1. Introduction

2. Background and Related Work

2.1. Feature Extraction Domains

There are three primary domains for feature extraction: the time domain, frequency domain, and time-frequency domain. These distinct domains are employed to capture unique insights into signal behaviour:

2.1.1. Time Domain Analysis

Traditional Statistical Features (TSFs), shown in Table 1, are essential time domain measures derived from vibration or time series data. These features, including mean, Root Mean Square (RMS), kurtosis, peak-to-peak, Variance (Var), Standard Deviation (Std), shape factor, peaking factor, pulse factor, and margin factor, provide insights into statistical properties and amplitude variations of vibration signals [9]. Crest factor (CF), derived from time domain data, is a ratio representing the peak value of a signal to its RMS value and serves as a feature for assessing high-energy events or abnormal vibrations [21]. These features collectively capture the temporal characteristics of signals, enabling the examination of behaviour over time. Analyzing vibration signals in the time domain is crucial for understanding signal dynamics and detecting anomalies or faults [9]. Peak and RMS share the same unit and amplitude, while other factors are dimensionless [20]. RMS is widely used among these features due to its practical significance. Table 1 presents the formulas for TSFs [21,23,24]:

2.1.2. Frequency Domain Analysis

The extraction of features from the frequency domain can provide insights into the periodic components and harmonic structures within the data. The frequency domain analysis of vibration signals involves examining the amplitude changes with respect to different frequencies [11]. These features capture frequency-specific aspects of the signal and contribute to a better understanding of the vibration behaviour.[9].

Commonly used frequency domain features, such as Root Mean Square Frequency (RMSF), Center Frequency (CF), Mean Square Frequency (MSF), Frequency Variance (FV), and Root Frequency Variance (RVF), offer valuable information about the signal’s characteristics and power distribution within the frequency domain [9]. Standard harmonic features focus on the integral multiples of the fundamental frequency and have been widely applied. For instance, Total Harmonic Distortion (THD) has been utilized to characterize the frequency content of the vibration signal and is often represented as a percentage shown in Table 2 [25,26]. The Signal-to-Noise Ratio (SNR) and SINAD (Signal-to-Noise and Distortion Ratio) demonstrate the integration of SNR and SINAD with time domain features shown in Table 2; both are measured in decibels (dB), focusing on the signal's temporal properties and frequency domain features, especially in gearbox fault analysis [27].

On the other hand, computing spectral features from a signal involves transforming the signal from the time domain to the frequency domain and extracting various features that characterize the signal's frequency content, which provides a comprehensive overview of the entire frequency content of the signal [28]. The Autoregressive (AR) model is widely utilized for spectral estimation. Several techniques exist to compute the AR coefficients from data samples, including the Yule-Walker method, Burg’s method, and the covariance method [29]. In Rotating machine fault diagnosis, the AR model, especially with the forward-backwards approach, has demonstrated superior classification results, overcoming the limitations of traditional methods [30]. The AR model with linear prediction captures the deterministic signal component, while the residual part contains noise and bearing fault impulses. It can effectively cancel noise for bearings diagnosis when the optimal order is chosen [31]. Each technique has strengths and weaknesses; the periodogram lacks precision, Welch’s method struggles with rapid shifts, and the multitaper method is computationally intense [6,32]. The AR model’s model with an order of $p$, represented as AR(P), can be expressed in Equation 1 [29]:

$$x\left[n\right]=\sum _{p=1}^p{a}_{p}x\left[n-p\right]+e\left[n\right]$$

(1)

where, $x\left[n\right]$ represents the current value of the signal, $a_p$ is the autoregressive coefficients that need to be estimated, and $e\left[n\right]$is the random noise term. The current value of the signal $x\left[n\right]$ is influenced by its past values $x\left[n-p\right]$weighted by the corresponding autoregressive coefficients $a_p$. The sum is taken over $p$ terms, where $p$ ranges from 1 to the order of the AR model. The noise term $e\left[n\right]$ accounts for any unpredictability or randomness in the signal that is not captured by the past values. The AR coefficients $a_p$would be estimated using any of the following approaches to predict the behaviour of the signal based on its past values, like the Yule-Walker method, Burg's method, the covariance method or the forward-backwards approach to calculate the values of $a_p$, and these calculated coefficients [6,29]. The resulting spectral features from the AR model, such as Peak Amplitude, Peak Frequency, and Band Power, are shown in Table 2.

