【ML從入門到入土系列05】支持向量機

1 理論

SVM基本形式：有監督二分類線性分類模型；擴展形式：有監督二分類非線性分類模型，有監督多分類（線性/非線性）分類模型，有監督線性迴歸模型（SVR），基於核函數的SVM/SVR

1.1 線性SVM

與邏輯迴歸對比圖如下：

多分類SVM的核心思想是OVO，其流程如下圖

1.2 非線性SVM

如何將線性支持向量機擴展爲非線性支持向量機？

• 將線性內積（線性核函數）替換爲$K\left(x_{i}, x_{j}\right)$，如下圖所示
  
• 新模型在變換後的空間仍然是線性模型
• 新模型在原空間相對於x是非線性模型
• 計算複雜度較小：只需計算核矩陣$K\left(x_{i}, x_{j}\right)$

2 代碼

import numpy as np
from numpy import linalg
import cvxopt
import cvxopt.solvers
import pylab as pl

# 線性核
def linear_kernel(x1, x2):
    return np.dot(x1, x2)

# 多項式核
def polynomial_kernel(x, y, p=3):
    return (1 + np.dot(x, y)) ** p

# 創建svm類
class soft_margin_svm(object):

    def __init__(self, kernel=linear_kernel, C=None):
        self.kernel = kernel
        self.C = C
        if self.C is not None:
            self.C = float(self.C)

    def fit(self, X, y):
        n_samples, n_features = X.shape

        K = np.zeros((n_samples, n_samples))
        for i in range(n_samples):
            for j in range(n_samples):
                K[i, j] = self.kernel(X[i], X[j])

        P = cvxopt.matrix(np.outer(y, y) * K)
        q = cvxopt.matrix(np.ones(n_samples) * -1)
        A = cvxopt.matrix(y, (1, n_samples))
        b = cvxopt.matrix(0.0)

        if self.C is None:
            G = cvxopt.matrix(np.diag(np.ones(n_samples) * -1))
            h = cvxopt.matrix(np.zeros(n_samples))
        else:
            tmp1 = np.diag(np.ones(n_samples) * -1)
            tmp2 = np.identity(n_samples)
            G = cvxopt.matrix(np.vstack((tmp1, tmp2)))
            tmp1 = np.zeros(n_samples)
            tmp2 = np.ones(n_samples) * self.C
            h = cvxopt.matrix(np.hstack((tmp1, tmp2)))

        solution = cvxopt.solvers.qp(P, q, G, h, A, b)

        a = np.ravel(solution['x'])

        sv = a > 1e-5
        ind = np.arange(len(a))[sv]
        self.a = a[sv]
        self.sv = X[sv]
        self.sv_y = y[sv]
        print("%d support vectors out of %d points" % (len(self.a), n_samples))

        self.b = 0
        for n in range(len(self.a)):
            self.b += self.sv_y[n]
            self.b -= np.sum(self.a * self.sv_y * K[ind[n], sv])
        self.b /= len(self.a)

        if self.kernel == linear_kernel:
            self.w = np.zeros(n_features)
            for n in range(len(self.a)):
                self.w += self.a[n] * self.sv_y[n] * self.sv[n]
        else:
            self.w = None

    def project(self, X):
        if self.w is not None:
            return np.dot(X, self.w) + self.b
        else:
            y_predict = np.zeros(len(X))
            for i in range(len(X)):
                s = 0
                for a, sv_y, sv in zip(self.a, self.sv_y, self.sv):
                    s += a * sv_y * self.kernel(X[i], sv)
                y_predict[i] = s
            return y_predict + self.b

    def predict(self, X):
        return np.sign(self.project(X))


if __name__ == "__main__":

    def gen_non_lin_separable_data():
        mean1 = [-1, 2]
        mean2 = [1, -1]
        mean3 = [4, -4]
        mean4 = [-4, 4]
        cov = [[1.0, 0.8], [0.8, 1.0]]
        X1 = np.random.multivariate_normal(mean1, cov, 50)
        X1 = np.vstack((X1, np.random.multivariate_normal(mean3, cov, 50)))
        y1 = np.ones(len(X1))
        X2 = np.random.multivariate_normal(mean2, cov, 50)
        X2 = np.vstack((X2, np.random.multivariate_normal(mean4, cov, 50)))
        y2 = np.ones(len(X2)) * -1
        return X1, y1, X2, y2

