4. Preliminary results

#### 4.1. Phase space reconstruction and investigation of chaotic behavior

Existence of chaotic behavior in the time series is shown in Figure 2. However, the results are not entirely based on the proof of having chaotic behavior, as the figure only shows possible low-dimensional chaotic behavior. Theoretically, several methods are well known for investigating the chaotic behavior such as lag time calculation method (e.g., average mutual information (AMI), Autocorrelation function (ACF)), correlation dimension, largest Lyapunov exponent, etc.). This study investigates the chaotic behavior by applying ACF and correlation dimension. Having chaotic behavior allows using ACF to calculate the lag time of the time series. The value of lag time is considered as the first approach of ACF to 0 (Figure 2).

The results show 83-days as the lag time of the time series. Therefore, 83-day is used to design combination of inputs as phase space for the time series. In this study, the difference between 1st day and 83rd day is used as delay period for phase space reconstruction varying embedding dimensions from 1 to 10 (m1: Dt; m2: Dt,Dt-τ; m10: Dt,…,Dt-<sup>10</sup>τ). It should be noticed that

Figure 2. (a) Autocorrelation function (τ); (b) reconstructed phase space by (τ and 2τ -day lag time).

several methods were introduced in literature to calculate the value of optimum embedding dimension which may be more than 10 for the used time series in this study. This study aims at showing the performance of embedding dimension and reconstructed phase space, where m is only considered 1 to 10. Figure 2 shows the value of ACF for the demand series and reconstructed phase space (τ = 83). Figure 3a shows the relation between C(r) and r and (3b) correlation exponent by varying m. Figure 3b shows that the value of correlation exponent increases by m and as m = 17, the correlation exponent reaches a specific value (Ce = 3.41). This constant value of Ce at m = 17 indicates the existence of the deterministic behavior of the time series.

Statistical indices for the fitness values showed m = 1 for 1-day delay and m = 4 for the reconstructed phase space with the value of (CD = 0.9565, RMSE = 3642.89 and MAE = 50.42) and (CD = 0.9572, RMSE = 3636.34 and MAE = 51.04), respectively. However, the difference between the two models is not considerable, in the large value of demand in long-term this difference can come into account. Figure 4 shows the comparison of observed and demand

Table 2. Fitness values for MLR and PSR-MLR methods in different embedding dimensions (bolded lines are the most

/day) MAE m CD RMSE(m<sup>3</sup>

Application of Wavelet Decomposition and Phase Space Reconstruction in Urban Water Consumption Forecasting:…

1 0.9565 3642.89 50.42 1 0.9565 3642.89 50.42 0.9565 3804.14 52.14 2 0.9565 3804.14 52.14 0.9468 14106.70 112.82 3 0.9570 5319.51 66.90 0.9473 13174.97 108.82 4 0.9572 3636.34 51.04 0.9505 3724.99 49.81 5 0.9568 4167.55 56.45 0.9503 3746.33 50.09 6 0.9569 5907.90 71.65 0.9503 3747.49 50.10 7 0.9565 4370.03 58.86 0.9493 6058.34 70.88 8 0.9566 4581.10 60.89 0.9505 3736.33 50.02 9 0.9566 5023.16 64.71 0.9506 3738.35 50.07 10 0.9566 4327.34 58.48

/day) MAE

141

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ANN is another approach to model the demand values which represented in Section 3.4. ANN's structures have different hidden layer neurons (HLN) from 1 to 20 with 200 epochs

Dtþ<sup>1</sup> ¼ �0:00854Dt � 0:0366Dtþ<sup>τ</sup> � 0:0128Dtþ2<sup>τ</sup> þ 0:9427Dtþ3τ: (13)

values. Moreover, the suggested equation for the best result by MLR is given by:

Figure 4. The performance of MLR and PSR-MLR in comparison with observed values.

MLR, τ = 1 PSR-MLR, τ = 83

m CD RMSE(m<sup>3</sup>

accurate values).

