使用python实现非线性SVM算法

非线性SVM算法可以使用核函数来解决非线性分类问题。下面是使用Python实现非线性SVM算法的步骤：

1. 导入需要的库和数据集

from sklearn.datasets import make_circles
import matplotlib.pyplot as plt
import numpy as np

X, y = make_circles(n_samples=100, noise=0.1, factor=0.1, random_state=42)

2. 可视化数据集

plt.scatter(X[:, 0], X[:, 1], c=y)
plt.show()

3. 定义核函数

这里我们使用径向基函数（RBF）作为核函数。

def rbf_kernel(x1, x2, gamma):
    return np.exp(-gamma * np.linalg.norm(x1 - x2) ** 2)

4. 定义非线性SVM分类器

class NonlinearSVM:
    def __init__(self, gamma=1.0):
        self.gamma = gamma
        
    def fit(self, X, y):
        n_samples, n_features = X.shape
        
        # 计算Gram矩阵
        K = np.zeros((n_samples, n_samples))
        for i in range(n_samples):
            for j in range(n_samples):
                K[i, j] = rbf_kernel(X[i], X[j], self.gamma)
                
        # 定义二次规划问题
        from cvxopt import matrix, solvers
        P = matrix(np.outer(y, y) * K)
        q = matrix(np.ones(n_samples) * -1)
        A = matrix(y, (1, n_samples))
        b = matrix(0.0)
        G = matrix(np.diag(np.ones(n_samples) * -1))
        h = matrix(np.zeros(n_samples))
        
        # 求解二次规划问题
        solvers.options['show_progress'] = False
        sol = solvers.qp(P, q, G, h, A, b)
        alphas = np.array(sol['x']).reshape(n_samples)
        
        # 提取支持向量
        threshold = 1e-5
        sv_indices = alphas > threshold
        self.support_vectors = X[sv_indices]
        self.support_vector_labels = y[sv_indices]
        self.support_vector_alphas = alphas[sv_indices]
        
        # 计算偏置项
        self.b = np.mean(
            [y_k - self.predict(x_k, self.support_vectors, self.support_vector_labels, self.support_vector_alphas) 
             for (y_k, x_k) in zip(self.support_vector_labels, self.support_vectors)])
        
    def predict(self, x, support_vectors, support_vector_labels, support_vector_alphas):
        y_predict = 0
        for i in range(len(support_vector_labels)):
            y_predict += support_vector_alphas[i] * support_vector_labels[i] * rbf_kernel(x, support_vectors[i], self.gamma)
        y_predict += self.b
        return np.sign(y_predict)

5. 训练模型并可视化决策边界

clf = NonlinearSVM(gamma=1.0)
clf.fit(X, y)

# 可视化决策边界
def plot_contours(ax, clf, xx, yy, **params):
    Z = clf.predict(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)
    out = ax.contourf(xx, yy, Z, **params)
    return out

xx, yy = np.meshgrid(np.linspace(-1.5, 1.5, 100), np.linspace(-1.5, 1.5, 100))
fig, ax = plt.subplots()
plot_contours(ax, clf, xx, yy, cmap=plt.cm.coolwarm, alpha=0.8)
plt.scatter(X[:, 0], X[:, 1], c=y)
plt.show()

完整代码如下：

from sklearn.datasets import make_circles
import matplotlib.pyplot as plt
import numpy as np

X, y = make_circles(n_samples=100, noise=0.1, factor=0.1, random_state=42)
plt.scatter(X[:, 0], X[:, 1], c=y)
plt.show()

def rbf_kernel(x1, x2, gamma):
    return np.exp(-gamma * np.linalg.norm(x1 - x2) ** 2)

class NonlinearSVM:
    def __init__(self, gamma=1.0):
        self.gamma = gamma
        
    def fit(self, X, y):
        n_samples, n_features = X.shape
        
        # 计算Gram矩阵
        K = np.zeros((n_samples, n_samples))
        for i in range(n_samples):
            for j in range(n_samples):
                K[i, j] = rbf_kernel(X[i], X[j], self.gamma)
                
        # 定义二次规划问题
        from cvxopt import matrix, solvers
        P = matrix(np.outer(y, y) * K)
        q = matrix(np.ones(n_samples) * -1)
        A = matrix(y, (1, n_samples))
        b = matrix(0.0)
        G = matrix(np.diag(np.ones(n_samples) * -1))
        h = matrix(np.zeros(n_samples))
        
        # 求解二次规划问题
        solvers.options['show_progress'] = False
        sol = solvers.qp(P, q, G, h, A, b)
        alphas = np.array(sol['x']).reshape(n_samples)
        
        # 提取支持向量
        threshold = 1e-5
        sv_indices = alphas > threshold
        self.support_vectors = X[sv_indices]
        self.support_vector_labels = y[sv_indices]
        self.support_vector_alphas = alphas[sv_indices]
        
        # 计算偏置项
        self.b = np.mean(
            [y_k - self.predict(x_k, self.support_vectors, self.support_vector_labels, self.support_vector_alphas) 
             for (y_k, x_k) in zip(self.support_vector_labels, self.support_vectors)])
        
    def predict(self, x, support_vectors, support_vector_labels, support_vector_alphas):
        y_predict = 0
        for i in range(len(support_vector_labels)):
            y_predict += support_vector_alphas[i] * support_vector_labels[i] * rbf_kernel(x, support_vectors[i], self.gamma)
        y_predict += self.b
        return np.sign(y_predict)

clf = NonlinearSVM(gamma=1.0)
clf.fit(X, y)

# 可视化决策边界
def plot_contours(ax, clf, xx, yy, **params):
    Z = clf.predict(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)
    out = ax.contourf(xx, yy, Z, **params)
    return out

xx, yy = np.meshgrid(np.linspace(-1.5, 1.5, 100), np.linspace(-1.5, 1.5, 100))
fig, ax = plt.subplots()
plot_contours(ax, clf, xx, yy, cmap=plt.cm.coolwarm, alpha=0.8)
plt.scatter(X[:, 0], X[:, 1], c=y)
plt.show()