1.06支撑向量机思维导图

2.今日份实验

2.1 scikit-learn中的SVM

import numpy as np  #导包
import matplotlib.pyplot as plt

from sklearn import datasets
iris = datasets.load_iris()
X = iris.data
y = iris.target
X = X[y<2,:2]      #y<2即取y=0和y=1的两个类别，并只取前两个特征
y = y[y<2]

plt.scatter(X[y==0,0], X[y==0,1], color='red')
plt.scatter(X[y==1,0], X[y==1,1], color='blue')
plt.show()
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运行结果如下所示：

2.2 训练SVC模型

• 数据标准化操作

from sklearn.preprocessing import StandardScaler
standardScaler = StandardScaler()
standardScaler.fit(X)
X_standard = standardScaler.transform(X)
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from sklearn.svm import LinearSVC
svc = LinearSVC(C=1e9) #C值越大，容错能力就越小，就越趋向于Hard Margin SVM
svc.fit(X_standard, y)
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• 传入模型和边界，定义决策边界函数

def plot_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]
    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)
    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
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plot_decision_boundary(svc, axis=[-3, 3, -3, 3])
plt.scatter(X_standard[y==0,0], X_standard[y==0,1])
plt.scatter(X_standard[y==1,0], X_standard[y==1,1])
plt.show()
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运行结果如下所示：

• 当C=0.01时，SVM的容错能力变大

svc2 = LinearSVC(C=0.01)
svc2.fit(X_standard, y)
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plot_decision_boundary(svc2, axis=[-3, 3, -3, 3])
plt.scatter(X_standard[y==0,0], X_standard[y==0,1])
plt.scatter(X_standard[y==1,0], X_standard[y==1,1])
plt.show()
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运行结果如下图所示：

• 求Margin对应的上下两根直线

svc.coef_
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svc.intercept_ #截距
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• 绘制SVC的决策边界，包括上下两根线

def plot_svc_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]

    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
    
    w = model.coef_[0]     #取出第0个元素
    b = model.intercept_[0]
    
    # w0*x0 + w1*x1 + b = 0
    # => x1 = -w0/w1 * x0 - b/w1  #x1是纵坐标轴
    plot_x = np.linspace(axis[0], axis[1], 200)
    up_y = -w[0]/w[1] * plot_x - b/w[1] + 1/w[1]  #上边界直线方程
    down_y = -w[0]/w[1] * plot_x - b/w[1] - 1/w[1]  #下边界直线方程
    
    up_index = (up_y >= axis[2]) & (up_y <= axis[3])  #过滤操作
    down_index = (down_y >= axis[2]) & (down_y <= axis[3])
    plt.plot(plot_x[up_index], up_y[up_index], color='black')
    plt.plot(plot_x[down_index], down_y[down_index], color='black')
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plot_svc_decision_boundary(svc, axis=[-3, 3, -3, 3])
plt.scatter(X_standard[y==0,0], X_standard[y==0,1])
plt.scatter(X_standard[y==1,0], X_standard[y==1,1])
plt.show()
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运行结果如下所示：

• 当C=0.01时的情况

plot_svc_decision_boundary(svc2, axis=[-3, 3, -3, 3])
plt.scatter(X_standard[y==0,0], X_standard[y==0,1])
plt.scatter(X_standard[y==1,0], X_standard[y==1,1])
plt.show()
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运行结果如下所示：

2.3 SVM中使用多项式特征

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

x,y = datasets.make_moons()#利用月亮函数生成数据集
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x.shape #数据集有100个样本，每个样本有2个特征
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y.shape
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plt.scatter(x[y==0,0],x[y==0,1])
plt.scatter(x[y==1,0],x[y==1,1])
plt.show()
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运行结果如下所示：

• 为数据几增加噪音后的情况

X, y = datasets.make_moons(noise=0.15, random_state=666)#为数据集增加噪音

plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()
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运行结果如下所示：

2.4 用多项式特征的SVM

#总结：先升维，再计算
from sklearn.preprocessing import PolynomialFeatures,StandardScaler
from sklearn.svm import LinearSVC
from sklearn.pipeline import Pipeline

def PolynomialSVC(degree,C=1.0):
    return Pipeline([
        ("poly",PolynomialFeatures(degree=degree)),
        ("std_scaler",StandardScaler()),
        ("linearSVC",LinearSVC(C=C))
    ])
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poly_svc = PolynomialSVC(degree=3)
poly_svc.fit(X, y)
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#绘制图像
def plot_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]

    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
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plot_decision_boundary(poly_svc, axis=[-1.5, 2.5, -1.0, 1.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()
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运行结果如下所示：

2.5 使用多项式核函数的SVM

from sklearn.svm import SVC

def PolynomialKernelSVC(degree, C=1.0):
    return Pipeline([
        ("std_scaler", StandardScaler()),
        ("kernelSVC", SVC(kernel="poly", degree=degree, C=C,coef0=0))
    ])
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poly_kernel_svc = PolynomialKernelSVC(degree=3)
poly_kernel_svc.fit(X, y)
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plot_decision_boundary(poly_kernel_svc, axis=[-1.5, 2.5, -1.0, 1.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()
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运行结果如下所示：