用python语言编写实现支持向量机算法的代码（要求提供至少5类核函数）

好的，以下是用 Python 语言实现支持向量机算法的代码，并提供了五种常见的核函数：

import numpy as np
from cvxopt import matrix, solvers

class SVM:
    def __init__(self, kernel='linear', C=1.0, degree=3, gamma='scale', coef0=0.0):
        self.kernel = kernel
        self.C = C
        self.degree = degree
        self.gamma = gamma
        self.coef0 = coef0
    
    def fit(self, X, y):
        n_samples, n_features = X.shape
        
        if self.gamma == 'scale':
            self.gamma = 1 / (n_features * X.var())
        elif self.gamma == 'auto':
            self.gamma = 1 / n_features
        
        if self.kernel == 'linear':
            K = np.dot(X, X.T)
        elif self.kernel == 'poly':
            K = (self.gamma * np.dot(X, X.T) + self.coef0)**self.degree
        elif self.kernel == 'rbf':
            X2 = np.sum(X**2, axis=1).reshape(-1, 1)
            K = np.exp(-self.gamma * (X2 - 2 * np.dot(X, X.T) + X2.T))
        elif self.kernel == 'sigmoid':
            K = np.tanh(self.gamma * np.dot(X, X.T) + self.coef0)
        elif self.kernel == 'laplacian':
            X2 = np.sum(X**2, axis=1).reshape(-1, 1)
            K = np.exp(-self.gamma * np.sqrt(X2 - 2 * np.dot(X, X.T) + X2.T))
        else:
            raise ValueError('Invalid kernel')
        
        P = matrix(np.outer(y, y) * K)
        q = matrix(-1 * np.ones(n_samples))
        G = matrix(np.vstack((np.eye(n_samples)*-1, np.eye(n_samples))))
        h = matrix(np.hstack((np.zeros(n_samples), np.ones(n_samples) * self.C)))
        A = matrix(y.reshape(1, -1))
        b = matrix(np.zeros(1))
        
        solvers.options['show_progress'] = False
        solution = solvers.qp(P, q, G, h, A, b)
        
        self.alpha = np.array(solution['x']).reshape(-1)
        self.sv_idx = self.alpha > 1e-5
        self.support_vectors = X[self.sv_idx]
        self.support_vectors_y = y[self.sv_idx]
        self.alpha = self.alpha[self.sv_idx]
        
        if self.kernel == 'linear':
            self.w = np.dot(self.alpha * self.support_vectors_y, self.support_vectors)
            self.b = np.mean(self.support_vectors_y - np.dot(self.support_vectors, self.w))
        else:
            self.b = np.mean([self.support_vectors_y[i] - np.sum(self.alpha * self.support_vectors_y * K[self.sv_idx[i], self.sv_idx]) for i in range(len(self.alpha))])
    
    def predict(self, X):
        if self.kernel == 'linear':
            return np.sign(np.dot(X, self.w) + self.b)
        else:
            y_pred = np.zeros(len(X))
            for i in range(len(X)):
                s = 0
                for alpha, sv_y, sv in zip(self.alpha, self.support_vectors_y, self.support_vectors):
                    if self.kernel == 'linear':
                        kernel = np.dot(X[i], sv)
                    elif self.kernel == 'poly':
                        kernel = (self.gamma * np.dot(X[i], sv) + self.coef0)**self.degree
                    elif self.kernel == 'rbf':
                        kernel = np.exp(-self.gamma * np.linalg.norm(X[i] - sv)**2)
                    elif self.kernel == 'sigmoid':
                        kernel = np.tanh(self.gamma * np.dot(X[i], sv) + self.coef0)
                    elif self.kernel == 'laplacian':
                        kernel = np.exp(-self.gamma * np.abs(np.linalg.norm(X[i] - sv)))
                    else:
                        raise ValueError('Invalid kernel')
                    s += alpha * sv_y * kernel
                y_pred[i] = s
            return np.sign(y_pred + self.b)

这里提供了五种核函数：

1. 线性核函数：$K(x_i, x_j) = x_i^T x_j$
2. 多项式核函数：$K(x_i, x_j) = (\gamma x_i^T x_j + r)^d$
3. 高斯核函数（RBF核函数）：$K(x_i, x_j) = \exp(-\gamma ||x_i - x_j||^2)$
4. sigmoid核函数：$K(x_i, x_j) = \tanh(\gamma x_i^T x_j + r)$
5. 拉普拉斯核函数：$K(x_i, x_j) = \exp(-\gamma ||x_i - x_j||)$

其中，$\gamma$、$d$和$r$为超参数，分别代表高斯核函数的带宽、多项式核函数的阶数和常数项、sigmoid核函数的斜率和常数项。在实际使用中，可以通过交叉验证等方法确定这些超参数的值。