## 4 Generation of the Deep Neural Network
### Performance of the DNN
The fourth parameter is the kurtosis of a sample:  
\[k=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_{i}-\overline{x}}{\sigma}\right)^{4} \tag{5}\]  
The kurtosis is a measure for how outlier-prone a distribution is. For example, a normal distribution has a kurtosis of \(k=3\). The higher the kurtosis, the more the data spreads around the mode of a distribution. The variance, skewness, and kurtosis are also known as the second, third, and fourth central moments of a distribution. Figure 8 shows that, in particular, the kurtosis and skewness vary for the measured geometrical parameters and the SCF. In contrast, the parameters are similar on the left and right side of the weld seam. Interestingly, three parameters are left- and three are right-skewed. From a visual assessment, some histograms could be described by a normal or lognormal distribution; however, even for such a small step size of 0.15 mm, the amount of data is limited and not yet representative for longer weld seams. Thus, in the next step, the data along the full weld seam of about 1 m length are assessed. Table 2 presents the skewness and kurtosis of different typical two-parameter distribution functions that are compared to the investigated parameters. In this table, \(\sigma\) and \(\alpha\) are the shape parameters of the Lognormal, Gamma, and Inverse Gamma distributions, respectively. In Figure 9, exemplary histograms for three welding methods are shown, as it would be impossible to directly compare the change of weld parameters along the weld seam. Again, differences between weld parameters are clearly visible. In addition, differences between the welding methods become obvious. This supports findings by [PERSON] et al. [12], who presented a comparison of distributions for weld toe radii and flank angles obtained from literature. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution** & **Skewness** & **Kurtosis** \\ \hline Normal & 0 & 3 \\ Exponential & 2 & 9 \\ Lognormal & \(\left(e^{x^{2}}+2\right)\sqrt{e^{x^{2}}-1}\) & \(e^{4x^{2}}+2e^{3x^{2}}+3e^{2x^{2}}-3\) \\ Gamma & \(2/\sqrt{\alpha}\) & \(3+6/\alpha\) \\ Inverse gamma & \(\frac{4\sqrt{\alpha-2}}{\alpha-3}\) for \(\alpha>3\) & \(\frac{6(5\alpha-11)}{(\alpha-3)(\alpha-4)}+3\) for \(\alpha>3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Skewness and kurtosis of a number of typical distribution functions. Interestingly, while, for example, the histogram for the weld toe angle of the flux-cored arc welded plate is only slightly skewed, see Figure 9b, and somewhat resembles a normal distribution, the histograms of the other two plates, see Figure 9a,b, are significantly more skewed. Thus, below, the measured sample parameters are used next to determine whether the weld parameters can be described by well-known distribution functions. To this end, a skewness-kurtosis plot is created to compare these two important features of the measured samples with theoretical distributions, see Figure 10. As reference, the distributions presented in Table 2 are used. As skewness values may be either positive or negative, the square of skewness is typically used in such plots. The typical range for such a plot is \(s^{2}<4\) and \(k<9\); nevertheless, in some cases, data falls outside this range, see [60]. In fact, for every weld parameter, a number of measures for the different plates fall outside this range. Thus, the range was increased to accommodate data that is only slightly outside this range; however, there are still a number of measurements that are outside this range. This might be related to a number of reasons, which are subsequently discussed, but first the main observations based on Figure 10 are listed:  
* The measured sample skewness and kurtosis of the weld toe radii is close to the line representing the lognormal, the gamma, and the inverse gamma distribution. Interestingly, the weld toe radii also show, on average, the smallest scatter of kurtosis. * The majority of plates have weld toe angles with distributions that are only slightly skewed.