机器学习-支撑向量机SVM

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 datasetsiris = datasets.load_iris()X = iris.data
y = iris.targetX = 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 StandardScalerstandardScaler = StandardScaler()
standardScaler.fit(X)
X_standard = standardScaler.transform(X)

svm

from sklearn.svm import LinearSVCsvc = 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 ListedColormapcustom_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 ListedColormapcustom_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/w1plot_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 datasetsX, 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 Pipelinedef 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 ListedColormapcustom_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 SVCdef 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 = 1.0return np.exp(-gamma * (x-l)**2)
l1, l2 = -1, 1X_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 datasetsX, 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 SVCdef 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 ListedColormapcustom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)

plot_decision_boundary(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()

svc_gamma100 = RBFKernelSVC(gamma=100)
svc_gamma100.fit(X, y)

plot_decision_boundary(svc_gamma100, 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()

svc_gamma10 = RBFKernelSVC(gamma=10)
svc_gamma10.fit(X, y)
plot_decision_boundary(svc_gamma10, 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()

svc_gamma05 = RBFKernelSVC(gamma=0.5)
svc_gamma05.fit(X, y)
plot_decision_boundary(svc_gamma05, 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()

svc_gamma01 = RBFKernelSVC(gamma=0.1)
svc_gamma01.fit(X, y)
plot_decision_boundary(svc_gamma01, 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()

SVM思路解决回归问题

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasetsboston = datasets.load_boston()
X = boston.data
y = boston.target
from sklearn.model_selection import train_test_splitX_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)

from sklearn.svm import LinearSVR
from sklearn.svm import SVR
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipelinedef StandardLinearSVR(epsilon=0.1):return Pipeline([('std_scaler', StandardScaler()),('linearSVR', LinearSVR(epsilon=epsilon))])

svr = StandardLinearSVR()
svr.fit(X_train, y_train)

svr.score(X_test, y_test)