Table 2. Frequency domain features.

Table 2. Frequency domain features.

Parameter		Formula	Description
Harmonic Features	THD	$\sqrt{\frac{\left(\sum _{\mathit{i}=2}^{\mathit{N}}{\mathit{A}}_{\mathit{i}}^2\right)}{{\mathit{A}}_{\mathit{I}}}}$	Frequency domain, measuring the distortion caused by harmonics in the signal.
	SNR	$10{\log }_{10}\left(\frac{P_{signal}}{P_{noise}}\right)$	Compares the level of a desired signal to the level of background noise.
	SINAD	$10{\log }_{10}\left(\frac{P_{signal}}{P_{noise}+P_{distortion}}\right)$	A measure of signal quality compares the level of desired signal to the level of background noise and harmonics.
Spectral Features	Peak amplitude	$\left|x_{f-peak}\right|$	Represents the highest point (or peak) of the signal's waveform when viewed in the frequency domain.
	Peak frequency	$f-peak$	Corresponds to the frequency component that is most prominent or dominant in the signal.
	Band power	${\sum }_{f-start}^{f-end}{\left|x\left(f\right)\right|}^2$	Quantifies the total energy within a specific frequency range, providing insights into the distribution of signal energy across the spectrum.

Where, ${\mathit{A}}_{\mathit{I}}$denotes the amplitude of the fundamental frequency in the spectrum, while ${\mathit{A}}_{\mathit{i}}$ represents the amplitude of the i-th harmonic. In the formulas for SNR and SINAD, $P_{signal}$ refers to the power of the signal, $P_{noise}$​ to the power of the noise, and $P_{distortion}$​ to the power of harmonic distortion. In the peak amplitude, $x_{f-peak}$ represents the complex value of the signal in the frequency domain at bin $f-peak$, and “|” indicates the magnitude of a complex number. $x\left(f\right)$ is the complex value of the signal in the frequency domain at frequency bin 'f'. The term ${\left|x\left(f\right)\right|}^2$signifies the squared magnitude of $x\left(f\right).$ Σ symbolizes the summation over all frequency bins 'f' within the specified frequency range [$f-start$, $f-end$].

2.1.3. Time-Frequency Domain Analysis

Time-frequency domain analysis combines time and frequency information to understand the signal's frequency band over a specific time interval [33]. It provides a localised breakdown of the signal by considering smaller time segments. By examining the signal’s frequency content over time, these techniques can reveal hidden aspects and better capture the characteristics of non-stationary data [10]. One of the most famous techniques is wavelet transform, which offers the unique advantage of localising this frequency information in time, exemplified by the term "timescale (frequency) based analysis method." The wavelet transform predominantly employs three mother wavelet functions: Amor, Bump, and Morse, which have become staple choices in signal pre-processing [32]. The Wavelet scalogram based on CWT is a foundational tool in wavelet-based fault diagnostics [6]. Another pivotal technique is WSE, which merges the wavelet transform, singular value decomposition, and Shannon entropy. WSE adeptly extracts fault characteristics even from noise-laden and subtle fault transients, cementing its efficacy and superiority in fault detection [32,33].

3. Methodology

5.

Data Segmentation and Load-Dependent Subfiles Creation: time and frequency domain features are extracted from the segmented data, focusing on assessing feature variations during faults and their sensitivity to load changes.

6.

Time and Frequency Domain Feature Extraction from Data Segmentation: generate a load-dependent time and frequency feature set, where an initial load-dependent feature set is created for use in the following step.

7.

Significant Load-Dependent Feature Selection and Validation: select and validate the most significant load-dependent features using an iterative one-way ANOVA approach. Then, validate this feature set by assessing the accuracy of different classifiers.

4. Results and Discussion

4.2. Phase 2: Customised Load Adaptive Framework (CLAF) for IM Fault Classification

This section comprehensively explores time-frequency feature analysis across various fault types. The investigation encompasses CWT applied to vibration signals, wherein diverse mother wavelet functions are examined. The selection of the optimal wavelet function is determined through the utilisation of Wavelet Singular Entropy. This phase aims to develop the Customised Load Adaptive Framework (CLAF) for the MFPT-bearing dataset.

4.2.1. Step1: CWT Signal Encoding and Optimal Technique Selection

This entails selecting an optimal Continuous Wavelet Transform (CWT) for the MFPT-bearing dataset using Wavelet Singular Entropy (WSE) to create a time-frequency representation of the vibration signal. This representation serves multiple purposes, including feature extraction, denoising, and pattern recognition.

