After the qualitative modeling of the river water, by using seven approaches, the performance of the models was investigated. These approaches are performance analysis with 1) scattered plot, 2) response plot, 3) performance indicators, 4) discrepancy ratio, and 5) error distribution. Finally, the Uncertainty Analysis by Wilson Score Method (UA-WSM) is performed for models. The results are presented in order.
5.1. Scattered Plot
These plots show the correlation rate between input and output values based on a linear relationship.
According to the scattered plots in Fig. 4, the best correlation of the outputs with the observed values was identified for model GAELM(EC). Also, model GAELM(DO) was ranked second in correlation. ELM(DO) and ELM(EC) models were ranked following places, respectively.
5.2. Response Plot
After the simulation, by drawing the response plots, we can check the accuracy of the prediction. Indeed, the closer actual values to predicted values show higher modeling precision.
Based on the above plots, the most accurate prediction was assigned to model GAELM(EC) due to the most significant adaptation between predicted and actual values. Also, model GAELM(DO) was ranked second in modeling accuracy. Meanwhile, model ELM(DO) provided higher accuracy than model ELM(EC).
5.3. Performance metric
After the DO and EC parameters simulation, the performance indicators results are shown in Table 1. The MSE, RMSE, and R indices were chosen to evaluate the models' performance. Based on statistical indicators, which are written in Table.1, the GAELM(EC) model had the best performance indices with MSE, RMSE, and R equal to 0.0170, 0.1304, and 0.9284, respectively. The GAELM(DO)models also performed well and earned second place.
Table 1
No.
|
Model
|
MSE
|
RMSE
|
R
|
1
|
ELM-EC
|
0.2350
|
0.4848
|
0.5053
|
2
|
GAELM-EC
|
0.0170
|
0.1304
|
0.9284
|
3
|
ELM-DO
|
0.5694
|
0.7546
|
0.3717
|
4
|
GAELM-DO
|
0.2600
|
0.5099
|
0.6506
|
5.4. Discrepancy ratio
The discrepancy ratio (D.R) parameter is obtained by allocate the predicted on observed value. The D.R has two primary components. First, line D.R = 1, and the second is D.R points. More concentration of these points on line D.R = 1 will indicate higher accuracy in modeling (Poursaeid et al. 2021). Based on Fig. 7, and according to this concept, it can be noted that the GAELM(EC) model has the best performance in modeling. Also, the ELM(EC) model ranked in second place related to model accuracy. The GAELM(Do) and ELM(DO) models earned further positions.
5.5. Error distribution
This chart is expressed based on the percentage of prediction error. Meanwhile, calculating the prediction error, the error percentage is calculated. Then, the percentage is categorized as less than 5%, 5%-10%, and more than 10%. According to the error distribution graphs in Fig. 8, the GAELM(EC) model was the most accurate. Then, the GAELM(DO) model had a better performance in prediction error. Moreover, the ELM(DO) and ELM(EC) models were ranked next places.
5.2. Uncertainty Analysis (UA-WSM)
Uncertainty analysis is an essential part of regression problems. In this part, the basic parameters of UA-WSM are calculated. The parameters of this analysis are mean error (µ), standard deviation (σ), the width of uncertainty bandwidth (WUB), and confidence band. The ± 1.64σ causes to reach the 95% confidence bandwidth denoted by PEI_95%. Also, in this analysis, the µ and σ are calculated based on Eq. (13) to (15) (Bonakdari et al. 2020; Poursaeid et al. 2020b; Poursaeid, Poursaeed, et al. 2022a)
Table 2. UAWSM results
No.
|
Model
|
WUB
|
𝛍
|
𝛔
|
95% PEI
|
Performance
|
Upper bound
|
Lower bound
|
1
|
ELM-EC
|
0.0626
|
0.4046
|
0.2692
|
0.4673
|
0.3420
|
Overestimation
|
2
|
GAELM-EC
|
0.0692
|
0.0100
|
0.1313
|
0.0792
|
-0.0592
|
Overestimation
|
3
|
ELM-DO
|
0.3028
|
-0.2192
|
0.7281
|
0.0836
|
-0.5221
|
Underestimation
|
4
|
GAELM-DO
|
0.1420
|
0.0508
|
0.5212
|
0.1928
|
-0.0913
|
Overestimation
|
After the operation, the ELM(DO) had an underestimation performance. Also, other models had overestimation performance. Also, the GAELM(EC) was the best and most accurate model according to its least mean error value (mean error = 0.0100). Another model's results are presented in Table 2.