非线性支持向量机可通过引入核函数将数据映射到高维特征空间，使原本非线性可分的数据变得线性可分。以下是在给定代码基础上补充代码，实现一个径向基核函数（RBF），并使用该核函数将数据集中四个样本转化为线性可分的数据的示例： ```python import numpy as np from sklearn.metrics.pairwise import rbf_kernel # 定义径向基核函数 def RBFKernelFunction(xInput, xSample, gamma): def NormCalculation(xInput, xSample): NormResult = np.linalg.norm(xInput - xSample) return NormResult ** 2 return np.exp((-1 * gamma) * NormCalculation(xInput, xSample)) # 假设我们有四个样本 x = np.array([[1, 2], [2, 3], [3, 4], [4, 5]]) y = np.array([1, 1, -1, -1]) # 定义核函数参数 gamma = 0.5 # begin # 计算核矩阵 kernel_matrix = np.zeros((x.shape[0], x.shape[0])) for i in range(x.shape[0]): for j in range(x.shape[0]): kernel_matrix[i, j] = RBFKernelFunction(x[i], x[j], gamma) # 这里可以使用核矩阵进行后续的训练，以下是一个简单示例 # 定义凸优化所需的参数 from cvxopt import matrix, solvers C = 1.0 k = kernel_matrix l = np.inner(y, y) p = matrix(l * k) q = matrix(np.ones(x.shape[0]) * -1) A = matrix(y.reshape(1, -1)) b = matrix(0.) g = matrix(np.vstack((np.eye(x.shape[0]) * -1, np.eye(x.shape[0])))) h = matrix(np.vstack((np.zeros(len(y)).reshape(-1, 1), np.ones(len(y)).reshape(-1, 1) * C))) # 求解函数 solution = solvers.qp(p, q, g, h, A, b) # 获得拉格朗日系数a a = np.ravel(solution['x']) # 获得最优w与b w_best = np.sum(a.reshape(-1, 1) * y[:, np.newaxis] * x, axis=0) b_best = 0 for j in range(x.shape[0]): b_best += y[j] - np.sum(y * a.reshape(-1, 1) * np.inner(x, x[j].T).reshape(-1, 1)) b_best = b_best / x.shape[0] # end print("最优w:", w_best) print("最优b:", b_best) ``` ###