# Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna)

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Problem Statement

#### 1.3. Objective

## 2. Materials and Methods

#### 2.1. Phase Space Reconstruction

_{t}is a vector of the consumption data of {d

_{t}}

_{t=1,…,N}, N is the number of recorded consumption data points, $\tau $ is the lag time and m is the number of embedding dimension that generally varies from 1 to 10 or 1 to 20 [61,66,88,89]. In the case when m is greater than the minimum embedding dimension, the trajectory of reconstructed vectors can display the true state of the chaotic system. Indeed, the lag time ($\tau $) is arbitrary as the data are often assumed to have infinite precision. The lag time should not be too small given the difference between various elements of the delay vectors and, it should not be too large as this can result in low coordinate correlation [90]. If the dynamics of the system can be reduced to a set of deterministic laws, trajectories will converge towards a subset of the phase-space with a fractional dimension called the attractor [89]. The lag-embedding method is sensitive to both embedding parameters of $\tau $ and m. Average mutual information (AMI) is a well-known method for estimating the lag time [91,92]. This research employed AMI to estimate the lag time [91]:

_{t}) and P(d

_{t+τ}) are the marginal probabilities for measurements d

_{t}and d

_{t+τ}and P (d

_{t}, d

_{t+τ}) is their joint probability. The optimal τ minimizes the value of the function I (d

_{t}, d

_{t+τ}) for t = τ. AMI considers the first local minimum as the lag time [91,93].

#### 2.2. Correlation Dimension

_{i}is the ith state vector, and r is the radius of a sphere with the content of X

_{i}or X

_{j}as the center. C

_{m}(r) is proportional to r for stochastic time series, whereas for chaotic time series, it scales with r as

_{e}can be considered as the correlation dimension of the attractor in the system. But, if ${c}_{e}$ is not stable as a function of embedding dimensions, this system can be considered non-chaotic [54,96,97].

#### 2.3. Nonlinear Local Approximation

_{T}, by

_{j}is the vector of dimension m at the current state of the system at time j and X

_{j+T}is the vector of dimension m at the future state of the system at time j + T. NLA entails the subdivision of the f

_{T}domain into many subsets. In other words, the dynamics of the system were described step-by-step locally in the phase-space [98]. In an m-dimensional space, estimating the change of trajectory with time would lead to forecasting. Considering the relation between two states ${\overrightarrow{X}}_{t}$ and ${\overrightarrow{X}}_{t+p}$, the behaviour at a future time (p) on the attractor was forecasted by the mapping $\overrightarrow{F}$ as [66]:

#### 2.4. Largest Lyapunov Exponent

_{n}

_{0}was selected and all the points in the neighborhood of Y

_{n}, with the closer distance of r to that point were found. r is the radius of a sphere with the content of Y

_{n}as the center. The procedure was repeated for N points in the route to find a stretch factor S:

_{n0}. The plot of S verses N consists of linear and nonlinear components. The literature provides two methods for calculating the largest Lyapunov exponent (λ

_{max}). Shang et al. [102] determined λ

_{max}as the slope of the linear part of the curve, while Rosenstein et al. [101] determined λ

_{max}as the average of the slope of the first part and that of the second part [89,102]. This research applied the second approach to study different temporal scales. The prediction horizon (Δt) is a time in which the consumption dataset sustains its dynamics in the most accurate forecasting; Δt was obtained as the inverse of the largest Lyapunov exponent:

#### 2.5. Gene Expression Programming

#### 2.6. Multiple Linear Regression

_{1}, X

_{2}, …, X

_{n}). In other words, MLR finds a relationship with a linear combination of independent explanatory variables (e.g., reconstructed phase space with a different number of embedding dimensions for recorded values of water consumption) and response variable (time ahead values of water consumption) by

_{i}is the output consumption value, X

_{i}is the explanatory input variables, dependent upon the number of embedding dimensions, where ${\alpha}_{0}$ is constant, ${\alpha}_{p}$ are slope coefficients for each input variables and $\u03f5$ is the unexplained residual value.

#### 2.7. Models Selection Criteria

_{t}, and the recorded and forecasted values, R and F, respectively. $\overline{R}$ and $\overline{F}$ are the mean values of the recorded and forecasted water consumption, respectively. The range of CC is −1 and 1, where larger positive value of CC and smaller value for RMSE and MAE indicates better model performance in forecasting accuracy.

#### 2.8. Test Case

## 3. Results

**Lag Time:**Figure 3a,b presents the time variation of water consumption dataset for a six-year period (from January 2011 to December 2016) and the average 24-h consumption based on City of Kelowna’s Utility report. The data were parsed into hourly maximum, minimum, and mean values for a 24-h period (Figure 3b). Figure 3b also presents a boxplot of the total mean, minimum, and maximum consumption value distributions for the six years period. Figure 3b indicates that the average consumption and minimum values have low frequency, demonstrating the highly deterministic behavior of the data. To calibrate the models, the data splinted into two folds; (1) 80% for training period (1st January 2016 to 31st December 2015); (2) 20% for test period (1st January 2016 to 31st December 2016). Water consumption estimation and forecasting of the average and minimum values are not as complex as compared to the peak consumption values. However, the maximum values exhibit a non-linear behavior which does not follow a specific pattern (i.e., appears to be random) unlike average and minimum values. Further, the maximum consumption values in a water distribution system are very important. This is because of estimation of peak consumption to supply customers’ consumption and optimize WDS pipeline to make WDS more reliable, such as managing pipeline failures, improving peak pressure, reducing leakage, etc.

