Gene Expression Programming Coupled with Unsupervised Learning: A Two-Stage Learning Process in Multi-Scale, Short-Term Water Demand Forecasts
Abstract
:1. Introduction
- Evaluation of GEP as an alternative to other black-box models used in the literature that have not been explored by other researchers in the field of short-term water demand forecasting
- Investigation of coupling time series clustering with GEP in short-term water demand forecasts to reduce the adverse effect of seasonality and holidays/working days on performance of proposed forecast models
- Proposing a suitable sampling frequency for WDS operators through a time-scale modeling process
2. Model Development
2.1. Unsupervised Learning: K-Means Clusters
2.2. Average Mutual Information
2.3. Gene Expression Programing
- Creating a random initial population of chromosomes
- Expressing chromosomes in a tree diagram with subsets
- Comparing the new offspring solutions based on fitness criteria
- Keeping the best solution, followed by reproduction methods like replication, mutation, recombination, etc.
3. Study Area and Data Collection
- 149,639 junctions
- 118,950 pipes
- 26 pumping stations
- 501 wells and well pumps
- 33 storage tanks
- 95 booster pumps
- 36,295 valves
- 602 check valves
- Total base demand 7.5 ± 4.2 (m3/s)
4. Results
5. Conclusions
Supplementary Materials
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
Model ID | Training | Test | Model ID | Training | Test | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
MAE | RMSE | R2 | MAE | RMSE | R2 | MAE | RMSE | R2 | MAE | RMSE | R2 | ||
K1HT1OP1 | 0.27 | 0.36 | 0.88 | 0.27 | 0.33 | 0.91 | K4HT1OP1 | 0.20 | 0.29 | 0.92 | 0.25 | 0.35 | 0.84 |
K1HT1OP2 | 0.31 | 0.42 | 0.84 | 0.29 | 0.38 | 0.86 | K4HT1OP2 | 0.23 | 0.32 | 0.90 | 0.25 | 0.36 | 0.82 |
K1HT1OP3 | 0.35 | 0.52 | 0.75 | 0.32 | 0.48 | 0.78 | K4HT1OP3 | 0.31 | 0.41 | 0.84 | 0.25 | 0.31 | 0.90 |
K1HT3OP1 | 0.50 | 0.64 | 0.64 | 0.53 | 0.40 | 0.65 | K4HT3OP1 | 0.47 | 0.55 | 0.70 | 1.63 | 1.48 | 0.38 |
K1HT3OP2 | 0.54 | 0.68 | 0.67 | 0.55 | 0.72 | 0.62 | K4HT3OP2 | 0.61 | 0.73 | 0.49 | 1.52 | 1.53 | 0.32 |
K1HT3OP3 | 0.41 | 0.51 | 0.74 | 0.44 | 0.52 | 0.75 | K4HT3OP3 | 0.46 | 0.56 | 0.69 | 1.01 | 1.41 | 0.24 |
K1HT6OP1 | 0.23 | 0.30 | 0.91 | 0.29 | 0.36 | 0.89 | K4HT6OP1 | 0.42 | 0.49 | 0.76 | 1.28 | 1.75 | 0.13 |