4.3. Performance of artificial neural network

#### 4.2. Multilinear regression

Excel 2010 was used to implement MLR model. The train period was used to derive regression coefficient from getting the value of variables in the linear equation. The availability of trained equation, helped in testifying the last year data as the test period. In the first fold, the 1-day delay was considered for m 1 to 10, and second fold applied 83-day delay. Table 2 shows the results of both MLR and PSR-MLR in the test period.

Figure 3. (a) The relation between correlation function C(r) and r by various m; (b) Saturation of correlation dimension Ce (m) with embedding dimensions.

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Table 2. Fitness values for MLR and PSR-MLR methods in different embedding dimensions (bolded lines are the most accurate values).

Statistical indices for the fitness values showed m = 1 for 1-day delay and m = 4 for the reconstructed phase space with the value of (CD = 0.9565, RMSE = 3642.89 and MAE = 50.42) and (CD = 0.9572, RMSE = 3636.34 and MAE = 51.04), respectively. However, the difference between the two models is not considerable, in the large value of demand in long-term this difference can come into account. Figure 4 shows the comparison of observed and demand values. Moreover, the suggested equation for the best result by MLR is given by:

$$D\_{l+1} = -0.00854D\_l - 0.0366D\_{l+7} - 0.0128D\_{l+2\tau} + 0.9427D\_{l+3\tau} \,\text{.}\tag{13}$$

#### 4.3. Performance of artificial neural network

several methods were introduced in literature to calculate the value of optimum embedding dimension which may be more than 10 for the used time series in this study. This study aims at showing the performance of embedding dimension and reconstructed phase space, where m is only considered 1 to 10. Figure 2 shows the value of ACF for the demand series and reconstructed phase space (τ = 83). Figure 3a shows the relation between C(r) and r and (3b) correlation exponent by varying m. Figure 3b shows that the value of correlation exponent increases by m and as m = 17, the correlation exponent reaches a specific value (Ce = 3.41). This constant value of

Excel 2010 was used to implement MLR model. The train period was used to derive regression coefficient from getting the value of variables in the linear equation. The availability of trained equation, helped in testifying the last year data as the test period. In the first fold, the 1-day delay was considered for m 1 to 10, and second fold applied 83-day delay. Table 2 shows the

Figure 3. (a) The relation between correlation function C(r) and r by various m; (b) Saturation of correlation dimension Ce

Ce at m = 17 indicates the existence of the deterministic behavior of the time series.

Figure 2. (a) Autocorrelation function (τ); (b) reconstructed phase space by (τ and 2τ -day lag time).

4.2. Multilinear regression

140 Wavelet Theory and Its Applications

(m) with embedding dimensions.

results of both MLR and PSR-MLR in the test period.

ANN is another approach to model the demand values which represented in Section 3.4. ANN's structures have different hidden layer neurons (HLN) from 1 to 20 with 200 epochs

Figure 4. The performance of MLR and PSR-MLR in comparison with observed values.

for each model. Table 3 represents the result of ANN for both 1-day delay and PSR values. The results in the table for each m, are extracted from the result of various HLN and epochs. Figure 5 shows the example for selecting m = 3 among (20 200 = 4000). This calculation has been done for all m from 1 to 10 for both 1-day delay and PSR. (4000 10 2 = 80,000) number of calculations where the best 10 values have been selected (Table 3).

MAE = 47.13), respectively. Regarding the results, PSR-ANN mostly dominates in all embedding dimensions for the fitness accuracy indices. Figure 6 shows the comparison of observed and demand values in the test period for both ANN and PSR-ANN in m = 6 and 3, respectively.

