# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Data Processing Methods
#### 2.1.3 The Empirical Model Decomposition Approach  
The EMD approach is an adaptive decomposition method for non-stationary signals proposed by [PERSON] et al. (2010). in 1998. Using an iterative screening process, the approach decomposes the original non-stationary signal with poor performance and contaminants into a sequence of new, improved sequence of signals representing the oscillatory part of the original signal, which is also called the intrinsic mode functions (IMF). By layer-by-layer deconstructing the components of different feature scales in the signal, it is easier to extract the distinctive information parameters of the condition monitoring signal, which has gained significant attention in equipment fault diagnosis. The steps of EMD are as follows:  
1. After calculating all local extremes of the original signal \(x(t)\), a cubic spline function is used to link all local maxima as the upper envelope \(e_{+}(t)\), followed by a cubic spline function to connect all local minima as the lower envelope \(e_{-}(t)\). The mean envelope \(e(t)\) is then determined between the upper and lower envelopes. Next, subtract \(e(t)\) from the original signal \(x(t)\) to obtain a new signal \(m_{1}(t)\). \[m_{1}(t)=x(t)-e(t)\] (4)
2. If \(m_{1}(t)\) satisfies the IMF criteria (Stein  
3. Subtract \(c_{1}(t)\) from the original signal \(x(t)\) to obtain the new signal \(r_{1}(t)\) as: \[r_{1}(t)=x(t)-c_{1}(t)\] (6) As \(r_{1}(t)\) still contains some IMF, it is treated as a new original input signal and steps a-c are repeated to find the \(n\) components until the iterative process stops when \(r_{n}\) is a monotonic function or a constant. \[\left\{ \begin{array}{l}r_{1}(t)=x(t)-c_{1}(t)\\ r_{2}(t)=r_{1}(t)-c_{2}(t)\\ \vdots\\ r_{n}(t)=r_{n-1}(t)-c_{n}(t)\end{array} \right.\] (7) where \(c_{1}(t)\), \(c_{2}(t)\), \(\cdots\), \(c_{n}(t)\) represent the information of the signal from high to low frequencies. This completes the entire EMD process, where the original signal \(x\) is split into several components and a sum of residual functions. \[x(t)=\sum_{i=1}^{n}c_{i}(t)+r_{n}(t)\] (8)  
The various order components characterize the different frequency components of the signal and the residual function represents the average trend of the signal. EMD is also a popular method for time-frequency domain signal analysis. Compared to wavelet transform techniques, it overcomes the problem of non-adaptive basis functions and eliminates the requirement for pre-determined basis functions, making it more adaptive to non-linear and non-stationary signals. This has won the approach a great deal of attention in the field of equipment condition monitoring [66; 67; 68; 69; 70; 71; 72; 73]. In [74], the envelope analysis of the IMF obtained by EMD decomposition of the vibration signal of the hydraulic pump casing was carried out to derive the envelope spectrum of the first three IMFs containing the main information about the fault and used to implement fault diagnosis. The experiments show that the method can effectively diagnose the slipper loosing, swashplate wearing and valve plate wearing faults of hydraulic pump. In [75], EMD is used to extract the IMFs of the different frequency bands of the induction motor current. The dominating IMF is obtained for the analysis of motor faults by decomposing the current signals in different motor operating modes. The results show that the diagnosis of broken rotor bars faults in induction motors can be accurately done by evaluating the variation of the amplitude of the IMF oscillation, under varied loads. [PERSON] et al. [76], used EMD to decompose the stator current of an induction motor into multiple IMFs with specific frequency bands, and the IMF in the first row with the fastest response to the fault was used for the detection of motor stator faults. Subsequently, to distinguish between motor current fluctuations caused by normal operation and those caused by fault conditions, the instantaneous frequency (IF) variation of the IMFs is used as a criterion for fault detection.