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作业及代码：/s/1L-Tbo3flzKplAof3fFdD1w 密码:oin0

本次作业的理论部分：吴恩达机器学习（七）支持向量机

编程环境：Jupyter Notebook

1. 线性 SVM

任务

观察惩罚项系数 C 对决策边界的影响，数据集：data/ex6data1.mat

在理论部分，我们得到SVM的代价函数为：

J(θ)=C∑i=1m[y(i)cost1(θTx(i))+(1−y(i))cost0(θTx(i))]+12∑j=1nθj2J(\theta)=C \sum_{i=1}^{m} \left[y^{(i)} cos t_{1}(\theta^{T} x^{(i)})+(1-y^{(i)}) cos t_{0}(\theta^{T} x^{(i)})\right]+\frac{1}{2} \sum_{j=1}^{n} \theta_{j}^{2} J(θ)=Ci=1∑m​[y(i)cost1​(θTx(i))+(1−y(i))cost0​(θTx(i))]+21​j=1∑n​θj2​其中C为误差项惩罚系数，C越大，容错率越低，越易过拟合。

import numpy as npimport scipy.io as sioimport matplotlib.pyplot as pltdata = sio.loadmat('./data/ex6data1.mat')X,y = data['X'],data['y']def plot_data():plt.scatter(X[:,0],X[:,1],c = y.flatten(), cmap ='jet')plt.xlabel('x1')plt.ylabel('y1')plot_data() # 绘制原始数据

由图可知，左上角的那个数据点为异常点（误差点）。

Scikit-learn ，kernel=‘linear’

简称 sklearn，参考官方中文文档：

提供了很多机器学习的库，本次作业主要也是用它来解决SVM的问题

C = 1

from sklearn.svm import SVC svc1 = SVC(C=1,kernel='linear') #实例化分类器，C为误差项惩罚系数，核函数选择线性核svc1.fit(X,y.flatten())#导入数据进行训练>>> svc1.score(X,y.flatten())#分类器的准确率> 0.9803921568627451# 绘制决策边界def plot_boundary(model):x_min,x_max = -0.5,4.5y_min,y_max = 1.3,5xx,yy = np.meshgrid(np.linspace(x_min,x_max,500),np.linspace(y_min,y_max,500))z = model.predict(np.c_[xx.flatten(),yy.flatten()])zz = z.reshape(xx.shape)plt.contour(xx,yy,zz)plot_boundary(svc1)plot_data()

C = 100

svc100 = SVC(C=100,kernel='linear')svc100.fit(X,y.flatten())>>>svc100.score(X,y.flatten())> 1.0#绘制决策边界plot_boundary(svc100)plot_data()

结论

误差项惩罚系数C越大，容错率越低，越易过拟合。

2. 非线性 SVM

任务

使用高斯核函数解决线性不可分问题，并观察 σ\sigmaσ 取值对模型复杂度的影响。数据集：data/ex6data2.mat

高斯核函数公式：

K(x1,x2)=exp⁡{−∥x1−x2∥22σ2}K\left(\boldsymbol{x}_{1}, \boldsymbol{x}_{2}\right)=\exp \left\{-\frac{\left\|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}\right\|^{2}}{2 \sigma^{2}}\right\} K(x1​,x2​)=exp{−2σ2∥x1​−x2​∥2​}

data = sio.loadmat('./data/ex6data2.mat')X,y = data['X'],data['y']def plot_data():plt.scatter(X[:,0],X[:,1],c = y.flatten(), cmap ='jet')plt.xlabel('x1')plt.ylabel('y1')plot_data() # 绘制原始数据

Scikit-learn ，kernel=‘rbf’

σ=1\sigma = 1σ=1 ，注意：sklearn中的σ\sigmaσ表示为gammer，高斯核表示为rbf

svc1 = SVC(C=1,kernel='rbf',gamma=1) #实例化分类器，C为误差项惩罚系数，核函数选择高斯核svc1.fit(X,y.flatten())#导入数据进行训练>>> svc1.score(X,y.flatten())#分类器的准确率> 0.8088064889918888# 绘制决策边界def plot_boundary(model):x_min,x_max = 0,1y_min,y_max = 0.4,1xx,yy = np.meshgrid(np.linspace(x_min,x_max,500),np.linspace(y_min,y_max,500))z = model.predict(np.c_[xx.flatten(),yy.flatten()])zz = z.reshape(xx.shape)plt.contour(xx,yy,zz)plot_boundary(svc1)plot_data()

