# Diagnosis Methodology Based on Deep Feature Learning for Fault Identification in Metallic, Hybrid and Ceramic Bearings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Autoencoder-Based Deep Feature Learning

**x**. The encoder procedure aims to perform the mapping of the input

**x**into a hidden representation which is denoted by the hidden layer or encoder vector

**h**. Therefore, the hidden layer

**h**contains the numerical information and maps that lead to achieve the connection of the input layer

**x**with the hidden layer

**h**through n sparse-activated neurons and non-linear transformation following Equation (1):

**x**from the information contained in the hidden layer

**h**; thus, an output decoder layer vector

**y**is obtained during the decoder procedure, which is achieved by following Equation (2):

**y**as the achieved reconstruction of the corresponding input

**x**. Therefore, the aim of the AE feature learning process lies in the optimization of the parameters $\mathsf{\theta}=\left\{{W}_{d},{B}_{e},{W}_{d},{B}_{d}\right\}$, where the minimization of the reconstruction error between the input and the output feature vectors,

**x**and

**y**, respectively, is accomplished in terms of the mean squared error (MSE) by following Equation (3):

_{2}as the regularization term to the cost or fitness function. Specifically, L

_{2}is denominated as the weight decay regularization term, which is added to avoid the emergence of large weights in the weight matrix $W$, thus, such a term is formulated as Equation (4) depicts:

_{2}regularization term that takes control over the weight decay and $\beta $ is the corresponding parameter to the sparsity regularization term.

## 3. Deep Feature Learning Based Methodology

#### 3.1. Multi-Domain Feature Calculation

**T**∈ ℝ

^{T}. On the other hand, for the FD analysis, the FFT technique is applied to each segmented part of the signals in order to obtain its representative frequency spectrum. Subsequently, a numerical set of 14 statistical features is calculated from each one of the obtained frequency spectrums. Additionally, six characteristic-bearing fault-related frequency components are estimated, that is, the first and second harmonic corresponding to the outer, inner and ball bearing faults. Therefore, in the FD analysis, a resulting set of F = 20 features are calculated from each frequency spectrum, and a representative F-dimensional frequency-domain feature matrix,

**F**∈ ℝ

^{F}, is calculated. Finally, for the TFD analysis, the acquired signals are, first, processed by means of one of the most suitable signal-decomposition techniques, the empirical mode decomposition (EMD), which allows obtaining a set of sub-signals containing the main oscillatory modes included in the original one. Thereby, the EMD technique is applied to each one of the segmented parts of the available signals and, following the related literature, the two first intrinsic modes functions (IMF), are taken into account for being characterized, since they contain the most significant information in terms of characteristic fault patterns. Later, the FFT technique is applied to each IMF to compute their corresponding frequency spectrum. The estimation of the same set of 14 statistical frequency-features is carried out for each IMF’s frequency spectrum. Therefore, for the TFD analysis, a set of TF = 28 features are estimated for each acquisition. As a result, a representative TF-dimensional time–frequency domain feature matrix,

**TF**∈ ℝ

^{T}

^{F}, is calculated.

#### 3.2. SAE Feature Learning

**T**,

_{i}**F**, and

_{i}**TF**, for i = 1, 2,...N. Thus, it is proposed to consider as many SAE structures as available feature matrices to extract; for each matrix, a reduced set of features is considered while preserving most of the characteristic information that allows a proper reconstruction of the original signal. In this sense, it must be noted that performing a specific domain-based feature learning procedure over the extracted patterns leads to maximizing the characterization capabilities from each matrix. Indeed, the SAE seeks for an optimum codification in terms of posterior reconstruction error minimization, thus leading to the preservation of the underlying physical behavior of the signal from an unsupervised point of view.

