Outline

1.算法思想

2.概念解释

3.Sklearn Code

Part 1 算法思想：

• 目的：依据原有数据将不同类别用一条‘决策边界“分离开，数据的分类根据位置在边界两侧中的一侧进行分类。在保证一定error在上限以下，margin(support vector距离边界的距离)越大越好
• 优势：support vector非常重要，以至于其他的训练数据可以忽略。而这也是SVM的优势，使用少量的support vector数据，训练速度极快。

Part 2 概念解释：

linear kernel：处理线性边界进行分类的问题。

C：The C parameter controls how much error is allowed. A large C allows for little error and creates a hard margin. A small C allows for more error and creates a soft margin.惩罚参数越小，容忍性就越大。

s1.jpeg

polynomial kernel：A polynomial kernel transforms points into three dimensions

degree：2

rbf：rbf kernel transforms points into infinite dimensions

gamma：If gamma is large, the training data is more relevant, and as a result overfitting can occur.

Part 3 Sklearn Code：

利用以下代码，可绘制分类边界和支持向量

from sklearn.svm import SVC
# SVM 函数
clf = SVC(kernel='linear')
clf.fit(X, y)

# 最佳函数：为什么这是最佳函数？？？待回答
w = clf.coef_[0]
a = -w[0] / w[1]
y_3 = a*x_fit - (clf.intercept_[0]) / w[1]
# 最大边距 下届
b_down = clf.support_vectors_[0]
y_down = a* x_fit + b_down[1] - a * b_down[0]
# 最大边距 上届
b_up = clf.support_vectors_[-1]
y_up = a* x_fit + b_up[1] - a * b_up[0]

# 画散点图
X, y = make_blobs(n_samples=60, centers=2, random_state=0, cluster_std=0.4)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap=plt.cm.Paired)
# 画函数
plt.plot(x_fit, y_3, '-c')
# 画边距
plt.fill_between(x_fit, y_down, y_up, edgecolor='none', color='#AAAAAA', alpha=0.4)
# 画支持向量
plt.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1], edgecolor='b',
 s=80, facecolors='none')

3.1 线性核代码：

#linear

from sklearn.svm import SVC
from graph import points, labels

classifier = SVC(kernel = 'linear')
classifier.fit(points, labels)
print(classifier.predict([[3, 4], [6, 7]]))

3.2 多项式核代码：

#polynomial kernel：将二维数据升级为三维数据

from sklearn.datasets import make_circles
from sklearn.svm import SVC
from sklearn.model_selection import train_test_split

#Makes concentric circles
points, labels = make_circles(n_samples=300, factor=.2, noise=.05, random_state = 1)

#Makes training set and validation set.
training_data, validation_data, training_labels, validation_labels = train_test_split(points, labels, train_size = 0.8, test_size = 0.2, random_state = 100)

classifier = SVC(kernel = "linear", random_state = 1)
classifier.fit(training_data, training_labels)
print(classifier.score(validation_data, validation_labels))
print(training_data[0])
new_training = [[2 ** 0.5 * pt[0] * pt[1], pt[0] ** 2, pt[1] ** 2] for pt in training_data]
new_validation = [[2 ** 0.5 * pt[0] * pt[1], pt[0] ** 2, pt[1] ** 2] for pt in validation_data]

#classifier=SVC(kernel="poly",degree=2)
#classifier.fit(training_data,training_labels)


classifier.fit(new_training, training_labels)
print(classifier.score(new_validation, validation_labels))

*超平面的补充：

# 如果我们遇到这样的数据集，没有办法利用线性分类器进行分类

from sklearn.datasets.samples_generator import make_circles
# 画散点图
X, y = make_circles(100, factor=.1, noise=.1, random_state=2019)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap=plt.cm.Paired)
clf = SVC(kernel='linear').fit(X, y)
# 最佳函数
x_fit = np.linspace(-1.5, 1.5)
w = clf.coef_[0]
a = -w[0] / w[1]
y_3 = a*X - (clf.intercept_[0]) / w[1]
plt.plot(X, y_3, '-c')

s2.png

#我们可以将二维（低维）空间的数据映射到三维（高维）空间中。 此时，我们便可以通过一个超平面对数据进行划分。所以，我们映射的目的在于使用 SVM 在高维空间找到超平面的能力。

from mpl_toolkits.mplot3d import Axes3D
# 数据映射
r = np.exp(-(X[:, 0] ** 2 + X[:, 1] ** 2))
ax = plt.subplot(projection='3d')
ax.scatter3D(X[:, 0], X[:, 1], r, c=y, s=50, cmap=plt.cm.Paired)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
x_1, y_1 = np.meshgrid(np.linspace(-1, 1), np.linspace(-1, 1))
z = 0.01*x_1 + 0.01*y_1 + 0.5
ax.plot_surface(x_1, y_1, z, alpha=0.3)

s3.png

#在SVC中，我们可以用高斯核函数来实现这以功能：kernel='rbf'

# 画图
X, y = make_circles(100, factor=.1, noise=.1, random_state=2019)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap=plt.cm.Paired)
clf = SVC(kernel='rbf')
clf.fit(X, y)
ax = plt.gca()
x = np.linspace(-1, 1)
y = np.linspace(-1, 1)
x_1, y_1 = np.meshgrid(x, y)
P = np.zeros_like(x_1)
for i, xi in enumerate(x):
    for j, yj in enumerate(y):
        P[i, j] = clf.decision_function(np.array([[xi, yj]]))
ax.contour(x_1, y_1, P, colors='k', levels=[-1, 0, 0.9], alpha=0.5,
 linestyles=['--', '-', '--'])
plt.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1], edgecolor='b',
 s=80, facecolors='none');

s4.png

3.3高斯核代码：

#rbf

from data import points, labels
from sklearn.model_selection import train_test_split
from sklearn.svm import SVC

training_data, validation_data, training_labels, validation_labels = train_test_split(points, labels, train_size = 0.8, test_size = 0.2, random_state = 100)

classifier = SVC(kernel = "rbf", gamma = 0.1)
classifier.fit(training_data, training_labels)
print(classifier.score(validation_data, validation_labels))