A Hybrid Grey System Model Based on Stacked Long Short-Term Memory Layers and Its Application in Energy Consumption Forecasting
Abstract
:1. Introduction
2. Methodology
2.1. The General Formulation of a Grey System Model
2.2. The Proposed Hybrid Grey System Model Based on Stacked LSTM Layers
2.3. Adam Algorithm for Training the Proposed Model
Algorithm 1: Adam algorithm for training the hybrid grey system model | |||
Input: (Equation (24)), learning rate l, max_iteration | |||
Output: (Initialize the parameter set) | |||
(Initialize the exponential decay rate) | |||
(Initialize the exponential decay rate) | |||
(Initialize the small constant) | |||
(Initialize the moment) | |||
(Initialize the moment) | |||
(Initialize the number of iterations) | |||
1 | while do | ||
2 | iteration = iteration + 1; | ||
3 | Equation (26); (Calculate the objective function gradient) | ||
4 | Equation (29); (Calculate the first and second moment) | ||
5 | Equation (30); (Calculate the bias-corrected first and second moment) | ||
6 | Equation (31); (Update the parameters) | ||
7 | end | ||
8 | return (Resulting parameters) |
2.4. Grid Search Algorithm for Tuning Parameters of the Proposed Model
2.5. The Proposed Complete Forecasting Process
Algorithm 2: Complete forecasting process of the GM-ResNet | |||
Input: Training input: | |||
Number of neurons L, learning rate l and max_iteration | |||
Number of LSTM layers z; | |||
Output: | |||
; | |||
1 | Equation (32); (Use GreySLstm-M1 or M2 to select the best ) | ||
2 | while do | ||
3 | ← Algorithm 1; (Use Algorithm 1 to train the model) | ||
4 | end | ||
5 | for to do | ||
6 | ← Equations (22) and (8); (Forecast by using the response function and 1-IAGO) | ||
7 | end | ||
Result: |
3. Application
3.1. Data Collection
3.2. Selection of Comparison Models and Assessment Criteria
3.3. Case 1: Henan’s Coal Consumption
3.4. Case 2: Henan’s Electricity Consumption
3.5. Case 3: Henan’s Gasoline Consumption
3.6. Discussions
Analysis of the Performance of the Models on Real-World Cases
3.7. Analysis of the Indicator Optimization of the Proposed Model
Evaluating GreySLstm Performance with Different Numbers of LSTM Layers
4. Conclusions
4.1. Paper Structure Overview
4.2. Main Findings and Contributions
4.3. Analysis of Potential Limitations of the Model
4.4. Recommendations for Model Enhancement
4.5. Future Research Directions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Name | Reference | Year | Model Structure | Parameter |
---|---|---|---|---|
GM | [51] | 2010 | / | |
NGM | [52] | 2014 | / | |
DGM | [53] | 2018 | / | |
NDGM | [3] | 2013 | / | |
BernoulliGM | [54] | 2008 | b | |
FGM | [55] | 2013 | ||
FNGM | [56] | 2018 | ||
FNDGM | [57] | 2023 | ||
FDGM | [58] | 2014 | ||
FBernoulliGM | [59] | 2019 | b | |
NIPGM | [60] | 2015 | ||
NIPNGM | [61] | 2020 | ||
NIPDGM | [62] | 2017 | ||
NIPNDGM | [63] | 2008 | ||
NIPBernoulliGM | [64] | 2020 | b |
Full Name | Abbreviation | Reference | Year |
---|---|---|---|
Support vector regression | svr | [65] | 1996 |
Long Short-Term Memory | lstm | [45] | 2000 |
Random forest regression | rf | [66] | 2001 |
Multilayer perceptron | mlp | [67] | 2009 |
Extreme gradient boosting | xgb | [68] | 2015 |