**Correlation Dimension:**Figure 5a plots the results for correlation integral versus log (r) for different dimensions (m) in the range of 1 to 20 for daily consumption time series. The correlation exponents, C

_{e}(m), were determined and the results were plotted in Figure 5b to evaluate the presence of chaos in the dataset. As demonstrated in Figure 5b, the correlation exponent increases with the embedding dimension up to a certain value and then remains steady. The figure reveals that the slope of larger embedding would become constant for all temporal scales.

_{e}< 2 logN (with N as the number of data point) was satisfied for the chaotic temporal scales [106].

**Nonlinear Local Approximation and Largest Lyapunov Exponent:**The first objective in this research was to investigate the impact of PSR in the accuracy of forecasted values. Forecasted values were evaluated for the embedding dimensions (ranging from 2 to 10) at lag times 1 and 17 days to forecast 1-day-ahead and highlight the effect of embedding dimension and performance of models. For m = 2, for a lag time of 1 day, two variables D

_{t}and D

_{t}

_{−1}, and for m = 2, for a lag time of 17 days, two variables D

_{t}and D

_{t}

_{−17,}were used as input variables to forecast 1-day ahead (D

_{t}

_{+1}). For m = 3, for the lag time of 1 and 17 days, three variables D

_{t}, D

_{t}

_{−1}, D

_{t}

_{−2}and D

_{t}, D

_{t}

_{−17}, D

_{t}

_{−34}were used as input variables to predict 1-day-ahead. Moreover, phase space was reconstructed for m > 3. Table 3 presents a summary of the 1-day ahead forecasted values that reconstructed phase-space in dimensions ranging from 2 to 20 for τ = 1 and 17 days. The overall average of fitness values for all embedding dimensions CC > 0.96, RMSE < 4200 (m3/day) (8% of average daily consumption) and MAE < 49. These results can be considered as reasonable performance of the model. Table 3 reveals the optimum embedding dimension for the most accurate forecasted value in bold. Using CC, RMSE, and MAE, the optimum embedding dimension (m

_{opt}) was found to be 18 and 19 for the lag time of 1 and 17 days, respectively. The results of nonlinear local approximation (NLA) identify the closest results of correlation exponent in optimum embedding dimension (m = 17 for correlation dimension of daily value). The similar values for the embedding dimension of daily values imply that m = 17 or 18 is the optimum dimension of the system to be used as the models’ input. The second objective was to evaluate the performance of selected m and τ. The optimum models for m = 18, τ = 1 and m = 19, τ = 17 have been applied to forecast 2-, 4-, 7-, 14-, 30- and 60-day lead time water consumption. Figure 6 compares the NLA with observed data using bolded values in Table 3.

**Phase Space Reconstructed GEP (PSR-GEP) and Multiple Linear Regression (MLR) Models:**One of the most important steps in developing an accurate model is the selection of the input variables. Input variable selection challenges the effect of the number of inputs in models’ performance. In other words, there are diminishing returns on performance based on the number of input variables selected [42]. The combinations were selected in a way that included daily consumption data with lag times of τ = 1 and 17 days. Different combinations of the time series of daily consumption were used to structure a policy for input dataset criteria. Combinations of D

_{t}, D

_{t}

_{−1}, D

_{t}

_{−2}, …, D

_{t}

_{−20}variables were used as input data with D

_{t}

_{+1}(1-day time ahead) as output of the GEP and MLR model and combinations of D

_{t}, D

_{t}

_{−τ}, D

_{t}

_{−2τ}, …, D

_{t}

_{−20τ}variables were used as input data with D

_{t+1}as output of the PSR-GEP and PSR-MLR model (forecasting of 1 day time ahead). The combinations of arithmetic functions of $\left\{+,-,\times ,x,{x}^{2},\sqrt{x},\mathrm{log}x\right\}$ and $\left\{+,-,\times ,\right\}$ were used for PSR-GEP and PSR-MLR models, respectively. Ultimately, the best combination was selected using the criteria of CC, RMSE, and MAE. Table 4 reveals the 20 combinations of inputs and their performance with GEP and PSR-GEP models. In the table, the three criteria indicate the fourth combination of inputs (m = 4) with the lag time of 1-day, and (m = 8) with the lag time of 17 days as the best combinations of input data (reconstructed phase space) for GEP and PSR-GEP. The study reveals the following relationship for both GEP and PSR-GEP, respectively:

