# CEEMDAN-IHO-SVM: A Machine Learning Research Model for Valve Leak Diagnosis
## 2 Materials and Methods  
### The CEEMDAN Algorithm  
Empirical Mode Decomposition (EMD) [19] and its optimization methods are adaptive processing techniques well-suited for analyzing nonlinear, unsteady signals. These methods rely on the characteristic time scale of the signal itself, allowing for decomposition into multiple IMFs and a residual component that reflects the overall trend of the signal. EMD optimization methods, such as EEMD, Complete Ensemble Empirical Mode De-composition (CEEMD), and CEEMDAN, are all derived from EMD, inheriting its advantages, including intuitiveness and adaptability.  
CEEMDAN is an advanced ensemble empirical mode decomposition algorithm that incorporates adaptive noise. It builds upon the EMD and EEMD algorithms to address limitations inherent in these earlier methods. During EMD decomposition, waveform aliasing often occurs, leading to modal aliasing--a phenomenon where signal interactions are difficult to distinguish. EEMD was introduced as a noise-assisted data analysis method to mitigate modal aliasing in EMD by adding white noise to the original signal in each iteration before performing EMD decomposition. CEEMDAN further refines this approach by decomposing both the original signal plus white noise and the original signal minus white noise using EMD, followed by averaging the resulting IMFs to eliminate added noise, thereby enhancing decomposition accuracy and stability. However, residual white noise can still affect subsequent processing and analysis in both EEMD and CEEMD. CEEMDAN addresses this issue by more effectively suppressing noise interference and reducing the number of iterations and computational complexity.  
The subsequent section illustrates the decomposition process of CEEMDAN applied to the signals.  
First, Gaussian white noise is added to the original signals.  
\[x_{i}(t)=x(t)+\varepsilon\delta_{i}(t) \tag{1}\]  
where \(\varepsilon\) is the Gaussian white noise weighting factor and \(\delta_{i}(t)\) is the Gaussian white noise.  
Next, EMD decomposition of the sequence \(x_{i}(t)\) is performed, and the average value of the first modal components obtained from the decomposition is taken as the first-order component of CEEMDAN.  
\[IMF_{1}(t)=\frac{1}{K}\sum_{i-1}^{K}IMF_{1}^{i}(t) \tag{2}\]  
\[r_{1}(t)=x(t)-IMF_{1}(t) \tag{3}\]  
where \(IMF_{1}(t)\) (Intrinsic Mode Function) denotes the first modal component obtained from the CEEMDAN decomposition and \(r_{1}(t)\) denotes the residual signal after the first decomposition.  
After introducing specific Gaussian white noise to the \(j\)th order residual signal obtained from the decomposition process, the EMD decomposition is further applied to derive a new component and residual signal.  
\[IMF_{j}(t)=\frac{1}{K}\sum_{i-1}^{K}E_{1}\big{(}r_{j-1}(t)+\varepsilon_{j-1}E_{j- 1}(\delta_{i}(t))\big{)} \tag{4}\]  
\[r_{j}(t)=r_{j-1}(t)-IMF_{j}(t) \tag{5}\]  
where \(IMF_{j}(t)\) denotes the \(j\)th order modal component; \(E_{j-1}(.)\) denotes the \((j-1)\)th \(IMF\) component after EMD decomposition; \(\varepsilon_{j-1}\) denotes the weight coefficient of the noise added by CEEMDAN for the \((j-1)\)th iteration; and \(r_{j}(t)\) denotes the \(j\)th stage's residual signal.  
The decomposition process of the CEEMDAN algorithm concludes when the residual signals obtained from EMD become monotonic or meet other predetermined criteria.  
The flowchart of CEEMDAN decomposition is shown in Figure 1.