Convolution neural network | cnn | [69] | 2015 |
Gated recurrent unit | gru | [70] | 2017 |
Convolutional LSTM | convlstm | [71] | 2017 |
CNN-LSTM | cnnlstm | [72] | 2019 |
Full Name | Metrics | Equation |
---|---|---|
Root-mean-square error | RMSE | |
Mean absolute error | MAE | |
Mean Absolute Percentage Error | MAPE | |
Theil’s inequality coefficient | TIC | |
Theil’s U1 statistic | U1 | |
Theil’s U2 statistic | U2 |
Model | GreySLstm-M1 | GreySLstm-M2 | gru | rf | xgb | lstm | svr | |
---|---|---|---|---|---|---|---|---|
Training | RMSE | 490.3060 | 960.1925 | 928.6231 | 870.6966 | 1574.2322 | 811.1070 | 1566.8574 |
MAE | 339.5045 | 609.2220 | 658.7882 | 518.2737 | 1352.1988 | 529.3345 | 1367.6741 | |
MAPE | 1.8857 | 2.9844 | 4.6797 | 2.7050 | 11.1550 | 3.6196 | 11.9076 | |
TIC | 0.0133 | 0.0260 | 0.0252 | 0.0238 | 0.0432 | 0.0221 | 0.0425 | |
U1 | 0.0133 | 0.0260 | 0.0252 | 0.0238 | 0.0432 | 0.0221 | 0.0425 | |
U2 | 0.0266 | 0.0522 | 0.0505 | 0.0473 | 0.0855 | 0.0441 | 0.0851 | |
Test | RMSE | 928.6467 | 5323.7461 | 2507.0439 | 3073.1162 | 1661.8960 | 1651.8883 | 2308.4474 |
MAE | 910.8103 | 5108.5371 | 2329.0432 | 2799.0660 | 1172.6112 | 1436.7536 | 2166.6322 | |
MAPE | 4.1222 | 23.2868 | 10.6819 | 12.8810 | 5.5687 | 6.6503 | 9.7389 | |
TIC | 0.0209 | 0.1066 | 0.0532 | 0.0645 | 0.0362 | 0.0357 | 0.0540 | |
U1 | 0.0209 | 0.1066 | 0.0532 | 0.0645 | 0.0362 | 0.0357 | 0.0540 | |
U2 | 0.0414 | 0.2373 | 0.1117 | 0.1370 | 0.0741 | 0.0736 | 0.1029 | |
Model | cnn | mlp | cnnlstm | convlstm | GM | NGM | DGM | |
Training | RMSE | 2430.8773 | 3915.1753 | 861.2289 | 812.6766 | 3013.4610 | 2491.0085 | 3017.3331 |
MAE | 2159.5672 | 3154.0362 | 597.2083 | 566.9696 | 2640.8209 | 2156.8366 | 2655.5472 | |
MAPE | 14.2518 | 17.5495 | 3.1941 | 3.5721 | 17.3731 | 14.6886 | 17.6296 | |
TIC | 0.0665 | 0.1152 | 0.0236 | 0.0221 | 0.0815 | 0.0691 | 0.0815 | |
U1 | 0.0665 | 0.1152 | 0.0236 | 0.0221 | 0.0815 | 0.0691 | 0.0815 | |
U2 | 0.1321 | 0.2127 | 0.0468 | 0.0442 | 0.1637 | 0.1353 | 0.1639 | |
Test | RMSE | 11,761.7003 | 4033.4668 | 3381.0791 | 5542.6099 | 16,836.7866 | 9023.2598 | 16,728.2120 |
MAE | 11,319.1136 | 3203.2006 | 3175.0204 | 5061.8062 | 16,073.4176 | 8607.2024 | 15,971.1954 | |
MAPE | 51.5327 | 14.9912 | 14.5368 | 23.2690 | 73.2872 | 39.2627 | 72.8201 | |
TIC | 0.2092 | 0.0839 | 0.0704 | 0.1110 | 0.2756 | 0.1687 | 0.2743 | |
U1 | 0.2092 | 0.0839 | 0.0704 | 0.1110 | 0.2756 | 0.1687 | 0.2743 | |
U2 | 0.5243 | 0.1798 | 0.1507 | 0.2471 | 0.7505 | 0.4022 | 0.7456 | |
Model | NDGM | BernoulliGM | FGM | FNGM | FNDGM | FDGM | FBernoulliGM | |
Training | RMSE | 2403.2727 | 5532.4287 | 4248.0164 | 4070.0333 | 2318.3493 | 4003.4205 | 6202.7378 |
MAE | 2120.1498 | 4954.7318 | 3574.8920 | 3599.8241 | 1948.7597 | 3384.7620 | 5537.3872 | |
MAPE | 15.0650 | 36.3972 | 30.5036 | 26.0696 | 12.7429 | 28.3588 | 41.8137 | |
TIC | 0.0656 | 0.1669 | 0.1146 | 0.1150 | 0.0623 | 0.1086 | 0.1903 | |
U1 | 0.0656 | 0.1669 | 0.1146 | 0.1150 | 0.0623 | 0.1086 | 0.1903 | |
U2 | 0.1306 | 0.3006 | 0.2308 | 0.2211 | 0.1260 | 0.2175 | 0.3370 | |
Test | RMSE | 9081.4995 | 1671.9496 | 3807.3586 | 6217.6065 | 11,715.2465 | 3689.2512 | 2269.5763 |
MAE | 8708.0001 | 1383.7527 | 3298.5991 | 5447.6971 | 11,347.7484 | 3197.6934 | 2226.4167 | |