## 4. Discussion

_{opt}= 19 provided more accurate predictions than m

_{opt}= 18 (Table 3). Figure 8 compares observed values to the forecasted results, which highlights the negligible error in the results of PSR-NLA and GEP during the test period. Regarding the average fitness results for all dimensions, the performance of GEP is better than PSR-GEP, while the application of PSR is shown in forecasting with equal dimension and different lead times. Figure 8a shows the greatest accuracy from the results shown by PSR-NLA compared to the other models. Figure 8b shows the value of residual for forecasted consumption by PSR-NLA in two forecasting horizons. The performance of PSR-NLA in forecasting high-frequency values is shown in Figure 8b. The figure reveals that the optimum embedding dimension of 19 was more acceptable than the embedding dimension of 17 determined by the correlation function (Figure 5b).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Yousefi, P. Integrated Management Plan of Water Distribution Systems: Forecasting Approach. Ph.D. Thesis, University of British Columbia, Kelowna, BC, Canada, 2020. [Google Scholar] [CrossRef]
- Xenochristou, M.; Kapelan, Z.; Hutton, C. Using Smart Demand-Metering Data and Customer Characteristics to Investigate Influence of Weather on Water Consumption in the UK. J. Water Resour. Plan. Manag.
**2020**, 146, 4019073. [Google Scholar] [CrossRef] - Water Conflict—World’s Water. Available online: https://www.worldwater.org/water-conflict/ (accessed on 14 February 2019).
- Billings, R.B.; Jones, C.V. Forecasting Urban Water Demand; American Water Works Association: Denver, CO, USA, 2008. [Google Scholar]
- Ghalehkhondabi, I.; Ardjmand, E.; Young, W.A.; Weckman, G.R. Water demand forecasting: Review of soft computing methods. Environ. Monit. Assess.
**2017**, 189, 313. [Google Scholar] [CrossRef] - Sastri, T.; Valdes, J.B. Rainfall Intervention Analysis for On-Line Applications. J. Water Resour. Plan. Manag.
**2008**, 115, 397–415. [Google Scholar] [CrossRef] - Odan, F.K.; Reis, L.F.R. Hybrid Water Demand Forecasting Model Associating Artificial Neural Network with Fourier Series. J. Water Resour. Plan. Manag.
**2012**, 138, 245–256. [Google Scholar] [CrossRef] [Green Version] - Iwanek, M.; Kowalska, B.; Hawryluk, E.; Kondraciuk, K. Distance and time of water effluence on soil surface after failure of buried water pipe. Laboratory investigations and statistical analysis. Eksploat. I Niezawodn. Maint. Reliab.
**2016**, 18, 278–284. [Google Scholar] [CrossRef] - Ghiassi, M.; Zimbra, D.K.; Saidane, H. Urban Water Demand Forecasting with a Dynamic Artificial Neural Network Model. J. Water Resour. Plan. Manag.
**2008**, 134, 138–146. [Google Scholar] [CrossRef] - Jayawardena, A.W.; Gurung, A.B. Noise reduction and prediction of hydrometeorological time series: Dynamical systems approach vs. stochastic approach. J. Hydrol.
**2000**, 228, 242–264. [Google Scholar] [CrossRef] - Lisi, F.; Villi, V. CHAOTIC FORECASTING OF DISCHARGE TIME SERIES: A CASE STUDY. J. Am. Water Resour. Assoc.
**2001**, 37, 271–279. [Google Scholar] [CrossRef] - Cominola, A.; Giuliani, M.; Piga, D.; Castelletti, A.; Rizzoli, A.E. Benefits and challenges of using smart meters for advancing residential water demand modeling and management: A review. Environ. Model. Softw.
**2015**, 72, 198–214. [Google Scholar] [CrossRef] [Green Version] - Oshima, N. Information Integration Type Chaos Theory-Based Demand Forecasting for Predictive Control of Waterworks. Water Purify Technol.
**2015**, 164, 6–12. [Google Scholar] - Jain, A.; Ormsbee, L.E. Short-term water demand forecast modeling techniques—Conventional methods versus AI. J. Am. Water Work Assoc.
**2002**, 94, 64–72. [Google Scholar] [CrossRef] - Kame’enui, A.E. Water Demand Forecasting in the Puget Sound Region: Short and long-Term Models. 2003, pp. 1–97. Available online: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.405&rep=rep1&type=pdf (accessed on 18 February 2019).
- Herrera, M.; Torgo, L.; Izquierdo, J.; Pérez-García, R. Predictive models for forecasting hourly urban water demand. J. Hydrol.
**2010**, 387, 141–150. [Google Scholar] [CrossRef] - Yousefi, P.; Naser, G.; Mohammadi, H. Surface Water Quality Model: Impacts of Influential Variables. J. Water Resour. Plan. Manag.
**2018**, 144, 4018015. [Google Scholar] [CrossRef] - Shabani, S.; Yousefi, P.; Adamowski, J.; Naser, G. Intelligent Soft Computing Models in Water Demand Forecasting. In Water Stress in Plants; IntechOpen: London, UK, 2016. [Google Scholar] [CrossRef] [Green Version]
- Miaou, S.-P. A stepwise time series regression procedure for water demand model identification. Water Resour. Res.
