# Performance of Bearing Ball Defect Classification Based on the Fusion of Selected Statistical Features

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## Abstract

**:**

## 1. Introduction

## 2. Paper Contribution

## 3. The Fault Diagnosis Methodology

#### 3.1. Preprocessing and Feature Extraction and Selection

- The mean and the skewness had very poor detection performance
- The kurtosis had a very low sensitivity to the ball fault level.

**Variance**: $IM{F}_{2}$, $IM{F}_{3}$ and $IM{F}_{4}$; denoted as $vIM{F}_{j}$**KLD**: $IM{F}_{2}$, $IM{F}_{3}$, $IM{F}_{4}$ and $IM{F}_{6}$; denoted as $kIM{F}_{j}$.

#### 3.2. Feature Analysis

#### 3.2.1. Principal Component Analysis (PCA)

- Kernel-based techniques: kernel principal component analysis (KPCA) and support vector machine (SVM);
- Deterministic systematic exploration techniques: K-nearest neighbours (KNN) and decision tree (DT);
- Probabilistic systematic exploration techniques: naive Bayes classifiers (NB).

#### 3.2.2. Kernel Principal Component Analysis (KPCA)

- The polynomial kernel defined as ( $p\in {\mathbb{R}}^{+}$ is the kernel’s order):$$K(x,y)={\left(\langle x,y\rangle +1\right)}^{p}$$
- The Gaussian kernel defined as ( $\gamma \in {\mathbb{R}}^{+}$ is the standard deviation of the kernel):$$K(x,y)=exp\left(-\frac{{\parallel x-y\parallel}^{2}}{2{\gamma}^{2}}\right)$$

#### 3.2.3. Support Vector Machine (SVM)

#### 3.2.4. K-Nearest Neighbours (KNN)

- Euclidean distance (Euc). It is defined by:$$Euc(A,B)=\sqrt{\sum _{i=1}^{m}{({x}_{i}-{y}_{i})}^{2}}$$
- City block distance (CB) given as:$$CB(A,B)=\sum _{i=1}^{m}|{x}_{i}-{y}_{i}|$$

#### 3.2.5. Decision Tree (DT)

#### 3.2.6. Naive Bayes (NB)

## 4. Results and Discussions

#### 4.1. Experimental Data

- No-load condition (${L}_{0}$): $0\%$ of the nominal load;
- Half-loaded condition (${L}_{1}$): $50\%$ of the nominal load;
- Fully loaded condition (${L}_{2}$): $100\%$ of the nominal load;
- Overloaded condition (${L}_{3}$): $150\%$ of the nominal load;
- Combination of all the load conditions: (${L}_{n}$).

- H: corresponding to the healthy behaviour (no fault);
- ${F}_{1}$: faulty case with a severity of $0.007$ inch;
- ${F}_{2}$: faulty case with a severity of $0.014$ inch;
- ${F}_{3}$: faulty case with a severity of $0.021$ inch.

#### 4.2. Experimental Validation

#### 4.2.1. Linear Classification with PCA

#### 4.2.2. Kernel-Based Classifiers

#### 4.2.3. Classification Results Based on the Systematic Data Exploration Strategy

- Under the single-load condition, all the three classifiers exhibited good performance despite a low testing accuracy rate of $96.5\%$ for the NB classifier;
- Under the combined-load condition, the performance of the NB classifier was severely degraded with $82.3\%$ and $81.92\%$ for the training accuracy rate and the testing accuracy rate, respectively.

- ■
- Case study with four features
- □
- KLD and variance of $IM{F}_{2}$ and $IM{F}_{4}$, denoted as $C4$;

In this case study, the KLD and the variance of the selected IMFs ($IM{F}_{2}$ and $IM{F}_{4}$) were merged together for each load condition as in the following matrix.$$\left\{\begin{array}{cccc}kIM{F}_{2{,}_{1}}& vIM{F}_{2{,}_{1}}& kIM{F}_{4{,}_{1}}& vIM{F}_{4{,}_{1}}\\ kIM{F}_{2{,}_{2}}& vIM{F}_{2{,}_{2}}& kIM{F}_{4{,}_{2}}& vIM{F}_{4{,}_{2}}\\ \vdots & \vdots & \vdots & \vdots \\ kIM{F}_{2{,}_{900}}& vIM{F}_{2{,}_{900}}& kIM{F}_{4{,}_{900}}& vIM{F}_{4{,}_{900}}\end{array}\right\}$$ - ■
- Case study with two features
- □
- KLD and variance of $IM{F}_{2}$, denoted as $C{2}_{1}$;
- □
- KLD and variance of $IM{F}_{4}$, denoted as $C{2}_{2}$;
- □
- Variances of $IM{F}_{2}$ and $IM{F}_{4}$, denoted as $C{2}_{3}$;
- □
- KLD of $IM{F}_{2}$ and $IM{F}_{4}$, denoted as $C{2}_{4}$.

- ■
- Case study with one feature
- □
- Variance of $IM{F}_{2}$, denoted as $C{1}_{1}$;
- □
- Variance of $IM{F}_{4}$, denoted as $C{1}_{2}$;
- □
- KLD of $IM{F}_{2}$, denoted as $C{1}_{3}$;
- □
- KLD of $IM{F}_{4}$, denoted as $C{1}_{4}$.

- For both training and testing steps, whatever the load condition or the used classifier, case $C4$ with the combination of KLD and variance for $IM{F}_{2}$ and $IM{F}_{4}$ offers the best performance.
- This analysis shows that it is possible to adapt to each case and meet the application requirements. Taking the example of load ${L}_{3}$, we can choose to work with either four features ($C4$), two features ($C{2}_{1}$) or even one feature ($C{1}_{1}$: variance of $IM{F}_{2}$) to reach $100\%$ of classification accuracy. This flexibility can address the computation time that is strongly linked to the number of used features corresponding to the input’s dimension of the classification system.
- Finally, we can conclude that in our study, the KNN classifier offers the most efficient combinations of features.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Testbed of the CWRU for bearing defects [42] and the components of REBs: (

**a**) Photo of the test bench, (

**b**) Structural description of the bench.

**Figure 4.**The three-dimensional principal subspace for bearing ball fault data under the different load conditions.

**Figure 5.**KPCA kernel function hyperparameters adjustment under fully loaded condition. (

**a**) Gaussian kernel width parameter regularisation; (

**b**) polynomial kernel degree parameter regularisation.

**Figure 6.**KPCA scatter plot under the all-load-conditions combination. (

**a**) Results with Gaussian Kernel ($\gamma =0.01$); (

**b**) results with polynomial kernel ($p=6$).

**Figure 7.**Classification time computation for the $C4$ feature selection. (

**a**) Training time evaluation; (

**b**) testing time evaluation.

IMF Rank | SNR RDP (%) |
---|---|

1 | 5 |

2 | 11.7 |

3 | 21.6 |

4 | 20 |

5 | 17.3 |

6 | 22.3 |

7 | 28.2 |

8 | 34.8 |

9 | 44.9 |

10 | 51.8 |

11 | 59.2 |

12 | 62.1 |

13 | 68.2 |

14 | 68.9 |

15 | 59.1 |

16 | 62.1 |

17 | 83 |

18 | 81.1 |

AUC for Mean | AUC for Variance | AUC for Skewness | AUC for Kurtosis | AUC for KLD | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

IMF | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ |

2 | $0.971$ | $0.982$ | 1 | $0.998$ | 1 | $0.996$ | 1 | 1 | $0.7$ | $0.838$ | $0.835$ | $0.65$ | 1 | $0.955$ | $0.996$ | $0.878$ | 1 | 1 | 1 | 1 |

3 | $0.853$ | $0.968$ | $0.602$ | $0.836$ | 1 | $0.998$ | 1 | 1 | $0.816$ | $0.996$ | $0.911$ | $0.962$ | $0.824$ | $0.845$ | $0.784$ | $0.638$ | 1 | 1 | 1 | 1 |

4 | $0.937$ | $0.82$ | $0.801$ | $0.84$ | 1 | $0.968$ | 1 | 1 | $0.882$ | $0.733$ | $0.872$ | $0.602$ | 1 | $0.975$ | 1 | $0.993$ | 1 | 1 | 1 | 1 |

5 | $0.586$ | $0.773$ | $0.999$ | $0.729$ | 1 | $0.967$ | 1 | 1 | $0.587$ | $0.554$ | $0.674$ | $0.963$ | $0.926$ | $0.966$ | 1 | 1 | 1 | 1 | 1 | 1 |

6 | $0.889$ | $0.71$ | $0.979$ | $0.953$ | 1 | $0.995$ | 1 | 1 | $0.776$ | $0.681$ | $0.634$ | $0.705$ | 1 | $0.769$ | $0.917$ | $0.913$ | 1 | 1 | 1 | 1 |

7 | $0.649$ | $0.634$ | $0.783$ | $0.958$ | 1 | $0.962$ | 1 | 1 | $0.565$ | $0.51$ | $0.632$ | $0.6$ | $0.968$ | $0.862$ | $0.994$ | $0.733$ | $0.666$ | $0.457$ | $0.842$ | $0.911$ |

8 | $0.517$ | $0.623$ | $0.637$ | $0.651$ | 1 | $0.972$ | 1 | 1 | $0.552$ | $0.687$ | $0.674$ | $0.622$ | $0.534$ | $0.733$ | $0.917$ | $0.976$ | $0.49$ | $0.45$ | $0.901$ | $0.5$ |

_{0}: no-load condition, L

_{1}: half-load condition, L

_{2}: full-load condition, L

_{3}: overload condition.