MAPE | 39.6817 | 6.0014 | 15.2810 | 25.1836 | 51.5896 | 14.8132 | 9.9161 | |
TIC | 0.1695 | 0.0363 | 0.0791 | 0.1235 | 0.2084 | 0.0768 | 0.0482 | |
U1 | 0.1695 | 0.0363 | 0.0791 | 0.1235 | 0.2084 | 0.0768 | 0.0482 | |
U2 | 0.4048 | 0.0745 | 0.1697 | 0.2771 | 0.5222 | 0.1644 | 0.1012 | |
Model | NIPGM | NIPNGM | NIPDGM | NIPNDGM | NIPBernoulliGM | |||
Training | RMSE | 4317.6857 | 3141.6617 | 4127.4823 | 2429.1336 | 3457.6183 | ||
MAE | 3704.4585 | 2639.5239 | 3531.0537 | 1970.3929 | 2963.5155 | |||
MAPE | 31.4827 | 16.0038 | 29.7322 | 11.6506 | 23.9604 | |||
TIC | 0.1171 | 0.0886 | 0.1121 | 0.0656 | 0.0996 | |||
U1 | 0.1171 | 0.0886 | 0.1121 | 0.0656 | 0.0996 | |||
U2 | 0.2346 | 0.1707 | 0.2243 | 0.1320 | 0.1879 | |||
Test | RMSE | 5153.0724 | 11,050.5598 | 3907.2315 | 13,174.3111 | 23,210.8592 | ||
MAE | 4600.4453 | 10,461.0367 | 3395.0718 | 12,732.8356 | 18,689.1563 | |||
MAPE | 21.2189 | 47.7791 | 15.7211 | 57.9115 | 87.2516 | |||
TIC | 0.1042 | 0.1995 | 0.0810 | 0.2286 | 0.6128 | |||
U1 | 0.1042 | 0.1995 | 0.0810 | 0.2286 | 0.6128 | |||
U2 | 0.2297 | 0.4926 | 0.1742 | 0.5872 | 1.0346 |
Model | GreySLstm-M1 | GreySLstm-M2 | gru | rf | xgb | lstm | svr | |
---|---|---|---|---|---|---|---|---|
Training | RMSE | 37.4968 | 63.6828 | 488.3888 | 34.8961 | 145.2803 | 295.4851 | 181.7269 |
MAE | 28.8925 | 42.0938 | 220.5750 | 25.8150 | 121.5278 | 150.3027 | 156.6424 | |
MAPE | 1.6761 | 2.0229 | 32.1378 | 1.4577 | 11.4320 | 20.5718 | 14.6458 | |
TIC | 0.0096 | 0.0163 | 0.1259 | 0.0090 | 0.0378 | 0.0763 | 0.0468 | |
U1 | 0.0096 | 0.0163 | 0.1259 | 0.0090 | 0.0378 | 0.0763 | 0.0468 | |
U2 | 0.0193 | 0.0328 | 0.2514 | 0.0180 | 0.0748 | 0.1521 | 0.0935 | |
Test | RMSE | 115.3096 | 200.4789 | 421.2057 | 523.4443 | 772.2424 | 457.9253 | 129.5320 |
MAE | 95.7251 | 148.0640 | 377.4322 | 480.7118 | 743.9410 | 414.9213 | 108.6905 | |
MAPE | 2.7584 | 3.9858 | 10.3856 | 13.2748 | 20.7229 | 11.4350 | 3.1498 | |
TIC | 0.0162 | 0.0288 | 0.0627 | 0.0791 | 0.1215 | 0.0685 | 0.0180 | |
U1 | 0.0162 | 0.0288 | 0.0627 | 0.0791 | 0.1215 | 0.0685 | 0.0180 | |
U2 | 0.0325 | 0.0564 | 0.1186 | 0.1474 | 0.2174 | 0.1289 | 0.0365 | |
Model | cnn | mlp | cnnlstm | convlstm | GM | NGM | DGM | |
Training | RMSE | 189.6547 | 118.7663 | 105.7004 | 47.3233 | 231.5212 | 165.9904 | 232.7358 |
MAE | 142.9024 | 93.4915 | 75.3073 | 35.2646 | 192.6853 | 132.2427 | 193.8591 | |
MAPE | 14.3022 | 6.1338 | 4.0891 | 2.3983 | 14.3749 | 9.9145 | 14.5544 | |
TIC | 0.0490 | 0.0304 | 0.0267 | 0.0122 | 0.0586 | 0.0435 | 0.0589 | |
U1 | 0.0490 | 0.0304 | 0.0267 | 0.0122 | 0.0586 | 0.0435 | 0.0589 | |
U2 | 0.0976 | 0.0611 | 0.0544 | 0.0244 | 0.1192 | 0.0854 | 0.1198 | |
Test | RMSE | 293.7241 | 331.2288 | 124.8711 | 448.6507 | 1213.7337 | 277.8937 | 1215.1447 |
MAE | 270.8711 | 310.8389 | 104.1500 | 392.9850 | 1163.1175 | 246.7314 | 1164.8605 | |
MAPE | 7.6876 | 8.8156 | 2.9615 | 10.9248 | 32.5157 | 6.9568 | 32.5664 | |
TIC | 0.0398 | 0.0447 | 0.0175 | 0.0598 | 0.1464 | 0.0378 | 0.1466 | |
U1 | 0.0398 | 0.0447 | 0.0175 | 0.0598 | 0.1464 | 0.0378 | 0.1466 | |
U2 | 0.0827 | 0.0933 | 0.0352 | 0.1263 | 0.3417 | 0.0782 | 0.3421 | |
Model | NDGM | BernoulliGM | FGM | FNGM | FNDGM | FDGM | FBernoulliGM | |