_{i}_{2}regularization term, (ii) the coefficient for the sparsity regularization term and (iii) the parameter for sparsity proportion. Thus, the optimization of the three mentioned hyperparameters is performed by following the next steps:

**Step 1:**Initialization of the population: the chromosomes of the GA are initially defined with a logical vector containing three elements, where each element represents each one of the hyperparameters. Subsequently, a random initialization of the population is performed by assigning a specific value to each particular hyperparameter; in fact, the values assigned to each hyperparameter are within a predefined range of values. Once the initialization of the population is achieved, the procedure continues in step 2.**Step 2:**Evaluation of the population: in this step the fitness function is evaluated based on the minimization of the reconstruction error between the input and the output features. Specifically, the minimization of the reconstruction error is evaluated in terms of the MSE value following Equation (3), as mentioned in Section 2. Thereby, the optimization problem to be solved by the GA involves the search of those specific hyperparameter values that leads a high-performance feature mapping. Then, once the whole population is evaluated under a wide range of values, the condition of best hyperparameter values is analyzed and the procedure continues in step 4.**Step 3:**Mutation operation: the mutation of the GA produces a new population by means of the roulette wheel selection; the newly generated population takes into account the choice of the best fitness value achieved by the previously evaluated population. Moreover, a mutation operation that is based on the Gaussian distribution is applied during the generation of the new population. Subsequently, the procedure continues in step 2.**Step 4:**Stop criteria: there are two stop criteria for the GA: (i) the obtention of a reconstruction MSE value lower than a predefined threshold, 5%, and/or, (ii) reaching the maximum number of iterations, 1000. In the case of the first stop criterion, (i), the procedure is repeated iteratively until those optimal hyperparameter values are found until the GA evolves, then the procedure continues in step 3.

^{T}

_{i}, SAE

^{F}

_{i}, SAE

^{TF}

_{i}, respectively, SAE

^{T}

_{i}, SAE

^{F}

_{i}, SAE

^{TF}

_{i}∈ ℝ

^{C}.

#### 3.3. Data Fusion and Fault Diagnosis

_{1}(z), λ

_{2}(z), λ

_{C}(z), where the λ

_{j}(z) represents the probability of j-th class. The calculation of λ

_{j}(z) is defined following:

## 4. Experimental Validation

#### 4.1. Pulley-Belt Electromechanical System

#### 4.2. Rolling Bearing CWRU Dataset

## 5. Results and Validation

#### 5.1. Evaluation of the Diagnostic Model in the Pulley-Belt Electromechanical System

**T**that contains 30 statistical features represented with 300 samples was achieved; meanwhile, the feature matrix achieved from the stator current

_{1}**T**consists of 15 statistical features with 300 samples. Subsequently, when the FD analysis is performed, from each segmented part of each available signal, the corresponding frequency spectrum is estimated by means of applying the FFT technique and then, from each resulting spectrum, the proposed set of 14 statistical features is calculated to characterize each spectrum into a set of representative numerical values. As a result, a characteristic feature matrix

_{2}**F**with 28 statistical features and 300 samples is carried out from both vibrations signals, while the characteristic matrix

_{1}**F**estimated from the stator current contains 14 statistical features with 300 samples. Additionally, for each available signal, six fault-related frequency features are taken into account; these additional features are the two first harmonic frequencies associated with three possible sources of fault in the bearing (i.e., outer, inner and ball bearing faults); thereby, these fault-related frequency features are represented by its corresponding amplitude resulting from each frequency spectrum. Accordingly, for the FD analysis, the resulting feature matrix

_{2}**F**of both vibration signals has a total of 40 features with 300 samples, and the feature matrix

_{1}**F**for the stator current has a total of 20 features with 300 samples. Then, during the TFD analysis, each segmented part is evaluated by the EMD technique in order to obtain the intrinsic modes associated with nonlinearities of the signals; thus, from each segmented part of the available signals, the two first intrinsic mode functions (i.e., IMF

_{2}_{1}and IMF

_{2}) are considered as the two main intrinsic modes. Consecutively, the FFT technique is applied over each intrinsic mode and the resulting frequency spectra are then individually characterized by the estimation of the proposed set of 14 statistical features. Therefore, a feature matrix

**TF**with 56 statistical features and 300 samples is achieved for both vibration signals, whereas, for the stator current a feature matrix

_{1}**TF**with 28 features and 300 samples is obtained.

_{2}**T**,

_{1}**F**and

_{1}**TF**, respectively; whereas,

_{1}**T**,

_{2}**F**,

_{2}**TF**are the resulting feature matrices from the analysis of the stator current in the TD, FD and TFD, respectively. Additionally, it should be clarified that these representative feature matrices are also estimated for each one of the supply frequencies considered (i.e., 5 Hz, 15 Hz, 50 Hz and 60 Hz). Following the proposed method, for each one of the bearing technologies and for each one of the available signals, the whole resulting feature matrices, considering all supply frequencies and taking into account the assessed bearing conditions, are grouped according to the domain of analysis. For example, in Table 3 the grouping of the feature matrices that are estimated from the vibrations signals is summarized; the feature matrices are grouped according to the domain of analysis, as it can be appreciated, and each corresponding grouping considers the bearing conditions (i.e., HC and BD conditions). These grouped matrices are represented by

_{2}**T**,

_{1 group}**F**,

_{1 group}**TF**, where Equation (40) gives the detail of a specific set of grouped matrices for

_{1 group}**T**. Thus, the same grouping is applied to feature matrices that are estimated from analyzing the stator current in the TD, FD and TFD for each one of the supply frequencies tested in the VFD and the obtained grouped matrices are

_{1 group}**T**,

_{2 group}**F**and

_{2 group}**TF**.

_{2 group}_{2}Regularization parameter about (2 ± 1) × 10

^{−5}, by considering values about (5 ± 3) × 10

^{−5}in the Sparsity Regularization parameter and by considering the Sparsity Proportion with values about (0.5 ± 0.2).

#### 5.2. Evaluation of the Diagnostic Model in the CWRU Database

**T**,

_{1}**F**and

_{1}**TF**, where each feature matrix comprises of 15, 20 and 28 features with 150 samples. Accordingly, for each one of the three resulting feature matrices, the total number of 150 samples corresponding to each tested condition is divided for training and testing purposes; in this sense, a random number of 100 samples are adopted for training and the remaining 50 samples for testing.

_{1}**T**,

_{i}**F**

**, and**

_{i}**TF**

**, for i = 1,...N. Thus, the SAE structures for TD, FD and TFD analysis are SAE**

_{i}^{T}

_{i}, SAE

^{F}

_{i}and SAE

^{TF}

_{i}, respectively. In this regard, the automatic optimization of the hyperparameters of the SAE network is carried out through the GA, aiming to minimize the MSE reconstruction value. Thereby, in Table 9 the achieved hyperparameters obtained through the application of the proposed tuning strategy are summarized; the average MSE value obtained during the optimization procedure shows values about (2 ± 1) × 10

^{−4}that depict a high-performance characterization in the feature learning by the means of the SAE structure. Aiming to prove the effectiveness of the SAE-based feature learning approach, from Figure 12a–d the original and reconstructed feature patterns that represent assessed bearing conditions, NC, IF, OF and BF in different domains are shown, respectively. As it can be appreciated, the reconstructed patterns match properly with the original ones. Figure 12a shows the patterns corresponding to the HC represented in TD and their corresponding reconstruction obtained by the characterization of the SAE. As can be seen qualitatively, the representing feature patterns are correctly characterized, presenting an error of only about 5.07%. On the other hand, Figure 12b–d show the characteristic feature patterns of the vibration signals related to the IF, OF and BF bearing conditions represented in TFD, TD and FD, respectively. The reconstruction error achieved during the feature learning of the considered bearing fault conditions are: 1.99%, 6.08% and 2.43%, correspondingly.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Flow chart of the proposed data-driven diagnosis methodology based on deep feature learning for fault diagnosis applied to the diagnosis and identification of bearing faults for different bearing technologies, such as metallic, hybrid and ceramic bearings, in electromechanical systems.

**Figure 4.**Flow chart of the laboratory electromechanical system based on a pulley-belt system and its wiring to perform the data acquisition.

