Hierarchical Meta-Learning in Time Series Forecasting for Improved Interference-Less Machine Learning
Abstract
:1. Introduction
2. Background
2.1. Motivation for Noise Elimination in Time Series Data
2.2. NAR and NARX
2.3. Moving Average
2.4. Empirical Mode Decomposition
- STEP 1:
- Let hi(t) = y(t), and i = 1.
- STEP 2:
- Find some local minima and maxima in hi(t).
- STEP 3:
- Connect all identified maxima and minima by a cubic spline as the upper envelope upi(t) and lower envelope lowi(t), and calculate local mean as mi(t) = [upi(t) + lowi(t)]/2.
- STEP 4:
- Update as hi(t) = hi(t) − mi(t).
- STEP 5:
- Ensure hi(t) fulfils the requirement of IMF. If not, then redo STEP 2 to STEP 5. If done, then IMFi(t) = hi(t), i = i + 1 and hi(t) = y(t) − IMFi−1(t).
- STEP 6:
- Check if hi(t) is a monotonic function or not. If it is not then redo STEP 2 to STEP 5 for the next IMF. If it is monotonic then hi(t) = rc(t) and end the EMD process.
3. Method
3.1. Dataset and Sample Selection
3.2. Algorithms and Learning process
- (1)
- The traditional approach, in which the time series is directly used by the NAR neural network for training. We denote this as NAR.
- (2)
- The NARX neural network is presented with the original time series as input, and an additional pre-processed meta-information from the moving average technique. Denoted as NARX-MA.
- (3)
- Finally, the NARX neural network inputs contain both the meta-information output of the EMD de-noising algorithm (see Section 3.4 for definition) and the original time series. Referred to as NARX-EMDv2 in this paper.
3.3. NARX-MA Algorithm (Moving Average)
3.4. NARX-EMDv2 Algorithm (EMD de-Noising Version 2)
- STEP 1:
- Run the EMD routine on the time series data using normal parameters in [13]. Let N be the total number of IMFs produced and set the number of neural network hidden neuron to 75% of N (rounded up). [The parameters used are: resolution (qResol) = 40 dB, residual energy (qResid) = 40 dB, and gradient step size (qAlfa) = 1. The tolerance level is determined by resolution (qResol) which terminates the IMF computation [13].]
- STEP 2:
- Based on use-case, set the input and feedback delays [i.e., number of past values presented to the artificial neural network model for prediction].
- STEP 3:
- Create N combinations of all IMF whilst progressively excluding higher frequency IMF in each combination.
- STEP 4:
- Create a modified input series x(t) from the remaining IMF in each combination which will either be summed [as a single input] or included directly [as multiple inputs] into the input [of the artificial neural network].
- STEP 5:
- Train the NARX model with each combination and test the performance to reveal a combination that contributes to noise reduction.
- STEP 6:
- If noise reduction requirement is not met, decrease N and repeat from STEP 3.
Algorithm 1: EMDdenoiseV2 |
Input: Time Series Data, x; Heuristic Data Division Value, d. |
Output: Denoised meta_info; |
1 sequentially divide x into d approximately equal partitions |
2 for each partition p to dth |
3 | run EMD routine |
4 | initialize sub-NARX ANN and train by exempting one IMF at a time |
5 | identify IMF elimination that contributes to lowest MSE |
6 | sum valid IMF to create meta_info for pth partition |
7 end |
8 append meta_info from each partition sequentially to form series length equal to x. |