Training | RMSE | 152.1839 | 301.4826 | 268.4222 | 147.0146 | 136.9840 | 472.5171 | 263.7041 |
MAE | 131.1087 | 266.4801 | 225.9895 | 110.9666 | 104.1567 | 376.2808 | 232.4938 | |
MAPE | 10.3028 | 21.8860 | 19.2908 | 5.8802 | 6.6519 | 37.5129 | 18.9998 | |
TIC | 0.0392 | 0.0822 | 0.0662 | 0.0381 | 0.0348 | 0.1151 | 0.0713 | |
U1 | 0.0392 | 0.0822 | 0.0662 | 0.0381 | 0.0348 | 0.1151 | 0.0713 | |
U2 | 0.0783 | 0.1552 | 0.1382 | 0.0757 | 0.0705 | 0.2432 | 0.1357 | |
Test | RMSE | 340.3275 | 116.3608 | 773.9608 | 480.4346 | 473.8145 | 301.5636 | 149.3024 |
MAE | 315.9793 | 107.1247 | 751.6470 | 459.6381 | 457.4457 | 270.4788 | 123.7991 | |
MAPE | 8.9202 | 3.0525 | 21.1297 | 12.9606 | 12.9383 | 7.4586 | 3.3959 | |
TIC | 0.0459 | 0.0163 | 0.0984 | 0.0635 | 0.0627 | 0.0442 | 0.0212 | |
U1 | 0.0459 | 0.0163 | 0.0984 | 0.0635 | 0.0627 | 0.0442 | 0.0212 | |
U2 | 0.0958 | 0.0328 | 0.2179 | 0.1353 | 0.1334 | 0.0849 | 0.0420 | |
Model | NIPGM | NIPNGM | NIPDGM | NIPNDGM | NIPBernoulliGM | |||
Training | RMSE | 214.9696 | 138.8300 | 232.3187 | 125.2973 | 272.0905 | ||
MAE | 169.2131 | 109.0311 | 189.8683 | 90.8680 | 238.6010 | |||
MAPE | 12.6166 | 6.0049 | 15.0377 | 5.2743 | 20.0758 | |||
TIC | 0.0537 | 0.0361 | 0.0577 | 0.0319 | 0.0736 | |||
U1 | 0.0537 | 0.0361 | 0.0577 | 0.0319 | 0.0736 | |||
U2 | 0.1107 | 0.0715 | 0.1196 | 0.0645 | 0.1401 | |||
Test | RMSE | 862.1933 | 404.1645 | 790.7576 | 429.9505 | 197.6275 | ||
MAE | 836.4000 | 384.1215 | 768.0924 | 412.9963 | 150.0556 | |||
MAPE | 23.4874 | 10.8571 | 21.5902 | 11.6904 | 4.0554 | |||
TIC | 0.1085 | 0.0540 | 0.1004 | 0.0572 | 0.0283 | |||
U1 | 0.1085 | 0.0540 | 0.1004 | 0.0572 | 0.0283 | |||
U2 | 0.2427 | 0.1138 | 0.2226 | 0.1211 | 0.0556 |
Model | GreySLstm-M1 | GreySLstm-M2 | gru | rf | xgb | lstm | svr | |
---|---|---|---|---|---|---|---|---|
Training | RMSE | 5.5247 | 8.7590 | 30.3996 | 9.4015 | 29.0164 | 17.2873 | 24.9984 |
MAE | 3.3122 | 5.5920 | 19.1020 | 6.4205 | 25.1756 | 13.5572 | 23.0586 | |
MAPE | 1.4747 | 2.2650 | 12.0591 | 2.9254 | 15.7324 | 7.8731 | 13.2535 | |
TIC | 0.0128 | 0.0202 | 0.0711 | 0.0221 | 0.0666 | 0.0405 | 0.0593 | |
U1 | 0.0128 | 0.0202 | 0.0711 | 0.0221 | 0.0666 | 0.0405 | 0.0593 | |
U2 | 0.0257 | 0.0407 | 0.1413 | 0.0437 | 0.1348 | 0.0803 | 0.1162 | |
Test | RMSE | 76.9153 | 96.7842 | 141.9118 | 278.8786 | 318.5326 | 94.8583 | 287.8411 |
MAE | 65.5358 | 77.5446 | 126.7438 | 265.2736 | 306.6917 | 85.9361 | 254.5826 | |
MAPE | 9.5421 | 11.1199 | 18.0005 | 38.6305 | 44.9456 | 12.3212 | 36.1240 | |
TIC | 0.0566 | 0.0691 | 0.1168 | 0.2592 | 0.3079 | 0.0751 | 0.2640 | |
U1 | 0.0566 | 0.0691 | 0.1168 | 0.2592 | 0.3079 | 0.0751 | 0.2640 | |
U2 | 0.1142 | 0.1437 | 0.2108 | 0.4142 | 0.4731 | 0.1409 | 0.4275 | |
Model | cnn | mlp | cnnlstm | convlstm | GM | NGM | DGM | |
Training | RMSE | 89.2788 | 39.4255 | 35.1136 | 34.6697 | 40.1285 | 177.1090 | 40.0240 |
MAE | 73.5071 | 29.2930 | 26.1866 | 26.6884 | 32.8573 | 122.6439 | 32.7378 | |
MAPE | 43.4380 | 14.4692 | 13.2231 | 13.9887 | 17.4882 | 54.4627 | 17.4793 | |
TIC | 0.2081 | 0.0922 | 0.0822 | 0.0811 | 0.0954 | 0.2982 | 0.0949 | |
U1 | 0.2081 | 0.0922 | 0.0822 | 0.0811 | 0.0954 | 0.2982 | 0.0949 | |
U2 | 0.4149 | 0.1832 | 0.1632 | 0.1611 | 0.1865 | 0.8230 | 0.1860 | |
