## 2 Basic Theory
### Support Vector Machine Classification Principles  
SVM, a significant classification algorithm in the traditional machine learning field, was initially proposed by [PERSON] in 1963 [13], However, the commonly seen form of SVM nowadays (i.e., soft-margin) was introduced by [PERSON] et al. in 1993 and published in 1995 [14]. SVM has been regarded as the most successful and high-performing algorithm in machine learning until deep learning emerged as a dominant force in 2012. By incorporating VC-dimensionality theory and the principle of structural risk minimization, SVM aims to find an optimal classification hyperplane that minimizes structural risk. Compared to traditional machine learning algorithms, SVM exhibits exceptional performance and finds broad applications. The fundamental concept behind SVM is to identify the best hyperplane for performing classification tasks. This hyperplane not only accurately separates two sets of samples (even when empirical risk is zero), but also maximizes the margin between classes. Consequently, SVM demonstrates superior robustness and generalization capabilities while maintaining high accuracy levels for classification tasks. The working principle of a support vector machine can be understood as follows: given training samples, it constructs a decision surface represented by a hyperplane that maximizes separation between positive and negative examples [15]. Figure 1 illustrates a schematic diagram depicting support vector machine classification.  
The general model of SVM is described as follows [16]: Given the sample set \((x_{i},y_{i})\), \(i=1,2,\ldots,l\). \(x\in R^{l},y_{i}\in\{+1,-1\}\), the classification decision function constructed by SVM is defined as follows:  
\[f(x)=\text{sgn}\{(w\cdot x+b)\}=\text{sgn}\Bigg{(}\sum_{i=1}^{l}\alpha_{i}y_{i }K(x_{i},x)+b^{*}\Bigg{)} \tag{11}\]Here, \(K(x_{i},x)\) is a symmetric kernel function satisfying the Mercer condition. The task of constructing a nonlinear categorical decision function becomes solving the following optimization problem:  
\[\left\{\begin{array}{l}\max\!W(\alpha)=\sum\limits_{i=1}^{l}\alpha_{i}-\frac{1 }{2}\sum\limits_{i,j=1}^{l}\alpha_{i}\alpha_{j}y_{i}y_{j}K(x_{i},x_{j})\\ \text{s.t.}\sum\limits_{i=1}^{l}\alpha_{i}y_{i}=0\quad 0\leq\alpha_{i}\leq C, \quad i=1,2,\ldots,l\end{array}\right.\]  
The vector corresponding to the coefficients \(\alpha_{i}\
eq 0\) is known as the support vector, and \(C\) is a constant that reflects the degree of penalty for misclassified samples. SVM training is solving a convex quadratic programming problem with boundary constraints and linear equation constraints.  
Aiming at the difficulties of SVM methods in dealing with high-dimensional large-scale data and the sensitivity to anomalous samples, we propose a DOA-SVM classification method based on the preprocessing of neighborhood rough set data. This method preprocesses the training dataset in two main ways. On the one hand, we use the outlier detection algorithm based on isolated forests [17] proposed by [PERSON] et al. in 2008 to eliminate those outlier samples or noisy data that are mixed with other categories of samples. On the other hand, we use neighborhood rough sets for attribute simplification based on attribute importance. Finally, the support vector machine is optimized with the dingo optimization algorithm to build a support vector machine model (DOA-SVM).