**1990**, 26, 1887–1897. [Google Scholar] [CrossRef] - Jain, A.; Kumar Varshney, A.; Chandra Joshi, U. Short-Term Water Demand Forecast Modelling at IIT Kanpur Using Artificial Neural Networks. Water Resour. Manag.
**2001**, 15, 299–321. [Google Scholar] [CrossRef] - Gato, S.; Jayasuriya, N.; Roberts, P. Temperature and rainfall thresholds for base use urban water demand modelling. J. Hydrol.
**2007**, 337, 364–376. [Google Scholar] [CrossRef] - Bougadis, J.; Adamowski, K.; Diduch, R. Short-term municipal water demand forecasting. Hydrol. Process.
**2005**, 19, 137–148. [Google Scholar] [CrossRef] - Adamowski, J.; Fung Chan, H.; Prasher, S.O.; Ozga-Zielinski, B.; Sliusarieva, A. Comparison of multiple linear and nonlinear regression, autoregressive integrated moving average, artificial neural network, and wavelet artificial neural network methods for urban water demand forecasting in Montreal, Canada. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Zhou, S.L.; McMahon, T.A.; Walton, A.; Lewis, J. Forecasting daily urban water demand: A case study of Melbourne. J. Hydrol.
**2000**, 236, 153–164. [Google Scholar] [CrossRef] - Mukhopadhyay, A.; Akber, A.; Al-Awadi, E. Analysis of freshwater consumption patterns in the private residences of Kuwait. Urban. Water.
**2001**, 3, 53–62. [Google Scholar] [CrossRef] - Dos Santos, C.C.; Pereira Filho, A.J. Water Demand Forecasting Model for the Metropolitan Area of São Paulo, Brazil. Water Resour. Manag.
**2014**, 28, 4401–4414. [Google Scholar] [CrossRef] - Brekke, L.; Larsen, M.D.; Ausburn, M.; Takaichi, L. Suburban Water Demand Modeling Using Stepwise Regression. J. Am. Water Works Assoc.
**2002**, 94, 65–75. [Google Scholar] [CrossRef] - Polebitski, A.S.; Palmer, R.N. Seasonal Residential Water Demand Forecasting for Census Tracts. J. Water Resour. Plan. Manag.
**2010**, 136, 27–36. [Google Scholar] [CrossRef] - Lee, S.-J.; Wentz, E.A.; Gober, P. Space–time forecasting using soft geostatistics: A case study in forecasting municipal water demand for Phoenix, Arizona. Stoch. Environ. Res. Risk Assess.
**2010**, 24, 283–295. [Google Scholar] [CrossRef] - Adamowski, J.; Karapataki, C. Comparison of Multivariate Regression and Artificial Neural Networks for Peak Urban Water-Demand Forecasting: Evaluation of Different ANN Learning Algorithms. J. Hydrol. Eng.
**2010**, 15, 729–743. [Google Scholar] [CrossRef] [Green Version] - Cutore, P.; Campisano, A.; Kapelan, Z.; Modica, C.; Savic, D. Probabilistic prediction of urban water consumption using the SCEM-UA algorithm. Urban. Water J.
**2008**, 5, 125–132. [Google Scholar] [CrossRef] - Adamowski, J.F. Peak Daily Water Demand Forecast Modeling Using Artificial Neural Networks. J. Water Resour. Plan. Manag.
**2008**, 134, 119–128. [Google Scholar] [CrossRef] [Green Version] - Zhou, T.; Wang, F.; Yang, Z. Comparative Analysis of ANN and SVM Models Combined with Wavelet Preprocess for Groundwater Depth Prediction. Water
**2017**, 9, 781. [Google Scholar] [CrossRef] [Green Version] - Firat, M.; Yurdusev, M.A.; Turan, M.E. Evaluation of Artificial Neural Network Techniques for Municipal Water Consumption Modeling. Water Resour. Manag.
**2009**, 23, 617–632. [Google Scholar] [CrossRef] - Xu, Y.; Zhang, J.; Long, Z.; Chen, Y. A Novel Dual-Scale Deep Belief Network Method for Daily Urban Water Demand Forecasting. Energies
**2018**, 11, 1068. [Google Scholar] [CrossRef] [Green Version] - Msiza, I.S.; Nelwamondo, F.V.; Marwala, T. Artificial neural networks and support vector machines for water demand time series forecasting. In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Montreal, QC, Canada, 7–10 October 2007; pp. 638–643. [Google Scholar] [CrossRef] [Green Version]
- Msiza, I.S.; Nelwamondo, F.V.; Marwala, T. Water demand prediction using artificial neural networks and support vector regression. J. Comput.
**2008**, 3, 1–8. [Google Scholar] [CrossRef] - Shabani, S.; Yousefi, P.; Naser, G. Support Vector Machines in Urban Water Demand Forecasting Using Phase Space Reconstruction. Procedia Eng.
**2017**, 186, 537–543. [Google Scholar] [CrossRef] - Yousefi, P.; Shabani, S.; Mohammadi, H.; Naser, G. Gene Expression Programing in Long Term Water Demand Forecasts Using Wavelet Decomposition. Procedia Eng.
**2017**, 186, 544–550. [Google Scholar] [CrossRef] - Shabani, S. Water Demand Forecasting: A Flexible Approach. Ph.D. Thesis, University of British Columbia, Kelowna, BC, Canada, 2018. [Google Scholar] [CrossRef]
- Ambrosio, J.K.; Brentan, B.M.; Herrera, M.; Luvizotto, E.; Ribeiro, L.; Izquierdo, J. Committee Machines for Hourly Water Demand Forecasting in Water Supply Systems. Math. Probl. Eng.
