
1 | R2 | 97.80% | 98.47% | 98.67% | 96.71% | 90.00% | 89.35% | 89.97% | 87.71% | |
5 | MSE | 0.0045 | 0.0051 | 0.0043 | 0.0092 | 0.0257 | 0.0267 | 0.0268 | 0.0241 | |
5 | RMSE | 0.0669 | 0.0712 | 0.0654 | 0.096 | 0.1604 | 0.1635 | 0.1636 | 0.1552 | |
5 | MAE | 0.0475 | 0.0491 | 0.0448 | 0.0669 | 0.1089 | 0.1106 | 0.1114 | 0.1028 | |
5 | R2 | 93.29% | 92.40% | 93.60% | 86.19% | 61.45% | 59.91% | 59.90% | 63.89% | |
10 | MSE | 0.013 | 0.0134 | 0.0124 | 0.0202 | 0.0386 | 0.0407 | 0.0532 | 0.0364 | |
10 | RMSE | 0.1139 | 0.1158 | 0.1113 | 0.1421 | 0.1964 | 0.2017 | 0.2307 | 0.1907 | |
10 | MAE | 0.0734 | 0.076 | 0.0698 | 0.0942 | 0.1359 | 0.1372 | 0.1552 | 0.1258 | |
10 | R2 | 80.95% | 80.31% | 81.81% | 70.34% | 43.39% | 40.28% | 21.85% | 46.61% | |
15 | MSE | 0.0178 | 0.0188 | 0.0165 | 0.0229 | 0.0394 | 0.0448 | 0.0467 | 0.0379 | |
15 | RMSE | 0.1334 | 0.1372 | 0.1284 | 0.1512 | 0.1986 | 0.2116 | 0.2162 | 0.1947 | |
15 | MAE | 0.0857 | 0.0877 | 0.0815 | 0.0992 | 0.1363 | 0.1432 | 0.1496 | 0.1265 | |
15 | R2 | 72.53% | 70.95% | 74.57% | 64.72% | 39.16% | 30.93% | 27.90% | 41.53% |
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Dataset | No. of Samples | Mean | Standard Deviation | Minimum | Maximum |
---|---|---|---|---|---|
White | 15,360 | 0.0673 | 0.5269 | −2.6461 | 2.3082 |
El-Centro | 15,360 | 0.0441 | 0.1509 | −1.2667 | 1.0134 |
RG (artificial wave) | 15,360 | −0.0260 | 0.2602 | −1.7171 | 1.5201 |
Dataset | Point | Metrics | EEMD-BiGRU | EEMD-GRU | EEMD-LSTM | EEMD-DNN | BiGRU | GRU | LSTM | DNN |
---|---|---|---|---|---|---|---|---|---|---|
White | 1 | MSE | 0.0060 | 0.0056 | 0.0052 | 0.0101 | 0.0247 | 0.0284 | 0.0289 | 0.0325 |
1 | RMSE | 0.0774 | 0.0746 | 0.0722 | 0.1007 | 0.1571 | 0.1685 | 0.1699 | 0.1801 | |
1 | MAE | 0.0585 | 0.0566 | 0.0543 | 0.0787 | 0.1243 | 0.1322 | 0.1338 | 0.1414 | |
1 | R2 | 97.92% | 98.06% | 98.19% | 96.47% | 91.42% | 90.13% | 89.96% | 88.72% | |
El-Centro | 1 | MSE | 0.0003 | 0.0003 | 0.0003 | 0.0006 | 0.003 | 0.003 | 0.0034 | 0.0036 |
1 | RMSE | 0.018 | 0.0179 | 0.0178 | 0.0244 | 0.0547 | 0.0551 | 0.058 | 0.0601 | |
1 | MAE | 0.0143 | 0.0144 | 0.0143 | 0.0191 | 0.0432 | 0.0436 | 0.0458 | 0.0455 | |
1 | R2 | 98.63% | 98.65% | 98.66% | 97.49% | 87.40% | 87.17% | 85.80% | 84.78% | |
RG | 1 | MSE | 0.0016 | 0.0011 | 0.001 | 0.0024 | 0.0072 | 0.0077 | 0.0072 | 0.0089 |
1 | RMSE | 0.0399 | 0.0333 | 0.031 | 0.0488 | 0.085 | 0.0877 | 0.0851 | 0.0942 | |
1 | MAE | 0.0286 | 0.0237 | 0.0226 | 0.0358 | 0.0628 | 0.0662 | 0.0638 | 0.0698 | |
