*2.2. Fault Description*

The running gears system is equipped with many sensors to keep track of the actual status. The real data used in this study is based on data collected by a railway department in a specific year and then classified and processed to obtain the fault signals of gears. This paper uses the matrix to describe the data set for research purposes. This paper uses the matrix *Zw* to describe the data set as followings

$$Z\_w = [q\_w(1), q\_w(2), \dots, q\_w(8)] \tag{1}$$

where *Zw* ∈ *R N*×*m* with *N* samplings collected from *m* sensors. In this application *m* = 8, *N* = 2000. Furthermore, *Zw* can be rewritten as

$$\begin{aligned} Z\_w &= \begin{bmatrix} X\_x & Y\_y \end{bmatrix} \\ X\_x(k) &= \begin{bmatrix} q\_{w\nu}(1), q\_{w\nu}(2), q\_{w\nu}(3), q\_{w\nu}(4) \end{bmatrix} \\ Y\_y(k) &= \begin{bmatrix} q\_{w\nu}(5), q\_{w\nu}(6), q\_{w\nu}(7), q\_{w\nu}(8) \end{bmatrix} \end{aligned} \tag{2}$$

where *qw*(*i*) represents the data collected for the *i*th sensor. The data subset *Xx* is the input matrix, and *Y* is the output matrix. In the paper, we use types of faults as follows: (1)

Bogie 1 failure; (2) Bogie 2 failure; (3) Motor drive side bearing failure; (4) Non-drive side bearing failure; (5) Motor side big gear failure; (6) Wheel side pinion failure; (7) Wheel side motor big gear failure; (8) Motor side pinion failure. Moreover, in the process of data collection, the data collected from the same carriage in a train is selected. Without loss of generality, after splitting data into the two groups, we added fault data with labels to form experimental training data. Therefore, the fault data can be represented as

$$q\_w(\eta) = \begin{bmatrix} \ \times & f\_{\text{wt}} \ \end{bmatrix}^T, \eta = \mathbf{1}, \mathbf{2} \cdot \cdot \cdot \ , \eta \tag{3}$$

**Remark 1.** *Divide the data into the two groups: (1) qw*(1) *to qw*(4) *in one group as input; (2) qw*(5) *to qw*(8) *in another group as output. We added fault data with labels to form experimental data.*

**Remark 2.** *In this study, all the FD models were constructed and compiled within the software environment of MATLAB, and all the experiments were executed and evaluated in a PC in CPU mode.*

### *2.3. Objective and Design Issues*

The FD models for moving gear parts were often error-prone due to the scalability and complexity issues of signals. Our CCA-JITL FD model solved many challenges as below:

