## 2 Materials and Methods
### Machine Learning Condition Indicators
The Kurtogram returns the signal's most impulsive bandwidth and its centre frequency. Later, [PERSON] proposed an optimised version of the Kurtogram called the Fast Kurtogram [18]. This version reduces the number of variants for which the filter parameters are calculated without affecting the result's accuracy by applying the filter bank approach. The main drawback of SK is that it is susceptible to non-Gaussian random components external to the mechanical system, such as noise or interference of various sorts. [PERSON]'s CPW [19] can be applied to a raw time signal to enhance the presence of REB's fault's impulse responses in the envelope spectrum and remove both unwanted components' harmonics and sidebands. Earlier in this section, REB fault signals have been referred to as repetitive and not periodic, as they are not strictly periodic: they are second-order cyclostationary. This means that their second-order statistic, the variance, is periodic but not the impulses themselves. The repetitive impulses of REB failures are not precisely periodic due to the rolling elements' slightly random location in the cage-free space and the non-exact cage's rotational speed due to slippage. This is why the REB's fault information is not altered by applying CPW, as the cyclostationary components of a signal do not represent significant peaks in the cepstral domain. A detailed explanation of CPW can be found in the paper by [PERSON] et al. [20]. In summary, when a signal is demodulated correctly, a series of impulse responses will appear in the envelope spectrum spaced at the fault's characteristic frequency. When a fault affects a specific REB part, the impacts with the fault will excite a particular frequency band in the form of repetitive impulses. Suppose this band (including the transfer path from the place of impact to the sensor) is selected for demodulation. In that case, the impulse responses in the envelope spectrum spaced at the actual fault's characteristic frequency will have a greater amplitude than those spaced at other fault's characteristic frequencies, as depicted in Figure 2. Computing CPW to a raw time signal enhances cyclostationary components' responses by filtering out periodic components' harmonics and sidebands  
Figure 2: Envelope spectrum of a raw time signal. The signal belongs to the CWRU’s bearing database, fan end REB, 0.014\({}^{\prime\prime}\) depth, centred-to-the-load outer race fault, tested at a motor speed of 1797 rpm. for datasets containing more than two classes will be applied to answer these questions: one-way ANOVA [26] and the Kruskal-Wallis test [27]. Comparing the means of three or more unrelated groups within a dependent variable (feature), one-way ANOVA determines whether there are statistically significant differences between them. A non-parametric alternative to the one-way ANOVA test is the Kruskal-Wallis test. Unlike one-way ANOVA, where means are compared, the Kruskal-Wallis test contrasts the samples' distribution to determine whether they belong to the same population. Both tests make assumptions that the data of each group in every dependent variable must meet. The Kruskal-Wallis test is known to be used if one-way ANOVA's assumptions are not met for each class within every dependent variable, since they are more potent than those of the Kruskal-Wallis test. One-way ANOVA tests the null hypothesis that all group means are equal (\(H_{0}:\mu_{1}=\mu_{2}=\cdots=\mu_{J}\)) against the alternative hypothesis that at least one group mean is different from the others (\(H_{1}:\mu_{j}\
eq\mu_{j}\) for at least one \(j^{\prime}\) and \(j\)) in a one-way layout, where \(i=1,\ldots,I\) is the observation number and \(j=1,\ldots,J\) is the group number. The one-way ANOVA result is the ratio of across-group variation to within-group variation \(F\). If \(F\) is larger than the critical value of the F-distribution [28] with \((J-1,J(I-1))\) degrees of freedom and a significance level of \(\alpha\), the null hypothesis is rejected.