## 3 Results and Discussion
### Soft Computing Predictive Models  
This paper applied two ensemble approaches on the two most commonly used ANN models for predicting the peak uplift resistance of pipelines buried in sand. The ensemble techniques employed in this research include boosting and bagging techniques. ANNMLP and ANN-RBF were also developed to estimate the peak uplift resistance of pipes buried in sand. Two standard models (STANDARDANNMLP and STANDARDANNRBF), without applying the ensemble approaches, were developed to benchmark the ANN-ensemble models. Before the models' development, the data were divided into train and test partitions with an 80:20 proportion. The results of these models were then analyzed using a number of performance criteria, including the mean absolute error (MAE), correlation coefficient (R), and gains. The expression for calculating the correlation coefficient (R) and mean absolute error (MAE) are presented below:  
\[R=\sqrt{1-\frac{\Sigma_{i}(y_{i}-f_{i})^{2}}{\Sigma_{i}(y_{i}-\overline{y})^{2}}} \tag{1}\]  
\[MAE=\frac{1}{N}\sum_{j=1}^{N}\abs{y_{i}-y_{j}} \tag{2}\]  
where \(y\) are the measured values, \(\overline{y}\) and \(y^{\prime}\) show are the mean and predicted values of \(y\) and \(N\) is the total number of data. This research initially applied standard ANN models based on MLP and RBF structures to a database of 44 laboratory test data. In these two models, a single hidden layer, with a maximum training time of 5 min, and an over-fit prevention of 30% were used. It is worth mentioning that the neural network modelling internally divides samples into a model building set and an over-fit prevention set. The structure of these models are shown in Figure 7.  
Figure 6: The effects of geo-grid length and number of layers on the peak uplift resistance of pipes buried in sand. Thereafter, bagging and boosting techniques were applied to the trained ANN, to assess the effectiveness of these techniques when applied to standard ANN models. For these four models (ANNMLP-BOOSTED, ANNMLP-BAGGED, ANNRF-BOOSTED, and ANNRFB-BAGGED), the \"mean\" value was set as the combination rule for the target variable. In addition, the quantity of component models for boosting and bagging was established as in [10]. The measured and predicted peak uplift resistances of the buried pipes for all six models are shown in Figure 8.  
Figure 7: The structures of the standard ANN models based on MLP and RBF approaches to predict the peak uplift resistance values. The performance of these models was assessed using the criteria mentioned above. In addition, a simple ranking system was deployed to obtain a comprehensive assessment. In this system, each model performance is evaluated based on R and MAE within each partition. Then, each model is ranked according to the value of each criterion. Thus, for R, the highest rank is assigned to a model which obtains the highest value; and for RMSE, the highest rank should be given to a model which acquires the lowest value. It should be noted that, in this case, the highest value is six because this research deployed six models. The cumulative rank of each model is calculated using the following expression:  
\[\centering\text{\emph{Cumulative ranking}}\ =\ \sum(\alpha_{tr}+\alpha_{te}+\beta_{tr}+ \beta_{te})\@add@centering \tag{3}\]  
where \(\alpha\) shows the ranking of R, \(\beta\) represents the ranking of MAE, with the subscript \"\(tr\)\" and \"\(te\)\" denoting training and testing values respectively. Table 3 shows the results of each performance criterion and the rankings and cumulative ranking of each model. As can be seen, the ANNMLP-BOOSTED model obtained the highest cumulative, training R, and training and testing MAE rankings. Alternatively, the ANNRFB-BAGGED model achieved the lowest cumulative ranking in addition to the testing and training of R and MAE rankings. For the total ranking of the training phase, the highest-ranking was achieved by ANNMLP-BOOSTED, while both ANNMLP-BOOSTED and ANNMLP-BAGGED achieved the highest and similar testing phase total ranking.