**Figure 5.**Set of damaged bearing used to be experimentally tested in the IM of an electromechanical system: (

**a**) metallic damaged bearing, (

**b**) hybrid damaged bearing and (

**c**) full ceramic damaged bearing.

**Figure 6.**Vibration patterns acquired during the evaluation of the outer race faulty condition in: (

**a**) the metallic bearing, (

**b**) the hybrid bearing and (

**c**) the full ceramic bearing.

**Figure 7.**Frequency spectra obtained by applying the fast Fourier transform to the raw vibration signals of all considered bearing technologies: metallic, hybrid and full ceramic.

**Figure 8.**Qualitative representation of the original and reconstructed set of features for healthy metallic bearing at 5 Hz. (

**a**) TD feature vibrations, (

**b**) FD feature vibrations, (

**c**) TFD feature vibrations.

**Figure 9.**Two-dimensional visual representations of the data distribution achieved by the T-SNE technique over the resulting feature spaces mapped by their corresponding SAE of the metallic bearing when analyzing: (

**a**) vibrations signals in FD, (

**b**) stator current signature in FD, (

**c**) vibrations signals in TFD and (

**d**) stator current signature in TFD.

**Figure 10.**Achieved T-SNE representation of the data distribution into a 2D space of the resulting feature spaces mapped by their corresponding SAE when analyzing the stator current signature in the TD for each different bearing technology: (

**a**) metallic, (

**b**) hybrid and (

**c**) full ceramic.

**Figure 11.**Resulting 2D representation of the data distribution of the mapped feature spaces by their corresponding SAE when all feature spaces domains are considered under the proposed fusion scheme (TD + FD + TFD) of all available signals (vibrations and stator current) are taken into account by the T-SNE for each different bearing technology: (

**a**) metallic, (

**b**) hybrid and (

**c**) full ceramic.

**Figure 12.**Comparison between the input feature patterns and its corresponding reconstruction obtained using the deep SAE-based feature extraction model on: (

**a**) the NC represented in TD, (

**b**) IF represented in TFD, (

**c**) OF represented in TD and (

**d**) OF represented in FD.

**Figure 13.**Feature visualization via t-SNE for: (

**a**) time domain, (

**b**) frequency domain (

**c**) time–frequency domain and (

**d**) fusion scheme. The colors used to represent the clusters shown in the resulting projections belong to the NC (●), BF (●), IF (●) and OF (●) conditions, where the different load conditions for each bearing condition are represented by the markers

**△**,

**+**,

**○**and

*****, respectively.

**Table 1.**Proposed set of statistical time domain features estimated during the signal processing in the TD analysis, where each y(j) value belongs to each sample point of the processed signal for j = 1, 2, 3…P, with a total number P samples.

Statistical Feature | Mathematical Equation | |
---|---|---|

Mean | ${t}_{1}=\frac{1}{P}{\displaystyle \sum}_{j=1}^{P}\left|{y}_{j}\right|$ | (9) |

Maximum value | ${t}_{2}=max\left(x\right)$ | (10) |

Root mean square | ${t}_{3}=\sqrt{\frac{1}{P}{\displaystyle \sum}_{j=1}^{P}{\left({y}_{j}\right)}^{2}}$ | (11) |

Square root mean | ${t}_{4}={\left(\frac{1}{P}{\displaystyle \sum}_{j=1}^{P}\sqrt{\left|{y}_{j}\right|}\right)}^{2}$ | (12) |

Standard deviation | ${t}_{5}=\sqrt{\frac{1}{P}{\displaystyle \sum}_{j=1}^{P}{\left({y}_{j}-{t}_{1}\right)}^{2}}$ | (13) |

Variance | ${t}_{6}=\frac{1}{P}{\displaystyle \sum}_{j=1}^{P}{\left({y}_{j}-{t}_{1}\right)}^{2}$ | (14) |

RMS Shape factor | ${t}_{7}=\frac{{t}_{3}}{\frac{1}{P}{{\displaystyle \sum}}_{j=1}^{P}\left|{y}_{j}\right|}$ | (15) |

SRM Shape factor | ${t}_{8}=\frac{{t}_{4}}{\frac{1}{P}{{\displaystyle \sum}}_{j=1}^{P}\left|{y}_{j}\right|}$ | (16) |

Crest factor | ${t}_{9}=\frac{{t}_{2}}{{t}_{3}}$ | (17) |

Latitude factor | ${t}_{10}=\frac{{t}_{2}}{{t}_{4}}$ | (18) |

Impulse factor | ${t}_{11}=\frac{{t}_{2}}{\frac{1}{P}{{\displaystyle \sum}}_{j=1}^{P}\left|{y}_{j}\right|}$ | (19) |

Skewness | ${t}_{12}=\frac{{{\displaystyle \sum}}^{}\left[{\left({y}_{j}-{t}_{1}\right)}^{3}\right]}{{t}_{5}{}^{3}}$ | (20) |

Kurtosis | ${t}_{13}=\frac{{{\displaystyle \sum}}^{}\left[{\left({y}_{j}-{t}_{1}\right)}^{4}\right]}{{t}_{5}{}^{4}}$ | (21) |

Fifth moment | ${t}_{14}=\frac{{{\displaystyle \sum}}^{}\left[{\left({y}_{j}-{t}_{1}\right)}^{5}\right]}{{t}_{5}{}^{5}}$ | (22) |

Sixth moment | ${t}_{15}=\frac{{{\displaystyle \sum}}^{}\left[{\left({y}_{j}-{t}_{1}\right)}^{6}\right]}{{t}_{5}{}^{6}}$ | (23) |

**Table 2.**Proposed set of statistical time domain features estimated from the frequency spectra during the signal processing in the FD analysis, where each z(i) is a spectrum for i = 1, 2,…,Q, and Q is the total number of lines for an estimated spectrum; f

_{q}(i) is the frequency value of the i-th spectrum line.

Statistical Feature | Mathematical Equation | |
---|---|---|

Mean | ${f}_{1}=\frac{1}{Q}{\displaystyle \sum}_{i=1}^{Q}z\left(i\right)$ | (24) |

Variance | ${f}_{2}=\frac{1}{Q-1}{\displaystyle \sum}_{\mathrm{i}=1}^{Q}{\left(z\left(i\right)-{f}_{1}\right)}^{2}$ | (25) |

Third moment | ${f}_{3}=\frac{1}{Q{\left(\sqrt{{f}_{2}}\right)}^{3}}{\displaystyle \sum}_{i=1}^{Q}{\left(z\left(i\right)-{f}_{1}\right)}^{3}$ | (26) |

Fourth moment | ${f}_{4}=\frac{1}{Q{\left(\sqrt{{f}_{2}}\right)}^{2}}{\displaystyle \sum}_{i=1}^{Q}{\left(z\left(i\right)-{f}_{1}\right)}^{4}$ | (27) |

Grand mean | ${f}_{5}=\frac{{{\displaystyle \sum}}_{i=1}^{Q}{f}_{q}\left(i\right)z\left(i\right)}{{{\displaystyle \sum}}_{i=1}^{Q}z\left(i\right)}$ | (28) |

Standard deviation 1 | ${f}_{6}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{Q}{\left({f}_{q}\left(i\right)-{f}_{5}\right)}^{2}z\left(i\right)}{Q}}$ | (29) |

C Factor | ${f}_{7}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{Q}{f}_{q}{\left(i\right)}^{2}z\left(i\right)}{{{\displaystyle \sum}}_{i=1}^{Q}z\left(i\right)}}$ | (30) |

D Factor | ${f}_{8}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{Q}{f}_{q}{\left(i\right)}^{4}z\left(i\right)}{{{\displaystyle \sum}}_{i=1}^{Q}{f}_{q}{\left(i\right)}^{2}z\left(i\right)}}$ | (31) |

E Factor | ${f}_{9}=\frac{{{\displaystyle \sum}}_{i=1}^{Q}{f}_{q}{\left(i\right)}^{2}z\left(i\right)}{\sqrt{{{\displaystyle \sum}}_{i=1}^{Q}z\left(i\right){{\displaystyle \sum}}_{i=1}^{Q}{f}_{q}{\left(i\right)}^{4}z\left(i\right)}}$ | (32) |