K1HT6OP2 | 0.32 | 0.40 | 0.84 | 0.28 | 0.33 | 0.91 | K4HT6OP2 | 0.38 | 0.46 | 0.81 | 0.58 | 0.76 | 0.28 |
K1HT6OP3 | 0.17 | 0.23 | 0.95 | 0.22 | 0.28 | 0.93 | K4HT6OP3 | 0.27 | 0.35 | 0.88 | 1.01 | 1.28 | 0.07 |
K1HT12OP1 | 0.30 | 0.42 | 0.82 | 0.22 | 0.33 | 0.88 | K4HT12OP1 | 0.30 | 0.44 | 0.80 | 1.02 | 1.18 | 0.09 |
K1HT12OP2 | 0.34 | 0.49 | 0.77 | 0.29 | 0.38 | 0.84 | K4HT12OP2 | 0.35 | 0.48 | 0.76 | 0.56 | 0.62 | 0.55 |
K1HT12OP3 | 0.31 | 0.45 | 0.79 | 0.28 | 0.37 | 0.84 | K4HT12OP3 | 0.35 | 0.50 | 0.75 | 0.83 | 1.03 | 0.15 |
K1HT24OP1 | 0.54 | 0.73 | 0.47 | 0.55 | 0.66 | 0.38 | K4HT24OP1 | 0.40 | 0.52 | 0.72 | 1.12 | 1.82 | 0.26 |
K1HT24OP2 | 0.51 | 0.72 | 0.48 | 0.54 | 0.66 | 0.25 | K4HT24OP2 | 0.41 | 0.48 | 0.78 | 1.35 | 1.88 | 0.14 |
K1HT24OP3 | 0.56 | 0.75 | 0.43 | 0.71 | 0.54 | 0.11 | K4HT24OP3 | 0.36 | 0.44 | 0.80 | 1.30 | 1.88 | 0.27 |
K2HT1OP1 | 0.20 | 0.29 | 0.93 | 0.22 | 0.27 | 0.94 | K5HT1OP1 | 0.28 | 0.35 | 0.89 | 0.23 | 0.28 | 0.90 |
K2HT1OP2 | 0.18 | 0.27 | 0.93 | 0.19 | 0.24 | 0.95 | K5HT1OP2 | 0.24 | 0.31 | 0.91 | 0.20 | 0.24 | 0.92 |
K2HT1OP3 | 0.22 | 0.30 | 0.91 | 0.22 | 0.28 | 0.93 | K5HT1OP3 | 0.22 | 0.30 | 0.92 | 0.22 | 0.28 | 0.92 |
K2HT3OP1 | 0.70 | 0.82 | 0.32 | 0.77 | 0.87 | 0.30 | K5HT3OP1 | 0.71 | 0.83 | 0.30 | 0.62 | 0.74 | 0.25 |
K2HT3OP2 | 0.69 | 0.81 | 0.34 | 0.74 | 0.86 | 0.33 | K5HT3OP2 | 0.71 | 0.84 | 0.32 | 0.63 | 0.75 | 0.26 |
K2HT3OP3 | 0.63 | 0.75 | 0.43 | 0.67 | 0.79 | 0.43 | K5HT3OP3 | 0.71 | 0.83 | 0.30 | 0.62 | 0.73 | 0.27 |
K2HT6OP1 | 0.43 | 0.51 | 0.74 | 0.45 | 0.55 | 0.72 | K5HT6OP1 | 0.49 | 0.60 | 0.64 | 0.52 | 0.59 | 0.63 |
K2HT6OP2 | 0.34 | 0.45 | 0.80 | 0.35 | 0.43 | 0.87 | K5HT6OP2 | 0.45 | 0.55 | 0.71 | 0.44 | 0.51 | 0.66 |
K2HT6OP3 | 0.36 | 0.45 | 0.80 | 0.34 | 0.42 | 0.89 | K5HT6OP3 | 0.34 | 0.48 | 0.77 | 0.34 | 0.46 | 0.80 |
K2HT12OP1 | 0.40 | 0.53 | 0.72 | 0.34 | 0.45 | 0.78 | K5HT12OP1 | 0.70 | 0.78 | 0.35 | 0.64 | 0.80 | 0.35 |
K2HT12OP2 | 0.43 | 0.56 | 0.69 | 0.45 | 0.54 | 0.80 | K5HT12OP2 | 0.70 | 0.78 | 0.35 | 0.64 | 0.80 | 0.35 |
K2HT12OP3 | 0.45 | 0.59 | 0.65 | 0.40 | 0.46 | 0.81 | K5HT12OP3 | 0.68 | 0.72 | 0.38 | 0.58 | 0.75 | 0.44 |
K2HT24OP1 | 0.57 | 0.82 | 0.34 | 0.50 | 0.63 | 0.13 | K5HT24OP1 | 0.63 | 0.81 | 0.33 | 0.81 | 0.95 | 0.08 |
K2HT24OP2 | 0.58 | 0.80 | 0.34 | 0.41 | 0.52 | 0.12 | K5HT24OP2 | 0.64 | 0.83 | 0.29 | 0.78 | 1.05 | 0.06 |
