目录
一、SVM案例:线性支持向量机
SVM:支持向量机
支持向量基本原理
例子
Support Vector Machines: 最小化 雷区
训练一个基本的SVM
对比实验
二、软间隔C值对结果的影响
引入核函数的SVM
高维核变换
调节SVM参数: Soft Margin问题
调节C参数
三、模型复杂度的权衡
四、人脸识别实例
Example: Face Recognition
1、下载数据
2、降维后再SVM
3、使用grid search cross-validation来选择我们的参数
4、测试
5、查看
6、热度图
与传统算法进行对比,看看SVM究竟能带来什么样的效果
软间隔的作用,这么复杂的算法肯定会导致过拟合现象,如何来进行解决呢?
核函数的作用,如果只是做线性分类,好像轮不到SVM登场了,核函数才是它的强大之处!
导包
%matplotlib inline
#为了在notebook中画图展示
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
import seaborn as sns; sns.set()
#随机来点数据
#其中 cluster_std是数据的离散程度
from sklearn.datasets.samples_generator import make_blobs
X, y = make_blobs(n_samples=50, centers=2,
random_state=0, cluster_std=0.60)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
#随便的画几条分割线,哪个好来这?
xfit = np.linspace(-1, 3.5)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
for m, b in [(1, 0.65), (0.5, 1.6), (-0.2, 2.9)]:
plt.plot(xfit, m * xfit + b, '-k')
#限制一下X的取值范围
plt.xlim(-1, 3.5);
xfit = np.linspace(-1, 3.5)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
for m, b, d in [(1, 0.65, 0.33), (0.5, 1.6, 0.55), (-0.2, 2.9, 0.2)]:
yfit = m * xfit + b
plt.plot(xfit, yfit, '-k')
plt.fill_between(xfit, yfit - d, yfit + d, edgecolor='none',
color='#AAAAAA', alpha=0.4)
plt.xlim(-1, 3.5);
#分类任务
from sklearn.svm import SVC
#线性核函数 相当于不对数据进行变换
model = SVC(kernel='linear')
model.fit(X, y)
绘图函数(模板)
#绘图函数
def plot_svc_decision_function(model, ax=None, plot_support=True):
if ax is None:
ax = plt.gca()
xlim = ax.get_xlim()
ylim = ax.get_ylim()
# 用SVM自带的decision_function函数来绘制
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)
# 绘制决策边界
ax.contour(X, Y, P, colors='k',
levels=[-1, 0, 1], alpha=0.5,
linestyles=['--', '-', '--'])
# 绘制支持向量
if plot_support:
ax.scatter(model.support_vectors_[:, 0],
model.support_vectors_[:, 1],
s=300, linewidth=1, alpha=0.2);
ax.set_xlim(xlim)
ax.set_ylim(ylim)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(model)
这条线就是我们希望得到的决策边界啦
观察发现有3个点做了特殊的标记,它们恰好都是边界上的点
它们就是我们的support vectors(支持向量)
在Scikit-Learn中, 它们存储在这个位置 support_vectors_
(一个属性)
model.support_vectors_
观察可以发现,只需要支持向量我们就可以把模型构建出来
接下来我们尝试一下,用不同多的数据点,看看效果会不会发生变化
分别使用60个和120个数据点
def plot_svm(N=10, ax=None):
X, y = make_blobs(n_samples=200, centers=2,
random_state=0, cluster_std=0.60)
X = X[:N]
y = y[:N]
model = SVC(kernel='linear', C=1E10)
model.fit(X, y)
ax = ax or plt.gca()
ax.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
ax.set_xlim(-1, 4)
ax.set_ylim(-1, 6)
plot_svc_decision_function(model, ax)
# 分别对不同的数据点进行绘制
fig, ax = plt.subplots(1, 2, figsize=(16, 6))
fig.subplots_adjust(left=0.0625, right=0.95, wspace=0.1)
for axi, N in zip(ax, [60, 120]):
plot_svm(N, axi)
axi.set_title('N = {0}'.format(N))
from sklearn.datasets.samples_generator import make_circles
# 绘制另外一种数据集
X, y = make_circles(100, factor=.1, noise=.1)
#看看这回线性和函数能解决嘛
clf = SVC(kernel='linear').fit(X, y)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(clf, plot_support=False);
#加入了新的维度r
from mpl_toolkits import mplot3d
r = np.exp(-(X ** 2).sum(1))
# 可以想象一下在三维中把环形数据集进行上下拉伸
def plot_3D(elev=30, azim=30, X=X, y=y):
ax = plt.subplot(projection='3d')
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')
plot_3D(elev=45, azim=45, X=X, y=y)
#加入高斯核函数
clf = SVC(kernel='rbf')
clf.fit(X, y)
#这回厉害了!
