Support Vector Machine
离分类样本尽可能远
Soft Margin SVM
scikit-learn中的SVM
和kNN一样，要做数据标准化处理！
涉及距离！
加载数据集
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]
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)
svm
from sklearn.svm import LinearSVC
svc = LinearSVC(C=1e9)
svc.fit(X_standard, y)
可视化
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)
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)
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()
绘制上下对应的两条线
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]
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]
down_y = -w[0]/w[1] * plot_x - b/w[1] - 1/w[1]
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')
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()
SVM中使用多项式特征
生成数据集
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
X, y = datasets.make_moons()
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()
X, y = datasets.make_moons(noise=0.15, random_state=666)
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()
使用多项式特征的SVM
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)
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)
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()
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))
])
poly_kernel_svc = PolynomialKernelSVC(degree=3)
poly_kernel_svc.fit(X, y)
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()
什么是核函数
多项式核函数
高斯核函数
RBF核 Radial Basis Function Kernel
将每一个样本点映射到一个无穷维的特征空间
多项式特征
高斯核
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(-4, 5, 1)
y = np.array((x >= -2) & (x <= 2), dtype='int')
plt.scatter(x[y==0], [0]*len(x[y==0]))
plt.scatter(x[y==1], [0]*len(x[y==1]))
plt.show()
高斯核
def gaussian(x, l):
gamma = 1.0
return np.exp(-gamma * (x-l)**2)
l1, l2 = -1, 1
X_new = np.empty((len(x), 2))
for i, data in enumerate(x):
X_new[i, 0] = gaussian(data, l1)
X_new[i, 1] = gaussian(data, l2)
plt.scatter(X_new[y==0,0], X_new[y==0,1])
plt.scatter(X_new[y==1,0], X_new[y==1,1])
plt.show()
scikit-learn中的高斯核函数
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
X, y = datasets.make_moons(noise=0.15, random_state=666)
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()
预处理
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline
from sklearn.svm import SVC
def RBFKernelSVC(gamma):
return Pipeline([
("std_scaler", StandardScaler()),
("svc", SVC(kernel="rbf", gamma=gamma))
])
svc = RBFKernelSVC(gamma=1)
svc.fit(X, y)
可视化
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(