Application of Wavelet Decomposition and Phase Space Reconstruction in Urban Water Consumption Forecasting:…

GEP preliminarily investigates the relationship between input and output as discussed in Section 3.5. Unlike the other models in this study, 1-day ahead is output, and various combinations of input in terms of m are considered as input variables. The arithmetic operations used in this study are {+, �, �, x, x2, √x}, and GEP applies them to fit the best accuracy between input and output variables. Further details of GEP initial term values are in following of [14, 38, 59] to extract the GEP model for both 1-day delay and PSR. The results are shown in the Table 4

According to the Table 4, there is not much difference among the different m. But the difference in PSR-GEP results can be considered as a proof of sensitivity to the initial values of specific time lags where the variations of the results for different m are more than 1-day delay. There is not a significant difference in the results in this study comparing to other alternative models, especially PSR-ANN is not an advantage of GEP. However, extracting the mathematical equation through GEP is one of advantage of GEP comparing to other artificial models. As a result of given model, equation for m = 3 (PSR-GEP) can calculate the demand value for 1-day

Dtþ<sup>τ</sup> þ Dtþ2<sup>τ</sup>

Although, variety of other arithmetic operations may have been applied here but focusing on the aim of study, only simple known operations were applied to extract the GEP equation. The results of PSR-GEP and alternative ones prove the advantage of PSR to improve the accuracy of the models. Statistical indices for the fitness values showed m = 2 for 1-day delay and m = 3 for the reconstructed phase space with the value of (CD = 0.9497, RMSE = 3609.82, and

<sup>p</sup> <sup>þ</sup> Dt � <sup>7</sup>:<sup>0838</sup> (14)

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143

Dtþ<sup>1</sup> <sup>¼</sup> <sup>0</sup>:<sup>0529</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Figure 6. The performance of ANN and PSR-ANN in comparison with observed values.

The results showed (Dt, Dt <sup>+</sup> <sup>τ</sup>, Dt + 2τ) as the best input combination for the models.

4.4. Performance of gene expression programming

for the test period.

ahead by:

Selection of ANN structures are represented in Table 3 for the test period. Statistical indices for the fitness values showed m = 6 for 1-day delay and m = 3 for PSR, with the values of (CD = 0.9520, RMSE = 3535.66 and MAE = 47.58) and (CD = 0.9578, RMSE = 3330.53 and


Table 3. Fitness values for ANN and PSR-ANN in different embedding dimensions \* m3 /day). (bolded lines are the most accurate values).

Figure 5. The results of ANN for τ = 83 PSR by various HLN and epochs.

MAE = 47.13), respectively. Regarding the results, PSR-ANN mostly dominates in all embedding dimensions for the fitness accuracy indices. Figure 6 shows the comparison of observed and demand values in the test period for both ANN and PSR-ANN in m = 6 and 3, respectively. The results showed (Dt, Dt <sup>+</sup> <sup>τ</sup>, Dt + 2τ) as the best input combination for the models.

#### 4.4. Performance of gene expression programming

for each model. Table 3 represents the result of ANN for both 1-day delay and PSR values. The results in the table for each m, are extracted from the result of various HLN and epochs. Figure 5 shows the example for selecting m = 3 among (20 200 = 4000). This calculation has been done for all m from 1 to 10 for both 1-day delay and PSR. (4000 10 2 = 80,000) number

Selection of ANN structures are represented in Table 3 for the test period. Statistical indices for the fitness values showed m = 6 for 1-day delay and m = 3 for PSR, with the values of (CD = 0.9520, RMSE = 3535.66 and MAE = 47.58) and (CD = 0.9578, RMSE = 3330.53 and

m Structure Epoch CD RMSE\* MAE m Structure Epoch CD RMSE\* MAE 1-5-1 110 0.9505 3611.56 48.34 1 1-6-1 150 0.9573 3369.55 47.63 1-3-1 140 0.9509 3602.25 48.25 2 1-4-1 20 0.9568 3369.50 47.83 1-16-1 20 0.9514 3554.13 48.06 3 1-2-1 120 0.9578 3330.53 47.13 1-16-1 170 0.9516 3550.99 47.90 4 1-2-1 70 0.9575 3333.67 47.25 1-3-1 160 0.9513 3561.33 47.98 5 1-3-1 110 0.9578 3340.36 47.15 6 1-9-1 50 0.9520 3535.66 47.58 6 1-3-1 40 0.9572 3340.16 47.46 1-3-1 100 0.9511 3563.14 48.08 7 1-3-1 100 0.9570 3348.68 47.80 1-8-1 20 0.9510 3570.70 47.84 8 1-2-1 150 0.9573 3333.88 47.24 1-4-1 200 0.9511 3566.00 47.69 9 1-2-1 140 0.9571 3338.53 47.89 1-3-1 100 0.9515 3546.72 47.94 10 1-4-1 10 0.9539 3518.86 49.28

m3

/day). (bolded lines are the most

of calculations where the best 10 values have been selected (Table 3).