K2HT24OP3 | 0.58 | 0.80 | 0.34 | 0.41 | 0.52 | 0.13 | K5HT24OP3 | 0.67 | 0.79 | 0.36 | 0.80 | 1.05 | 0.08 |
K3HT1OP1 | 0.19 | 0.24 | 0.95 | 0.18 | 0.23 | 0.93 | K6HT1OP1 | 0.33 | 0.43 | 0.81 | 0.30 | 0.38 | 0.82 |
K3HT1OP2 | 0.21 | 0.28 | 0.93 | 0.21 | 0.27 | 0.92 | K6HT1OP2 | 0.29 | 0.39 | 0.85 | 0.25 | 0.34 | 0.86 |
K3HT1OP3 | 0.19 | 0.25 | 0.94 | 0.20 | 0.25 | 0.93 | K6HT1OP3 | 0.26 | 0.36 | 0.88 | 0.22 | 0.28 | 0.91 |
K3HT3OP1 | 0.60 | 0.73 | 0.46 | 0.55 | 0.67 | 0.45 | K6HT3OP1 | 0.42 | 0.57 | 0.67 | 0.41 | 0.55 | 0.64 |
K3HT3OP2 | 0.60 | 0.75 | 0.46 | 0.56 | 0.69 | 0.45 | K6HT3OP2 | 0.45 | 0.58 | 0.70 | 0.48 | 0.60 | 0.63 |
K3HT3OP3 | 0.47 | 0.56 | 0.74 | 0.43 | 0.53 | 0.73 | K6HT3OP3 | 0.45 | 0.59 | 0.66 | 0.46 | 0.59 | 0.61 |
K3HT6OP1 | 0.71 | 0.87 | 0.33 | 0.62 | 0.87 | 0.52 | K6HT6OP1 | 0.33 | 0.43 | 0.81 | 0.40 | 0.52 | 0.84 |
K3HT6OP2 | 0.58 | 0.69 | 0.54 | 0.50 | 0.62 | 0.58 | K6HT6OP2 | 0.45 | 0.55 | 0.70 | 0.47 | 0.51 | 0.77 |
K3HT6OP3 | 0.58 | 0.73 | 0.50 | 0.61 | 0.78 | 0.35 | K6HT6OP3 | 0.38 | 0.48 | 0.79 | 0.40 | 0.45 | 0.89 |
K3HT12OP1 | 0.48 | 0.59 | 0.65 | 0.36 | 0.49 | 0.24 | K6HT12OP1 | 0.68 | 0.92 | 0.14 | 0.66 | 0.89 | 0.19 |
K3HT12OP2 | 0.41 | 0.50 | 0.75 | 0.38 | 0.47 | 0.37 | K6HT12OP2 | 0.73 | 0.94 | 0.12 | 0.73 | 0.92 | 0.17 |
K3HT12OP3 | 0.40 | 0.50 | 0.75 | 0.35 | 0.47 | 0.27 | K6HT12OP3 | 0.65 | 0.89 | 0.22 | 0.73 | 0.95 | 0.12 |
K3HT24OP1 | 0.42 | 0.52 | 0.73 | 0.46 | 0.54 | 0.37 | K6HT24OP1 | 0.30 | 0.44 | 0.80 | 0.26 | 0.36 | 0.68 |
K3HT24OP2 | 0.53 | 0.63 | 0.63 | 0.40 | 0.56 | 0.10 | K6HT24OP2 | 0.32 | 0.46 | 0.78 | 0.26 | 0.37 | 0.67 |
K3HT24OP3 | 0.42 | 0.52 | 0.72 | 0.37 | 0.51 | 0.37 | K6HT24OP3 | 0.31 | 0.45 | 0.80 | 0.28 | 0.39 | 0.67 |
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(a). MAE * (Mean ± Standard Deviation) | |||||
Operator | Head Time = 1 | Head Time = 3 | Head Time = 6 | Head Time = 12 | Head Time = 24 |
GEP-operator_1 | 0.2409 ± 0.0388 | 0.7497 ± 0.4449 | 0.5943 ± 0.3548 | 0.5399 ± 0.2932 | 0.6163 ± 0.3039 |
GEP-operator_2 | 0.2319 ± 0.0391 | 0.7448 ± 0.3885 | 0.4386 ± 0.1096 | 0.5061 ± 0.1660 | 0.6242 ± 0.3952 |
GEP-operator_3 | 0.2387 ± 0.0444 | 0.6031 ± 0.2228 | 0.4854 ± 0.2857 | 0.5276 ± 0.2211 | 0.6433 ± 0.3808 |
(b). RMSE * (Mean ± Standard Deviation) | |||||
Operator | Head Time = 1 | Head Time = 3 | Head Time = 6 | Head Time = 12 | Head Time = 24 |
