# Multiparametric Analysis of a Gravity Retaining Wall
## 4 Sensitivity Analysis  
The optimal design of the gravity retaining wall was determined for different input data. The five main input data are the height of the retained ground (\(H_{s}\)), the variable surcharge load (\(q_{Q,k}\)), the slope angle (\(\beta\)), the shear angle of the soil (\(\varphi_{k}=\varphi_{rot}=\varphi_{found}\)) and the interaction coefficient between the soil and the retaining wall (\(k=k_{ret}=k_{found}\)). The main objective of this multiparametric analysis was to use these main attributes to predict other continuous attributes, such as the optimal cost of the retaining wall (\(COST\)), the utilization of the bearing capacity condition (\(V_{d}/R_{d}\)), the utilization of sliding condition (\(H_{Ed}/H_{Rd}\)) and the utilization of the eccentricity condition (\(e_{B}/e_{max}\)). Before applying the predictive model, the dataset was divided into a training dataset (odd indexed samples) and a checking dataset (even indexed samples). The \"exhsrch\" function included in MATLAB was used to perform an exhaustive search within the available inputs to select the set of inputs that have the greatest impact on the optimal cost of the gravity retaining wall and the utilization of multiple constraints. For the \"exhsrch\" function, the predictive models were built for each parameter combination and trained for an epoch; the achieved performance was then reported. The leftmost input variable in Figure 12 is the most relevant in terms of output, while it has the lowest root-mean-square error (RMSE). The \(RMSE\) is defined as follows:  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}(\hat{x}_{i}-x_{i})^{2}}{n}} \tag{7}\]  
where \(\hat{x}_{i}\) are the predicted values and \(x_{i}\) are the values obtained by the optimization procedure (\(COST\), \(V_{d}/R_{d}\), \(H_{Ed}/H_{Rd}\), \(e_{B}/e_{max}\)). The prediction models are often subject to the problem of overfitting. However, in this simple prediction model, it can be seen that the training and checking errors are comparable, which means that there is no overfitting. It should be noted that the main objective of this prediction model is to find the inputs that have the greatest impact on the output, not to build the prediction model with minimum training error. It also examines the combination of two inputs that have the greatest influence on the output. The result of the parametric analysis clearly shows that the most important parameter for the optimal cost of a gravity retaining wall is the height of the retained ground, followed by the shear angle of the soil, the interaction coefficient, the slope angle and the variable surcharge load. The shear angle of the soil is most relevant to the bearing capacity and eccentricity condition, while the interaction coefficient is most relevant to the sliding condition. The parametric analysis also shows that the \"height of the retained ground\" and the \"shear angle of the soil\" form the optimal combination of two inputs that are most relevant to the optimal cost and bearing capacity condition. However, the combination of \"shear angle of soil\" and \"interaction coefficient\" is most relevant for the sliding and eccentricity condition.