    def split_train(X1, y1, X2, y2):
        X1_train = X1[:90]
        y1_train = y1[:90]
        X2_train = X2[:90]
        y2_train = y2[:90]
        X_train = np.vstack((X1_train, X2_train))
        y_train = np.hstack((y1_train, y2_train))
        return X_train, y_train

    def split_test(X1, y1, X2, y2):
        X1_test = X1[90:]
        y1_test = y1[90:]
        X2_test = X2[90:]
        y2_test = y2[90:]
        X_test = np.vstack((X1_test, X2_test))
        y_test = np.hstack((y1_test, y2_test))
        return X_test, y_test

    def plot_margin(X1_train, X2_train, clf):
    
        def f(x, w, b, c=0):
            return (-w[0] * x - b + c) / w[1]

        pl.plot(X1_train[:, 0], X1_train[:, 1], "ro")
        pl.plot(X2_train[:, 0], X2_train[:, 1], "bo")
        pl.scatter(clf.sv[:, 0], clf.sv[:, 1], s=100, c="g")

        a0 = -4;
        a1 = f(a0, clf.w, clf.b)
        b0 = 4;
        b1 = f(b0, clf.w, clf.b)
        pl.plot([a0, b0], [a1, b1], "k")

        a0 = -4;
        a1 = f(a0, clf.w, clf.b, 1)
        b0 = 4;
        b1 = f(b0, clf.w, clf.b, 1)
        pl.plot([a0, b0], [a1, b1], "k--")

        a0 = -4;
        a1 = f(a0, clf.w, clf.b, -1)
        b0 = 4;
        b1 = f(b0, clf.w, clf.b, -1)
        pl.plot([a0, b0], [a1, b1], "k--")

        pl.axis("tight")
        pl.show()


    def plot_contour(X1_train, X2_train, clf):
        pl.plot(X1_train[:, 0], X1_train[:, 1], "ro")
        pl.plot(X2_train[:, 0], X2_train[:, 1], "bo")
        pl.scatter(clf.sv[:, 0], clf.sv[:, 1], s=100, c="g")

        X1, X2 = np.meshgrid(np.linspace(-6, 6, 50), np.linspace(-6, 6, 50))
        X = np.array([[x1, x2] for x1, x2 in zip(np.ravel(X1), np.ravel(X2))])
        Z = clf.project(X).reshape(X1.shape)
        pl.contour(X1, X2, Z, [0.0], colors='k', linewidths=1, origin='lower')
        pl.contour(X1, X2, Z + 1, [0.0], colors='grey', linewidths=1, origin='lower')
        pl.contour(X1, X2, Z - 1, [0.0], colors='grey', linewidths=1, origin='lower')

        pl.axis("tight")
        pl.show()


    def test_non_linear():
        X1, y1, X2, y2 = gen_non_lin_separable_data()
        X_train, y_train = split_train(X1, y1, X2, y2)
        X_test, y_test = split_test(X1, y1, X2, y2)

        clf = soft_margin_svm(polynomial_kernel)
        clf.fit(X_train, y_train)

        y_predict = clf.predict(X_test)
        correct = np.sum(y_predict == y_test)
        print("%d out of %d predictions correct" % (correct, len(y_predict)))

        plot_contour(X_train[y_train == 1], X_train[y_train == -1], clf)


    def test_soft():
        X1, y1, X2, y2 = gen_lin_separable_overlap_data()
        X_train, y_train = split_train(X1, y1, X2, y2)
        X_test, y_test = split_test(X1, y1, X2, y2)

        clf = soft_margin_svm(C=1000.1)
        clf.fit(X_train, y_train)

        y_predict = clf.predict(X_test)
        correct = np.sum(y_predict == y_test)
        print("%d out of %d predictions correct" % (correct, len(y_predict)))

        plot_contour(X_train[y_train == 1], X_train[y_train == -1], clf)

if __name__ == '__main__':
    test_non_linear()

3 參考

理論：周志華《機器學習》，李航《統計學習方法》
代碼：https://github.com/luwill/machine-learning-code-writing