## 3 TF Transforms and Results
### Time-Frequency Representations Robustness Analysis  
Due to the different mathematical foundations of the analyzed transformations, the information about the studied material (after the transformation) may ultimately be expressed in a different way in the given TF distribution. In consequence, it reflects in different properties of the TFR for a subsequent processing method. Therefore, one or two parameters describing the properties of these representation (e.g., relating to statistical parameters) may not be able to provide complete information about the observed properties changes. In such a situation, the need of using one of the methods of the Exploratory Data Analysis (EDA) is arising. The Principal Component Analysis (PCA) is one of such a method that enables the assessment of information variability in the entire database. Therefore, it was applied to evaluate the variance of vector, consisting of multiple TF parameters values gathered within the entire range of analyzed cases. The PCA leads to the linear transformation of the dataset so that the most variability of the input vector is presented in the new vector by fewer of its features called principal components _pc_ (being a linear combination of the original parameters). In this case, the PCA analysis provides data on how the information is distributed over the new parameters (_pc_s) vector. As a result, the greatest variability is provided by the first component, and the higher the index number of the _pc_, the lower rate of variability it explains. Several results can be expected when using this data analysis method. If most of the data variance is expressed with the first or a few first principal components, then there is a large correlation between the parameters inside a vector. In the second case, when a high variability rate is reached under the term of several first principal components, then this may refer to the situation in which each of the underlying features provides unique information. However, on the other hand, it may also be the result of a case where the parameters are correlated, although they are supplemented by the interfering components that add variability to the input TF parameters vector. During the analysis, eleven TF parameters were taken into account, such as skewness, standard deviation, power density and entropy of the TFR (according to [19]). In Figure 7, the first principal component for each of the transformations is presented. The total variance of data within the parameters vector is shown in Figure 8.  
Figure 6: Result of the \(r^{2}\) coefficient for \(BN_{TF\_CM}\) and \(BN_{TF\_MEAN}\) parameters.  
As it can be seen, the linear combination of parameters composing the first principal component for the STFT explain more than 90% of the variation. The rest of the transformation does not reach the level of 80% explained by the first \(pc\). The worst transformation in  
Figure 8: Explanation of variability by first three principal components for each transformation. The green horizontal line marks the 90% level of variability in dataset.  
Figure 7: First principal component for each of the transformations.  
this case is the WSST. In reference to the given results, in the case of transformations other than the STFT, the convergence rate of information gathered in the TFR is lower. This, in the context of the obtained distributions for individual parameters, can be explained by the lower level of readability of the useful information about the tested material (see the characteristic curves with local parameter deviations presented in Figure 5). Additionally, by looking at the first component, it can be noticed that there is a lot of noisy information. The high rate of variation expressed by the first \(pc\) obtained for the STFT also proves the high robustness of this transformation to disturbing factors, hindering the ability to read the correct information about the tested material.