线性可分 SVM

import numpy as np
import matplotlib.pyplot as plt
from sklearn.svm import SVC # "Support vector classifier"

# 定义函数plot_svc_decision_function用于绘制分割超平面和其两侧的辅助超平面
def plot_svc_decision_function(model, ax=None, plot_support=True):
    """Plot the decision function for a 2D SVC"""
    if ax is None:
        ax = plt.gca()
    xlim = ax.get_xlim()
    ylim = ax.get_ylim()

    # 创建网格用于评价模型
    x = np.linspace(xlim[0], xlim[1], 30)
    y = np.linspace(ylim[0], ylim[1], 30)
    Y, X = np.meshgrid(y, x)
    xy = np.vstack([X.ravel(), Y.ravel()]).T
    P = model.decision_function(xy).reshape(X.shape)

    #绘制超平面
    ax.contour(X, Y, P, colors='k',
               levels=[-1, 0, 1], alpha=0.5,
               linestyles=['--', '-', '--'])

    #标识出支持向量
    if plot_support:
        ax.scatter(model.support_vectors_[:, 0], model.support_vectors_[:, 1], s=300, linewidth=1,  edgecolors='blue', facecolors='none');
    ax.set_xlim(xlim)
    ax.set_ylim(ylim)


# 用make_blobs生成样本数据
from sklearn.datasets.samples_generator import make_blobs
X, y = make_blobs(n_samples=50, centers=2, random_state=0, cluster_std=0.60)

# 用线性核函数的SVM来对样本进行分类
model = SVC(kernel='linear')
model.fit(X, y)

# 将样本数据绘制在直角坐标中
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
# 在直角坐标中绘制出分割超平面、辅助超平面和支持向量
plot_svc_decision_function(model)
plt.show()

结果

线性 SVM

不同类型样本发生了重叠的情况

import numpy as np
import matplotlib.pyplot as plt
from sklearn.svm import SVC # "Support vector classifier"

# 定义函数plot_svc_decision_function用于绘制分割超平面和其两侧的辅助超平面
def plot_svc_decision_function(model, ax=None, plot_support=True):
    """Plot the decision function for a 2D SVC"""
    if ax is None:
        ax = plt.gca()
    xlim = ax.get_xlim()
    ylim = ax.get_ylim()

    # 创建网格用于评价模型
    x = np.linspace(xlim[0], xlim[1], 30)
    y = np.linspace(ylim[0], ylim[1], 30)
    Y, X = np.meshgrid(y, x)
    xy = np.vstack([X.ravel(), Y.ravel()]).T
    P = model.decision_function(xy).reshape(X.shape)

    #绘制超平面
    ax.contour(X, Y, P, colors='k',
               levels=[-1, 0, 1], alpha=0.5,
               linestyles=['--', '-', '--'])

    #标识出支持向量
    if plot_support:
        ax.scatter(model.support_vectors_[:, 0], model.support_vectors_[:, 1], s=300, linewidth=1,  edgecolors='blue', facecolors='none');
    ax.set_xlim(xlim)
    ax.set_ylim(ylim)


# 用make_blobs生成样本数据
from sklearn.datasets.samples_generator import make_blobs
# X, y = make_blobs(n_samples=50, centers=2, random_state=0, cluster_std=0.60)
X, y = make_blobs(n_samples=100, centers=2, random_state=0, cluster_std=0.9)

# 用线性核函数的SVM来对样本进行分类
model = SVC(kernel='linear')
model.fit(X, y)

# 将样本数据绘制在直角坐标中
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
# 在直角坐标中绘制出分割超平面、辅助超平面和支持向量
plot_svc_decision_function(model)
plt.show()

结果

提高惩罚系数C

SVC 类，有一个 C 参数，对应的是错误项（Error Term）的惩罚系数。这个系数设置得越高，容错性也就越小，分隔空间的硬度也就越强。

import numpy as np
import matplotlib.pyplot as plt
from sklearn.svm import SVC # "Support vector classifier"

# 定义函数plot_svc_decision_function用于绘制分割超平面和其两侧的辅助超平面
def plot_svc_decision_function(model, ax=None, plot_support=True):
    """Plot the decision function for a 2D SVC"""
    if ax is None:
        ax = plt.gca()
    xlim = ax.get_xlim()
    ylim = ax.get_ylim()

    # 创建网格用于评价模型
    x = np.linspace(xlim[0], xlim[1], 30)
    y = np.linspace(ylim[0], ylim[1], 30)
    Y, X = np.meshgrid(y, x)
    xy = np.vstack([X.ravel(), Y.ravel()]).T
    P = model.decision_function(xy).reshape(X.shape)

    #绘制超平面
    ax.contour(X, Y, P, colors='k',
               levels=[-1, 0, 1], alpha=0.5,
               linestyles=['--', '-', '--'])

    #标识出支持向量
    if plot_support:
        ax.scatter(model.support_vectors_[:, 0], model.support_vectors_[:, 1], s=300, linewidth=1,  edgecolors='blue', facecolors='none');
    ax.set_xlim(xlim)
    ax.set_ylim(ylim)


