Coursera吴恩达机器学习课程 总结笔记及作业代码——第7周支持向量机

1.1 Optimization objective

先回顾一下逻辑回归的相关概念
hθ(x)=11+e−θTx

IF y=1, we want hθ(x)≈1, θTx≫0
IF y=0, we want hθ(x)≈0, θTx≪0

其CostFunction为：
J(θ)=1m[∑mi=1y(i)(−loghθ(x(i)))+(1−y(i))(−(log(1−hθ(x(i)))))]+λ2m∑mj=1θ2j

我们看下在SVM中对costfunction的改变

将其中log函数部分换成了蓝色折线所代表的cost函数。

costFunction也相应的改变为
J(θ)=1m[∑mi=1y(i)Cost1(θTx(i))+(1−y(i))Cost0(θTx(i))]+λ2m∑mj=1θ2j

在SVM中，我们常常用C代替λ
J(θ)=C∑mi=1[y(i)Cost1(θTx(i))+(1−y(i))Cost0(θTx(i))]+12∑mj=1θ2j

1.2 Large Margin Intuition

和逻辑回归相比较

IF y=1, we want θTx≥1 (not just ≥0)
IF y=0, we want θTx≤−1 (not just ≤0)

当C非常大时，我们希望蓝色的这部分为0

即min 12∑ni=1θ2j
θTx(i)≥+1 　 if y(i)=1
θTx(i)≤−1 　 if y(i)=0
归结起来为一个条件极值问题

SVM通过寻找分类中淡黄色背景的那条线作为边界，而不是其余满足条件的边界，因此SVM又被称为大间隔分类器。

1.3 The mathematics behind large margin classification

下面我们从数学角度看一下SVM
通过简化问题，我们知道要求的最小值为||θ||的最小值，即θ的范数最小值

下面看一下限制条件代表的含义，通过高中数学，我们知道两个向量相乘的几何含义如下

通过上面可知，我们要求||θ||的最小值，因此我们希望p(i)尽量大。

假如选择了下面图中的绿色线作为边界，我们会发现p(i)比较小，这样不能得出||θ||的最小值

如果选择下面的绿色线作为边界，我们可以得到较小的||θ||值

这样我们就从直观上感受了SVM作为大间距分类器的效果。

1.4 Kernels

之前的课程中我们讲解了使用多项式解决非线性拟合问题

在这里我们通过引入核函数来解决这个问题。
假设函数hθ(x)=θ0+θ1f1+θ2f2+θ3f3+⋯

给出几个向量l(i)作为landmarks

fi=similarity(x(i),l(i))=exp(−||x(i)−l(i)||22δ2)
exp中的函数为高斯核函数

If x(i)≈l(i)
　　　fi≈exp(−022δ2)≈1
If x(i) is far from l(i)
　　　fi≈exp(−(large　number)22δ2)≈0

通过下面的图我们可以看出δ2对函数图形的改变

关于landmarks我们应该怎么选取呢？
我们可以把x个数据集作为landmarks

这样，对于每一个训练集中的数据，我们都有一个m+1维向量与之对应。

在预测时



关于参数对算法的影响
大C:低偏差，高方差（对应低λ）
小C:高偏差，低方差（对应高λ）

大δ2：fi分布更平滑，高偏差，低方差
小δ2：fi分布更集中，低偏差，高方差

使用SVM步骤

SVM VS Logistic regression

2 程序代码

2.1 使用SVM构建分类器，进行非线性分类，最后选择不同的C值选出最好的边界线。
ex6.m

%% Machine Learning Online Class%  Exercise 6 | Support Vector Machines%%  Instructions%  ------------% %  This file contains code that helps you get started on the%  exercise. You will need to complete the following functions:%%     gaussianKernel.m%     dataset3Params.m%     processEmail.m%     emailFeatures.m%%  For this exercise, you will not need to change any code in this file,%  or any other files other than those mentioned above.%%% Initializationclear ; close all; clc%% =============== Part 1: Loading and Visualizing Data ================%  We start the exercise by first loading and visualizing the dataset. %  The following code will load the dataset into your environment and plot%  the data.%fprintf('Loading and Visualizing Data ...\n')% Load from ex6data1: % You will have X, y in your environmentload('ex6data1.mat');% Plot training dataplotData(X, y);fprintf('Program paused. Press enter to continue.\n');pause;%% ==================== Part 2: Training Linear SVM ====================%  The following code will train a linear SVM on the dataset and plot the%  decision boundary learned.%% Load from ex6data1: % You will have X, y in your environmentload('ex6data1.mat');fprintf('\nTraining Linear SVM ...\n')% You should try to change the C value below and see how the decision% boundary varies (e.g., try C = 1000)C = 1;model = svmTrain(X, y, C, @linearKernel, 1e-3, 20);visualizeBoundaryLinear(X, y, model);fprintf('Program paused. Press enter to continue.\n');pause;%% =============== Part 3: Implementing Gaussian Kernel ===============%  You will now implement the Gaussian kernel to use%  with the SVM. You should complete the code in gaussianKernel.m%fprintf('\nEvaluating the Gaussian Kernel ...\n')x1 = [1 2 1]; x2 = [0 4 -1]; sigma = 2;sim = gaussianKernel(x1, x2, sigma);fprintf(['Gaussian Kernel between x1 = [1; 2; 1], x2 = [0; 4; -1], sigma = %f :' ...         '\n\t%f\n(for sigma = 2, this value should be about 0.324652)\n'], sigma, sim);fprintf('Program paused. Press enter to continue.\n');pause;%% =============== Part 4: Visualizing Dataset 2 ================%  The following code will load the next dataset into your environment and %  plot the data. %fprintf('Loading and Visualizing Data ...\n')% Load from ex6data2: % You will have X, y in your environmentload('ex6data2.mat');% Plot training dataplotData(X, y);fprintf('Program paused. Press enter to continue.\n');pause;%% ========== Part 5: Training SVM with RBF Kernel (Dataset 2) ==========%  After you have implemented the kernel, we can now use it to train the %  SVM classifier.% fprintf('\nTraining SVM with RBF Kernel (this may take 1 to 2 minutes) ...\n');% Load from ex6data2: % You will have X, y in your environmentload('ex6data2.mat');% SVM ParametersC = 1; sigma = 0.1;% We set the tolerance and max_passes lower here so that the code will run% faster. However, in practice, you will want to run the training to% convergence.model= svmTrain(X, y, C, @(x1, x2) gaussianKernel(x1, x2, sigma)); visualizeBoundary(X, y, model);fprintf('Program paused. Press enter to continue.\n');pause;%% =============== Part 6: Visualizing Dataset 3 ================%  The following code will load the next dataset into your environment and %  plot the data. %fprintf('Loading and Visualizing Data ...\n')% Load from ex6data3: % You will have X, y in your environmentload('ex6data3.mat');% Plot training dataplotData(X, y);fprintf('Program paused. