Improved graph-regularized deep belief network with sparse features learning for fault diagnosis

Abstract

Vibration signals are widely used in fault diagnosis of rotating machinery in real-world situations. However, it is very challenging to extract effective fault features from noisy signals and construct an accurate diagnosis model. In this paper, we propose a novel Gaussian–Bernoulli deep belief network (GDBN) model for intelligent fault diagnosis, where improved graph regularization and sparse features learning are embedded in the GDBN smoothly. In particular, the improved graph regularization is added to the hidden layer of original and reconstructed data. Therefore, our model can not only transform the original data into features with improved separability, but also generate discriminant features from vibration signal. An unsupervised pre-training learning process followed by a supervised fine-tuning is implemented in proposed model to contribute the classification capabilities. The effectiveness and superiority of the proposed model have been validated by gearbox and bearing cases studies. The results illustrate that our model can learn effective discriminative features and the extracted features are more separable. Furthermore, the proposed model achieves the better diagnosis accuracy in comparison with that of other models.

My Appendix

From the energy function defined by Eq. (8), we can obtain the joint probability distribution of the states $({\mathbf {v}},{\mathbf {h}})$ [10]:

$$\begin{aligned} P({\mathbf {v}},{\mathbf {h}})=\frac{e^{-E({\mathbf {v}},{\mathbf {h}})}}{\mathop {\sum }\limits _{{\mathbf {v}}}\mathop {\sum }\limits _{{\mathbf {h}}}e^{-E({\mathbf {v}},{\mathbf {h}})}} \end{aligned}$$

(18)

where $Z=\mathop {\sum }\limits _{{\mathbf {v}}}\mathop {\sum }\limits _{{\mathbf {h}}}e^{-E({\mathbf {v}},{\mathbf {h}})}$.

When calculating the features of the original data, the features of the reconstruction layer data can be regarded as constants. Denote $\mathbf{{h}}_{ - m}^k\mathrm{{ = }}\left( {h_1^k,h_2^k, \cdots ,h_{m - 1}^k,h_{m + 1}^k, \cdots ,h_{{n_h}}^k} \right)$ is the removed vector of components from ${\mathbf{{h}}^k}$.

Considering the energy function of (14), Derive $p(v_{m}^{k}=v\mid {\mathbf {h}}^{k})$

$$\begin{aligned}&p\left( {\mathbf {v}}_{m}^{k}\mid {\mathbf {h}}^{k}\right) =\frac{e^{-\mathop \sum \limits _{i=1}^{n_{v}}\frac{\left( v_{i}^{k}-b_{i}\right) ^{2}}{2\sigma _{i}^{2}} +\mathop \sum \limits _{i=1}^{n_{v}}\mathop \sum \limits _{j=1}^{n_{h}}\frac{v_{i}^{k}}{\sigma _{i}^{2}}w_{ij}h_{i}^{k}}}{\mathop \prod \limits _{i=1}^{n_{v}}\sqrt{2\pi }\sigma _{i} e^{\frac{2b_{i}\mathop \sum \limits _{j=1}^{n_{h}}w_{ij}h_{j}^{k}+\left( \mathop \sum \limits _{j=1}^{n_{h}}w_{ij}h_{j}^{k}\right) ^{2}}{2\sigma _{i}^{2}}}} \nonumber \\&=\mathop \prod \limits _{i=1}^{n_{v}}\frac{1}{\sqrt{2\pi }\sigma _{i}}e^{-\frac{\left( v_{i}^{k}-b_{i}-\sigma _{i}\mathop \sum \limits _{j=1}^{n_{h}}\right) ^{2}}{2\sigma _{i}^{2}}}. \end{aligned}$$

(19)

Therefore,

$$\begin{aligned} p\left( v_{m}^{k}\mid {\mathbf {h}}^{k}\right) =\frac{1}{\sqrt{2\pi }\sigma _{m}}e^{-\frac{\left( v_{m}^{k}-b_{m}-\sigma _{m}\mathop \sum \limits _{j=1}^{n_{h}}w_{mj}h_{m}^{k}\right) ^{2}}{2\sigma _{m}^{2}}}. \end{aligned}$$

(20)

Derivate $p(h_{m}^{k}=1\mid {\mathbf {v}}^{k})$ :

$$\begin{aligned} p\left( {h_{m}^{k} = 1{\mid }{\mathbf{v}}^{k} } \right) & = & p\left( {h_{m}^{k} = 1{\mid }{\mathbf{h}}_{{ - m}}^{k} ,{\mathbf{v}}^{k} } \right) = \frac{{p\left( {h_{m}^{k} = 1,{\mathbf{h}}_{{ - m}}^{k} ,{\mathbf{v}}^{k} } \right)}}{{p\left( {{\mathbf{h}}_{{ - m}}^{k} ,{\mathbf{v}}^{k} } \right)}} \\ & = \frac{{\frac{1}{Z}e^{{ - E\left( {h_{m}^{k} = 1,{\mathbf{h}}_{{ - m}}^{k} ,{\mathbf{v}}^{k} } \right)}} }}{{\frac{1}{Z}e^{{ - E\left( {h_{m}^{k} = 1,{\mathbf{h}}_{{ - m}}^{k} ,{\mathbf{v}}^{k} } \right)}} + \frac{1}{Z}e^{{ - E\left( {h_{m}^{k} = 0,{\mathbf{h}}_{{ - m}}^{k} ,{\mathbf{v}}^{k} } \right)}} }} \\ & = \frac{1}{{1 + e^{{ - b_{m} - \sum\limits_{{i = 1}}^{{n_{v} }} {\frac{{v_{i}^{k} }}{{2\sigma _{i}^{2} }}} w_{{im}} + \left( {\frac{{\lambda _{2} }}{2}\mathop \sum \limits_{l} W_{{intra,kl}} \left( {1 - 2h_{m}^{l} } \right) - \frac{{\lambda _{2} }}{2}\mathop \sum \limits_{l} W_{{inter,kl}} \left( {1 - 2h_{m}^{l} } \right)} \right)}} }} \\ & = s\left( {b_{m} + \sum\limits_{{i = 1}}^{{n_{v} }} {\frac{{v_{i}^{k} }}{{2\sigma _{i}^{2} }}} w_{{im}} - } \right.\;\;\left. {\left( {\frac{{\lambda _{2} }}{2}\mathop \sum \limits_{l} W_{{intra,kl}} \left( {1 - 2h_{m}^{l} } \right) - \frac{{\lambda _{2} }}{2}\mathop \sum \limits_{l} W_{{inter,kl}} \left( {1 - 2h_{m}^{l} } \right)} \right)} \right) \\ \end{aligned}$$

(21)

The derivation process of reconstructed data is similar to the original data.