ANN, τ = 1 PSR-ANN, τ = 83

Table 3. Fitness values for ANN and PSR-ANN in different embedding dimensions \*

Figure 5. The results of ANN for τ = 83 PSR by various HLN and epochs.

accurate values).

142 Wavelet Theory and Its Applications

GEP preliminarily investigates the relationship between input and output as discussed in Section 3.5. Unlike the other models in this study, 1-day ahead is output, and various combinations of input in terms of m are considered as input variables. The arithmetic operations used in this study are {+, �, �, x, x2, √x}, and GEP applies them to fit the best accuracy between input and output variables. Further details of GEP initial term values are in following of [14, 38, 59] to extract the GEP model for both 1-day delay and PSR. The results are shown in the Table 4 for the test period.

According to the Table 4, there is not much difference among the different m. But the difference in PSR-GEP results can be considered as a proof of sensitivity to the initial values of specific time lags where the variations of the results for different m are more than 1-day delay. There is not a significant difference in the results in this study comparing to other alternative models, especially PSR-ANN is not an advantage of GEP. However, extracting the mathematical equation through GEP is one of advantage of GEP comparing to other artificial models. As a result of given model, equation for m = 3 (PSR-GEP) can calculate the demand value for 1-day ahead by:

$$D\_{t+1} = 0.0529 \sqrt{D\_{t+\tau} + D\_{t+2\tau}} + D\_t - 7.0838 \tag{14}$$

Although, variety of other arithmetic operations may have been applied here but focusing on the aim of study, only simple known operations were applied to extract the GEP equation. The results of PSR-GEP and alternative ones prove the advantage of PSR to improve the accuracy of the models. Statistical indices for the fitness values showed m = 2 for 1-day delay and m = 3 for the reconstructed phase space with the value of (CD = 0.9497, RMSE = 3609.82, and

Figure 6. The performance of ANN and PSR-ANN in comparison with observed values.


wavelet transforms were applied (Section 3.7.). As suggested by Nourani et al. [83], 3rd level

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145

Table 5 indicates the results of wavelet decomposition for the selected models in the previous section. As the table highlights, db4 and db2 are the transforms which resulted in the highest accuracy in W-MLR and W-PSR-MLR, with the value of (CD = 0.9697, RMSE = 2804.44 and MAE = 42.11) and (CD = 0.9745, RMSE = 2699.83 and MAE = 43.61), respectively. After implying the decomposed inputs for MLR and PSR-MLR for result comparison improved the results in both models. Also, sym4 and db2 are the transforms which resulted in the highest accuracy in W-ANN and W-PSR-ANN, with the value of (CD = 0.9915, RMSE = 1486.21 and MAE = 30.06) and (CD = 0.9756, RMSE = 2517.24, and MAE = 41.68), respectively. Also, calculations for W-ANN and W-PSR-ANN are done with HLN 1 to 20 and epochs 1 to 200, and the mentioned results in the table are selective of the highest among them. Unlike the results of MLR, W-ANN forecasted accurately than W-PSR-ANN which is the inversion of the results of ANN and PSR-ANN. However, wavelet decomposition improved the results of W-ANN and W-PSR-ANN comparing to the alternative without decomposition (Table 3). Moreover, db4 and db2 are the transforms which resulted in the highest accuracy in W-GEP and W-PSR- GEP, with the value of (CD = 0.9845, RMSE = 2027.28 and MAE = 36.62) and (CD = 0.9753, RMSE = 2532.21, and MAE = 41.69), respectively. Following the results of ANN method, W-GEP forecasted accurately than W-PSR-GEP. However, wavelet decomposition improved the results of W-GEP and W-PSR-

All PSR models resulted in the highest values which used the decomposed inputs by db2 transform. It is noticeable that PSR affects the inherent of the time series which the results of performance of all models are in common about improving the accuracy. Considering this fact,

decomposition is recommended for 2186 point data.