CWT Vibration Signal Time-Frequency Analysis

The analysis is initiated with the original MFPT-bearing dataset, categorized into health conditions: IRF, ORF, and Normal. The objective is to evaluate the capability of Continuous Wavelet Transform (CWT) in fault recognition, given its suitability for time-frequency analysis. CWT generates wavelet scalograms, which are 2D representations illustrating the local energy density across time and frequency, offering insights into system behaviour over time. Scalograms present time on the x-axis and scale on the y-axis, providing a comprehensive view of time-frequency domain characteristics compared to one-dimensional signals. The CWT effectively filters transient and non-smooth signal segments shown in Table 12. In Figure 8(a), 12 impulses in the inner vibration signal, corresponding to the bearing's IRF frequency, are observed. This results in 12 distinct peaks in the 2D time-frequency diagram in Table 12, with clearer patterns produced by Amor and Morse wavelets. Similarly, in Figure 8(b), eight peaks for outer faults are observed, with the most distinct pattern generated by Amor wavelets in Table 12. In contrast, in Figure 8(c), a lack of clear patterns or features is observed in the healthy signal, regardless of the wavelet used, refer to Table 12. The count of distinct peaks proves to be a valuable feature for distinguishing between inner race faults, outer race faults, and normal conditions. To quantitatively validate the selection of the optimal mother wavelet, Wavelet Singular Entropy (WSE) will be employed in the next section.

Wavelet Singular Entropy Analysis For Appropriate CWT Selection

A meticulous comparison of Wavelet Singular Entropy (WSE) scores identifies the most suitable mother wavelet function for fault scenarios. The largest WSE score indicates a more scattered signal with a less noticeable pattern, likely representing the healthy state, see Figure 8(c). WSE is a crucial quantitative measure for CWT, guiding the selection of effective wavelet foundations in wavelet analysis. The chosen mother wavelet significantly influences denoising, signal preservation, and feature extraction, enhancing the frequency spectrum of the denoised signal [6,43]. Average WSE was subsequently calculated in the process of selecting the optimal mother wavelet function by comparing WSE scores across different wavelet types [32]. The selection process involves evaluating (${WSE}_j$) scores across various mother wavelet functions:

$${WSE}_j={\sum }_{t=1}^n{\left|C_{fs}(t,j)\right|}^2.log\left({\left|C_{fs}(t,j)\right|}^2\right),$$

(9)

Here, $C_{fs}$ is the wavelet transform coefficient obtained from W, and fs (Hz) is the sampling frequency that determines the number of samples taken per second. The range of summation depends on the number of wavelet coefficients obtained from the transform and the chosen wavelet scale. Each coefficient corresponds to a specific scale, j, and time, t, capturing information about the signal's frequency content and time location [32,33].

Afterwards, $\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{W}\mathrm{S}\mathrm{E}\left(W\right)$ is calculated in Equation 10, where D represents the dataset (e.g., healthy, inner fault, outer fault), W represents the wavelet type (e.g., 'bump', 'morse', 'amor'), and N is the total number of datasets. Subsequently, the average mean WSE score ($\mathrm{A}\mathrm{v}\mathrm{g}\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{W}\mathrm{S}\mathrm{E}\left(W\right))$across all datasets for a specific wavelets determined in Equation 11:

$$\mathbf{M}\mathbf{e}\mathbf{a}\mathbf{n}\mathbf{W}\mathbf{S}\mathbf{E}\left(\mathit{W},\mathit{D}\right)=\frac{1}{\mathit{n}}\sum _{\mathit{j}=1}^{\mathit{n}}{\mathit{W}\mathit{S}\mathit{E}}_{\mathit{j}},$$

(10)

$$\mathbf{A}\mathbf{v}\mathbf{g}\mathbf{M}\mathbf{e}\mathbf{a}\mathbf{n}\mathbf{W}\mathbf{S}\mathbf{E}\left(\mathit{W}\right)=\frac{1}{\mathit{N}}{\sum }_{\mathit{D}}\mathit{M}\mathit{e}\mathit{a}\mathit{n}\mathit{W}\mathit{S}\mathit{E}\left(\mathit{W},\mathit{D}\right),$$

(11)

Table 13 scores provide valuable insights into energy distribution patterns in signals under different fault conditions, with two randomly chosen datasets assessed using WSE:

Bump:

For the inner fault, the Wavelet Singular Entropy (WSE) scores are low (0.017424 and 0.039571), indicating a more concentrated energy distribution and simpler signals. In contrast, the outer fault exhibits higher scores (2.0282 and 1.7431), suggesting a more complex energy distribution. In the normal condition, the scores are relatively low (1.4832 and 1.5995), indicating a simpler energy distribution.