1 | R2 | 97.80% | 98.47% | 98.67% | 96.71% | 90.00% | 89.35% | 89.97% | 87.71% |
Dataset | Point | Metrics | EEMD-BiGRU | EEMD-GRU | EEMD-LSTM | EEMD-DNN | BiGRU | GRU | LSTM | DNN |
---|---|---|---|---|---|---|---|---|---|---|
White | 1 | MSE | 0.006 | 0.0056 | 0.0052 | 0.0101 | 0.0247 | 0.0284 | 0.0289 | 0.0325 |
1 | RMSE | 0.0774 | 0.0746 | 0.0722 | 0.1007 | 0.1571 | 0.1685 | 0.1699 | 0.1801 | |
1 | MAE | 0.0585 | 0.0566 | 0.0543 | 0.0787 | 0.1243 | 0.1322 | 0.1338 | 0.1414 | |
1 | R2 | 97.92% | 98.06% | 98.19% | 96.47% | 91.42% | 90.13% | 89.96% | 88.72% | |
5 | MSE | 0.0228 | 0.0237 | 0.0221 | 0.0345 | 0.116 | 0.1287 | 0.1123 | 0.1007 | |
5 | RMSE | 0.1510 | 0.1539 | 0.1487 | 0.1857 | 0.3406 | 0.3588 | 0.3351 | 0.3173 | |
5 | MAE | 0.1137 | 0.1129 | 0.1109 | 0.1375 | 0.2571 | 0.2714 | 0.2536 | 0.2393 | |
5 | R2 | 91.66% | 91.34% | 91.91% | 87.39% | 57.59% | 52.93% | 58.93% | 63.18% | |
10 | MSE | 0.0793 | 0.0679 | 0.0675 | 0.0853 | 0.1498 | 0.1423 | 0.1459 | 0.1417 | |
10 | RMSE | 0.2815 | 0.2606 | 0.2599 | 0.2921 | 0.387 | 0.3772 | 0.382 | 0.3764 | |
10 | MAE | 0.2019 | 0.1849 | 0.1869 | 0.2183 | 0.299 | 0.2914 | 0.2954 | 0.2891 | |
10 | R2 | 71.15% | 75.29% | 75.42% | 68.95% | 45.48% | 48.22% | 46.90% | 48.43% | |
15 | MSE | 0.1218 | 0.1176 | 0.1145 | 0.1462 | 0.1856 | 0.1868 | 0.1794 | 0.1764 | |
15 | RMSE | 0.3490 | 0.343 | 0.3383 | 0.3824 | 0.4309 | 0.4322 | 0.4236 | 0.4201 | |
15 | MAE | 0.26 | 0.2561 | 0.2478 | 0.289 | 0.332 | 0.3335 | 0.3272 | 0.3258 | |
15 | R2 | 57.42% | 58.89% | 59.99% | 48.90% | 35.12% | 34.72% | 37.29% | 38.33% |
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Chen, Z.; Yuan, C.; Wu, H.; Zhang, L.; Li, K.; Xue, X.; Wu, L. An Improved Method Based on EEMD-LSTM to Predict Missing Measured Data of Structural Sensors. Appl. Sci. 2022, 12, 9027. https://doi.org/10.3390/app12189027
Chen Z, Yuan C, Wu H, Zhang L, Li K, Xue X, Wu L. An Improved Method Based on EEMD-LSTM to Predict Missing Measured Data of Structural Sensors. Applied Sciences. 2022; 12(18):9027. https://doi.org/10.3390/app12189027
Chicago/Turabian StyleChen, Zengshun, Chenfeng Yuan, Haofan Wu, Likai Zhang, Ke Li, Xuanyi Xue, and Lei Wu. 2022. "An Improved Method Based on EEMD-LSTM to Predict Missing Measured Data of Structural Sensors" Applied Sciences 12, no. 18: 9027. https://doi.org/10.3390/app12189027
APA StyleChen, Z., Yuan, C., Wu, H., Zhang, L., Li, K., Xue, X., & Wu, L. (2022). An Improved Method Based on EEMD-LSTM to Predict Missing Measured Data of Structural Sensors. Applied Sciences, 12(18), 9027. https://doi.org/10.3390/app12189027