Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna)
Abstract
:1. Introduction
1.1. Background
1.2. Problem Statement
1.3. Objective
2. Materials and Methods
2.1. Phase Space Reconstruction
2.2. Correlation Dimension
2.3. Nonlinear Local Approximation
2.4. Largest Lyapunov Exponent
2.5. Gene Expression Programming
2.6. Multiple Linear Regression
2.7. Models Selection Criteria
2.8. Test Case
3. Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Property | Daily | 2-Day | 4-Day | 7-Day | 14-Day | Monthly |
---|---|---|---|---|---|---|
Number of Data | 2186 | 1092 | 552 | 312 | 156 | 72 |
Max. value (m3) | 114,597.2 | 210,740.3 | 410,428.3 | 656,173.6 | 1,255,211 | 2,475,026 |
Min. value (m3) | 14124 | 31,477.3 | 69,655.5 | 124,112.9 | 252,704.1 | 557,066.8 |
Average (m3) | 43,046.4 | 86,102 | 170,332.4 | 301,357.3 | 602,714.7 | 1,291,944 |
Standard deviation (m3) | 20,074.5 | 39,897 | 79,304.3 | 136,626.8 | 268,733.2 | 552,701.5 |
Coefficient of variation | 0.46 | 0.46 | 0.46 | 0.45 | 0.44 | 0.42 |
Skew | 0.73 | 0.71 | 0.72 | 0.66 | 0.63 | 0.54 |
Kurtosis | −0.38 | −0.45 | −0.51 | −0.63 | −0.79 | −0.91 |
Time Scale | AMI | Ce | 2LogN > Ce |
---|---|---|---|
Daily | 17 | 3.50 | 6.67 |
2-Day | 12 | 3.37 | 6.07 |
4-Day | 10 | 3.74 | 5.48 |
7-Day | 6 | 3.94 | 4.98 |
14-Day | 3 | 3.83 | 4.38 |
30-Day | 2 | 3.49 | 3.71 |
NLA, τ = 1, T = 1 | PSR-NLA, τ = 17, T = 1 | τ = 1, m = 18 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
m | CC | RMSE * | MAE | m | CC | RMSE * | MAE | T | CC | RMSE * | MAE |
1 | 1 | 0.9842 | 2855.6 | 43.07 | |||||||
2 | 0.9759 | 3532.1 | 47.85 | 2 | 0.9751 | 3581.7 | 49.09 | 2 | 0.9783 | 3351.6 | 48.02 |
3 | 0.9771 | 3424.8 | 47.34 | 3 | 0.9773 | 3425.3 | 48.01 | 4 | 0.9331 | 5883.5 | 63.03 |
4 | 0.9762 | 3495.5 | 47.74 | 4 | 0.9789 | 3302.9 | 47.30 | 7 | 0.8932 | 7445.8 | 72.49 |
5 | 0.9785 | 3332.0 | 47.08 | 5 | 0.9785 | 3331.7 | 47.10 | 14 | 0.7877 | 10555.0 | 87.76 |
6 | 0.9795 | 3248.6 | 46.13 | 6 | 0.9795 | 3257.8 | 46.79 | 30 | 0.6735 | 13307.6 | 100.44 |
7 | 0.9802 | 3187.3 | 45.49 | 7 | 0.9829 | 2967.1 | 45.80 | 60 | 0.2523 | 20189.1 | 129.71 |
8 | 0.9805 | 3176.4 | 45.65 | 8 | 0.9838 | 2887.9 | 45.22 | τ = 17, m = 19 | |||
9 | 0.9806 | 3164.1 | 45.65 | 9 | 0.9849 | 2792.8 | 43.95 | T | CC | RMSE * | MAE |
10 | 0.9803 | 3193.6 | 45.09 | 10 | 0.9846 | 2828.2 | 44.52 | 1 | 0.9852 | 2772.8 | 43.83 |
11 | 0.9813 | 3098.7 | 44.56 | 11 | 0.9850 | 2792.2 | 43.95 | 2 | 0.9898 | 2295.7 | 39.59 |
12 | 0.9763 | 3495.1 | 48.11 | 12 | 0.9804 | 3189.4 | 46.69 | 4 | 0.9415 | 5504.2 | 61.47 |
13 | 0.9752 | 3578.0 | 48.11 | 13 | 0.9768 | 3457.4 | 48.32 | 7 | 0.9002 | 7211.2 | 71.46 |
14 | 0.9779 | 3378.9 | 47.14 | 14 | 0.9788 | 3303.6 | 47.65 | 14 | 0.8048 | 10147.5 | 86.61 |
15 | 0.9806 | 3169.7 | 46.06 | 15 | 0.9790 | 3290.2 | 47.23 | 30 | 0.6776 | 13265.1 | 95.31 |
16 | 0.9765 | 3491.0 | 47.24 | 16 | 0.9796 | 3250.2 | 46.70 | 60 | 0.4363 | 17784.9 | 118.53 |