Morse:

In the case of the inner fault, low scores (0.011188 and 0.022887) suggest simpler signals. Conversely, the outer fault displays higher scores (2.311 and 2.2253), indicating a more complex energy distribution. In the normal condition, the scores are relatively low (2.2357 and 2.836), suggesting a simpler energy distribution.

Amor:

For the inner fault, low scores (0.0090466 and 0.019031) indicate simpler signals. The outer fault, however, shows a positive score (0.61065), suggesting a more dispersed energy distribution. In the normal condition, higher scores (2.6529, 5.3807, and 15.826) indicate more complex energy distributions.

The mother of wavelet analysis can be summarised in Figure 9, where it shows the visual comparison; the "Amor" wavelet type shows relatively better discrimination between the healthy and faulty conditions, as it exhibits lower WSE scores for the faulty conditions compared to the healthy condition. However, Based on the analysis of the WSE scores, three wavelet coefficients were evaluated: Morse, Bump, and Amor. For the healthy dataset, the Morse coefficient had an average WSE score of 2.53585, the Bump coefficient had a score of 1.54135, and the Amor coefficient had the highest score of 10.60335, indicating a more dispersed energy distribution. When considering the inner fault dataset, the Morse, Bump, and Amor coefficients had average WSE scores of 0.0170375, 0.0284975, and 0.0140388, respectively. For the outer fault dataset, the average WSE scores were 2.26815, 1.88565, and 1.631775 for the Morse, Bump, and Amor coefficients, respectively. The results show that the Amor coefficient exhibited the highest average WSE score for the healthy dataset, suggesting a distinct energy distribution. This makes the Amor coefficient a potential candidate for identifying healthy conditions compared to faulty ones.

4.2.2. Step 2: CWT Energy Assessment For Each Load Factor

This section uses the data segmentation subfiles described in Table 8 in section 4.1.2. for further mean energy analysis per load factor for inner and outer race fault types per load factor i, calculate the wavelet energy values using the CWT technique. Let $x_i\left(t\right)$ represent the vibration signal for load factor i at time $t$. The CWT coefficients are denoted as $C_{i,j}\left(t\right)$, where j represents the selected wavelet scale [6,46]. Following these steps:

• Extract the vibration signal for load factor i: $x_i\left(t\right)$
• Perform the CWT on the vibration signal: $C_{i,j}\left(t\right)$, see Equation 12. The scale used in this study is 5.
• Calculate the wavelet energy $E_{wavelet,i}^j$​ for each scale j: $E_{wavelet,i}^j$, in Equation 13.

  $$C_{i,j}\left(t\right)=CWT({\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right),{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}}_{\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}},\mathrm{j}),$$

  $$E_{wavelet,i}^j={\sum }_t{\left|C_{i,j}\left(t\right)\right|}^2$$

Hence, the concept of "scale" j is crucial in understanding the CWT technique in wavelet analysis. The CWT is a method used to examine signals at various scales, allowing the detection of different frequency components in a signal with varying levels of detail. Each scale j corresponds to a specific width of the analyzing wavelet, a mathematical function used in the transformation process. Smaller scales represent narrower wavelets sensitive to high-frequency details, enabling the capture of rapid signal variations. Conversely, larger scales correspond to wider wavelets, capturing lower-frequency signal components with broader coverage but less fine detail. In equations involving wavelet analysis, such as ${\left|C_{i,j}\left(t\right)\right|}^2$, the squared absolute value of wavelet coefficients at a particular scale j and for a specific load factor i is calculated. This squared magnitude is summed across time t, resulting in the computation of the wavelet energy at that scale j. This energy measure provides valuable insights into the contribution of different frequency components to the overall energy content of the signal [46].

