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Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis

This research paper proposes two methods for fusing multiple wavelet coefficients in deep residual networks to improve fault diagnosis accuracy. The methods utilize concatenation and maximization techniques to combine wavelet coefficients from different frequencies. The effectiveness of these methods is demonstrated through experiments.

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Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis

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  1. Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis Minghang Zhao, Myeongsu Kang, Baoping Tang, Michael Pecht M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  2. Backgrounds • Accurate fault diagnosis is important to ensure the safety of automobiles and helicopters, long-term generation of electric power, and reliable operating of other electrical and mechanical systems. • Discrete wavelet packet transform (DWPT), an effective tool to decompose non-stationary vibration signals into various frequency bands, has been widely appliedfor machine fault diagnosis [1]. • Besides, the usage of deep learning methods is becoming more and more popular to automatically learn discriminative features from vibration signals for improving diagnostic accuracies [2]. M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  3. Motivations • However, there is still no consensus as to which wavelet (e.g., DB1, DB2, and DB3) can achieve an optimal performance in fault diagnosis. • Besides, different wavelets may be optimal for recognizing different kinds of faults under different working conditions. • It is very unlikely for one certain wavelet to be the most effective in recognizing all kinds of faults (such as bearing inner raceway faults, outer raceway faults, and rolling element faults). • Therefore, the fusion of multiple wavelets into deep neural networks has an potential to improve the accuracy of a fault diagnostic task which involves the recognition of various fault types. M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  4. Wavelet coefficients at the 1st decomposition level 2D matrices of wavelet coefficients at the th decomposition level Input Data Configuration thwavelet ··· Decomposition using variouswavelets 2ndwavelet 1stwavelet ··· • The wavelet coefficients at various frequency bands obtained using a certain wavelet can be stacked to be a 2D matrix; then, the 2D matrices derived from multiple wavelets can be formed to be a 3D matrix. Frequency band ··· ··· ··· ··· · ··· Time ··· · ··· ··· ··· M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  5. An Overview of Deep Residual Networks Input • The deep residual network (DRN) is an improved variant of convolutional neural networks (CNNs), which uses identity shortcuts to ease the difficulty of training [3]-[4]. Conv 3×3 BN, ReLU, Conv 3×3 BN BN, ReLU, Conv 3×3 ReLU Conv 3×3 BN, ReLU, Conv 3×3 A number of RBUs BN BN, ReLU, Conv 3×3 ReLU … BN, ReLU, GAP Conv 3×3 Fully connected output layer BN: Batch normalization ReLU: Rectifier linear unit Conv 3×3: Convolution with kernels in the size of 3×3 GAP: Global average pooling A residual building unit (RBU) A deep residual network M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  6. The First Developed Method • To achieve multiple wavelet coefficients fusion, a simple method is to concatenate these 2D matrices of wavelet coefficients and feed them into a DRN. • The method was named as “Multiple Wavelet Coefficients Fusion in a Deep Residual Network by Concatenation (MWCF-DRN-C)”. 2D matrix 1 BN, ReLU, Conv, m BN, ReLU, Conv, m, /2 A concatenation layer BN, ReLU, GAP (Dropout) Fully connected output layer A vibration signal + DWPTs BN, ReLU, Conv, 4m BN, ReLU, Conv, 4m Conv, m, /2 2D matrix 2 2D matrix 3 … … m: an indicator of the number of convolutional kernels 2D matrix N M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  7. The Second Developed Method • An individual convolutional layer with trainable parameters is applied to each 2D matrix of wavelet coefficients with the goal of converting the important wavelet coefficients to be large features. Then, the element-wise maximum features are chosen to be the output in the maximization layer [5]. • The method was named as “Multiple Wavelet Coefficients Fusion in a Deep Residual Network by Maximization (MWCF-DRN-M)”. Conv, m, /2 2D matrix 1 BN, ReLU, Conv, m, /2 BN, ReLU, Conv, m BN, ReLU, GAP (Dropout) Fully connected output layer BN, ReLU, Conv, 4m BN, ReLU, Conv, 4m A vibration signal + DWPTs A maximization layer Conv, m, /2 2D matrix 2 Conv, m, /2 2D matrix 3 … … … 2D matrix N Conv, m, /2 M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  8. Explanations on the Second Developed Method • The 2D matrices of wavelet coefficients are different representations of the same vibration signal. • It is unavoidable that these 2D matrices of wavelet coefficients contain much redundant/repetitive information. Conv, m, /2 2D matrix 1 Much redundancy BN, ReLU, Conv, m, /2 BN, ReLU, Conv, m BN, ReLU, GAP (Dropout) Fully connected output layer BN, ReLU, Conv, 4m BN, ReLU, Conv, 4m A vibration signal + DWPTs A maximization layer Conv, m, /2 2D matrix 2 Conv, m, /2 2D matrix 3 … … … 2D matrix N Conv, m, /2 M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  9. Explanations on the Second Developed Method • The maximization layer and the convolutional layers before it can be interpreted as a trainable feature selection process, which allows the important features to be passed to the subsequent layers while the relatively unimportant features being abandoned. Conv, m, /2 2D matrix 1 Trainable feature selection Much redundancy BN, ReLU, Conv, m, /2 BN, ReLU, Conv, m BN, ReLU, GAP (Dropout) Fully connected output layer BN, ReLU, Conv, 4m BN, ReLU, Conv, 4m A vibration signal + DWPTs A maximization layer Conv, m, /2 2D matrix 2 Conv, m, /2 2D matrix 3 … … … 2D matrix N Conv, m, /2 M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  10. Experimental Setup • A drivetrain dynamics simulator [6] was used to simulate the faults. • Experiments were conducted under the 10-fold cross-validation scheme. • Comparisons were made with the conventional CNN and DRN to demonstrate the efficacy of the developed MWCF-DRN-C and MWCF-DRN-M. M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  11. Results M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  12. Conclusions • The fusion of multiple wavelet coefficients in deep neural networks can be able to improve the fault diagnostic performance. • In the experimental result, the MWCF-DRN-M method was slightly better than the MWCF-DRN-C method by yielding a 0.80% improvement in terms of overall average testing accuracy. M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