Load Condition | PC | Eigenvalue | Variance Contribution (%) | Cumulative Variance (%) |
---|---|---|---|---|

${L}_{0}$ | 1 | 2.638 | 65.956 | 65.956 |

2 | 0.982 | 24.572 | 90.53 | |

3 | 0.277 | 6.932 | 97.46 | |

4 | 0.105 | 2.537 | 100 | |

${L}_{1}$ | 1 | 1.672 | 41.814 | 41.814 |

2 | 1.475 | 36.847 | 78.69 | |

3 | 0.482 | 12.071 | 90.76 | |

4 | 0.369 | 9.239 | 100 | |

${L}_{2}$ | 1 | 3.073 | 75.927 | 75.927 |

2 | 0.744 | 18.617 | 94.55 | |

3 | 0.136 | 3.422 | 97.97 | |

4 | 0.081 | 2.031 | 100 | |

${L}_{3}$ | 1 | 2.706 | 67.653 | 67.653 |

2 | 1.039 | 25.996 | 93.65 | |

3 | 0.172 | 4.31 | 97.96 | |

4 | 0.081 | 2.01 | 100 | |

${L}_{n}$ | 1 | 1.829 | 45.747 | 45.747 |

2 | 1.132 | 28.322 | 74.07 | |

3 | 0.806 | 20.151 | 94.22 | |

4 | 0.231 | 5.779 | 100 |

KPC | Variance Contribution | Cumulative Variance (%) | ||
---|---|---|---|---|

Gaussian Kernel | Polynomial Kernel | Gaussian Kernel | Polynomial Kernel | |

1 | $0.433$ | $0.967$ | $\mathbf{43.3}$ | $\mathbf{96.7}$ |

2 | $0.281$ | $0.23$ | $\mathbf{71.4}$ | $\mathbf{99}$ |

3 | $0.199$ | $0.007$ | $\mathbf{91.3}$ | $99.7$ |

4 | $0.087$ | $0.003$ | 100 | 100 |

Classifier | KNN | DT | NB | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Load (hp) | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ |

Training accuracy rate (%) | $98.2$ | $99.8$ | $99.1$ | 100 | 99 | $98.3$ | 100 | $98.6$ | 100 | $98.6$ | 98 | $99.3$ | 98 | $99.6$ | 82.3 |

Testing accuracy rate (%) | $97.42$ | $99.92$ | $98.91$ | $99.75$ | $98.83$ | $98.25$ | $99.92$ | $98.33$ | $99.75$ | $98.15$ | $\mathbf{96.5}$ | $99.42$ | $97.83$ | $99.83$ | 81.92 |

Training time (s) | $0.31$ | $0.9$ | $0.31$ | $0.3$ | $0.56$ | $0.28$ | $0.24$ | $0.25$ | $0.25$ | $1.15$ | $0.65$ | $0.75$ | $0.7$ | $0.67$ | 1 |

Testing time (s) | $0.03$ | $0.02$ | $0.03$ | $0.02$ | $0.15$ | $0.02$ | $0.02$ | $0.01$ | $0.02$ | $0.05$ | $0.05$ | $0.02$ | $0.02$ | $0.02$ | $0.05$ |

Features | Relevant $\mathbf{IMFs}$ | |||
---|---|---|---|---|

Variance | $IM{F}_{2}$ | $IM{F}_{3}$ | $IM{F}_{4}$ | |

KLD | $IM{F}_{2}$ | $IM{F}_{3}$ | $IM{F}_{4}$ | $IM{F}_{6}$ |

Classifier | KNN | DT | NB | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Load (hp) | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ |

Training accuracy rate (%) | 100 | 100 | $99.9$ | 100 | 100 | 100 | 100 | $99.9$ | 100 | 100 | $99.9$ | 100 | $99.9$ | 100 | 100 |

Testing accuracy rate (%) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | $99.83$ | 100 | $99.85$ | 100 | 100 |

Training time (s) | $0.89$ | $1.1$ | $1.31$ | $0.88$ | $0.69$ | $1.58$ | $1.04$ | $0.55$ | $0.56$ | $0.55$ | $1.8$ | $0.87$ | $1.24$ | $1.01$ | $1.53$ |