After that, the mean energy table for each fault was found. By combining the calculated wavelet energy values into a table for each load factor i. let let ${\mathit{E}}_{\mathit{w}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{e}\mathit{t},\mathit{i}}=\left[{\mathit{E}}_{\mathit{w}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{e}\mathit{t},\mathit{i}}^1,{\mathit{E}}_{\mathit{w}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{e}\mathit{t},\mathit{i}}^2,\dots ,{\mathit{E}}_{\mathit{w}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{e}\mathit{t},\mathit{i}}^{{\mathit{N}}_{\mathit{s}\mathit{c}\mathit{a}\mathit{l}\mathit{e}\mathit{s}}}\right],$ be the vector of wavelet energy values for load factor i. Then, calculate the mean wavelet energy wavelet, ${\overline{E}}_{wavelet,i}$for each load factor i by taking the average of the wavelet energy values across all scales:

$${\overline{E}}_{wavelet,i}=\frac{1}{N_{scales}}\sum _{j=1}^{N_{scales}}{E}_{wavelet,i}^j$$

(14)

Here, Building upon the foundation of wavelet energy, the mean wavelet energy ${\overline{E}}_{wavelet,i}$ is computed by averaging energy values over all scales. This metric provides a concise yet powerful representation of the energy behaviour post-fault for each load factor.

CWT Energy Assessment for Each Load Factor Using Optimal CWT Technique

In the assessment of mean energy values for IRF and ORF with load factor 270 as the normal condition shown in Table 14, the following observations are made: For inner bearings, load factor 270 (Normal Condition) exhibits a mean energy value of 5.7012, indicating lower energy content. Load factors 50, 100, and 150 have mean energy values ranging from 24.915 to 27.547, indicating relatively lower energy content, while load factors 200, 250, and 300 show mean energy values ranging from 32.199 to 36.147, suggesting higher energy content and a more pronounced presence of inner faults. Similarly, load factors 50, 100, 150, 200, and 250 for outer bearings have mean energy values ranging from 5.4309 to 7.6992, indicating relatively lower energy content than load factor 270. Load factor 270 (Normal Condition) has a mean energy value of 5.7012, representing normal or healthy conditions with lower energy content. Load factor 300 exhibits a mean energy value of 18.612, indicating a substantial 226.88% increase compared to normal conditions.

In summary, both ORF and IRF show notable increases in mean energy with distinct patterns. ORF exhibit the highest increase at load factor 300 (226.88%), while inner faults show higher increases, with the highest at load factor 250 (533.49%). The variability in increases ranges from 2.08% to 226.88% for outer faults and 337.68% to 533.49% for IRF. IRF generally display higher percentage increases than outer faults, providing insights for effective fault detection and system management.

Table 14. IRF and ORF CWT mean energy.

Table 14. IRF and ORF CWT mean energy.

	Inner Race Fault Type		Outer Race Fault Type
Load Factor (lbs)	MeanEnergy	Mean Energy Increase %	MeanEnergy	Mean Energy Increase %
50	25.549	347.70%	7.699	35.16%
100	27.547	383.65%	5.431	4.76%
150	24.915	337.68%	5.573	2.08%
200	33.742	491.88%	7.604	33.35%
250	36.147	533.49%	7.178	25.90%
270	5.7012	0% (baseline)	5.701	0% (baseline)
300	32.199	464.25%	18.612	226.88%

Two-Sample T-test for Significance Testing

In this study, a two-sample t-test was conducted using MATLAB R2023a to assess differences in mean CWT energy between the normal load condition (LoadFactor 270 lbs) and other loads (50, 100, 150, 200, 250, 300 lbs) for IRF in Figure 10 and ORF in Figure 11. Individual t-tests for each load factor determined whether the mean energy of the normal load significantly differed from other loads, with a significance level of 0.05. Results consistently demonstrated a clear and significant distinction in mean CWT energy between the normal condition and various loads. The null hypothesis (H0), suggesting no significant difference in CWT mean energy between LoadFactor 270 and other load factors, was rejected in favour of the alternative hypothesis (H1), indicating a substantial distinction. This finding held true for both IRF and ORF load factors, with low p-values, large sample sizes, substantial t-values, and confidence intervals, all supporting the robustness and reliability of these results.

