## 3 Artificial Neural Network
### Exploratory Data Analysis
#### 3.3.1 Searching for Correlation  
At this stage of the project, we only deal with training data. The first step in regression problems should be looking for the correlation between the features and the label (damping ratio, D). To do this, we calculate a linear correlation coefficient (Pearson correlation coefficient) between each pair of values (Table 4).  
Thanks to the correlation matrix, we can determine the degree of correlation of individual features with the damping ratio (Table 5).  
The values of the correlation coefficient range from \(-1\) to \(1\). Values close to \(1\) indicate a strong positive correlation, values close to \(-1\) say that there is a strong negative correlation, while values close to \(0\) indicate no linear correlation. However, it should be remembered that the Pearson correlation coefficient measures only a linear relationship. It may completely ignore the nonlinear relationship. The highest positive correlation with the damping factor is shown by the \(\gamma\), while the highest negative correlation with the \(\text{G}/\text{G}_{\text{max}}\) is shown by the \(\text{G}\) and \(\text{p}^{\prime}\).  
Two variables may also be related by a nonlinear relationship or may have a non-Gaussian distribution. In these cases, [PERSON]'s correlation coefficient can be useful to determine the strength and direction between the two data samples. [PERSON]'s rank correlation can also be used if there is a linear relationship between the variables, but will have slightly less power [35]. Table 6 presents the result of the [PERSON] correlation test.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**D** & \(\gamma\) & **Sa + Gr** & **OCR** & **IL** & **e\({}_{\mathbf{o}}\)** & **IP** & **Cl** & **LL** & **p\({}^{\prime}\)** & **Si** & **PL** & **G** & **G/Gmax** \\ \hline
1.00 & 0.62 & 0.39 & 0.11 & \(-\)0.04 & \(-\)0.29 & \(-\)0.30 & \(-\)0.32 & \(-\)0.40 & \(-\)0.40 & \(-\)0.48 & \(-\)0.48 & \(-\)0.65 & \(-\)0.71 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Degree of correlation ([PERSON]’s correlation coefficient) of individual features with the damping ratio.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline  & **Cl** & **Si** & **Sa + Gr** & **LL** & **PL** & **IP** & **IL** & **p\({}^{\prime}\)** & **e\({}_{\mathbf{o}}\)** & **OCR** & **Y** & **G** & **G/Gmax** & **D**  
From [PERSON]'s correlation, we see a moderately strong negative correlation of the damping ratio (D) with PL, Si, p\({}^{\prime}\), LL, Cl, and IP. We can observe that the other features are also characterized by a higher degree of correlation than in the [PERSON]'s correlation, which may indicate that the dependence of these features with D is nonlinear. On the other hand, a decrease of degree of correlation of values, such as the G/G\({}_{\text{max}}\) and \(\gamma\), can be noticed.  
Another way to check the correlation between the attributes is a scatter matrix. The scatter matrix of all numerical attributes would be unreadable (and unjustified). Figure 1 shows a scatter matrix for some of the most promising attributes. Both the color and size of the markers relate to the value of the damping ratio (D). The plot shows a matrix of scatter plots where each variable is plotted against all other variables, resulting in a grid of scatter plots. The diagonal of the plot shows the kernel density estimation (KDE) plot for each variable, which provides a visual representation of the distribution of each variable.  
The above charts show that the correlation between the shear strain (\(\gamma\)), normalized shear modulus (G/G\({}_{\text{max}}\)), mean effective stress (p\({}^{\prime}\)), and the damping ratio (D) is strong and the individual points are quite close to each other.