17 | 0.9810 | 3139.6 | 45.21 | 17 | 0.9825 | 2995.5 | 45.95 | ||||
18 | 0.9842 | 2855.6 | 43.07 | 18 | 0.9838 | 2894.0 | 45.21 | * m3 | |||
19 | 0.9685 | 4088.5 | 44.69 | 19 | 0.9852 | 2772.8 | 43.83 | ||||
20 | 0.9661 | 4209.2 | 45.29 | 20 | 0.9846 | 2833.1 | 44.43 | ||||
Tot | 0.9775 | 3394.2 | 46.30 | Tot | 0.9807 | 3142.9 | 46.34 | ||||
Best | 0.9842 | 2855.6 | 43.07 | Best | 0.9852 | 2772.8 | 43.83 | ||||
EM | 18 | 18 | 18 | EM | 19 | 19 | 19 |
GEP, τ = 1, T = 1 | PSR-GEP, τ = 17, T = 1 | τ = 1, m = 4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
m | CC | RMSE * | MAE | m | CC | RMSE * | MAE | T | CC | RMSE * | MAE |
1 | 1 | 0.9764 | 3486.6 | 47.83 | |||||||
2 | 0.9757 | 3543.7 | 48.14 | 2 | 0.9789 | 3636.9 | 48.57 | 2 | 0.9494 | 5112.2 | 57.92 |
3 | 0.9761 | 3517.8 | 47.91 | 3 | 0.9788 | 3644.6 | 48.59 | 4 | 0.9130 | 6716.4 | 67.48 |
4 | 0.9764 | 3486.6 | 47.83 | 4 | 0.9789 | 3647.1 | 48.56 | 7 | 0.8652 | 8376.4 | 76.57 |
5 | 0.9760 | 3519.0 | 47.95 | 5 | 0.9789 | 3635.5 | 48.68 | 14 | 0.7810 | 10,734.8 | 88.95 |
6 | 0.9760 | 3520.5 | 48.37 | 6 | 0.9788 | 3649.5 | 48.71 | 30 | 0.6548 | 13,649.0 | 97.03 |
7 | 0.9760 | 3500.9 | 47.91 | 7 | 0.9788 | 3649.0 | 48.70 | 60 | 0.2345 | 20411.3 | 130.23 |
8 | 0.9760 | 3521.4 | 47.97 | 8 | 0.9789 | 3631.6 | 48.62 | τ = 17, m = 8 | |||
9 | 0.9760 | 3511.9 | 47.89 | 9 | 0.9788 | 3653.9 | 48.63 | T | CC | RMSE * | MAE |
10 | 0.9760 | 3514.7 | 47.89 | 10 | 0.9788 | 3650.6 | 48.73 | 1 | 0.9789 | 3631.6 | 48.62 |
11 | 0.9760 | 3514.6 | 47.88 | 11 | 0.9788 | 3656.6 | 48.65 | 2 | 0.9553 | 5267.3 | 58.52 |
12 | 0.9760 | 3514.7 | 47.89 | 12 | 0.9787 | 3657.4 | 48.69 | 4 | 0.9227 | 6894.5 | 68.47 |
13 | 0.9760 | 3510.1 | 47.91 | 13 | 0.9789 | 3645.4 | 48.55 | 7 | 0.8713 | 8848.2 | 78.11 |
14 | 0.9760 | 3516.6 | 47.91 | 14 | 0.9787 | 3655.7 | 48.69 | 14 | 0.7782 | 11,571.8 | 91.84 |
15 | 0.9760 | 3510.4 | 47.86 | 15 | 0.9788 | 3650.5 | 48.61 | 30 | 0.6334 | 14,631.5 | 105.98 |
16 | 0.9760 | 3498.7 | 47.88 | 16 | 0.9789 | 3644.4 | 48.54 | 60 | 0.3864 | 18,670.2 | 126.20 |
17 | 0.9759 | 3515.1 | 47.89 | 17 | 0.9789 | 3638.9 | 48.56 | ||||
18 | 0.9760 | 3509.7 | 47.85 | 18 | 0.9789 | 3646.6 | 48.57 | * m3 | |||
19 | 0.9759 | 3514.8 | 47.93 | 19 | 0.9787 | 3650.6 | 48.74 | ||||
20 | 0.9759 | 3514.2 | 47.90 | 20 | 0.9789 | 3642.5 | 48.51 | ||||
Tot | 0.9759 | 3519.0 | 47.96 | Tot | 0.9788 | 3646.5 | 48.62 | ||||
Best | 0.9764 | 3486.6 | 47.83 | Best | 0.9789 | 3631.6 | 48.51 | ||||
EM | 4 | 4 | 4 | EM | 8 | 8 | 20 |
MLR, τ = 1, T = 1 | PSR-MLR, τ = 17, T = 1 | τ = 1, m = 17 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
m | CC | RMSE * | MAE | m | CC | RMSE * | MAE | T | CC | RMSE * | MAE |
1 | 1 | 0.9789 | 3638.9 | 48.56 | |||||||
2 | 0.7825 | 11658.6 | 92.37 | 2 | 0.9758 | 3763.4 | 50.48 | 2 | 0.9494 | 5139.4 | 58.65 |