4.2.3. Step 3: Customised Load Adaptive Framework

The Load Index, developed based on optimal CWT energy to capture the influence of load variations during fault occurrences, serves as a qualitative representation of the effects of varying loads on bearing behaviour. Subsequently, bearing faults were categorized into Load-Dependent subclasses, displaying distinct severity levels: mild, moderate, and severe, using CLAF. This comprehensive classification allows for understanding how varying loads contribute to the manifestation and progression of bearing faults with varying degrees of severity. This is built upon the previous steps:

Calculate Normalized Energy Values

For each load factor i, calculate normalized CWT energy values. $E_{normalized,i}^j$ By min-max scaling, the wavelet energy values $E_{wavelet,i}^j$ Between 0 and 1 Equation 15:

$$E_{normalized,i}^j=\frac{E_{wavelet,i}^j-\min \left(E_{wavelet}^j\right)}{\max \left(E_{wavelet,i}^j\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left(E_{wavelet,i}^j\right)},$$

(15)

In this normalized range, 0 represents the minimum energy value in the dataset, and 1 represents the maximum energy value in the dataset. All other energy values are linearly scaled within this range. Here, $\mathrm{m}\mathrm{i}\mathrm{n}\left(E_{wavelet}^j\right)$ represents the minimum wavelet energy value across all load factors and scales, and $\max \left(E_{wavelet,i}^j\right)$. Represents the maximum wavelet energy value across all load factors and scales.

Identify Normal Condition Indices

$I_{normal}$​ represents the indices corresponding to the normal condition. In the context of the analysis, it refers to the indices where the load factor is 270, which is considered the normal condition or baseline. These indices are used to calculate the deviation from the normal condition for each load factor and wavelet energy value.

In mathematical notation, ${\mathit{I}}_{\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}}$is a set of indices i for which the load factor is equal to 270:

$$I_{normal}=\left\{i|loadfactor=270\right\}$$

Quantify Deviation: Calculate Deviation from Normal Condition

$$D_i^j=\left\{\begin{array}{cc}{E}_{normalized,i}^j,\hfill & \hfill ifi\notin I_{normal}\\ 0,\hfill & \hfill otherwise\end{array}\right.$$

where, Deviations $D_i^j$ from the normal condition are calculated, highlighting differences between the normalized energy values and the baseline. When a load factor is not within $I_{normal}$, the corresponding normalized energy value $E_{normalized,i}^j$​ is considered. Otherwise, the deviation is set to zero.

Severity Of Changing Load: Threshold Setting

Define adjustable severity thresholds

Categorize the severity ${\mathrm{S}}_{\mathrm{i}}^{\mathrm{j}}$ based on the deviation magnitude ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{j}}$.and Threshold:

$$S_i^j=\left\{\begin{array}{cc}\text{'}Mild\text{'},\hfill & \hfill if{D}_i^j<mild\_threshold\\ \text{'}Moderate\text{'},\hfill & \hfill if{D}_i^j<mild\_threshold\\ \text{'}Severe\text{'},\hfill & \hfill otherwise\end{array}\right.$$

(12)

Hence, The severity of deviations $D_i^j$ is categorized to assess the impact post-fault. Adjustable severity thresholds differentiate between "Mild," "Moderate," and "Severe" conditions and then store severity as a cell array value. This step is vital in determining the gravity of the machinery's response to various fault scenarios. It is crucial for maintenance teams to allocate resources efficiently, plan timely interventions, and prevent potential escalations. In this paper, the author chooses the following thresholds:

% Define severity thresholds

mild_threshold = 0.2; % Adjust according to your application

moderate_threshold = 0.5; % Adjust according to your application

Categorize Severity ${\mathrm{S}}_{\mathrm{i}}^{\mathrm{j}}$ Based on The Deviation Magnitude ${\mathrm{D}}_{\mathrm{i}}^{\mathrm{j}}$

The normalised energy values allow us to effectively compare the energy levels of different load factors, as they are all scaled within the same range. However, it's essential to note that the normalised energy values are not directly related to the severity categorisation ('Mild', 'Moderate', or 'Severe'). The severity categorisation is based on the 'Deviation' column, which represents the deviation of each load factor's mean energy from the mean energy of the normal condition. Following are the inner and outer fault types after the assessment shown in Table 15 for both IRF and ORF:

Table 15. IRF and ORF Load-Dependent subclasses through CLAF.

Table 15. IRF and ORF Load-Dependent subclasses through CLAF.