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(clf)
plt.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1],
s=300, lw=1, facecolors='none');
使用这种核支持向量机,我们学习一个合适的非线性决策边界。这种核变换策略在机器学习中经常被使用!
# 这份数据集中cluster_std稍微大一些,这样才能体现出软间隔的作用
X, y = make_blobs(n_samples=100, centers=2,
random_state=0, cluster_std=0.8)
plt.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
c=10和0.1
#加大游戏难度的数据集
X, y = make_blobs(n_samples=100, centers=2,
random_state=0, cluster_std=0.8)
fig, ax = plt.subplots(1, 2, figsize=(16, 6))
fig.subplots_adjust(left=0.0625, right=0.95, wspace=0.1)
# 选择两个C参数来进行对别实验,分别为10和0.1
for axi, C in zip(ax, [10.0, 0.1]):
model = SVC(kernel='linear', C=C).fit(X, y)
axi.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(model, axi)
axi.scatter(model.support_vectors_[:, 0],
model.support_vectors_[:, 1],
s=300, lw=1, facecolors='none');
axi.set_title('C = {0:.1f}'.format(C), size=14)
gamma值对结果的影响
X, y = make_blobs(n_samples=100, centers=2,
random_state=0, cluster_std=1.1)
fig, ax = plt.subplots(1, 2, figsize=(16, 6))
fig.subplots_adjust(left=0.0625, right=0.95, wspace=0.1)
# 选择不同的gamma值来观察建模效果
for axi, gamma in zip(ax, [10.0, 0.1]):
model = SVC(kernel='rbf', gamma=gamma).fit(X, y)
axi.scatter(X[:, 0], X[:, 1], c=y, s=50, cmap='autumn')
plot_svc_decision_function(model, axi)
axi.scatter(model.support_vectors_[:, 0],
model.support_vectors_[:, 1],
s=300, lw=1, facecolors='none');
axi.set_title('gamma = {0:.1f}'.format(gamma), size=14)
gamma值小,简单模型。
As an example of support vector machines in action, let's take a look at the facial recognition problem. We will use the Labeled Faces in the Wild dataset, which consists of several thousand collated photos of various public figures. A fetcher for the dataset is built into Scikit-Learn.
#读取数据集
from sklearn.datasets import fetch_lfw_people
faces = fetch_lfw_people(min_faces_per_person=60)
#看一下数据的规模
print(faces.target_names)
print(faces.images.shape)
from sklearn.svm import SVC
from sklearn.decomposition import PCA
from sklearn.pipeline import make_pipeline
#降维到150维
pca = PCA(n_components=150, whiten=True, random_state=42)
svc = SVC(kernel='rbf', class_weight='balanced')
#先降维然后再SVM
model = make_pipeline(pca, svc)
from sklearn.model_selection import train_test_split
Xtrain, Xtest, ytrain, ytest = train_test_split(faces.data, faces.target,
random_state=40)
from sklearn.model_selection import GridSearchCV
param_grid = {'svc__C': [1, 5, 10],
'svc__gamma': [0.0001, 0.0005, 0.001]}
grid = GridSearchCV(model, param_grid)
%time grid.fit(Xtrain, ytrain)
print(grid.best_params_)
model = grid.best_estimator_
yfit = model.predict(Xtest)
yfit.shape
from sklearn.metrics import classification_report
print(classification_report(ytest, yfit,
target_names=faces.target_names))
from sklearn.metrics import confusion_matrix
mat = confusion_matrix(ytest, yfit)
sns.heatmap(mat.T, square=True, annot=True, fmt='d', cbar=False,
xticklabels=faces.target_names,
yticklabels=faces.target_names)
plt.xlabel('true label')
plt.ylabel('predicted label');