Test | RMSE | 462.0330 | 249.8680 | 104.4885 | 221.5902 | 168.8275 | 1536.1246 | 170.6696 |
MAE | 453.9507 | 244.4299 | 95.2879 | 210.2770 | 164.6643 | 1349.0963 | 166.6187 | |
MAPE | 67.3986 | 36.2082 | 14.6379 | 30.5910 | 24.8887 | 193.0584 | 25.1626 | |
TIC | 0.5208 | 0.2275 | 0.0833 | 0.1959 | 0.1426 | 0.5397 | 0.1444 | |
U1 | 0.5208 | 0.2275 | 0.0833 | 0.1959 | 0.1426 | 0.5397 | 0.1444 | |
U2 | 0.6862 | 0.3711 | 0.1552 | 0.3291 | 0.2508 | 2.2815 | 0.2535 | |
Model | NDGM | BernoulliGM | FGM | FNGM | FNDGM | FDGM | FBernoulliGM | |
Training | RMSE | 34.0744 | 43.1875 | 37.6800 | 41.5449 | 42.8570 | 50.4337 | 36.0213 |
MAE | 27.1748 | 33.1724 | 29.2709 | 34.4108 | 33.4238 | 40.2398 | 28.0154 | |
MAPE | 14.6721 | 15.7012 | 16.7802 | 18.6444 | 16.0451 | 22.6073 | 16.1310 | |
TIC | 0.0798 | 0.1050 | 0.0849 | 0.0978 | 0.1039 | 0.1174 | 0.0817 | |
U1 | 0.0798 | 0.1050 | 0.0849 | 0.0978 | 0.1039 | 0.1174 | 0.0817 | |
U2 | 0.1583 | 0.2007 | 0.1751 | 0.1931 | 0.1992 | 0.2344 | 0.1674 | |
Test | RMSE | 197.0548 | 232.0179 | 217.0482 | 193.0278 | 241.6070 | 311.1789 | 362.4447 |
MAE | 144.3595 | 229.0458 | 159.5841 | 189.3324 | 238.4823 | 305.2174 | 260.6160 | |
MAPE | 20.1963 | 34.4236 | 22.2412 | 28.5918 | 35.7358 | 45.2670 | 35.9641 | |
TIC | 0.1337 | 0.2076 | 0.1436 | 0.1666 | 0.2182 | 0.3001 | 0.2202 | |
U1 | 0.1337 | 0.2076 | 0.1436 | 0.1666 | 0.2182 | 0.3001 | 0.2202 | |
U2 | 0.2927 | 0.3446 | 0.3224 | 0.2867 | 0.3588 | 0.4622 | 0.5383 | |
Model | NIPGM | NIPNGM | NIPDGM | NIPNDGM | NIPBernoulliGM | |||
Training | RMSE | 40.0408 | 59.9252 | 33.8145 | 32.3140 | 37.7739 | ||
MAE | 31.1999 | 42.8442 | 25.8625 | 28.0125 | 29.7247 | |||
MAPE | 18.2299 | 24.0522 | 14.8879 | 16.1756 | 17.5189 | |||
TIC | 0.0895 | 0.1439 | 0.0790 | 0.0760 | 0.0850 | |||
U1 | 0.0895 | 0.1439 | 0.0790 | 0.0760 | 0.0850 | |||
U2 | 0.1861 | 0.2785 | 0.1571 | 0.1502 | 0.1755 | |||
Test | RMSE | 187.9752 | 107.3958 | 946.1340 | 7738.0291 | 443.3291 | ||
MAE | 139.3251 | 87.2050 | 690.9050 | 4938.1195 | 311.9407 | |||
MAPE | 19.4785 | 12.5335 | 94.8007 | 664.1561 | 42.8325 | |||
TIC | 0.1268 | 0.0757 | 0.4286 | 0.8693 | 0.2582 | |||
U1 | 0.1268 | 0.0757 | 0.4286 | 0.8693 | 0.2582 | |||
U2 | 0.2792 | 0.1595 | 1.4052 | 11.4929 | 0.6585 |
Model | vs. GreySLstm-M2 | vs. gru | vs. rf | vs. xgb | vs. lstm | vs. svr | vs. cnn |
---|---|---|---|---|---|---|---|
RMSE | 82.5565 | 62.9585 | 69.7816 | 44.1213 | 43.7827 | 59.7718 | 92.1045 |
MAE | 82.1708 | 60.8934 | 67.4602 | 22.3263 | 36.6064 | 57.9619 | 91.9533 |
MAPE | 82.2981 | 61.4092 | 67.9978 | 25.9755 | 38.0148 | 57.6727 | 92.0008 |
TIC | 80.3937 | 60.6812 | 67.6062 | 42.2685 | 41.4517 | 61.3125 | 90.0117 |
U1 | 80.3937 | 60.6812 | 67.6062 | 42.2685 | 41.4517 | 61.3125 | 90.0117 |
U2 | 82.5565 | 62.9585 | 69.7816 | 44.1213 | 43.7827 | 59.7718 | 92.1045 |
Model | vs. mlp | vs. cnnlstm | vs. convlstm | vs. GM | vs. NGM | vs. DGM | vs. NDGM |
RMSE | 76.9765 | 72.5340 | 83.2453 | 94.4844 | 89.7083 | 94.4486 | 89.7743 |
MAE | 71.5656 | 71.3132 | 82.0062 | 94.3334 | 89.4180 | 94.2972 | 89.5405 |
MAPE | 72.5025 | 71.6429 | 82.2846 | 94.3753 | 89.5010 | 94.3392 | 89.6118 |
TIC | 75.0942 | 70.3242 | 81.1786 | 92.4163 | 87.6138 | 92.3802 | 87.6714 |