σ=50\sigma = 50σ=50

σ=1000\sigma = 1000σ=1000

结论

σ\sigmaσ 值越大，模型复杂度越高，同时也越易过拟合

σ\sigmaσ 值越小，模型复杂度越低，同时也越易欠拟合

3. 寻找最优参数 C 和 σ\sigmaσ

数据集：data/ex6data3.mat

mat = sio.loadmat('data/ex6data3.mat')X, y = mat['X'], mat['y']# 训练集Xval, yval = mat['Xval'], mat['yval']# 验证集def plot_data():plt.scatter(X[:,0],X[:,1],c = y.flatten(), cmap ='jet')plt.xlabel('x1')plt.ylabel('y1')plot_data() # 绘制原始数据

# C 和 σ 的候选值Cvalues = [3, 10, 30, 100,0.01, 0.03, 0.1, 0.3,1 ] #9gammas = [1 ,3, 10, 30, 100,0.01, 0.03, 0.1, 0.3] #9# 获取最佳准确率和最优参数best_score = 0best_params = (0,0)for c in Cvalues:for gamma in gammas:svc = SVC(C=c,kernel='rbf',gamma=gamma)svc.fit(X,y.flatten())# 用训练集数据拟合模型score = svc.score(Xval,yval.flatten()) # 用验证集数据进行评分if score > best_score:best_score = scorebest_params = (c,gamma)>>> print(best_score,best_params)> 0.965 (3, 30)

注意：获取到的最优参数组合不只有一组，更改候选值的顺序，最佳参数组合及其对应的决策边界也会改变

svc2 = SVC(C=3,kernel='rbf',gamma=30)def plot_boundary(model):x_min,x_max = -0.6,0.4y_min,y_max = -0.7,0.6xx,yy = np.meshgrid(np.linspace(x_min,x_max,500),np.linspace(y_min,y_max,500))z = model.predict(np.c_[xx.flatten(),yy.flatten()])zz = z.reshape(xx.shape)plt.contour(xx,yy,zz)plot_boundary(svc2)plot_data()

4. 垃圾邮件过滤问题

注意：data/spamTrain.mat是对邮件进行预处理后（自然语言处理）获得的向量

# training datadata1 = sio.loadmat('data/spamTrain.mat')X, y = data1['X'], data1['y']# Testing datadata2 = sio.loadmat('data/spamTest.mat')Xtest, ytest = data2['Xtest'], data2['ytest']>>> X.shape,y.shape # 样本数为4000> ((4000, 1899), (4000, 1))>>> X # 每一行代表一个邮件样本，每个样本有1899个特征，特征为1表示在跟垃圾邮件有关的语义库中找到相关单词> array([[0, 0, 0, ..., 0, 0, 0],[0, 0, 0, ..., 0, 0, 0],[0, 0, 0, ..., 0, 0, 0],...,[0, 0, 0, ..., 0, 0, 0],[0, 0, 1, ..., 0, 0, 0],[0, 0, 0, ..., 0, 0, 0]], dtype=uint8)>>> y # 每一行代表一个邮件样本，等于1表示为垃圾邮件> array([[1],[1],[0],...,[1],[0],[0]], dtype=uint8)# 候选的 C值Cvalues = [3, 10, 30, 100,0.01, 0.03, 0.1, 0.3,1 ] # 获取最佳准确率和最优参数best_score = 0best_param = 0for c in Cvalues:svc = SVC(C=c,kernel='linear')svc.fit(X,y.flatten())# 用训练集数据拟合模型score= svc.score(Xtest,ytest.flatten()) # 用验证集数据进行评分if score > best_score:best_score = scorebest_param = c>>> print(best_score,best_param)> 0.99 0.03# 带入最佳参数svc = SVC(0.03,kernel='linear')svc.fit(X,y.flatten())score_train= svc.score(X,y.flatten())score_test= svc.score(Xtest,ytest.flatten())>>> print(score_train,score_test)> 0.99425 0.99