  13. References • R. Yan, R. X. Gao, and X. Chen, “Wavelets for fault diagnosis of rotary machines: A review with applications,” Signal Process., vol. 96, pp. 1–15, 2014. • M. Zhao, M. Kang, B. Tang, and M. Pecht, “Deep Residual Networks With Dynamically Weighted Wavelet Coefficients for Fault Diagnosis of Planetary Gearboxes,” IEEE Transactions on Industrial Electronics, vol. 65, no. 5, pp. 4290–4300, 2018. • K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proc. IEEE Conf. Comput. Vision Pattern Recognit., Seattle, WA, USA, Jun. 27–30, 2016, pp. 770–778. • K. He, X. Zhang, S. Ren, and J. Sun, “Identity mappings in deep residual networks,” in Computer Vision—ECCV 2016 (Lecture Notes in Computer Science 9908), B. Leibe, J. Matas, N. Sebe, and M. Welling, Eds., Cham, Switzerland: Springer, 2016, pp. 630–645. • Z. Liao and C. Gustavo, “A deep convolutional neural network module that promotes competition of multiple-size filters,” Pattern Recognit., vol. 71, pp. 94–105, 2017. • Drivetrain Diagnostics Simulator. SpectraQuest, Richmond, VA, USA, [Online]. Available: http://spectraquest.com/drivetrains/details/dds/ M. Zhao, M. Kang, B. Tang, M. Pecht, "Multiple Wavelet Coefficients Fusion in Deep Residual Networks for Fault Diagnosis," IEEE Transactions on Industrial Electronics, vol. 66, no. 6, pp. 4696-4706, 2019.

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