9 return meta_info |
4. Results
5. Discussions
5.1. Performance on Non-Linear Autoregressive Neural Networks
5.2. Performance on Long Short-Term Memory Neural Network
5.3. Potential Application
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | LSTM | LSTM-MA | LSTM-EMDv2 | LSTM | LSTM-MA | LSTM-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 8.588 | 24.525 | 8.019 | 8.541 | 8.462 | 8.093 | 8.548 | 19.212 | 7.979 | 7.945 | 8.611 | 8.008 |
Avg. Training Error | 0.1102562 | 0.0414285 | 0.1166426 | 0.0454050 | 0.1012255 | 0.1214790 | 0.0533478 | 0.0355548 | 0.0640460 | 0.0447802 | 0.0521724 | 0.0941153 |
Avg. Test Error ± standard deviation | 0.1485295 ±0.0504588 | 0.0975778 ±0.0262894 | 0.1430151 ±0.0508515 | 0.0971847 ±0.0254819 | 0.1389314 ±0.0397565 | 0.2434674 ±0.0399610 | 0.0606585 ±0.0320497 | 0.0277204 ±0.0077224 | 0.0666034 ±0.0353700 | 0.0272315 ±0.0099083 | 0.0573101 ±0.0283412 | 0.0905659 ±0.0267799 |
Minimum Test Error | 0.0456379 | 0.0314639 | 0.0302435 | 0.0458273 | 0.0450077 | 0.1499620 | 0.0029272 | 0.0112347 | 0.0017439 | 0.0054977 | 0.0038684 | 0.0398468 |
p-value vs. best | 4.20 × 10−59 | 6.08 × 10−63 | 9.90 × 10−56 | 2.32 × 10−64 | 1.32 × 10−68 | 8.59 × 10−118 | 4.54 × 10−19 | 6.99 × 10−01 | 2.83 × 10−21 | best | 3.17 × 10−19 | 2.88 × 10−55 |
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | LSTM | LSTM-MA | LSTM-EMDv2 | LSTM | LSTM-MA | LSTM-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 28.97 | 14.047 | 26.731 | 9.322 | 11.165 | 14.471 | 25.122 | 14.363 | 24.434 | 14.118 | 10.443 | 14.044 |
Avg. Training Error | 0.1328087 | 0.1319767 | 0.1446349 | 0.1762698 | 0.1340337 | 0.1248077 | 0.1173348 | 0.1148004 | 0.1161796 | 0.1103128 | 0.0902734 | 0.0916366 |
Avg. Test Error ± standard deviation | 0.2399828 ±0.0261120 | 0.2326393 ±0.0289071 | 0.2931989 ±0.0783809 | 0.3847042 ±0.1012203 | 0.2736516 ±0.0599508 | 0.2270046 ±0.0092213 | 0.1970842 ±0.0038476 | 0.1904281 ±0.0024817 | 0.1925080 ±0.0089283 | 0.1775317 ±0.0041043 | 0.1476351 ±0.0187093 | 0.2451787 ±0.0128595 |
Minimum Test Error | 0.2170601 | 0.2168433 | 0.2207024 | 0.2125954 | 0.2065012 | 0.2056850 | 0.1874098 | 0.1834852 | 0.1709400 | 0.1698466 | 0.0876853 | 0.2181468 |
p-value vs. best | 3.01 × 10−72 | 5.09 × 10−62 | 1.80 × 10−43 | 1.35 × 10−57 | 2.61 × 10−49 | 1.28 × 10−92 | 3.88 × 10−65 | 1.27 × 10−56 | 8.78 × 10−54 | 4.26 × 10−36 | best | 6.25 × 10−102 |
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | LSTM | LSTM-MA | LSTM-EMDv2 | LSTM | LSTM-MA | LSTM-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 8.611 | 14.114 | 9.332 | 13.055 | 14.9695 | 11.737 | 9.95 | 14.079 | 9.667 | 11.7 | 8.717 | 14.801 |
Avg. Training Error | 0.1801327 | 0.1110482 | 0.2093012 | 0.1134647 | 0.3633793 | 0.1815522 | 0.1286195 | 0.0688271 | 0.2099519 | 0.1354106 | 0.2999538 | 0.1037100 |
Avg. Test Error ± standard deviation | 0.0559024 ±0.0109374 | 0.0320430 ±0.0024095 | 0.0852827 ±0.0300794 | 0.0439690 ±0.0263654 | 0.2403362 ±0.0383493 | 0.0501263 ±0.0092680 | 0.0461984 ±0.0277388 | 0.0215419 ±0.0157497 | 0.0846316 ±0.0376178 | 0.0512063 ±0.0184436 | 0.1679626 ±0.0297202 | 0.0413049 ±0.0068129 |
Minimum Test Error | 0.0320886 | 0.0275861 | 0.0280625 | 0.0183325 | 0.1163166 | 0.0346829 | 0.0151599 | 0.0062204 | 0.0420030 | 0.0337207 | 0.0699817 | 0.0239597 |