U1 | 75.0942 | 70.3242 | 81.1786 | 92.4163 | 87.6138 | 92.3802 | 87.6714 |
U2 | 76.9765 | 72.5340 | 83.2453 | 94.4844 | 89.7083 | 94.4486 | 89.7743 |
Model | vs. BernoulliGM | vs. FGM | vs. FNGM | vs. FNDGM | vs. FDGM | vs. FBernoulliGM | vs. NIPGM |
RMSE | 44.4573 | 75.6092 | 85.0642 | 92.0732 | 74.8283 | 59.0828 | 81.9788 |
MAE | 34.1782 | 72.3880 | 83.2808 | 91.9736 | 71.5166 | 59.0908 | 80.2017 |
MAPE | 31.3125 | 73.0239 | 83.6314 | 92.0096 | 72.1720 | 58.4291 | 80.5729 |
TIC | 42.4376 | 73.5743 | 83.0796 | 89.9698 | 72.7863 | 56.6203 | 79.9418 |
U1 | 42.4376 | 73.5743 | 83.0796 | 89.9698 | 72.7863 | 56.6203 | 79.9418 |
U2 | 44.4573 | 75.6092 | 85.0642 | 92.0732 | 74.8283 | 59.0828 | 81.9788 |
Model | vs. NIPNGM | vs. NIPDGM | vs. NIPNDGM | vs. NIPBernoulliGM | |||
RMSE | 91.5964 | 76.2326 | 92.9511 | 95.9991 | |||
MAE | 91.2933 | 73.1726 | 92.8468 | 95.1265 | |||
MAPE | 91.3724 | 73.7791 | 92.8819 | 95.2755 | |||
TIC | 89.5263 | 74.1977 | 90.8570 | 96.5897 | |||
U1 | 89.5263 | 74.1977 | 90.8570 | 96.5897 | |||
U2 | 91.5964 | 76.2326 | 92.9511 | 95.9991 |
Model | vs. GreySLstm-M2 | vs. gru | vs. rf | vs. xgb | vs. lstm | vs. svr | vs. cnn |
---|---|---|---|---|---|---|---|
RMSE | 42.4830 | 72.6239 | 77.9710 | 85.0682 | 74.8191 | 10.9799 | 60.7422 |
MAE | 35.3488 | 74.6378 | 80.0868 | 87.1327 | 76.9293 | 11.9287 | 64.6603 |
MAPE | 30.7929 | 73.4399 | 79.2205 | 86.6889 | 75.8773 | 12.4241 | 64.1183 |
TIC | 43.9111 | 74.2029 | 79.5609 | 86.6971 | 76.4041 | 10.4043 | 59.4026 |
U1 | 43.9111 | 74.2029 | 79.5609 | 86.6971 | 76.4041 | 10.4043 | 59.4026 |
U2 | 42.4830 | 72.6239 | 77.9710 | 85.0682 | 74.8191 | 10.9799 | 60.7422 |
Model | vs. mlp | vs. cnnlstm | vs. convlstm | vs. GM | vs. NGM | vs. DGM | vs. NDGM |
RMSE | 65.1873 | 7.6571 | 74.2986 | 90.4996 | 58.5059 | 90.5106 | 66.1181 |
MAE | 69.2043 | 8.0892 | 75.6415 | 91.7700 | 61.2027 | 91.7823 | 69.7053 |
MAPE | 68.7097 | 6.8578 | 74.7508 | 91.5166 | 60.3493 | 91.5298 | 69.0766 |
TIC | 63.8041 | 7.4294 | 72.9519 | 88.9578 | 57.2180 | 88.9685 | 64.7390 |
U1 | 63.8041 | 7.4294 | 72.9519 | 88.9578 | 57.2180 | 88.9685 | 64.7390 |
U2 | 65.1873 | 7.6571 | 74.2986 | 90.4996 | 58.5059 | 90.5106 | 66.1181 |
Model | vs. BernoulliGM | vs. FGM | vs. FNGM | vs. FNDGM | vs. FDGM | vs. FBernoulliGM | vs. NIPGM |
RMSE | 0.9034 | 85.1014 | 75.9989 | 75.6636 | 61.7628 | 22.7678 | 86.6260 |
MAE | 10.6414 | 87.2646 | 79.1738 | 79.0740 | 64.6090 | 22.6771 | 88.5551 |
MAPE | 9.6337 | 86.9452 | 78.7168 | 78.6801 | 63.0165 | 18.7709 | 88.2557 |
TIC | 1.0207 | 83.5747 | 74.5354 | 74.1933 | 63.3875 | 23.8799 | 85.0928 |
U1 | 1.0207 | 83.5747 | 74.5354 | 74.1933 | 63.3875 | 23.8799 | 85.0928 |
U2 | 0.9034 | 85.1014 | 75.9989 | 75.6636 | 61.7628 | 22.7678 | 86.6260 |
Model | vs. NIPNGM | vs. NIPDGM | vs. NIPNDGM | vs. NIPBernoulliGM | |||
RMSE | 71.4696 | 85.4178 | 73.1807 | 41.6531 | |||
MAE | 75.0795 | 87.5373 | 76.8218 | 36.2069 | |||
MAPE | 74.5934 | 87.2237 | 76.4042 | 31.9820 | |||
TIC | 70.0377 | 83.8895 | 71.7290 | 42.8803 | |||
U1 | 70.0377 | 83.8895 | 71.7290 | 42.8803 | |||
U2 | 71.4696 | 85.4178 | 73.1807 | 41.6531 |
Model | vs. GreySLstm-M2 | vs. gru | vs. rf | vs. xgb | vs. lstm | vs. svr | vs. cnn |
---|---|---|---|---|---|---|---|
