Detailed SVM Examples (Including Classification and Regression) with Implementation Approaches

Support Vector Machine (SVM) is a powerful supervised learning algorithm applicable to both classification and regression problems. Its core concept involves finding an optimal hyperplane that maximizes the margin between different classes. Below are typical application scenarios and implementation approaches for SVM in classification and regression tasks.

### 1. SVM Classification Example Consider a binary classification problem aiming to distinguish between two classes of data points. SVM implements classification through the following steps: Data Preparation: Collect labeled training data, such as points in a two-dimensional feature space. Kernel Function Selection: Choose appropriate kernel functions (linear, polynomial, or Gaussian RBF) based on data distribution characteristics. Model Training: Optimize the hyperplane to maximize the margin between positive and negative samples through quadratic programming optimization. Prediction: Classify new data points by determining which side of the hyperplane they reside on using the decision function sign(w·x + b). SVM classification is particularly suitable for high-dimensional data applications like text classification and image recognition.

### 2. SVM Regression Example (SVR) SVM can also handle regression problems through Support Vector Regression (SVR), which aims to fit data while minimizing prediction error and model complexity. Data Preparation: Input features with continuous target values. Parameter Tuning: Select appropriate kernel functions and regularization parameter (C) to control fitting flexibility using epsilon-insensitive loss function. Training and Prediction: Optimize the regression function to fit data within an acceptable error margin through convex optimization. SVR is suitable for robust regression scenarios like financial forecasting and medical data analysis where outlier resistance is crucial.

### 3. SMO Algorithm and MATLAB Implementation Sequential Minimal Optimization (SMO) is an efficient algorithm for solving SVM optimization problems. MATLAB provides built-in functions `fitcsvm` (for classification) and `fitrsvm` (for regression), but manual SMO implementation follows these steps: Initialize Lagrange multipliers (α) with zeros or small random values. Select two multipliers for optimization using heuristic selection strategies (first choice: max violation pair). Update multipliers and calculate threshold (b) using analytical solution for two-variable subproblem. Iterate optimization until convergence criteria are met (KKT conditions satisfied within tolerance). For complete SMO MATLAB source code, refer to open-source machine learning libraries (like LIBSVM's MATLAB interface) or search relevant implementations on platforms like GitHub.

### Summary SVM demonstrates excellent performance in both classification and regression tasks, especially when dealing with complex nonlinear relationships. The SMO algorithm provides efficient SVM solution methods, with MATLAB offering built-in support while manual implementation enhances algorithmic understanding and customization capabilities.