# An Evolutionary Neural Network Approach for Slopes Stability Assessment
## 3 Methodology
### Sensitivity Analysis  
A model's interpretability is of paramount importance, particularly from an engineering point of view, to obtain a detailed understanding of what has been learned by the models. Indeed, data-driven models, namely those based on ANN algorithm, are usually designed as black box models. Explainable artificial intelligence (xAI) [65; 66] is a well-established domain with a thriving community that has developed a number of highly effective methods to explain and interpret the predictions of complex machine learning models, such as deep neural networks. In this study, a comprehensive sensitivity analysis (SA) [67] was conducted on the final models (i.e., the Pareto optimal solutions). SA is a simple method that was implemented following the training phase and measured the model's responses when a given input was modified, thereby allowing one to quantify the relative importance of each attribute.  
The SA approach is inspired by experimental design [68], which is often used to identify causal relationships. There are other approaches to obtain knowledge from trained ANNs, such as the extraction of rules from trained ANNs. However, such approaches tend to discretize or simplify the predictive model's responses, leading to rules that do not accurately represent the original model or that consider just single-input interactions. The advantage of the SA approach is that it accurately measures the trained ANN's responses (even if the class decision boundaries are complex) under a computationally efficient mapping approach. In effect, and in accordance with the findings of [PERSON] and [PERSON] [69], in this research, we employed a global sensitivity analysis (GSA) technique that is capable of detecting interactions among all combinations of input attributes. In this context, \(F\) inputs are changed at same time. \(F\) can range from one, as in a one-dimensional SA, denoted as 1-D, to \(I\), as in an \(I\)-D SA. Iteratively, each input assume different values withing its range by \(L\) levels. The other inputs are held constant at the baseline, \(b\). Seven levels (\(L=7\)) were adopted in this research, which allowed an interesting amount of detail to be added at a reasonable computational effort. For the baseline \(b\), the average value of each input was considered.  
In short, the ANN was initially calibrated to the entire dataset. The fitted model was then subjected to the GSA algorithm, and the respective sensitivity responses were stored. Next, significant visualization techniques were computed using these responses. Specifically, the input importance bar graph illustrates the relative impact of each input variable (from 0% to 100%). The rationale behind SA is that the input is more significant the greater the changes observed in the output. Following the recommendation of [PERSON] and [PERSON] [69], the gradient metric was chosen to measure this effect:  
\[g_{a}=\sum_{j=2}^{L}|\hat{y}_{a,j}-\hat{y}_{a,j-1}|/(L-1) \tag{1}\]where \(a\) denotes the input variable under analysis and \(y_{a,j}\) is the sensitivity response for \(x_{a,j}\). We note that in [69], other SA measures were tested, including range and variance. However, in that study, it was shown that the gradient measure is more theoretically sound and provided better experimental evidence when compared with the range and variance measures (e.g., the variance tended to overemphasize the importance of the most relevant input, while the range provided an identical value for linear or high-frequency wave responses that produced the same range). Having computed the gradient for all inputs, the relative importance (\(R_{a}\)) was then calculated using:  
\[R_{a}=g_{a}/\sum_{i=1}^{I}g_{i}\cdot 100(\%) \tag{2}\]