RMSE | 20.5290 | 45.8006 | 72.4198 | 75.8532 | 18.9155 | 73.2785 | 83.3528 |
MAE | 15.4864 | 48.2927 | 75.2950 | 78.6314 | 23.7390 | 74.2575 | 85.5632 |
MAPE | 14.1890 | 46.9899 | 75.2991 | 78.7697 | 22.5557 | 73.5852 | 85.8423 |
TIC | 18.1348 | 51.5610 | 78.1745 | 81.6272 | 24.6273 | 78.5708 | 89.1368 |
U1 | 18.1348 | 51.5610 | 78.1745 | 81.6272 | 24.6273 | 78.5708 | 89.1368 |
U2 | 20.5290 | 45.8006 | 72.4198 | 75.8532 | 18.9155 | 73.2785 | 83.3528 |
Model | vs. mlp | vs. cnnlstm | vs. convlstm | vs. GM | vs. NGM | vs. DGM | vs. NDGM |
RMSE | 69.2176 | 26.3887 | 65.2894 | 54.4415 | 94.9929 | 54.9332 | 60.9676 |
MAE | 73.1883 | 31.2234 | 68.8336 | 60.2004 | 95.1422 | 60.6672 | 54.6024 |
MAPE | 73.6466 | 34.8124 | 68.8075 | 61.6611 | 95.0574 | 62.0784 | 52.7533 |
TIC | 75.1257 | 32.0472 | 71.1178 | 60.3303 | 89.5159 | 60.8297 | 57.6863 |
U1 | 75.1257 | 32.0472 | 71.1178 | 60.3303 | 89.5159 | 60.8297 | 57.6863 |
U2 | 69.2176 | 26.3887 | 65.2894 | 54.4415 | 94.9929 | 54.9332 | 60.9676 |
Model | vs. BernoulliGM | vs. FGM | vs. FNGM | vs. FNDGM | vs. FDGM | vs. FBernoulliGM | vs. NIPGM |
RMSE | 66.8494 | 64.5630 | 60.1532 | 68.1651 | 75.2826 | 78.7787 | 59.0822 |
MAE | 71.3875 | 58.9334 | 65.3859 | 72.5196 | 78.5282 | 74.8535 | 52.9620 |
MAPE | 72.2804 | 57.0973 | 66.6266 | 73.2983 | 78.9205 | 73.4678 | 51.0122 |
TIC | 72.7439 | 60.6075 | 66.0292 | 74.0691 | 81.1451 | 74.3007 | 55.3960 |
U1 | 72.7439 | 60.6075 | 66.0292 | 74.0691 | 81.1451 | 74.3007 | 55.3960 |
U2 | 66.8494 | 64.5630 | 60.1532 | 68.1651 | 75.2826 | 78.7787 | 59.0822 |
Model | vs. NIPNGM | vs. NIPDGM | vs. NIPNDGM | vs. NIPBernoulliGM | |||
RMSE | 28.3814 | 91.8706 | 99.0060 | 82.6505 | |||
MAE | 24.8486 | 90.5145 | 98.6729 | 78.9909 | |||
MAPE | 23.8675 | 89.9346 | 98.5633 | 77.7223 | |||
TIC | 25.3015 | 86.7980 | 93.4915 | 78.0911 | |||
U1 | 25.3015 | 86.7980 | 93.4915 | 78.0911 | |||
U2 | 28.3814 | 91.8706 | 99.0060 | 82.6505 |
LSTM Layer Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
---|---|---|---|---|---|---|---|---|---|---|---|
Case 1 | RMSE | Train | 937.9942 | 851.0898 | 486.6038 | 844.6278 | 963.4058 | 3109.3818 | 10,351.4960 | 11,148.5194 | 11,152.6367 |
Test | 5132.2100 | 3611.1851 | 1627.7841 | 3030.8841 | 5331.7715 | 19,477.6190 | 64,645.9305 | 68,870.8128 | 68,889.6010 | ||
MAE | Train | 625.4415 | 521.0091 | 288.6944 | 547.3042 | 546.0693 | 1982.2757 | 6818.2956 | 7408.2859 | 7411.4835 | |
Test | 4841.1397 | 3463.4200 | 1190.5096 | 2618.6103 | 5128.7763 | 18,726.4847 | 62,154.8233 | 66,233.5773 | 66,251.8598 | ||
MAPE | Train | 3.1277 | 2.4315 | 1.3231 | 2.6728 | 2.4374 | 10.8775 | 39.6334 | 42.5158 | 42.5315 | |
Test | 22.1315 | 15.7899 | 5.6178 | 12.0999 | 23.3665 | 85.2647 | 282.9392 | 301.4895 | 301.5725 | ||
TIC | Train | 0.0254 | 0.0231 | 0.0132 | 0.0231 | 0.0260 | 0.0724 | 0.2276 | 0.2411 | 0.2412 | |
Test | 0.1027 | 0.0747 | 0.0354 | 0.0635 | 0.1066 | 0.2361 | 0.5946 | 0.6105 | 0.6106 | ||
U1 | Train | 0.0254 | 0.0231 | 0.0132 | 0.0231 | 0.0260 | 0.0724 | 0.2276 | 0.2411 | 0.2412 | |
Test | 0.1027 | 0.0747 | 0.0354 | 0.0635 | 0.1066 | 0.2361 | 0.5946 | 0.6105 | 0.6106 | ||
U2 | Train | 0.0510 | 0.0462 | 0.0264 | 0.0459 | 0.0523 | 0.1689 | 0.5624 | 0.6057 | 0.6060 | |