p-value vs. best | 4.82 × 10−43 | 4.65 × 10−10 | 1.47 × 10−45 | 8.34 × 10−12 | 3.52 × 10−118 | 3.37 × 10−36 | 6.70 × 10−13 | best | 1.12 × 10−35 | 8.37 × 10−26 | 6.01 × 10−103 | 1.19 × 10−23 |
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | LSTM | LSTM-MA | LSTM-EMDv2 | LSTM | LSTM-MA | LSTM-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 10.196 | 9.938 | 14.837 | 9.787 | 10.333 | 11.108 | 9.47 | 9.876 | 9.626 | 9.654 | 9.733 | 10.293 |
Avg. Training Error | 1.2221676 | 1.1722678 | 1.1575247 | 1.1340869 | 1.0323971 | 1.1280384 | 1.1382452 | 1.1299286 | 1.1250142 | 1.1185877 | 0.9930821 | 1.0977961 |
Avg. Test Error ± standard deviation | 0.8087027 ±0.0319960 | 0.7857503 ±0.0253995 | 0.7791352 ±0.0138709 | 0.7713591 ±0.0026795 | 0.6444975 ±0.0672950 | 0.7715180 ±0.0032700 | 0.7830149 ±0.0070671 | 0.7808031 ±0.0080943 | 0.6521218 ±0.0564859 | 0.7650546 ±0.0046042 | 0.6521219 ±0.0564861 | 0.7560338 ±0.0085550 |
Minimum Test Error | 0.7701950 | 0.7690272 | 0.7541847 | 0.7637071 | 0.4343486 | 0.7632934 | 0.7688600 | 0.7757211 | 0.4754810 | 0.7484215 | 0.4754810 | 0.7414659 |
p-value vs. best | 7.19 × 10−55 | 4.45 × 10−48 | 5.92 × 10−48 | 9.55 × 10−46 | best | 8.55 × 10−46 | 1.81 × 10−50 | 1.95 × 10−49 | 3.89 × 10−01 | 6.62 × 10−43 | 3.89 × 10−01 | 1.27 × 10−38 |
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Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | NAR | NARX-MA | NARX-EMDv2 | NAR | NARX-MA | NARX-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 2.113 | 1.551 | 2.766 | 3.493 | 3.041 | 4.269 | 2.456 | 2.288 | 3.157 | 5.394 | 3.224 | 7.332 |
Avg. Training Error | 0.0071252 | 0.0070175 | 0.0070970 | 0.0069618 | 0.0042749 | 0.0008638 | 0.0070956 | 0.0069004 | 0.0070665 | 0.0068903 | 0.0033542 | 0.0010875 |
Avg. Test Error ± standard deviation | 0.0016678 ±0.0009144 | 0.0037132 ±0.0011296 | 0.0017700 ±0.0010237 | 0.0040955 ±0.0010815 | 0.0007602 ±0.0011614 | 0.0004631 ±0.0003893 | 0.0013870 ±0.0010336 | 0.0042773 ±0.0021347 | 0.0014124 ±0.0013610 | 0.0046522 ±0.0020343 | 0.0007500 ±0.0013131 | 0.0008448 ±0.0015741 |
Minimum Test Error | 0.0003573 | 0.0009636 | 0.0004469 | 0.0022597 | 0.0000508 | 0.0000798 | 0.0003758 | 0.0015388 | 0.0004493 | 0.0019610 | 0.0000438 | 0.0000701 |
p-value vs. best | 1.81 × 10−25 | 1.87 × 10−68 | 6.71 × 10−25 | 6.34 × 10−79 | 1.67 × 10−02 | best | 1.39 × 10−14 | 5.01 × 10−42 | 2.46 × 10−10 | 9.11 × 10−50 | 3.84 × 10−02 | 2.02 × 10−02 |
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | NAR | NARX-MA | NARX-EMDv2 | NAR | NARX-MA | NARX-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 2.255 | 1.624 | 2.963 | 3.891 | 2.957 | 3.922 | 2.237 | 2.3 | 2.76 | 5.58 | 2.959 | 5.985 |
Avg. Training Error | 0.1116833 | 0.1005595 | 0.1087313 | 0.0949837 | 0.0394043 | 0.0558120 | 0.1085390 | 0.0960426 | 0.1074148 | 0.0879532 | 0.0398371 | 0.0561283 |
Avg. Test Error ± standard deviation | 0.3913484 ±0.7046393 | 0.8481608 ±0.4306742 | 0.3934549 ±1.1453019 | 0.8978903 ±1.1651971 | 0.0974777 ±0.0870907 | 0.6780297 ±0.5060787 | 0.2442256 ±0.0750881 | 0.6417264 ±0.4650035 | 0.2905302 ±0.1672015 | 0.6722254 ±0.6007605 | 0.0980438 ±0.0369733 | 0.6016824 ±0.4468190 |
Minimum Test Error | 0.1883577 | 0.2696093 | 0.1931820 | 0.2058434 | 0.0553965 | 0.1643110 | 0.1886322 | 0.2116681 | 0.1804997 | 0.1979736 | 0.0609981 | 0.1521699 |
p-value vs. best | 5.60 × 10−05 | 1.49 × 10−40 | 1.11 × 10−02 | 1.10 × 10−10 | best | 5.12 × 10−23 | 2.05 × 10−27 | 1.30 × 10−23 | 7.19 × 10−20 | 1.20 × 10−17 | 9.53 × 10−01 | 2.48 × 10−22 |