**2019**, 2019, 1–11. [Google Scholar] [CrossRef] [Green Version] - Yousefi, P.; Naser, G.; Mohammadi, H. Application of Wavelet Decomposition and Phase Space Reconstruction in Urban Water Consumption Forecasting: Chaotic Approach (Case Study). In Wavelet Theory and Its Applications; IntechOpen: London, UK, 2018. [Google Scholar] [CrossRef] [Green Version]
- Yousefi, P.; Naser, G.; Mohammadi, H. Hybrid Wavelet and Local Approximation Method for Urban Water Demand Forecasting—Chaotic Approach. In Proceedings of the WDSA Conference, Kingstone, ON, Canada, 23–25 July 2018. [Google Scholar]
- Azadeh, A.; Neshat, N.; Hamidipour, H. Hybrid Fuzzy Regression–Artificial Neural Network for Improvement of Short-Term Water Consumption Estimation and Forecasting in Uncertain and Complex Environments: Case of a Large Metropolitan City. J. Water Resour. Plan. Manag.
**2011**, 138, 71–75. [Google Scholar] [CrossRef] - Ahmadi, S.; Alizadeh, S.; Forouzideh, N.; Yeh, C.H.; Martin, R.; Papageorgiou, E. ICLA imperialist competitive learning algorithm for fuzzy cognitive map: Application to water demand forecasting. In Proceedings of the IEEE International Conference on Fuzzy Systems, Beijing, China, 6–11 July 2014; pp. 1041–1048. [Google Scholar] [CrossRef]
- Navarrete-López, C.; Herrera, M.; Brentan, B.; Luvizotto, E.; Izquierdo, J. Enhanced Water Demand Analysis via Symbolic Approximation within an Epidemiology-Based Forecasting Framework. Water.
**2019**, 11, 246. [Google Scholar] [CrossRef] [Green Version] - Yousefi, P.; Naser, G.; Mohammadi, H. Estimating High Resolution Temporal Scale of Water Demand Time Series—Disaggregation Approach (Case Study). In Proceedings of the 13th International Conference on Hydroinformatics (HIC 2018), Palermo, Italy, 1–6 July 2018; Volume 3, pp. 2408–2416. [Google Scholar] [CrossRef]
- Kozłowski, E.; Kowalska, B.; Kowalski, D.; Mazurkiewicz, D. Water demand forecasting by trend and harmonic analysis. Arch. Civ. Mech. Eng.
**2018**, 18, 140–148. [Google Scholar] [CrossRef] - Campisi-Pinto, S.; Adamowski, J.; Oron, G. Forecasting Urban Water Demand Via Wavelet-Denoising and Neural Network Models. Case Study: City of Syracuse, Italy. Water Resour. Manag.
**2012**, 26, 3539–3558. [Google Scholar] [CrossRef] - Casdagli, M. Chaos and Deterministic Versus Stochastic Non-Linear Modelling. J. R Stat. Soc. Ser. B
**1992**, 54, 303–328. [Google Scholar] [CrossRef] - Lorenz, E.N. Atmospheric Predictability as Revealed by Naturally Occurring Analogues. J. Atmos. Sci.
**2004**, 26, 636–646. [Google Scholar] [CrossRef] [Green Version] - Sivakumar, B.; Jayawardena, A.W.; Li, W.K. Hydrologic complexity and classification: A simple data reconstruction approach. Hydrol. Process.
**2007**, 21, 2713–2728. [Google Scholar] [CrossRef] - Ng, W.W.; Panu, U.S.; Lennox, W.C. Chaos based Analytical techniques for daily extreme hydrological observations. J. Hydrol.
**2007**, 342, 17–41. [Google Scholar] [CrossRef] - Regonda, S.K.; Sivakumar, B.; Jain, A. Temporal scaling in river flow: Can it be chaotic? Hydrol. Sci. J.
**2004**, 49, 373–385. [Google Scholar] [CrossRef] [Green Version] - Salas, J.D.; Kim, H.S.; Eykholt, R.; Burlando, P.; Green, T.R. Aggregation and Sampling in Deterministic Chaos: Implications for Chaos Identification in Hydrological Processes. June 2005. Available online: https://hal.archives-ouvertes.fr/hal-00302625/ (accessed on 29 July 2019).
- Elshorbagy, A.; Simonovic, S.P.; Panu, U.S. Estimation of missing streamflow data using principles of chaos theory. J. Hydrol.