G Factor | ${f}_{10}=\frac{{f}_{6}}{{f}_{5}}$ | (33) |

Third moment 1 | ${f}_{11}=\frac{{{\displaystyle \sum}}_{i=1}^{Q}{\left({f}_{q}-{f}_{5}\right)}^{3}z\left(i\right)}{Q{f}_{6}^{3}}$ | (34) |

Fourth moment 1 | ${f}_{12}=\frac{{{\displaystyle \sum}}_{i=1}^{Q}{\left({f}_{q}\left(i\right)-{f}_{5}\right)}^{4}z\left(i\right)}{Q{f}_{6}^{4}}$ | (35) |

H Factor | ${f}_{13}=\frac{{{\displaystyle \sum}}_{i=1}^{Q}{\left({f}_{q}\left(i\right)-{f}_{5}\right)}^{1/2}z\left(i\right)}{Q\sqrt{{f}_{6}}}$ | (36) |

J Factor | ${f}_{14}=\frac{\left({f}_{7}+{f}_{8}\right)}{{f}_{1}}$ | (37) |

**Table 3.**Representation of the grouping of the feature matrices computed by means of analyzing the vibration signals in TD, FD and TFD for each supply frequency for the considered bearing conditions; where the notations HC and BD represent the healthy condition and the bearing damaged condition.

Feature Domain | ||||
---|---|---|---|---|

Time | Frequency | Time–Frequency | ||

Related condition | Normal | T @5Hz@HC_{1} | F @5Hz@HC_{1} | TF @5Hz@HC_{1} |

T @15Hz@HC_{1} | F @15Hz@HC_{1} | TF @15Hz@HC_{1} | ||

T @50Hz@HC_{1} | F @50Hz@HC_{1} | TF @50Hz@HC_{1} | ||

T @60Hz@HC_{1} | F @60Hz@HC_{1} | TF @60Hz@HC_{1} | ||

Damaged | T @5Hz@BD_{1} | F @5Hz@BD_{1} | TF @5Hz@BD_{1} | |

T @15Hz@BD_{1} | F @15Hz@BD_{1} | TF @15Hz@BD_{1} | ||

T @50Hz@BD_{1} | F @50Hz@BD_{1} | TF @50Hz@BD_{1} | ||

T @60Hz@BD_{1} | F @60Hz@BD_{1} | TF @60Hz@BD_{1} | ||

Grouped matrices | T_{1 group} | F_{1 group} | TF_{1 group} |

**Table 4.**Parameters tuned by the GA for each SAE structure that is considered for each domain of analysis for the vibration signals used in the stacked AE; all technology materials.

Material | Feature Domain | Hyperparameters | ||
---|---|---|---|---|

L_{2}Regularization | Sparsity Regularizatio | Sparsity Proportion | ||

Metallic | Time (T_{1 group}) | 1.31 × 10^{−5} | 7.384 × 10^{−5} | 0.664 |

Frequency (F_{1 group}) | 1.616 × 10^{−5} | 5.761 × 10^{−5} | 0.655 | |

Time–Frequency (TF_{1 group}) | 3.250 × 10^{−5} | 7.164 × 10^{−5} | 0.779 | |

Ceramic | Time (T_{1 group}) | 2.090 × 10^{−5} | 8.037 × 10^{−5} | 0.132 |

Frequency (F_{1 group}) | 2.879 × 10^{−5} | 1.407 × 10^{−5} | 0.479 | |

Time–Frequency (TF_{1 group}) | 2.213 × 10^{−5} | 5.277 × 10^{−5} | 0.854 | |

Hybrid | Time (T_{1 group}) | 2.212 × 10^{−5} | 7.785 × 10^{−5} | 0.176 |

Frequency (F_{1 group}) | 1.276 × 10^{−5} | 2.406 × 10^{−5} | 0.560 | |

Time–Frequency (TF_{1 group}) | 1.064 × 10^{−5} | 4.435 × 10^{−5} | 0.240 |

**Table 5.**Parameters tuned by the GA for each SAE structure that is considered for each domain of analysis for the stator current signature used in the stacked AE; all technology materials.

