支持向量机（SVM、决策边界函数）

1.处理线性问题

1.1数据集预处理

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

iris = datasets.load_iris()
X = iris.data
y = iris.target

X = X[y<2, :2]        #只取前面两个特征
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()

#数据标准化
from sklearn.preprocessing import StandardScaler

standardScaler=StandardScaler()
standardScaler.fit(X)
X_standard=standardScaler.transform(X)

1.2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()]                 #c_[]将两个数组以列的形式拼接起来，形成矩阵。
    y_predict = model.predict(X_new)
    
    zz = y_predict.reshape(x0.shape)                      #通过训练好的模型，预测平面上这些点的分类 10     
    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)

1.3拟合数据，显示边界

from sklearn.svm import LinearSVC

svc = LinearSVC(C=1e9)            #Hard margin
print(svc.fit(X_standard, y))
      
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()


svc2 = LinearSVC(C=0.01)           #Soft margin
print(svc2.fit(X_standard, y))

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()

LinearSVC(C=1000000000.0, class_weight=None, dual=True, fit_intercept=True,intercept_scaling=1, loss='squared_hinge', max_iter=1000,multi_class='ovr', penalty='l2', random_state=None, tol=0.0001,verbose=0)

LinearSVC(C=0.01, class_weight=None, dual=True, fit_intercept=True,intercept_scaling=1, loss='squared_hinge', max_iter=1000,multi_class='ovr', penalty='l2', random_state=None, tol=0.0001,verbose=0)

软间隔对数据的包容性更大，所以会有少许分类错误的点。

1.4改编决策边界，画出间隔面的上下两个面

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()]                 #c_[]将两个数组以列的形式拼接起来，形成矩阵。
    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]
    b = model.intercept_[0]
    
    #w0*x0 + w1*x1 + b = 0
    #x1 = -w0/w1 *x0 -b/w1
    plot_x = np.linspace(axis[0], axis[1], 200)
    up_y = -w[0]/w[1] *plot_x - b/w[1] + 1/w[1]           #w0*x0 + w1*x1 + b = 1
    down_y = -w[0]/w[1] *plot_x - b/w[1] - 1/w[1]         #w0*x0 + w1*x1 + b = -1

    #防止超出绘图，进行过滤
    up_index = (up_y >= axis[2]) & (up_y <= axis[3])
    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')


#以上面hard margin为例，绘图
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()

2.处理非线性问题

 2.1加载数据集

#SVM使用多项式特征
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets

X, y = datasets.make_moons()    #X.shape=(100,2)    y.shape=(100,)

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()

#加点噪音扰乱一下
X, y = datasets.make_moons(noise=0.15, random_state=666)
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()

2.2拟合数据，显示边界

多项式特征可以理解为对现有特征的乘积，比如现在有特征A，特征B，特征C，那就可以得到特征A的平方（A^2），A*B，A*C，B^2，B*C以及C^2. 新生成的这些变量即原有变量的有机组合，换句话说，当两个变量各自与y的关系并不强时候，把它们结合成为一个新的变量可能更会容易体现出它们与y的关系。

Pipeline(list)

list 内的每一个元素为为管道的一步，每一步是一个元组。

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))
        ])

poly_svc = PolynomialSVC(degree=3)
poly_svc.fit(X, y)

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()

3.对决策边界函数的应用

以1.1中标准化的数据为例：

#KNN
from sklearn.neighbors import KNeighborsClassifier
knn = KNeighborsClassifier()         #n_neighbors越大，模型越简单，也意味着过拟合的程度越轻，决策边界越清晰
knn.fit(X_standard, y)

plot_decision_boundary(knn, 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()

#逻辑回归
from sklearn.linear_model import LogisticRegression
reg = LogisticRegression()
reg.fit(X_standard, y)

plot_decision_boundary(reg, 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()