GEP-operator_1 | 0.3087 ± 0.0563 | 0.7861 ± 0.3763 | 0.7731 ± 0.5048 | 0.6896 ± 0.3230 | 0.8272 ± 0.5226 |
GEP-operator_2 | 0.3048 ± 0.0632 | 0.8595 ± 0.3398 | 0.5274 ± 0.1483 | 0.6215 ± 0.2052 | 0.8401 ± 0.5591 |
GEP-operator_3 | 0.3116 ± 0.0842 | 0.7627 ± 0.3338 | 0.6104 ± 0.3661 | 0.6718 ± 0.2800 | 0.8149 ± 0.5710 |
(c). R2 (Mean ± Standard Deviation) | |||||
Operator | Head Time = 1 | Head Time = 3 | Head Time = 6 | Head Time = 12 | Head Time = 24 |
GEP-operator_1 | 0.8900 ± 0.0498 | 0.4455 ± 0.1681 | 0.6221 ± 0.2732 | 0.4227 ± 0.3275 | 0.3174 ± 0.2151 |
GEP-operator_2 | 0.8906 ± 0.0491 | 0.4332 ± 0.1593 | 0.6776 ± 0.2314 | 0.5131 ± 0.2665 | 0.2229 ± 0.2288 |
GEP-operator_3 | 0.8962 ± 0.0569 | 0.5077 ± 0.2248 | 0.6551 ± 0.3585 | 0.4395 ± 0.3204 | 0.2727 ± 0.2255 |
(d). MAPE % (Mean ± Standard Deviation) | |||||
Operator | Head Time = 1 | Head Time = 3 | Head Time = 6 | Head Time = 12 | Head Time = 24 |
GEP-operator_1 | 0.900 ± 0.0113 | 1.2067 ± 0.0080 | 2.1450 ± 0.0133 | 1.4267 ± 0.0098 | 2.0667 ± 0.0106 |
GEP-operator_2 | 0.9400 ± 0.0110 | 1.4367 ±0.0113 | 2.3933 ± 0.0090 | 1.5950 ± 0.0012 | 2.0683 ± 0.0101 |
GEP-operator_3 | 0.8850 ± 0.0089 | 1.2033 ± 0.0115 | 2.1867 ± 0.0084 | 1.4667 ± 0.0091 | 2.0917 ± 0.0010 |
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Shabani, S.; Candelieri, A.; Archetti, F.; Naser, G. Gene Expression Programming Coupled with Unsupervised Learning: A Two-Stage Learning Process in Multi-Scale, Short-Term Water Demand Forecasts. Water 2018, 10, 142. https://doi.org/10.3390/w10020142
Shabani S, Candelieri A, Archetti F, Naser G. Gene Expression Programming Coupled with Unsupervised Learning: A Two-Stage Learning Process in Multi-Scale, Short-Term Water Demand Forecasts. Water. 2018; 10(2):142. https://doi.org/10.3390/w10020142
Chicago/Turabian StyleShabani, Sina, Antonio Candelieri, Francesco Archetti, and Gholamreza Naser. 2018. "Gene Expression Programming Coupled with Unsupervised Learning: A Two-Stage Learning Process in Multi-Scale, Short-Term Water Demand Forecasts" Water 10, no. 2: 142. https://doi.org/10.3390/w10020142
APA StyleShabani, S., Candelieri, A., Archetti, F., & Naser, G. (2018). Gene Expression Programming Coupled with Unsupervised Learning: A Two-Stage Learning Process in Multi-Scale, Short-Term Water Demand Forecasts. Water, 10(2), 142. https://doi.org/10.3390/w10020142