**2002**, 255, 123–133. [Google Scholar] [CrossRef] - Elshorbagy, A.; Simonovic, S.P.; Panu, U.S. Noise reduction in chaotic hydrologic time series: Facts and doubts. J. Hydrol.
**2002**, 256, 147–165. [Google Scholar] [CrossRef] - Sivakumar, B.; Wallender, W.W. Predictability of river flow and suspended sediment transport in the Mississippi River basin: A non-linear deterministic approach. Earth Surf. Process. Landforms.
**2005**, 30, 665–677. [Google Scholar] [CrossRef] - Zounemat-Kermani, M. Investigating Chaos and Nonlinear Forecasting in Short Term and Mid-term River Discharge. Water Resour. Manag.
**2016**, 30, 1851–1865. [Google Scholar] [CrossRef] - Ghorbani, M.A.; Khatibi, R.; Danandeh Mehr, A.; Asadi, H. Chaos-based multigene genetic programming: A new hybrid strategy for river flow forecasting. J. Hydrol.
**2018**, 562, 455–467. [Google Scholar] [CrossRef] - Sivakumar, B. A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers. J. Hydrol.
**2002**, 258, 149–162. [Google Scholar] [CrossRef] - Sivakumar, B.; Jayawardena, A.W. An investigation of the presence of low-dimensional chaotic behaviour in the sediment transport phenomenon. Hydrol. Sci. J.
**2002**, 47, 405–416. [Google Scholar] [CrossRef] - Ghorbani, M.; Khatibi, R.; Asadi, H.; Yousefi, P. Inter-Comparison of an Evolutionary Programming Model of Suspended Sediment Time-Series with Other Local Models. In Genetic Programming—New Approaches and Successful Applications; IntechOpen: London, UK, 2012. [Google Scholar] [CrossRef] [Green Version]
- Petkov, B.H.; Vitale, V.; Mazzola, M.; Lanconelli, C.; Lupi, A. Chaotic behaviour of the short-term variations in ozone column observed in Arctic. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 26, 238–249. [Google Scholar] [CrossRef] - Ghorbani, M.A.; Kisi, O.; Aalinezhad, M. A probe into the chaotic nature of daily streamflow time series by correlation dimension and largest Lyapunov methods. Appl. Math. Model.
**2010**, 34, 4050–4057. [Google Scholar] [CrossRef] - Khatibi, R.; Ghorbani, M.A.; Aalami, M.T.; Kocak, K.; Makarynskyy, O. Dynamics of hourly sea level at Hillarys Boat Harbour, Western Australia: A chaos theory perspective. Ocean Dyn.
**2011**, 61, 1797–1807. [Google Scholar] [CrossRef] - Rodriguez-Iturbe, I.; Febres De Power, B.; Sharifi, M.B.; Georgakakos, K.P. Chaos in rainfall. Water Resour. Res.
**1989**, 25, 1667–1675. [Google Scholar] [CrossRef] - Jayawardena, A.W.; Lai, F. Analysis and prediction of chaos in rainfall and stream flow time series. J. Hydrol.
**1994**, 153, 23–52. [Google Scholar] [CrossRef] - Sivakumar, B.; Berndtsson, R.; Olsson, J.; Jinno, K.; Kawamura, A. Dynamics of monthly rainfall-runoff process at the Gota basin: A search for chaos. Hydrol. Earth Syst. Sci.
**2000**, 4, 407–417. [Google Scholar] [CrossRef] [Green Version] - Maskey, M.L.; Puente, C.E.; Sivakumar, B. Temporal downscaling rainfall and streamflow records through a deterministic fractal geometric approach. J. Hydrol.
**2019**, 568, 447–461. [Google Scholar] [CrossRef] - Wang, J.; Shi, Q. Short-term traffic speed forecasting hybrid model based on Chaos–Wavelet Analysis-Support Vector Machine theory. Transp. Res. Part. C Emerg. Technol.
**2013**, 27, 219–232. [Google Scholar] [CrossRef] - Ravi, V.; Pradeepkumar, D.; Deb, K. Financial time series prediction using hybrids of chaos theory, multi-layer perceptron and multi-objective evolutionary algorithms. Swarm Evol. Comput.
**2017**, 36, 136–149. [Google Scholar] [CrossRef] - Abdechiri, M.; Faez, K.; Amindavar, H.; Bilotta, E. The chaotic dynamics of high-dimensional systems. Nonlinear Dyn.
**2017**, 87, 2597–2610. [Google Scholar] [CrossRef] - Li, M.W.; Geng, J.; Han, D.F.; Zheng, T.J. Ship motion prediction using dynamic seasonal RvSVR with phase space reconstruction and the chaos adaptive efficient FOA. Neurocomputing
**2016**, 174, 661–680. [Google Scholar] [CrossRef] - Kalra, R.; Deo, M.C. Genetic programming for retrieving missing information in wave records along the west coast of India. Appl. Ocean. Res.
**2007**, 29, 99–111. [Google Scholar] [CrossRef] - Ustoorikar, K.; Deo, M.C. Filling up gaps in wave data with genetic programming. Mar. Struct.
**2008**, 21, 177–195. [Google Scholar] [CrossRef] - Gaur, S.; Deo, M.C. Real-time wave forecasting using genetic programming. Ocean. Eng.
**2008**, 35, 1166–1172. [Google Scholar] [CrossRef] - Aytek, A.; Kişi, Ö. A genetic programming approach to suspended sediment modelling. J. Hydrol.
**2008**, 351, 288–298. [Google Scholar] [CrossRef] - Ferreira, C. Gene Expression Programming in Problem Solving. In Soft Computing and Industry; Springer: London, UK, 2002; pp. 635–653. [Google Scholar] [CrossRef] [Green Version]
- Ferreira, C. Function Finding and the Creation of Numerical Constants in Gene Expression Programming. In Advances in Soft Computing; Springer: London, UK, 2003; pp. 257–265. [Google Scholar] [CrossRef] [Green Version]
- Nasseri, M.; Moeini, A.; Tabesh, M. Forecasting monthly urban water demand using Extended Kalman Filter and Genetic Programming. Expert Syst. Appl.