LoadFactor (lb)	Mean Energy		NormalizedEnergy		Deviation		Load-Dependent Subclasses
Fault Type	Inner	Outter	Inner	Outter	Inner	Outter	Inner	Outter
50	25.549	7.6992	0.14035	0.05758	0.1403	0.05758	{'Mild' }	{'Mild' }
100	27.547	5.4309	0.15053	0.023062	0.15053	0.023062	{'Mild' }	{'Mild' }
150	24.915	5.5728	0.14063	0.031372	0.14063	0.031372	{'Mild' }	{'Mild' }
200	33.742	7.6036	0.28444	0.092816	0.28444	0.092816	{'Moderate'}	{'Mild' }
250	36.147	7.1779	0.29911	0.061822	0.29911	0.061822	{'Moderate'}	{'Mild' }
270	5.7012	5.7012	0.00930	0.027659	0	0	{'Normal' }	{'Normal'}
300	32.199	18.612	0.23412	0.89814	0.23412	0.89814	{'Moderate'}	{'Severe'}

IRF Customised Load Factor Assessment:

Min-max scaling was employed to normalize the energy values, transforming the original energy values into a range of [0, 1]. In this normalized range, 0 signifies the minimum energy value in the dataset, while 1 represents the maximum energy value. All other energy values are linearly scaled within this range. The 'Normalized Energy' column in the provided table reflects the energy values post min-max scaling, where one corresponds to the maximum energy value. For instance, the energy value of 'LoadFactor' 250 is relatively the highest compared to other load factors in the dataset, evidenced by its proximity to 1 in the normalized range.

Conversely, 'LoadFactor' 50, 'LoadFactor' 100, and 'LoadFactor' 150 have normalized energy values around 0.14, indicating their energy values are closer to the lower end of the normalized range (0). These load factors exhibit lower energy values compared to others in the dataset. Notably, the normalized energy values do not directly correspond to severity categorization ('Mild,' 'Moderate,' or 'Severe'). The severity categorization is based on the 'Deviation' column, which represents the deviation of each load factor's mean energy from the mean energy of the normal condition.

ORF Type Customised Load Factor Assessment:

Long-duration operation at higher load factors for the ORF significantly influences degradation. Across load factors 50, 100, 150, 200, and 250, the mean energy values range from 5.4309 to 7.6992, indicating relatively lower energy content in the vibration signals compared to load factor 270, which represents the normal or healthy condition with a mean energy value of 6.0981. The normal condition exhibits relatively lower energy levels, as expected. However, load factor 300 stands out with a higher mean energy value of 18.612, suggesting that the associated outer fault condition has a notably higher energy content in the vibration signals than the other load factors. This detailed energy analysis provides valuable insights into the variations associated with different load factors and fault conditions, enhancing the understanding of the degradation process.

4.2.4. Step 4: CLAF Validation

The proposed CLAF is an early warning system that identifies potential issues based on the customised Load-Dependent fault subclasses. This approach enhances the efficiency of the CLAF, displaying time domain data grouped by the four framework classes. Time and frequency domain features are then extracted by creating a feature subset, training classifiers, and selecting optimal features based on accuracy. This dataset is detailed in section 5.1.4, "Classification and Features Selection using Second: Autoregressive (AR) Model (Order 15) and Peak Five," where 24 features from both time and frequency domains are generated within the 2500-25000 Hz frequency band. Each signal contributes five spectral peaks, resulting in five frequency features for each peak.

Following this, a one-way ANOVA test was conducted. Features below an ANOVA score of 26, referring to Figure 12, were excluded from further study. The aim of this step was to enhance the selection process by concentrating on features with a more significant impact. Observing the initial trial's high accuracy, the author systematically reduced the number of features, utilizing accuracy as a metric for efficient feature reduction. This reduction process was carried out gradually, guided by accuracy measures.

Subsequently, numerous classifiers were trained, and their performance was meticulously documented for each feature subset selection. A comprehensive evaluation of various classifier algorithms was then conducted, focusing on identifying combinations that optimised accuracy and minimized discrepancies among algorithms. The training dataset, comprising a total of 813 subfolders, was divided into 20% for testing, 20% for validation, and 60% for training. A five-fold cross-validation was executed, and testing accuracy was recorded for performance comparison. The process began with the top 20 features based on one-way ANOVA scores exceeding 26, followed by the top 17 features with scores exceeding 58.6, then the top 10 features with scores exceeding 161, followed by the top seven features with scores exceeding 215, and finally, the top five features with scores exceeding 240. Classifier performance for each subset was recorded, and in cases of equal accuracy, all classifiers were included in Table 16.

5. Conclusions

References

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