Test | 0.2288 | 0.1610 | 0.0726 | 0.1351 | 0.2377 | 0.8682 | 2.8815 | 3.0698 | 3.0707 | ||
Case 2 | RMSE | Train | 101.5587 | 94.1466 | 85.9299 | 118.1945 | 65.8725 | 343.2804 | 861.3843 | 1104.3757 | 1103.5302 |
Test | 317.5203 | 192.0910 | 149.4493 | 360.5283 | 177.3545 | 2044.3780 | 5160.0670 | 6546.0611 | 6542.0924 | ||
MAE | Train | 76.2756 | 69.4719 | 65.2312 | 85.1072 | 47.0495 | 202.2991 | 549.3752 | 699.6323 | 699.0246 | |
Test | 286.1201 | 168.4210 | 128.8550 | 314.3367 | 148.3387 | 1973.5665 | 4974.8283 | 6322.1106 | 6318.2444 | ||
MAPE | Train | 4.1573 | 4.0826 | 3.7143 | 4.5947 | 2.5382 | 10.6773 | 30.1723 | 36.3884 | 36.3614 | |
Test | 8.0050 | 4.6901 | 3.6203 | 8.6849 | 4.1085 | 55.1810 | 138.8855 | 176.5324 | 176.4243 | ||
TIC | Train | 0.0265 | 0.0245 | 0.0225 | 0.0313 | 0.0169 | 0.0807 | 0.1833 | 0.2274 | 0.2272 | |
Test | 0.0429 | 0.0276 | 0.0213 | 0.0493 | 0.0249 | 0.2058 | 0.4046 | 0.4814 | 0.4813 | ||
U1 | Train | 0.0265 | 0.0245 | 0.0225 | 0.0313 | 0.0169 | 0.0807 | 0.1833 | 0.2274 | 0.2272 | |
Test | 0.0429 | 0.0276 | 0.0213 | 0.0493 | 0.0249 | 0.2058 | 0.4046 | 0.4814 | 0.4813 | ||
U2 | Train | 0.0523 | 0.0485 | 0.0442 | 0.0608 | 0.0339 | 0.1767 | 0.4434 | 0.5685 | 0.5681 | |
Test | 0.0894 | 0.0541 | 0.0421 | 0.1015 | 0.0499 | 0.5756 | 1.4528 | 1.8430 | 1.8419 | ||
Case 3 | RMSE | Train | 38.5742 | 38.1353 | 31.8485 | 38.4768 | 38.4039 | 38.7443 | 38.8287 | 38.8505 | 38.6034 |
Test | 208.8081 | 235.0953 | 206.5127 | 207.9963 | 206.1128 | 210.7499 | 211.8525 | 212.1576 | 208.7974 | ||
MAE | Train | 30.9613 | 30.4737 | 25.0073 | 30.9819 | 30.8845 | 31.1706 | 31.2248 | 31.2383 | 31.0690 | |
Test | 152.8418 | 171.6049 | 150.9447 | 152.2497 | 150.8776 | 154.2628 | 155.0598 | 155.2788 | 152.8170 | ||
MAPE | Train | 18.5792 | 18.1695 | 14.8266 | 18.5924 | 18.5317 | 18.7202 | 18.7552 | 18.7637 | 18.6545 | |
Test | 21.2923 | 23.8549 | 21.0333 | 21.2110 | 21.0236 | 21.4869 | 21.5960 | 21.6260 | 21.2890 | ||
TIC | Train | 0.0867 | 0.0859 | 0.0720 | 0.0866 | 0.0864 | 0.0870 | 0.0872 | 0.0872 | 0.0868 | |
Test | 0.1395 | 0.1535 | 0.1381 | 0.1391 | 0.1380 | 0.1406 | 0.1412 | 0.1414 | 0.1395 | ||
U1 | Train | 0.0867 | 0.0859 | 0.0720 | 0.0866 | 0.0864 | 0.0870 | 0.0872 | 0.0872 | 0.0868 | |
Test | 0.1395 | 0.1535 | 0.1381 | 0.1391 | 0.1380 | 0.1406 | 0.1412 | 0.1414 | 0.1395 | ||
U2 | Train | 0.1793 | 0.1772 | 0.1480 | 0.1788 | 0.1785 | 0.1800 | 0.1804 | 0.1805 | 0.1794 | |
Test | 0.3101 | 0.3492 | 0.3067 | 0.3089 | 0.3061 | 0.3130 | 0.3147 | 0.3151 | 0.3101 |
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Hao, Y.; Ma, X. A Hybrid Grey System Model Based on Stacked Long Short-Term Memory Layers and Its Application in Energy Consumption Forecasting. Processes 2024, 12, 1749. https://doi.org/10.3390/pr12081749
Hao Y, Ma X. A Hybrid Grey System Model Based on Stacked Long Short-Term Memory Layers and Its Application in Energy Consumption Forecasting. Processes. 2024; 12(8):1749. https://doi.org/10.3390/pr12081749
Chicago/Turabian StyleHao, Yiwu, and Xin Ma. 2024. "A Hybrid Grey System Model Based on Stacked Long Short-Term Memory Layers and Its Application in Energy Consumption Forecasting" Processes 12, no. 8: 1749. https://doi.org/10.3390/pr12081749