# 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

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**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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**MDPI and ACS 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*, 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