3 | 0.9790 | 3762.3 | 49.89 | 3 | 0.9809 | 3336.0 | 48.17 | 4 | 0.9130 | 6701.6 | 68.26 |
4 | 0.9790 | 3792.3 | 50.19 | 4 | 0.9811 | 3443.9 | 49.41 | 7 | 0.8595 | 8511.9 | 77.68 |
5 | 0.9790 | 3814.1 | 50.42 | 5 | 0.9766 | 3905.5 | 52.04 | 14 | 0.7595 | 11204.4 | 91.66 |
6 | 0.9790 | 3958.0 | 52.11 | 6 | 0.9148 | 22846.5 | 145.73 | 30 | 0.6447 | 13707.5 | 101.02 |
7 | 0.9790 | 4133.6 | 54.24 | 7 | 0.9765 | 3568.2 | 48.60 | 60 | 0.2282 | 20211.2 | 129.01 |
8 | 0.9790 | 4187.1 | 54.85 | 8 | 0.9764 | 3811.2 | 51.08 | τ = 17, m = 3 | |||
9 | 0.9791 | 4432.7 | 57.65 | 9 | 0.9766 | 3568.2 | 48.71 | T | CC | RMSE * | MAE |
10 | 0.9792 | 5024.0 | 63.46 | 10 | 0.9766 | 3680.4 | 49.81 | 1 | 0.9809 | 3336.0 | 48.17 |
11 | 0.9792 | 5576.0 | 68.14 | 11 | 0.9767 | 3610.6 | 49.13 | 2 | 0.9555 | 5336.5 | 59.61 |
12 | 0.9792 | 5921.4 | 70.92 | 12 | 0.9767 | 3603.9 | 49.04 | 4 | 0.9232 | 6889.3 | 69.17 |
13 | 0.9793 | 6276.8 | 73.59 | 13 | 0.9766 | 3601.4 | 49.13 | 7 | 0.8723 | 8841.9 | 78.12 |
14 | 0.9793 | 7267.0 | 80.61 | 14 | 0.9767 | 3584.5 | 48.92 | 14 | 0.7790 | 11549.8 | 91.70 |
15 | 0.9794 | 9128.5 | 92.30 | 15 | 0.9769 | 3560.6 | 48.79 | 30 | 0.6344 | 14504.2 | 104.84 |
16 | 0.9794 | 10115.3 | 97.81 | 16 | 0.9769 | 3550.4 | 48.73 | 60 | 0.3859 | 18351.2 | 125.12 |
17 | 0.9789 | 3638.9 | 48.56 | 17 | 0.9769 | 3550.5 | 48.73 | ||||
18 | 0.9794 | 10114.2 | 97.80 | 18 | 0.9769 | 3560.4 | 48.81 | * m3 | |||
19 | 0.9794 | 10115.3 | 97.81 | 19 | 0.9768 | 3561.6 | 48.87 | ||||
20 | 0.9795 | 9618.8 | 95.07 | 20 | 0.9769 | 3610.5 | 49.39 | ||||
Tot | 0.9595 | 6709.6 | 72.00 | Tot | 0.9738 | 4579.2 | 54.20 | ||||
Best | 0.9795 | 3638.8 | 48.56 | Best | 0.9811 | 3336.0 | 48.17 | ||||
EM | 20 | 17 | 17 | EM | 4 | 3 | 3 |
Property | Observed | NLA τ = 1, m = 18 | PSR-NLA τ = 17, m = 19 | GEP τ =1, m = 4 | PSR-GEP τ = 17, m = 8 | MLR τ = 1, m =1 7 | PSR-MLR τ = 17, m = 3 |
---|---|---|---|---|---|---|---|
Max. value | 75,620.26 | ✓ | |||||
Min. value | 21,313.72 | ✓ | |||||
Average | 42,500.82 | ✓ | |||||
Standard deviation | 16,117.34 | ✓ | |||||
Coefficient of variation | 0.38 | ✓ | |||||
Skew | 0.43 | ✓ | |||||
Kurtosis | −1.13 | ✓ |
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Share and Cite
Yousefi, P.; Courtice, G.; Naser, G.; Mohammadi, H. Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna). Water 2020, 12, 753. https://doi.org/10.3390/w12030753
Yousefi P, Courtice G, Naser G, Mohammadi H. Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna). Water. 2020; 12(3):753. https://doi.org/10.3390/w12030753
Chicago/Turabian StyleYousefi, Peyman, Gregory Courtice, Gholamreza Naser, and Hadi Mohammadi. 2020. "Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna)" Water 12, no. 3: 753. https://doi.org/10.3390/w12030753
APA StyleYousefi, P., Courtice, G., Naser, G., & Mohammadi, H. (2020). Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna). Water, 12(3), 753. https://doi.org/10.3390/w12030753