附：邮件预处理

with open('data/emailSample1.txt', 'r') as f:sampe_email = f.read()print(sampe_email)'''预处理主要包括以下8个部分：1. 将大小写统一成小写字母；2. 移除所有HTML标签，只保留内容。3. 将所有的网址替换为字符串 “httpaddr”.4. 将所有的邮箱地址替换为 “emailaddr”5. 将所有dollar符号($)替换为“dollar”.6. 将所有数字替换为“number”7. 将所有单词还原为词源，词干提取8. 移除所有非文字类型9.去除空字符串‘’'''import numpy as npimport matplotlib.pyplot as pltfrom scipy.io import loadmatfrom sklearn import svmimport nltk.stem as nsimport redef preprocessing(email):# 1. 统一成小写email = email.lower()#2. 去除html标签email = re.sub('<[^<>]>', ' ', email)#3. 将网址替换为字符串 “httpaddr”.email = re.sub('(http|https)://[^\s]*', 'httpaddr', email ) #4. 将邮箱地址替换为 “emailaddr”email = re.sub('[^\s]+@[^\s]+', 'emailaddr', email)# 5.所有dollar符号($)替换为“dollar”.email = re.sub('[\$]+', 'dollar', email) # 6.匹配数字，将数字替换为“number”email = re.sub('[0-9]+', 'number', email) # 匹配一个数字， 相当于 [0-9]，+ 匹配1到多次# 7. 词干提取tokens = re.split('[ \@\$\/\#\.\-\:\&\*\+\=\[\]\?\!\(\)\{\}\,\'\"\>\_\<\;\%]', email)tokenlist=[]s = ns.SnowballStemmer('english')for token in tokens:# 8. 移除非文字类型email = re.sub('[^a-zA-Z0-9]', '', email)stemmed = s.stem(token)# 9.去除空字符串‘’if not len(token): continuetokenlist.append(stemmed) return tokenlistemail = preprocessing(sampe_email)def email2VocabIndices(email, vocab):"""提取存在单词的索引"""token = preprocessing(email)print(token)index = [i for i in range(len(token)) if token[i] in vocab]return index def email2FeatureVector(email):"""将email转化为词向量，n是vocab的长度。存在单词的相应位置的值置为1，其余为0"""df = pd.read_table('data/vocab.txt',names=['words'])vocab = df.values # return arrayvector = np.zeros(len(vocab)) # init vectorvocab_indices = email2VocabIndices(email, vocab) print(vocab_indices)# 返回含有单词的索引# 将有单词的索引置为1for i in vocab_indices:vector[i] = 1return vectorimport pandas as pdvector = email2FeatureVector(sampe_email)>>> print('length of vector = {}\nnum of non-zero = {}'.format(len(vector), int(vector.sum())))> ['anyon', 'know', 'how', 'much', 'it', 'cost', 'to', 'host', 'a', 'web', 'portal', '\n', '\nwell', 'it', 'depend', 'on', 'how', 'mani', 'visitor', 'you', 're', 'expect', '\nthis', 'can', 'be', 'anywher', 'from', 'less', 'than', 'number', 'buck', 'a', 'month', 'to', 'a', 'coupl', 'of', 'dollarnumb', '\nyou', 'should', 'checkout', 'httpaddr', 'or', 'perhap', 'amazon', 'ecnumb', '\nif', 'your', 'run', 'someth', 'big', '\n\nto', 'unsubscrib', 'yourself', 'from', 'this', 'mail', 'list', 'send', 'an', 'email', 'to', '\nemailaddr\n\n'][0, 1, 2, 3, 4, 5, 6, 7, 9, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 35, 36, 37, 39, 41, 42, 43, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61]length of vector = 1899num of non-zero = 46>>> vector.shape> (1899,)