## 6 Features Selection for Pattern Feature Vector Construction  
A statistical approach was employed to evaluate the correlation between segmented time domain, frequency domain and fractal features, on the one hand, and the measured tool flank wear values, on the other hand, using the Spearman correlation coefficient, \(r_{\rm S}\):  
\[r_{s}\ =\ \frac{cov(rg_{X},rg_{Y})}{\sigma_{rg_{X}}\sigma_{rg_{Y}}}\]  
where \(cov(rg_{X},rg_{Y})\) is the covariance of the rank variables (i.e., \(rg_{X}\) = features values, \(rg_{Y}\) = tool flank wear values) and \(\sigma_{rg_{X}}\sigma_{rg_{Y}}\) are the standard deviations of the rank variables.  
If \(0<r_{s}<0.3\), the correlation is weak; if \(0.3<r_{s}<0.7\), the correlation is adequate; if \(0.7<r_{s}<1\), a strong correlation occurs. Based on the \(r_{\rm S}\) value, the features displaying the highest correlation with tool wear level were selected as follows:  
* Time domain features--Three features were selected from the thrust force signals: thrust force average (F\({}_{\rm mean}\)), thrust force variance (F\({}_{\rm var}\)) and thrust force kurtosis (F\({}_{\rm kurt}\)), and one feature was selected from the torque signal: torque average (T\({}_{\rm mean}\)). All these features displayed a strong correlation with tool wear level.  
Figure 16: Regularization analysis curves: (**a**) rectangular kernel; (**b**) Gaussian kernel [33].  
* Frequency domain features--The following set of features was selected: Fpeak2x, Fpeak4x, Fpeak6x, Tpeak2x, and Tpeak4x. The first three features, extracted from the thrust force signal, displayed the highest correlation with tool wear level.
* Fractal analysis features--Three fractal features were selected from the thrust force signal: the \(G\) parameter in region 1 (F\({}_{\text{GIR}}\)), the \(G\) parameter in the Gaussian region (F\({}_{\text{G-G}}\)), the Index in region 2 (F\({}_{\text{Index,2R}}\)), exhibiting a strong correlation with tool wear. From the torque signal, one fractal feature was selected: the \(G\) parameter in the Gaussian region (T\({}_{\text{G-G}}\)), showing an adequate correlation with tool wear level.  
The Spearman correlation coefficient was utilized in order to select the features displaying the highest robustness in identifying the tool flank wear level. Accordingly, the selected time domain, frequency domain, and fractal features were used to construct sensor fusion pattern vectors (FPV), containing features from thrust force and torque signals to be then employed for tool wear diagnosis through ANN based machine learning. In particular:  
* For time domain, a FPV was constructed containing the selected statistical features of thrust force and torque signals plus the hole number: \(\text{FPV}_{\text{time}}=\left[\text{F}_{\text{mean}}  
on this subset during training, so that the ANN weights and biases that correspond to the lowest validation error are stored. The testing subset is used to evaluate the performance of the trained ANN. However, the given partition into training, validation and testing subsets can significantly affect the ANN pattern recognition performance in tool wear level evaluation. For this reason, a bootstrap resampling technique was applied to randomly generate the subsets several times with the aim to enhance the estimation of the ANN pattern recognition performance [34; 35]. Following the bootstrap procedure, from the original set of 60 FPVs, the training (42 FPV), validation (9 FPV) and testing (9 FPV) subsets were resampled 60 times with replacement. Thus, the overall pattern recognition performance in tool wear level evaluation for tool wear curve reconstruction was estimated by aggregating the recognition rates obtained by all 60 re-samplings.