# CEEMDAN-IHO-SVM: A Machine Learning Research Model for Valve Leak Diagnosis
## 2 Materials and Methods
### Fuzzy Entropy  
As nonlinear technologies evolve, many nonlinear dynamic methods based on estimations of statistical parameters have been applied to extract fault characteristics. The entropy features derived from information entropy serve as effective tools for characterizing nonlinear properties, which primarily include sample entropy [20], approximate entropy [21], and fuzzy entropy [22], among others. Approximate entropy offers a framework for analyzing the complexity of finite time series; however, it is a statistical measure that quantifies the regularity of a time series, which is characterized by poor statistical stability [23]. In instances where the time series is excessively brief, sample entropy utilizes a jumping self-similarity function to measure the complexity of the time series, resulting in inaccurate estimated values in real applications [24]. Sample entropy has lower calculation efficiency, especially for long time series [25], and may yield inaccurate entropy estimates or result in undefined entropy values [26]. Fuzzy entropy is a standardized metric for evaluating the uncertainty and complexity of fuzzy sets. It has been demonstrated that fuzzy entropy  
Figure 1: The flowchart of CEEMDAN.  
surpasses sample entropy in several respects, particularly due to its enhanced robustness to noise and its greater suitability for the analysis of short and noisy time series [27].  
Fuzzy entropy is derived from sample entropy, utilizing the degree of membership of elements within a fuzzy set as the probability density function to compute the entropy value. Subsequently, the fuzzy entropy value is determined in accordance with the principles of information entropy. Like approximate entropy and sample entropy, fuzzy entropy aims to quantify the likelihood of new patterns emerging within a sequence. Specifically, a higher fuzzy entropy value signifies an increased probability of the emergence of new patterns, thereby indicating a greater complexity within the sequence. This attribute renders fuzzy entropy a vital instrument for the analysis of uncertainty and complexity in the dynamic evolution of complex systems. The fundamental principles underlying fuzzy entropy are outlined below:  
(1) Perform phase space reconstruction on a set of time series \(X\) of length \(N\).  
\[\begin{split} Y(i)&=[x(i),x(i+1),\ldots,x(i+m-1)]\\ &-x_{0}(i),i=1,2,\ldots,N-m+1\\ \end{split} \tag{6}\]  
\[\begin{split} x_{0}(i)&=\frac{1}{m}\sum_{j=0}^{m-1}x (i+j)\\ \end{split} \tag{7}\]  
(2) Define the distance between \(Y(i)\) and \(Y(j)\).  
\[\begin{split} d_{i,j}^{m}&=d[Y(i),Y(j)]=\\ \max_{k\in(0,m-1)}&|(x(i+k)-x_{0}(i))-(x(j+k)-x_{0}( i))|\\ \end{split} \tag{8}\]  
(3) Introduce fuzzy affiliation degree to measure the similarity between \(Y(i)\) and \(Y(j)\).  
\[D_{i,j}^{m}(n,r)=f(d_{i,j}^{m},n,r) \tag{9}\]  
\[f(d_{i,j}^{m},n,r)=exp(-(d_{i,j}^{m})^{n}/r) \tag{10}\]  
(4) Define the \(\varnothing^{m}\) function.  
\[\begin{split}\varnothing^{m}(n,r)=\\ \frac{1}{N-m}\sum_{i=1}^{N-m}&\left(\frac{1}{N-m-1} \sum_{j=1,j\
eq i}^{N-m}D_{i,j}^{m}\right)\\ \end{split} \tag{11}\]  
(5) Similarly, define the \(\varnothing^{m+1}\) function.  
\[\begin{split}\varnothing^{m+1}(n,r)=\\ \frac{1}{N-m}\sum_{i=1}^{N-m}&\left(\frac{1}{N-m-1} \sum_{j=1,j\
eq i}^{N-m}D_{i,j}^{m+1}\right)\\ \end{split} \tag{12}\]  
(6) If the length \(N\) of the dataset is finite, the \(FuzzyEn(m,n,r,N)\) function is shown below.  
\[FuzzyEn(m,n,r,N)=\ln\varnothing^{m}(n,r)-\ln\varnothing^{m+1}(n,r) \tag{13}\]