**2011**, 38, 7387–7395. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] [Green Version] - Gutzler, D.S.; Nims, J.S. Interannual Variability of Water Demand and Summer Climate in Albuquerque, New Mexico. J. Appl. Meteorol.
**2006**, 44, 1777–1787. [Google Scholar] [CrossRef] - Donkor, E.A.; Mazzuchi, T.A.; Soyer, R.; Alan Roberson, J. Urban Water Demand Forecasting: Review of Methods and Models. J. Water Resour. Plan. Manag.
**2012**, 140, 146–159. [Google Scholar] [CrossRef] - Alvisi, S.; Franchini, M.; Marinelli, A. A short-term, pattern-based model for water-demand forecasting. J. Hydroinformatics
**2006**, 9, 39–50. [Google Scholar] [CrossRef] [Green Version] - Sivakumar, B.; Berndtsson, R.; Olsson, J.; Jinno, K. Evidence of chaos in the rainfall-runoff process. Hydrol. Sci. J.
**2001**, 46, 131–145. [Google Scholar] [CrossRef] - Takens, F. Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Warwick; Springer: Berlin/Heidelberg, Germany, 1981; pp. 366–381. [Google Scholar] [CrossRef]
- Sivakumar, B. Forecasting monthly flow dynamics in the western united states: A nonlinear dynamical approach. J. Environ. Model. Softw.
**2003**, 17, 721–728. [Google Scholar] [CrossRef] - Khatibi, R.; Sivakumar, B.; Ghorbani, M.A.; Kisi, O.; Koçak, K.; Farsadi Zadeh, D. Investigating chaos in river stage and discharge time series. J. Hydrol.
**2012**, 414–415, 108–117. [Google Scholar] [CrossRef] - Meng, Q.; Peng, Y. A new local linear prediction model for chaotic time series. Phys. Lett. Sect. A Gen. At. Solid State Phys.
**2007**, 370, 465–470. [Google Scholar] [CrossRef] - Fraser, A.M.; Swinney, H.L. Independent coordinates for strange attractors from mutual information. Phys. Rev. A
**1986**, 33, 1134–1140. [Google Scholar] [CrossRef] [PubMed] - Holzfuss, J.; Mayer-Kress, G. An Approach to Error-Estimation in the Application of Dimension Algorithms. In Dimensions and Entropies in Chaotic Systems; Springer: Berlin/Heidelberg, Germany, 2011; pp. 114–122. [Google Scholar] [CrossRef] [Green Version]
- Hegger, R.; Kantz, H.; Schreiber, T. Practical implementation of nonlinear time series methods: The TISEAN package. Chaos Interdiscip. J. Nonlinear Sci.
**1999**, 9, 413–435. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zounemat-Kermani, M.; Kisi, O. Time series analysis on marine wind-wave characteristics using chaos theory. Ocean. Eng.
**2015**, 100, 46–53. [Google Scholar] [CrossRef] - Grassberger, P.; Procaccia, I. Measuring the strangeness of strange attractors. Phys. D Nonlinear Phenom.
**1983**, 9, 189–208. [Google Scholar] [CrossRef] - Islam, M.N.; Sivakumar, B. Characterization and prediction of runoff dynamics: A nonlinear dynamical view. Adv. Water Resour.
**2002**, 25, 179–190. [Google Scholar] [CrossRef] - Tongal, H.; Berndtsson, R. Impact of complexity on daily and multi-step forecasting of streamflow with chaotic, stochastic, and black-box models. Stoch. Environ. Res. Risk Assess.
**2017**, 31, 661–682. [Google Scholar] [CrossRef] - Farmer, J.D.; Sidorowich, J.J. Predicting chaotic time series. Phys. Rev. Lett.
**1987**, 59, 845–848. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Itoh, K.-I. A method for predicting chaotic time-series with outliers. Electron. Commun. Jpn. Part III Fundam Electron. Sci.
**1995**, 78, 44–53. [Google Scholar] [CrossRef] - Porporato, A.; Ridolfi, L. Nonlinear analysis of river flow time sequences. Water Resour. Res.
**1997**, 33, 1353–1367. [Google Scholar] [CrossRef] - Rosenstein, M.T.; Collins, J.J.; De Luca, C.J. A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D Nonlinear Phenom.
**1993**, 65, 117–134. [Google Scholar] [CrossRef] - Shang, P.; Li, X.; Kamae, S. Chaotic analysis of traffic time series. Chaos Solitons Fractals
**2005**, 25, 121–128. [Google Scholar] [CrossRef] - Holland, J.H. Genetic algorithms and the optimal allocation of trials. In Evolutionary Computation: The Fossil Record; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1998; Volume 2, pp. 443–460. [Google Scholar] [CrossRef]
- Goldberg, D.E.; Holland, J.H. Genetic Algorithms and Machine Learning. Mach. Learn.
**1988**, 3, 95–99. [Google Scholar] [CrossRef] - Strategic Value Solution. Kelowna Integrated Water Suply Plan. Kelowna. 2017. Available online: https://www.kelowna.ca/city-services/water-wastewater/ (accessed on 26 February 2020).
- Ruelle, D. The Claude Bernard Lecture, 1989. Deterministic Chaos: The Science and the Fiction. Proc. R Soc. A Math. Phys. Eng. Sci.
**1990**, 427, 241–248. [Google Scholar] [CrossRef]