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | NAR | NARX-MA | NARX-EMDv2 | NAR | NARX-MA | NARX-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 1.622 | 0.993 | 1.720 | 3.449 | 1.210 | 4.019 | 1.546 | 2.366 | 2.482 | 4.959 | 1.491 | 4.194 |
Avg. Training Error | 0.0132433 | 0.0225547 | 0.0237647 | 0.0017928 | 0.0666883 | 0.0121078 | 0.0106071 | 0.0004086 | 0.0217637 | 0.0004789 | 0.0491124 | 0.0090752 |
Avg. Test Error ± standard deviation | 0.0043834 ±0.0140755 | 0.0037581 ±0.0022762 | 0.0115036 ±0.0677390 | 0.0009243 ±0.0018475 | 0.0298414 ±0.0555601 | 0.0083077 ±0.0398612 | 0.0041026 ±0.0108263 | 0.0004567 ±0.0001875 | 0.0052185 ±0.0094118 | 0.0004919 ±0.0003914 | 0.0264680 ±0.0613395 | 0.0057052 ±0.0298941 |
Minimum Test Error | 0.0007238 | 0.0017372 | 0.0004752 | 0.0003155 | 0.0020193 | 0.0008517 | 0.0003111 | 0.0002870 | 0.0002730 | 0.0002746 | 0.0006813 | 0.0003910 |
p-value vs. best | 6.04 × 10−03 | 1.39 × 10−32 | 1.06 × 10−01 | 1.30 × 10−02 | 3.68 × 10−07 | 5.14 × 10−02 | 9.66 × 10−04 | best | 1.08 × 10−06 | 4.21 × 10−01 | 3.73 × 10−05 | 8.22 × 10−02 |
Delay Window | lag: 5 | lag: 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Algorithm | NAR | NARX-MA | NARX-EMDv2 | NAR | NARX-MA | NARX-EMDv2 | ||||||
Division/ANN neurons | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 | 10 | 50 |
Avg. Training Time (s) | 0.616 | 0.451 | 0.637 | 0.797 | 0.682 | 0.748 | 0.591 | 0.667 | 0.640 | 1.077 | 0.591 | 1.032 |
Avg. Training Error | 1.0466828 | 0.8795638 | 1.0366650 | 0.8866556 | 0.5577968 | 0.7428110 | 1.0480104 | 0.8458651 | 1.0099425 | 0.8149923 | 0.6001350 | 0.8643662 |
Avg. Test Error ± standard deviation | 0.8579863 ±0.0516803 | 0.9902402 ±0.1252702 | 0.8677769 ±0.0810234 | 0.9540938 ±0.1115357 | 0.4721791 ±0.1063513 | 1.0481726 ±0.2227919 | 0.7809116 ±0.0359027 | 0.9983421 ±0.1287332 | 0.7878874 ±0.0559552 | 0.8587937 ±0.0783127 | 0.4165572 ±0.0513170 | 0.8576149 ±0.0742370 |
Minimum Test Error | 0.7707221 | 0.8026880 | 0.7634348 | 0.8174138 | 0.3335602 | 0.7699630 | 0.7152155 | 0.7854705 | 0.6834771 | 0.7091668 | 0.3318121 | 0.7032872 |
p-value vs. best | 2.20 × 10−129 | 7.32 × 10−101 | 4.92 × 10−109 | 2.15 × 10−103 | 5.16 × 10−06 | 1.64 × 10−69 | 4.78 × 10−126 | 3.92 × 10−100 | 4.19 × 10−112 | 2.41 × 10−109 | best | 4.79 × 10−112 |
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Share and Cite
Afolabi, D.; Guan, S.-U.; Man, K.L.; Wong, P.W.H.; Zhao, X. Hierarchical Meta-Learning in Time Series Forecasting for Improved Interference-Less Machine Learning. Symmetry 2017, 9, 283. https://doi.org/10.3390/sym9110283
Afolabi D, Guan S-U, Man KL, Wong PWH, Zhao X. Hierarchical Meta-Learning in Time Series Forecasting for Improved Interference-Less Machine Learning. Symmetry. 2017; 9(11):283. https://doi.org/10.3390/sym9110283
Chicago/Turabian StyleAfolabi, David, Sheng-Uei Guan, Ka Lok Man, Prudence W. H. Wong, and Xuan Zhao. 2017. "Hierarchical Meta-Learning in Time Series Forecasting for Improved Interference-Less Machine Learning" Symmetry 9, no. 11: 283. https://doi.org/10.3390/sym9110283
APA StyleAfolabi, D., Guan, S. -U., Man, K. L., Wong, P. W. H., & Zhao, X. (2017). Hierarchical Meta-Learning in Time Series Forecasting for Improved Interference-Less Machine Learning. Symmetry, 9(11), 283. https://doi.org/10.3390/sym9110283