# CEEMDAN-IHO-SVM: A Machine Learning Research Model for Valve Leak Diagnosis
## 2 Materials and Methods
### Support Vector Machine (SVM)  
As a powerful classification and regression model, SVM has been widely used across various domains. [PERSON] et al. [35] introduced an automated method for detecting valve leakage based on acoustic emission parameters and SVM, achieving an accuracy rate exceeding 98%. [PERSON] et al. [36] employed kernel principal component analysis (KPCA) in conjunction with SVM classifiers to ascertain leakage levels, attaining an accuracy rate of 96.75%. [PERSON] et al. [37] utilized particle swarm optimization combined with SVM (PSO-SVM) for the intelligent detection of water supply pipeline leaks, demonstrating superior performance compared to backpropagation neural networks (BPNN). [PERSON] et al. [38] proposed an SV-WTBSVM method for intelligent water supply pipeline leakage detection. Their findings indicated that this algorithm not only preserved the rapid training speed characteristic of TBSVM but also enhanced both classification accuracy and generalization capability. [PERSON] et al. [39] suggested the application of a convolutional neural network (CNN) model for fault feature extraction and dimensionality reduction, subsequently inputting these features into an SVM model for diagnostic purposes. The CNN-SVM model exhibited superior accuracy relative to other classification models. Originally, SVM was developed for binary classification challenges. However, when addressing multi-class classification issues, it becomes essential to devise an appropriate multi-class classifier. Several strategies for multi-class classification can be employed, including \"one-to-one\" or \"one-to-many\", both of which aim to convert the multi-class classification problem into a sequence of binary classification tasks. When addressing challenges associated with SVM, it is important to note that if the data are not linearly separable, different kernel functions can be employed in combination to classify nonlinear data. These kernel functions facilitate the mapping of data into a high-dimensional feature space, thereby rendering the data linearly separable A selection of commonly utilized kernel functions in SVM is presented in Table 1.  
Among the various kernel functions, the RBF kernel is one of the most frequently employed. In comparison to other kernel functions, the RBF kernel function exhibits a superior capacity for nonlinear feature mapping, increased model complexity, and the ability to accommodate more intricate models across a broader spectrum of applications. It is characterized by two significant input parameters:  
_C_: The regularization parameter, which serves as an indicator of the model's tolerance to error. A higher value of this parameter indicates a lower tolerance for errors, which may result in overfitting. Conversely, a value that is too low can lead to underfitting. Both excessively large and small values of the regularization parameter can adversely affect the model's generalization capability. \begin{table}
\begin{tabular}{c c} \hline \hline
**Kernel Function Type** & **Formula** \\ \hline Polynomial & \(k=\big{(}x_{i},x_{j}\big{)}=\big{(}x_{i},x_{j}+1\big{)}^{d}\) \\ Gaussian & \(k=(x,y)=exp\big{(}-\frac{\|x-y\|^{2}}{2\sigma^{2}}\big{)}\) \\ Laplace RBF & \(k(x,y)=exp\big{(}-\gamma-\frac{\|x-y\|^{2}}{\sigma}\big{)}\) \\ Gaussian RBF & \(k\big{(}x_{i},x_{j}\big{)}=exp\big{(}-\gamma\parallel x_{i}-x_{j}\parallel^{2} \big{)}\) \\ Sigmoid & \(k(x,y)=\tanh\big{(}ax^{T}y+c\big{)}\) \\ Hyperbolic Tangent & \(k\big{(}x_{i},x_{j}\big{)}=\tanh\big{(}kx_{i}x_{j}+c\big{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Kernel functions. \(\sigma\): The kernel function parameter, which plays a crucial role in determining the distribution of data once they have been mapped into a new feature space. A low value for this parameter tends to result in a greater clustering of data points, whereas a high value necessitates that points should be in close proximity to one another to be classified as belonging to the same group.