**Figure 3.**Time series plot of (

**a**) daily water consumption; (

**b**) 6-years consumption pattern within a 24-h period.

**Figure 4.**(

**a**) Average mutual information for the daily value (τ = 17); (

**b**) reconstructed phase space by (τ and 2τ-day lag time).

**Figure 5.**(

**a**) The relation between correlation integral C(r) and r by different embedding dimensions for daily temporal scale, (

**b**) saturation of correlation dimension C

_{e}(m) with embedding dimension m for different temporal scales.

**Figure 6.**Forecasted values for water consumption by NLA (nonlinear local approximation) and PSR-NLA (phase space reconstructed-NLA) methods in comparison with observed values (

**a**) consumption values time series (

**b**) scatter pilot.

**Figure 7.**Estimation of largest Lyapunov exponent for daily consumption (

**a**) for embedding dimension of 17, 18 and 19; (

**b**) Observed values of water consumption in the test period.

**Figure 8.**Performance of the models (

**a**) box-plot of forecasted values in comparison with observed values; (

**b**) Residual of PSR-NLA (best performance) in two forecasting horizons.

**Figure 9.**Performance of NLA and PSR-NLA in time ahead forecasting by the fitness functions of Correlation Coefficient (CC); Root Mean Square Error (RMSE); Mean Absolute Error (MAE) for lead time of 1-, 2-, 4-, 7-, 14-, 30- and 60-days.

**Table 1.**Characteristics of the temporal water consumption values in test case in whole urban scale.

Property | Daily | 2-Day | 4-Day | 7-Day | 14-Day | Monthly |
---|---|---|---|---|---|---|

Number of Data | 2186 | 1092 | 552 | 312 | 156 | 72 |

Max. value (m^{3}) | 114,597.2 | 210,740.3 | 410,428.3 | 656,173.6 | 1,255,211 | 2,475,026 |

Min. value (m^{3}) | 14124 | 31,477.3 | 69,655.5 | 124,112.9 | 252,704.1 | 557,066.8 |

Average (m^{3}) | 43,046.4 | 86,102 | 170,332.4 | 301,357.3 | 602,714.7 | 1,291,944 |

Standard deviation (m^{3}) | 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 |

**Table 2.**Average mutual information and correlation exponent values in different temporal resolutions.

Time Scale | AMI | C_{e} | 2LogN > C_{e} |
---|---|---|---|

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 |

**Table 3.**Fitness values for NLA and PSR-NLA methods in different embedding dimension lag time and lead time.

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 | * m^{3} | |||

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 |

**Table 4.**Fitness values for GEP (gene expression programming) and PSR-GEP (phase space reconstructed GEP) methods in different embedding dimension lag time and lead time.

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 | * m^{3} | |||

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 |

**Table 5.**Fitness values for MLR and PSR-MLR methods in different embedding dimension lag time and lead time.

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 | * m^{3} | |||

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 |

**Table 6.**Statistics comparison of observed and forecasted consumption in test period by the selected models.

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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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

**AMA Style**

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 Style**

Yousefi, 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