Material | Feature Domain | Hyperparameters | ||
---|---|---|---|---|

L_{2}Regularization | Sparsity Regularizatio | Sparsity Proportion | ||

Metallic | Time (T_{2 group}) | 2.277 × 10^{−5} | 7.108 × 10^{−5} | 0.3228 |

Frequency (F_{2 group}) | 1.077 × 10^{−5} | 8.433 × 10^{−5} | 0.2026 | |

Time–Frequency (TF_{2 group}) | 2.277 × 10^{−5} | 7.108 × 10^{−5} | 0.3228 | |

Ceramic | Time (T_{2 group}) | 1.675 × 10^{−5} | 6.944 × 10^{−5} | 0.4743 |

Frequency (F_{2 group}) | 1.320 × 10^{−5} | 4.897 × 10^{−5} | 0.546 | |

Time–Frequency (TF_{2 group}) | 1.821 × 10^{−5} | 4.950 × 10^{−5} | 0.678 | |

Hybrid | Time (T_{2 group}) | 2.228 × 10^{−5} | 2.824 × 10^{−5} | 0.332 |

Frequency (F_{2 group}) | 1.625 × 10^{−5} | 4.365 × 10^{−5} | 0.391 | |

Time–Frequency (TF_{2 group}) | 3.384 × 10^{−5} | 7.621 × 10^{−5} | 0.4227 |

**Table 6.**MSE error achieved by the GA during the tunning and optimization of hyperparameters of the considered SAE structures for each domain of analysis, all technology materials.

Material | Feature Domain | MSE Error | |
---|---|---|---|

Vibrations (i = 1) | Current (i = 2) | ||

Metallic | Time (T_{i group}) | 1.071 × 10^{−4} | 3.689 × 10^{−4} |

Frequency (F_{i group}) | 3.113 × 10^{−4} | 4.451 × 10^{−4} | |

Time–Frequency (TF_{i group}) | 4.348 × 10^{−4} | 1.137 × 10^{−4} | |

Ceramic | Time (T_{i group}) | 7.8202 × 10^{−5} | 1.990 × 10^{−4} |

Frequency (F_{i group}) | 1.894 × 10^{−4} | 2.786 × 10^{−4} | |

Time–Frequency (TF_{i group}) | 1.794 × 10^{−4} | 1.464 × 10^{−4} | |

Hybrid | Time (T_{i group}) | 6.550 × 10^{−5} | 1.995 × 10^{−4} |

Frequency (F_{i group}) | 3.543 × 10^{−4} | 2.778 × 10^{−4} | |

Time–Frequency (TF_{i group}) | 1.230 × 10^{−4} | 4.882 × 10^{−4} |

**Table 7.**Achieved classification ratios through the SoftMax layer for the particular evaluation of each feature domain and the proposed fusion scheme.

Material | Feature Domain | Proposed Method | PCA + NN | LDA + NN | |||
---|---|---|---|---|---|---|---|

Training | Test | Training | Test | Training | Test | ||

Metallic | Time (T_{1,2 group}) | 99.1% | 99.3% | 60.5% | 62.1% | 73.3% | 73.5% |

Frequency (F_{1,2 group}) | 96.5% | 95.8% | 68.6% | 70.7% | 78.3% | 79.9% | |

Time–Frequency (TF_{1,2 group}) | 95.3% | 96% | 78.9% | 80.8% | 71.5% | 65.8% | |

Fusion scheme (T_{1,2 group} + F_{1,2 group} + TF_{1,2 group}) | 100% | 100% | 81.2% | 83.1% | 62.5% | 62.7% | |

Ceramic | Time (T_{1,2 group}) | 97% | 96.5% | 69.5% | 70.8% | 69.0% | 71.9% |

Frequency (F_{1,2 group}) | 100% | 100% | 72.8% | 73.9% | 86.9% | 88.6% | |

Time–Frequency (TF_{1,2 group}) | 98.8% | 98.8% | 84.7% | 88.8% | 87.5% | 87.9% | |

Fusion scheme (T_{1,2 group} + F_{1,2 group} + TF_{1,2 group}) | 100% | 100% | 73.0% | 73.2% | 75.0% | 75.0% | |

