# Color Recurrence Plots for Bearing Fault Diagnosis

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Non-Uniform Embedding

#### 2.2. The Standard Recurrence Plot

#### 2.3. Recurrence Plots for Bearing Fault Diagnosis

## 3. The Color Recurrence Plot

#### 3.1. The Proposed Recurrence Plot Based on a Fixed Time Lag

#### 3.2. The Color Recurrence Plot Based on Non-Uniform Embedding

#### 3.3. Color Recurrence Plots Produced by Bearing Vibration Signals

## 4. The Description of the Data Set

## 5. Transfer Learning for Bearing Fault Diagnosis

#### 5.1. Transfer Learning Definition

_{++}or Java and expose him to programming in python. The person would be able to use the skills acquired in learning C

_{++}or Java and would be able to learn python much faster compared to a person who has no previous programming knowledge.

#### 5.2. Pretrained Network Selection

#### 5.3. Transfer Learning for AlexNet and SqueezeNet

#### 5.3.1. Transfer Learning for AlexNet

#### 5.3.2. Transfer Learning for SqueezeNet

`‘classificationLayer_preidction’`contain information on how to combine the features that the network uses for class probabilities. For retraining of the SqueezeNet the last two layers are replaced with a new set of layers to adapt to the new data set. In SqueezeNet, the last learnable layer is the final convolutional layer instead, which needs to be replaced by a new convolutional layer with a number of filters equal to 2 for the classification of the new data set with crack and without crack. Some learning parameters are then adjusted to, e.g., Weightlearnfactor and Biaslearnfactor. The initial learning rate parameter is also adjusted to slow down the learning rate of the new network to increase its prediction capabilities. Now, the new network is retrained with a new data set of images.

#### 5.3.3. Network Parameter Optimization

- Identify the most impacting network parameter through a sensitivity analysis.
- For sensitivity analysis, a parameter range is decided and then prediction is performed using the set of network parameter impact on prediction accuracy is checked.
- Several combinations of different network parameters are tested.
- Finally, the network parameters giving the best prediction accuracy are selected

## 6. Color Recurrence Plot Analysis Using Transfer Learning

## 7. Conclusions and Discussions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The standard recurrence plot generated by a normal bearing vibration signal without a fault (the dataset of Case Western Reserve University School of Engineering [33]). The threshold parameter $\epsilon $ is set to 0.253 which results in an equal proportion of black and white pixels.

**Figure 2.**The standard recurrence plot of a normal bearing vibration signal without a fault. The line graph shows the relationship between the percentage of black pixels in the recurrence plot from the threshold parameter $\epsilon $. The ratio between black and white pixels is equal at $\epsilon =0.253$. Part (

**A**) shows the standard recurrence plot at $\epsilon =0.21$; part (

**B**) at $\epsilon =0.144$; part (

**C**) at $\epsilon =0.295$.

**Figure 3.**Figure showing 3 different recurrence plot generation algorithms. Standard recurrence plot, Algorithm A corresponds to a recurrence relationship involving non-uniform time delay with time delay $\tau $ $+/-$ 1, Algorithm B corresponds to the recurrence relationship involving purely non-uniform time delays $\tau $.

**Figure 4.**A schematic diagram illustrating non-uniform embedding of a scalar time series into a 5-dimensional delay-coordinate space. ${X}_{1},\dots ,{X}_{5}$ denote the axes of the reconstructed phase space; ${\tau}_{1},\dots ,{\tau}_{4}$ denote time delays.

**Figure 5.**A schematic diagram illustrating the formation of the Color recurrence plot for a normal bearing test data without a fault. The embedding dimension $d=5$ yields ten different dichotomous recurrence plots based on a fixed time lag. The first column represents recurrence plots with a single time delay (${\tau}_{1}$, ${\tau}_{2}$, ${\tau}_{3}$, and ${\tau}_{4}$). The second column represents recurrence plots with double time delays (${\tau}_{1}+{\tau}_{2}$, ${\tau}_{2}+{\tau}_{3}$, and ${\tau}_{3}+{\tau}_{4}$). The third column represents triple time delays (${\tau}_{1}+{\tau}_{2}+{\tau}_{3}$, and ${\tau}_{2}+{\tau}_{3}+{\tau}_{4}$). The fourth column represents a recurrence plot with the maximal time delay (${\tau}_{1}+{\tau}_{2}+{\tau}_{3}+{\tau}_{4}$). The Color recurrence plot is produced by the arithmetic averaging of dichotomous plots; the number of different colors is 11. The Color recurrence plot is enlarged for clarity.

**Figure 6.**Color recurrence plots generated by vibration signals of a ball bearing. Part (

**A**) shows the Color recurrence plot produced by a normal bearing without a fault. Parts (

**B**–

**D**) show recurrent plots generated by the bearing with the ball fault, the inner race fault, and the outer race fault accordingly.

**Figure 10.**Figure showing application of network optimization workflow on AlexNet and SqueezeNet using the database of images and trial runs for initial learning rateparameter optimization. The blue box highlights the best case with optimum parameters selected from the experimental design.

**Figure 11.**Figure showing application of network optimization workflow on AlexNet and SqueezeNet using the database of images and trial runs with Initial Learning rate and Dropout Probability variations.

**Figure 13.**Figure showing conceptual workflow of how Alextnet or Squeezenet will be used for image classification using Color recurrence plots.

**Figure 14.**Figure showing training of SqueezeNet using Transfer learning on the Color recurrence plots combining the classes into a single classification problem.

**Figure 15.**Figure showing training of AlexNet using Transfer learning on the Color recurrence plots combining the classes into a single classification problem.

**Figure 16.**Figure showing Confusion Matrix for AlexNet and SqueezeNet using Transfer learning on the Color recurrence plots for the drive end bearing subset of the data set.

**Figure 17.**Figure showing Confusion Matrix for AlexNet and SqueezeNet using Transfer learning on the Color recurrence plots for the data set corresponding to the fan end bearing subset of the data set.

**Figure 18.**Figure showing Confusion Matrix for AlexNet and SqueezeNet using Transfer learning on the Color recurrence plots combining the classes into a single classification problem.

**Figure 19.**Figure showing confusion matrix using AlexNet and SqueezeNet for the fan end bearing data.

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**MDPI and ACS Style**

Petrauskiene, V.; Pal, M.; Cao, M.; Wang, J.; Ragulskis, M.
Color Recurrence Plots for Bearing Fault Diagnosis. *Sensors* **2022**, *22*, 8870.
https://doi.org/10.3390/s22228870

**AMA Style**

Petrauskiene V, Pal M, Cao M, Wang J, Ragulskis M.
Color Recurrence Plots for Bearing Fault Diagnosis. *Sensors*. 2022; 22(22):8870.
https://doi.org/10.3390/s22228870

**Chicago/Turabian Style**

Petrauskiene, Vilma, Mayur Pal, Maosen Cao, Jie Wang, and Minvydas Ragulskis.
2022. "Color Recurrence Plots for Bearing Fault Diagnosis" *Sensors* 22, no. 22: 8870.
https://doi.org/10.3390/s22228870