GEP comparing to the alternative without decomposition (Table 4).

Figure 8. Three level DWT of daily water demand time series of Kelowna City in 2016.

Table 4. Fitness values for GEP and PSR-GEP in different embedding dimensions (bolded lines are the most accurate values).

Figure 7. The performance of GEP and PSR-GEP in comparison with observed values.

MAE = 48.37) and (CD = 0.9569, RMSE = 3343.36, and MAE = 47.50), respectively. Figure 7 shows the comparison of observed and demand values in the test period for both GEP and PSR-GEP in m = 2 and 3, respectively.

#### 5. Wavelet decomposition and models' performance

The combination of models with wavelet decomposition is derived by adding the output of each wavelet to the input of the models. Figure 8 shows the example of the decomposed values for water demand time series by db2 transform function. To discrete the demand values, five wavelet transforms were applied (Section 3.7.). As suggested by Nourani et al. [83], 3rd level decomposition is recommended for 2186 point data.

Table 5 indicates the results of wavelet decomposition for the selected models in the previous section. As the table highlights, db4 and db2 are the transforms which resulted in the highest accuracy in W-MLR and W-PSR-MLR, with the value of (CD = 0.9697, RMSE = 2804.44 and MAE = 42.11) and (CD = 0.9745, RMSE = 2699.83 and MAE = 43.61), respectively. After implying the decomposed inputs for MLR and PSR-MLR for result comparison improved the results in both models. Also, sym4 and db2 are the transforms which resulted in the highest accuracy in W-ANN and W-PSR-ANN, with the value of (CD = 0.9915, RMSE = 1486.21 and MAE = 30.06) and (CD = 0.9756, RMSE = 2517.24, and MAE = 41.68), respectively. Also, calculations for W-ANN and W-PSR-ANN are done with HLN 1 to 20 and epochs 1 to 200, and the mentioned results in the table are selective of the highest among them. Unlike the results of MLR, W-ANN forecasted accurately than W-PSR-ANN which is the inversion of the results of ANN and PSR-ANN. However, wavelet decomposition improved the results of W-ANN and W-PSR-ANN comparing to the alternative without decomposition (Table 3). Moreover, db4 and db2 are the transforms which resulted in the highest accuracy in W-GEP and W-PSR- GEP, with the value of (CD = 0.9845, RMSE = 2027.28 and MAE = 36.62) and (CD = 0.9753, RMSE = 2532.21, and MAE = 41.69), respectively. Following the results of ANN method, W-GEP forecasted accurately than W-PSR-GEP. However, wavelet decomposition improved the results of W-GEP and W-PSR-GEP comparing to the alternative without decomposition (Table 4).

All PSR models resulted in the highest values which used the decomposed inputs by db2 transform. It is noticeable that PSR affects the inherent of the time series which the results of performance of all models are in common about improving the accuracy. Considering this fact,

Figure 8. Three level DWT of daily water demand time series of Kelowna City in 2016.

MAE = 48.37) and (CD = 0.9569, RMSE = 3343.36, and MAE = 47.50), respectively. Figure 7 shows the comparison of observed and demand values in the test period for both GEP and

The combination of models with wavelet decomposition is derived by adding the output of each wavelet to the input of the models. Figure 8 shows the example of the decomposed values for water demand time series by db2 transform function. To discrete the demand values, five

PSR-GEP in m = 2 and 3, respectively.

5. Wavelet decomposition and models' performance

Figure 7. The performance of GEP and PSR-GEP in comparison with observed values.