Testing time (s) | $0.19$ | $0.05$ | $0.03$ | $0.04$ | $0.06$ | $0.05$ | $0.03$ | $0.03$ | $0.02$ | $0.04$ | $0.07$ | $0.04$ | $0.02$ | $0.02$ | $0.05$ |

Ref | Fault Type | Ball | ||||||
---|---|---|---|---|---|---|---|---|

Load (hp) | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{3}$ | ${\mathit{L}}_{\mathit{n}}$ | Mean | ||

Algorithm | Testing Accuracy Rates (%) | |||||||

[6] | MPE | KNN | 93 | 99 | 100 | 100 | Not provided | 98 |

SVM | 81 | 99 | 100 | 98 | Not provided | 94.5 | ||

Logic regression | 96 | 99 | 100 | 100 | Not provided | 98.75 | ||

Backpropagation NN | 70 | 91 | 90 | 93 | Not provided | 86 | ||

Extreme learning Machine | 92 | 90 | 100 | 100 | Not provided | 97.5 | ||

Soft regression | 94 | 99 | 100 | 100 | Not provided | 98.25 | ||

Proposed technique | KLD and variance | KNN | 100 | 100 | 100 | 100 | 100 | 100 |

DT | 100 | 100 | 100 | 100 | 100 | 100 | ||

NB | 99.83 | 100 | 99.85 | 100 | 100 | 99.92 |

**Table 9.**Best feature combinations according to the classification accuracy under different load conditions.

KNN | DT | NB | ||||
---|---|---|---|---|---|---|

Load Condition | $\mathit{TrA}\phantom{\rule{0.166667em}{0ex}}(\%)$ | $\mathit{TsA}\phantom{\rule{0.166667em}{0ex}}(\%)$ | $\mathit{TrA}\phantom{\rule{0.166667em}{0ex}}(\%)$ | $\mathit{TsA}\phantom{\rule{0.166667em}{0ex}}(\%)$ | $\mathit{TrA}\phantom{\rule{0.166667em}{0ex}}(\%)$ | $\mathit{TsA}\phantom{\rule{0.166667em}{0ex}}(\%)$ |

${L}_{0}$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ |

$C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | |

$C{2}_{3}$ | $C{2}_{3}$ | |||||

${L}_{1}$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ |

$C{2}_{3}$ | $C{2}_{2}$ | $C{2}_{2}$ | $C{2}_{2}$ | $C{2}_{2}$ | $C{2}_{2}$ | |

$C{1}_{2}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | |

$C{2}_{4}$ | $C{1}_{2}$ | $C{1}_{2}$ | ||||

$C{1}_{2}$ | ||||||

${L}_{2}$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ |

$C{2}_{3}$ | $C{2}_{2}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | |

$C{1}_{2}$ | $C{2}_{3}$ | $C{1}_{2}$ | $C{1}_{2}$ | $C{1}_{2}$ | ||

$C{1}_{1}$ | ||||||

$C{1}_{2}$ | ||||||

${L}_{3}$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ |

$C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | $C{2}_{1}$ | |

$C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | $C{2}_{3}$ | |

$C{1}_{1}$ | $C{2}_{4}$ | $C{1}_{1}$ | $C{1}_{1}$ | $C{1}_{1}$ | $C{1}_{1}$ | |

$C{1}_{1}$ | ||||||

${L}_{n}$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ | $C4$ |

$C{2}_{1}$ | $C{2}_{1}$ | |||||

$C{2}_{3}$ | $C{2}_{2}$ | |||||

$C{2}_{3}$ | ||||||

$C{2}_{4}$ | ||||||

$C{1}_{1}$ | ||||||

$C{1}_{2}$ | ||||||

$C{1}_{4}$ |

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## Share and Cite

**MDPI and ACS Style**

Mezni, Z.; Delpha, C.; Diallo, D.; Braham, A.
Performance of Bearing Ball Defect Classification Based on the Fusion of Selected Statistical Features. *Entropy* **2022**, *24*, 1251.
https://doi.org/10.3390/e24091251

**AMA Style**

Mezni Z, Delpha C, Diallo D, Braham A.
Performance of Bearing Ball Defect Classification Based on the Fusion of Selected Statistical Features. *Entropy*. 2022; 24(9):1251.
https://doi.org/10.3390/e24091251

**Chicago/Turabian Style**

Mezni, Zahra, Claude Delpha, Demba Diallo, and Ahmed Braham.
2022. "Performance of Bearing Ball Defect Classification Based on the Fusion of Selected Statistical Features" *Entropy* 24, no. 9: 1251.
https://doi.org/10.3390/e24091251