A weighted DJP-MMD based deep transfer metric learning for the fault diagnosis of bearing under variable working conditions

Abstract

The change of working conditions not only makes the data distribution inconsistent, but also increases the diagnosis difficulty of fuzzy samples at the fault boundary. The traditional distance-based deep metric learning cannot effectively classify the fuzzy samples at the fault boundary. In the traditional transfer learning models, the maximum mean discrepancy (MMD) and joint maximum mean discrepancy only increase the transferability of same-class samples, and neglect the discriminability of different-class samples across different domains. The discriminative joint probability MMD (DJP-MMD) increases the transferability of same-class samples and the discriminability of different-class samples across different domains, but it only considers the global transferability of all fault classes, ignoring the different transferability of each same fault class. Therefore, a Yu norm-based deep transfer metric learning based on weighted DJP-MMD is proposed to further improve the diagnosis accuracy of bearings under variable working conditions. The deep transfer metric learning model adopts the Yu norm-based similarity instead of the distance-based similarity to effectively classify the data samples, especially those at the fault boundary, and uses the weighted DJP-MMD to measure the data distribution discrepancy between the source and target domains to increase the transferability of each same-class samples and discriminability of different-class samples across different domains. Through the fault diagnosis analysis on bearings under variable working conditions, the diagnosis results demonstrate that the proposed deep transfer metric learning model can diagnose bearing faults with higher accuracy, stronger generalization and anti-noise capabilities compared with other fault diagnosis methods based on transfer learning.

Graphical abstract

Keywords

Yu norm / weighted DJP-MMD / deep transfer metric learning / fault diagnosis / variable working conditions

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Nomenclature

Abbreviations
DA	Domain adaptation
DBN	Deep belief network
DDA	Deep domain adaptation
DJP-MMD	Discriminative joint probability maximum mean discrepancy
DML	Deep metric learning
DNN	Deep neural network
Euc_DML	Euclidean distance-based deep metric learning
Euc_DTML	Euclidean distance-based deep transfer metric learning
Euc_DTML(WDJP-MMD)	Euclidean distance-based deep transfer metric learning model based on WDJP-MMD
JMMD	Joint maximum mean discrepancy
JPD	Joint probability discrepancy
JP-MMD	Joint probability maximum mean discrepancy
MFA	Marginal fisher analysis
MMD	Maximum mean discrepancy
P	Precision
R	Recall
SGD	Stochastic gradient descent
SNR	Signal-to-noise ratio
TCA	Transfer component analysis
TCA+DBN	Deep belief network based on TCA
WDJP-MMD	Weighted discriminative joint probability maximum mean discrepancy
Yu_DML	Yu norm-based deep metric learning
Yu_DTML	Yu norm-based deep transfer metric learning
Yu_DTML(DJP-MMD)	Yu norm-based deep transfer metric learning model based on WDJP-MMD
Yu_DTML(JMMD)	Yu norm-based deep transfer metric learning model based on JMMD
Yu_DTML(MMD)	Yu norm-based deep transfer metric learning model based on MMD
Variables
b(m)	Bias of the mth layer
C	Number of fault classes
d_{f^{(n)}}^2( x_i,x_j)	Square of the Euclidean distance between f(n)(xi) and f(n)(xj)
Ds	Source domain
Dt	Target domain
f(n)(xi)	Corresponding features of xi at the nth layer
h(m)	mth layer of the network
JYu_DML	Loss function of deep metric learning model Yu_DML
JYu_DTML	Loss function of deep transfer metric learning model Yu_DTML based on WDJP-MMD
LDA	Loss function of deep domain adaptation
MD	JPD between different classes across different domains
MT	JPD on the same class between the source and target domains
M′T	JPD on each same-class between the source and target domains
N	Number of layers of DNN
P(n)	Number of neurons in the nth layer
P(X|Y)	Class conditional probability
P(Y)	Class prior probability
{\mathbb{R}}^d	Real numbers of dimension d
Sb	Inter-class separability
Sc	Intra-class compactness
W(m)	Learning weight of the mth layer
||W^{(n)}| {|}_{\text{F}}^2	Frobenius norm of the matrix W^{(n)}
X	Input data sample
{\mathit{Y}}_{\text{s}}	One-hot label matrix of the source domain dataset
{\mathit{Y}}_{\text{t}}	One-hot label matrix of a small number labeled samples of the target domain data set
ϕ	Nonlinear activation function
α	Free parameter used to balance the importance of intra-class compactness and inter-class separability
γ	Tunable regularization parameter
μ	Balance parameter
λc	Transferability weight of the fault class c
β	Weight of domain adaptation

Acknowledgements

This research was funded by the National Natural Science Foundation of China (Grant No. 51775391).

Conflict of Interest

The authors declare no conflict of interest.

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