GEP, τ = 1 PSR-GEP, τ = 83

/day) MAE m CD RMSE(m<sup>3</sup>

 0.9494 3621.87 48.59 1 0.9565 3363.46 48.03 2 0.9497 3609.82 48.37 2 0.9565 3357.00 47.82 0.9494 3633.87 48.42 3 0.9569 3343.36 47.50 0.9494 3637.74 48.42 4 0.9566 3359.53 47.95 0.9494 3639.05 48.43 5 0.9562 3372.70 48.04 0.9494 3619.77 48.60 6 0.9566 3359.64 48.08 0.9495 3630.44 48.38 7 0.9564 3365.04 47.95 0.9494 3634.41 48.42 8 0.9567 3353.24 47.62 0.9494 3628.46 48.42 9 0.9562 3370.08 48.05 0.9494 3631.12 48.40 10 0.9565 3356.68 47.84

Table 4. Fitness values for GEP and PSR-GEP in different embedding dimensions (bolded lines are the most accurate

/day) MAE

m CD RMSE(m<sup>3</sup>

144 Wavelet Theory and Its Applications

values).


Table 5. Fitness values for decomposition of selection of models for the test period (bolded lines are the most accurate values).

PSR can be introduced as a pre-processing method like wavelet decomposition; however, complexity and accuracy of PSR cannot be compared with the higher result of wavelet decomposition. Figure 9 shows the comparison of all selected models with highest accuracy (W-PSR-MLR, W-ANN, and W-GEP) in forecast of short-term water demand values.

This chapter presents the performance of two pre-processes methods in improving the accuracy of three models to forecast short-term urban water demand value in Kelowna City, BC, Canada. The first pre-process approach of PSR which is calculated by ACF method has improved the results of all models in this study. However, PSR does not improve the accuracy of models for entire dataset. Based on the behavior of time series, ACF or AMI (two lag time calculation methods) may have improved a non-deterministic dataset, but it seems in a chaotic dataset, PSR improves the performance of models in increasing accuracy with a proper number of embedding dimensions. Wavelet decomposition, the second pre-process method in the present study has also improved the accuracy of the models but, decomposition did not work on PSR based methods except MLR. It can be concluded that PSR and wavelet are in common with their outfits as two applicable pre-process methods. Also, PSR pre-processing is simpler than wavelet. Therefore, it is recommended to use PSR for the models. As per the results of this study it seems PSR works on a chaotic dataset which seems to be considered as disadvantage of PSR. Comparing the mentioned two pre-process methods, wavelet decomposition is significant to use, though, it is time-consuming and complex than PSR. Also, each transform functions have specific application where each of them can be used independently (e.g., seasonal,

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147

Figure 9. The performance of the W-models in comparison with observed values.

de-noising, peak points, etc.).

Figure 10. Residual values of the selected W-models.

The figure shows that the performance of W-ANN and W-GEP is better than W-PSR-MLR, while W-ANN's calculated values are more accurate than W-GEP in simulating peak points. This study eventually would suggest that these peak points are indication of critical issues related to water distribution system (pressure management, peak time demand, etc.) taking in account the performance of the models and simulations of highest and lowest values of demands. Therefore, it is recommended to evaluate models' performance in two separate parts as maximum values and minimum values along with evaluating criteria such as CD, RMSE, and MAE for the test period. The difference is not visible in Figure 9. Therefore focusing on Figure 10, it shows the performance of models by residual values in the test period.

In Figure 10 the residual values show the remarkable difference of performance of models. W-ANN values distributed in the area of (15%, +15%), unlike other two models. W-GEP dominates over W-PSR-MLR; however, the fitness criteria values for both are very close to each other (Tables 2 and 4).

Application of Wavelet Decomposition and Phase Space Reconstruction in Urban Water Consumption Forecasting:… http://dx.doi.org/10.5772/intechopen.76537 147

Figure 9. The performance of the W-models in comparison with observed values.

Figure 10. Residual values of the selected W-models.

