# Application of Machine Learning Tools for Long-Term Diagnostic Feature Data Segmentation
## 5 Simulated Data Analysis
### Extraction of Features from Simulated Signal for Non-Gaussian Noise Case  
By analogy, as for the Gaussian case, now the procedure is applied for simulated data with non-Gaussian noise. In Figure 14, all features are presented. It is clear that some of them are not suitable, so they are not considered for later stages. In Figure 15 (left top panel), three selected features are presented. One may conclude that the concept of moving windows and selected statistics as features works properly for Gaussian noise. However, for non-Gaussian noise, when a single outlier occurs, the effect generated by moving window and overlapping produces more artifacts than the original data contain. Hence, while statistics, such as sample mean, are sensitive to a single outlier, also the classical parameters responsible for variance (as sample variance) are not recommended for data with impulsive noise.  
In Figure 16, features are plotted in 3D space. Similar to the Gaussian case, regime 3 is clearly noticeable; however, regimes 1 and 2 can be distinguished as well. The impact of distortion of the features does not appear to be significant. In the next step, the data are processed by PCA. Again, the first component (PC1) contains the majority of information; see Figure 17. In Figure 15, one can see that the shape of PC1 is similar to the original data, and PC2 may be useful to recognize a third regime.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **1 st Point** & **2 nd Point** \\ \hline EM & 998 & 1500 \\ K-Means & 1240 & 1570 \\ BIRCH & 1390 & 1580 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Final changing points in case of Gaussian noise.  
Figure 13: The expansion of the clustering result to whole of data for simulated signal in presence of Gaussian noise: (**a**) results for EM algorithm, (**b**) results for K-Means algorithm, (**c**) results for BIRCH algorithm.  
Figure 16: Features space for simulation signal in presence of non-Gaussian noise.  
Figure 14: Statistical features obtained for non-Gaussian \(\alpha-\)stable simulation case.  
Figure 15: PCA of stochastic features for simulation signal in presence of non-Gaussian noise.  
In Figure 18, the PC1 time series and its kernel density is presented. One may notice a strong artifact around T = 100. However, based on the empirical probability density function, it is possible to establish threshold values to identify regimes in PC1 (thresholds at c.a PC1 = 0 and PC1 = 10).  
In Figure 19, the final results of clustering in the PC domain are demonstrated. The degradation process starts from the healthy stage (green points) to the warning and alarm stages (yellow and red points). As it can be seen in Figure 19, K-means and BIRCH detected regimes as the same while EM algorithm discover the second regime around (\(-3\), \(-2\)).  
Furthermore, K-means and BIRCH can separate regimes representing the warning and alarm stages, i.e., the yellow and red points. The results of EM for separating yellow and red points are acceptable. Finally, in Figure 20, the last results for simulated data with non-Gaussian noise are presented. It is clearly seen that the results include some mixture of classes that could not be seen in simulations with Gaussian data. Additionally, Figure 20 confirms that the results of clustering with K-means and BIRCH algorithms are correct. However, both of them detected the second stage with significant delay. The delay for EM clustering is less than K-means and BIRCH. The exact values of changing point (CP) are presented in Table 3.  
Figure 17: PCA score in case of simulation noise is non-Gaussian.  
Figure 18: PC1 time series, its Kernel density of features and final results of segmentation expressed by colors for simulation signal in presence of non-Gaussian noise.