Hybrid | Time (T_{1,2 group}) | 99.8% | 99.3% | 81.0% | 81.9% | 87.4% | 87.5% |

Frequency (F_{1,2 group}) | 100% | 100% | 84.2% | 84.4% | 87.9% | 88.0% | |

Time–Frequency (TF_{1,2 group}) | 99.9 | 99.8% | 81.1% | 86.7% | 82.8% | 84.0% | |

Fusion scheme (T_{1,2 group} + F_{1,2 group} + TF_{1,2 group}) | 100% | 100% | 75.2% | 75.3% | 70.3% | 71.4% |

**Table 8.**Performance metrics of the proposed method based on SAE and SoftMax layer for the particular evaluation of each feature domain and the proposed fusion scheme.

Material | Accuracy | Precision | Recall | F1 Score | |||||
---|---|---|---|---|---|---|---|---|---|

Training | Test | Training | Test | Training | Test | Training | Test | ||

Metallic | Time | 0.98 | 0.97 | 0.99 | 0.98 | 0.99 | 0.97 | 0.99 | 0.97 |

Frequency | 0.93 | 0.91 | 0.98 | 0.98 | 0.94 | 0.92 | 0.95 | 0.94 | |

Time–Frequency | 0.95 | 0.96 | 0.97 | 0.98 | 0.97 | 0.97 | 0.97 | 0.97 | |

Fusion scheme | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |

Ceramic | Time | 0.96 | 0.96 | 0.98 | 0.98 | 0.98 | 0.97 | 0.98 | 0.97 |

Frequency | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

Time–Frequency | 0.99 | 0.98 | 0.99 | 0.98 | 0.99 | 0.99 | 0.99 | 0.98 | |

Fusion scheme | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

Hybrid | Time | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 1 | 0.99 | 0.99 |

Frequency | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

Time–Frequency | 0.99 | 0.99 | 0.99 | 1 | 0.99 | 0.99 | 0.99 | 0.99 | |

Fusion scheme | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Feature Domain | Hyperparameters | ||
---|---|---|---|

L_{2} Regularization | Sparsity Regularization | Sparsity Proportion | |

Time (T_{1}) | 8.0 × 10^{−5} | 8.0 × 10^{−5} | 0.55 |

Frequency (F_{1}) | 8.0 × 10^{−5} | 8.0 × 10^{−5} | 0.45 |

Time–Frequency (TF_{1}) | 8.0 × 10^{−5} | 8.0 × 10^{−5} | 0.50 |

**Table 10.**Resulting classification ratios through the SoftMax layer for the particular evaluation of each feature domain and the proposed fusion scheme.

Feature Domain | Average Accuracy | |
---|---|---|

Training | Test | |

Time | 96.2% | 94.5% |

Frequency | 99.8% | 98.5% |

Time–Frequency | 98.4% | 96.8% |

Fusion Scheme | 100.0% | 99.8% |

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## Share and Cite

**MDPI and ACS Style**

Saucedo-Dorantes, J.J.; Arellano-Espitia, F.; Delgado-Prieto, M.; Osornio-Rios, R.A.
Diagnosis Methodology Based on Deep Feature Learning for Fault Identification in Metallic, Hybrid and Ceramic Bearings. *Sensors* **2021**, *21*, 5832.
https://doi.org/10.3390/s21175832

**AMA Style**

Saucedo-Dorantes JJ, Arellano-Espitia F, Delgado-Prieto M, Osornio-Rios RA.
Diagnosis Methodology Based on Deep Feature Learning for Fault Identification in Metallic, Hybrid and Ceramic Bearings. *Sensors*. 2021; 21(17):5832.
https://doi.org/10.3390/s21175832

**Chicago/Turabian Style**

Saucedo-Dorantes, Juan Jose, Francisco Arellano-Espitia, Miguel Delgado-Prieto, and Roque Alfredo Osornio-Rios.
2021. "Diagnosis Methodology Based on Deep Feature Learning for Fault Identification in Metallic, Hybrid and Ceramic Bearings" *Sensors* 21, no. 17: 5832.
https://doi.org/10.3390/s21175832