PSR can be introduced as a pre-processing method like wavelet decomposition; however, complexity and accuracy of PSR cannot be compared with the higher result of wavelet decomposition. Figure 9 shows the comparison of all selected models with highest accuracy (W-PSR-

Table 5. Fitness values for decomposition of selection of models for the test period (bolded lines are the most accurate values).

W-MLR CD 0.9612 0.9477 0.9697 0.9677 0.9694

W-PSR-MLR CD 0.9670 0.9745 0.9719 0.9745 0.9712

W-ANN CD 0.9868 0.9816 0.9861 0.9856 0.9915

W-PSR-ANN CD 0.9685 0.9756 0.9723 0.9752 0.9715

W-GEP CD 0.9721 0.9766 0.9845 0.9297 0.9255

W-PSR-GEP CD 0.9667 0.9753 0.9721 0.9748 0.9704

haar db2 db4 sym2 sym4

/day) 3168.62 3681.17 2804.44 2893.60 2816.06

/day) 3008.34 2699.83 2811.58 2699.83 2845.69

/day) 1853.11 2189.15 2136.25 1948.28 1486.21

/day) 2867.87 2517.24 2677.44 2547.89 2724.56

/day) 2698.16 2492.46 2027.28 4311.60 4429.89

/day) 2937.76 2532.21 2689.20 2555.82 2770.80

MAE 44.48 49.04 42.11 43.54 42.24

MAE 45.39 43.61 43.95 43.61 44.24

MAE 33.78 36.91 39.50 33.86 30.06

MAE 43.16 41.68 42.09 42.19 42.61

MAE 41.21 39.05 36.62 54.23 55.25

MAE 43.66 41.69 42.13 41.90 42.51

The figure shows that the performance of W-ANN and W-GEP is better than W-PSR-MLR, while W-ANN's calculated values are more accurate than W-GEP in simulating peak points. This study eventually would suggest that these peak points are indication of critical issues related to water distribution system (pressure management, peak time demand, etc.) taking in account the performance of the models and simulations of highest and lowest values of demands. Therefore, it is recommended to evaluate models' performance in two separate parts as maximum values and minimum values along with evaluating criteria such as CD, RMSE, and MAE for the test period. The difference is not visible in Figure 9. Therefore focusing on

In Figure 10 the residual values show the remarkable difference of performance of models. W-ANN values distributed in the area of (15%, +15%), unlike other two models. W-GEP dominates over W-PSR-MLR; however, the fitness criteria values for both are very close to each

MLR, W-ANN, and W-GEP) in forecast of short-term water demand values.

Models Fitness Transform functions

RMSE(m<sup>3</sup>

146 Wavelet Theory and Its Applications

RMSE(m<sup>3</sup>

RMSE(m<sup>3</sup>

RMSE(m<sup>3</sup>

RMSE(m<sup>3</sup>

RMSE(m<sup>3</sup>

Figure 10, it shows the performance of models by residual values in the test period.

other (Tables 2 and 4).

This chapter presents the performance of two pre-processes methods in improving the accuracy of three models to forecast short-term urban water demand value in Kelowna City, BC, Canada. The first pre-process approach of PSR which is calculated by ACF method has improved the results of all models in this study. However, PSR does not improve the accuracy of models for entire dataset. Based on the behavior of time series, ACF or AMI (two lag time calculation methods) may have improved a non-deterministic dataset, but it seems in a chaotic dataset, PSR improves the performance of models in increasing accuracy with a proper number of embedding dimensions. Wavelet decomposition, the second pre-process method in the present study has also improved the accuracy of the models but, decomposition did not work on PSR based methods except MLR. It can be concluded that PSR and wavelet are in common with their outfits as two applicable pre-process methods. Also, PSR pre-processing is simpler than wavelet. Therefore, it is recommended to use PSR for the models. As per the results of this study it seems PSR works on a chaotic dataset which seems to be considered as disadvantage of PSR. Comparing the mentioned two pre-process methods, wavelet decomposition is significant to use, though, it is time-consuming and complex than PSR. Also, each transform functions have specific application where each of them can be used independently (e.g., seasonal, de-noising, peak points, etc.).