# 用make_blobs生成样本数据
from sklearn.datasets.samples_generator import make_blobs
# X, y = make_blobs(n_samples=50, centers=2, random_state=0, cluster_std=0.60)
X, y = make_blobs(n_samples=100, centers=2, random_state=0, cluster_std=0.9)

# 用线性核函数的SVM来对样本进行分类
# model = SVC(kernel='linear')
model = SVC(kernel='linear', C=100.0)
model.fit(X, y)

# 将样本数据绘制在直角坐标中
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
# 在直角坐标中绘制出分割超平面、辅助超平面和支持向量
plot_svc_decision_function(model)
plt.show()

结果

完全线性不可分的数据

在完全线性不可分的数据中可以用RBF核在高维度空间分割样本

import numpy as np
import matplotlib.pyplot as plt
from sklearn.svm import SVC # "Support vector classifier"

# 定义函数plot_svc_decision_function用于绘制分割超平面和其两侧的辅助超平面
def plot_svc_decision_function(model, ax=None, plot_support=True):
    """Plot the decision function for a 2D SVC"""
    if ax is None:
        ax = plt.gca()
    xlim = ax.get_xlim()
    ylim = ax.get_ylim()

    # 创建网格用于评价模型
    x = np.linspace(xlim[0], xlim[1], 30)
    y = np.linspace(ylim[0], ylim[1], 30)
    Y, X = np.meshgrid(y, x)
    xy = np.vstack([X.ravel(), Y.ravel()]).T
    P = model.decision_function(xy).reshape(X.shape)

    #绘制超平面
    ax.contour(X, Y, P, colors='k',
               levels=[-1, 0, 1], alpha=0.5,
               linestyles=['--', '-', '--'])

    #标识出支持向量
    if plot_support:
        ax.scatter(model.support_vectors_[:, 0], model.support_vectors_[:, 1], s=300, linewidth=1,  edgecolors='blue', facecolors='none');
    ax.set_xlim(xlim)
    ax.set_ylim(ylim)


# 用make_blobs生成样本数据
# from sklearn.datasets.samples_generator import make_blobs
# X, y = make_blobs(n_samples=50, centers=2, random_state=0, cluster_std=0.60)
# X, y = make_blobs(n_samples=100, centers=2, random_state=0, cluster_std=0.9)

# 用make_circles生成样本数据
from sklearn.datasets.samples_generator import make_circles
X, y = make_circles(100, factor=.1, noise=.1)

from mpl_toolkits import mplot3d
def plot_3D(elev=30, azim=30, X=None, y=None):
    ax = plt.subplot(projection='3d')
    r = np.exp(-(X ** 2).sum(1))
    ax.scatter3D(X[:, 0], X[:, 1], r, c=y, s=50, cmap='autumn')
    ax.view_init(elev=elev, azim=azim)
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.set_zlabel('r')

# 用线性核函数的SVM来对样本进行分类
# model = SVC(kernel='linear')
# model = SVC(kernel='linear', C=10.0)
# 用 RBF 核对样本进行分类 再把惩罚系数再调高一百倍
model = SVC(kernel='rbf', C=100)
model.fit(X, y)

# 将样本数据绘制在直角坐标中
# plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')

# 将样本数据绘制在3维坐标中
plot_3D(X=X, y=y)
# 在直角坐标中绘制出分割超平面、辅助超平面和支持向量
plot_svc_decision_function(model)
plt.show()

结果

SVR

import numpy as np
from sklearn.svm import SVR
import matplotlib.pyplot as plt

# 生成样本数据
X = np.sort(5 * np.random.rand(40, 1), axis=0)
# 线性
# y = np.ravel(2*X + 3)
# 曲线
# y = np.polyval([2,3,5,2], X).ravel()
y = np.sin(X).ravel()

# 加入部分噪音
y[::5] += 3 * (0.5 - np.random.rand(8))

# 调用模型
svr_rbf = SVR(kernel='rbf', C=1e3, gamma=0.1)
svr_lin = SVR(kernel='linear', C=1e3)
svr_poly = SVR(kernel='poly', C=1e3, degree=2)
y_rbf = svr_rbf.fit(X, y).predict(X)
y_lin = svr_lin.fit(X, y).predict(X)
y_poly = svr_poly.fit(X, y).predict(X)

# 可视化结果
lw = 2
plt.scatter(X, y, color='darkorange', label='data')
plt.plot(X, y_rbf, color='navy', lw=lw, label='RBF model')
plt.plot(X, y_lin, color='c', lw=lw, label='Linear model')
plt.plot(X, y_poly, color='cornflowerblue', lw=lw, label='Polynomial model')
plt.xlabel('data')
plt.ylabel('target')
plt.title('Support Vector Regression')
plt.legend()
plt.show()

结果

从SVM到SVR
从SVM到SVR 代码