## 3 Application Results and Discussion
### Model Validity  
External validation (EV) is used for comparison of the results of estimated and experimental data. [PERSON] and [PERSON] [79] adopted new external validation criteria to assess the estimation precision of models according to the performance of validation data. EV means assessing the model performance with independent samples [80]. \[\sum_{i=1}^{n}\frac{t_{obs}\times t_{pre}}{t_{pre}^{2}} \tag{12}\]  
\[\frac{t_{obs}\times t_{pre}}{t_{obs}^{2}} \tag{13}\]  
where \(t_{obs}\) and \(t_{pre}\) represent the experimental and estimated target values, respectively. \[m=\left(R^{2}-R_{0}^{2}\right)/\ R^{2}<0.1 \tag{14}\]  
\[n=\left(R^{2}-R_{0}^{{}^{\prime}2}\right)/\ R^{2}<0.1 \tag{15}\]  
Furthermore, [PERSON] and [PERSON] [81] used \(R_{m}\) (calculated by Equation (14)), a stabilization criterion, for external predictability of the models [81]. They found that an \(R_{m}\) value less than 0.5 shows an appropriate simulation. \[R_{m}=R^{2}\times\left(1-\sqrt{|R^{2}-R_{0}^{2}|}\right)>0.5 \tag{16}\]  
The determination coefficients passing through the source between the estimated and experimental values (\(R_{0}^{2}\)), and conversely (\(R_{0}^{{}^{\prime}2}\)), are derived using the following equations:  
\[R_{0}^{2}=1-\sum_{i=1}^{n}t_{pre}^{2}(1-k)^{2}/\sum_{i=1}^{n}(t_{pre}-\overline {t_{pre}})^{2} \tag{17}\]  
\[R_{0}^{{}^{\prime}2}=1-\sum_{i=1}^{n}t_{obs}^{2}(1-k^{\prime})^{2}/\sum_{i=1}^ {n}\left(t_{obs}-\overline{t}_{obs}\right)^{2} \tag{18}\]  
The validation indicator and the related performance of CS, TS, and FS prediction obtained by the various models are presented in Table 6. According to this table, the RF models for CS, TS, and FS,  
Figure 9: Time series and residual plots of the testing phase of the classification-based regression methods for FS estimation. which yielded \(R_{m}=0.691\), \(R_{m}=0.834\), and \(R_{m}=0716\), respectively, satisfy the conditions and provide the best validation compared to the other models. In addition, the CART and M5 tree values for CS (\(R_{m}=0.187\)), TS (\(R_{m}=0.195\)), and FS were less than the required value for \(R_{m}\) (\(R_{m}>0.5\)). Thus, it is seen that RF shows highest validity for predicting the mechanical characteristics of RCCP. Monte-Carlo simulation (MCS)-based uncertainty analysis is used for determining the randomness of model uncertainty. This method was first used by [PERSON] and [PERSON] [82] in military projects for simulation of probabilistic events. It is well known that CS, TS, and FS contains various uncertainties, such as uncertainty of input variables and uncertainty of model parameters. For this purpose, an investigation of quantitative uncertainty associated with output prediction rate (E) was performed using the RF, M5 rule, M5p, and CHAID models. The MCS was performed for CS, TS, and FS values. The individual error of prediction was calculated for all the datasets (Equation (19)). Equations (20) and (21) are utilized for calculation of the mean (\(\overline{e}\)) and standard deviation (\(S_{e}\)) of the estimation error, respectively [76]:  
\[e_{i}=\log_{10}(t^{pre}_{\ i})-\log_{10}(t^{exp}_{\ i}) \tag{19}\]  
\[\frac{t_{obs}\times t_{pre}}{t_{obs}^{2}} \tag{13}\]0  
\[\frac{t_{obs}\times t_{pre}}{t_{obs}^{2}} \tag{13}\]1  
In the above equations, n is the dataset length, and \(t^{pre}\) and \(t^{obs}\) denote the estimated and experimental target values, respectively. A positive mean prediction denotes an overestimated prediction of the target variable, and a negative one denotes an underestimated value of the target variable compared to the observed values. Thus, a confidence band could be drawn around the predicted error value through application of [PERSON] score approach [83,84].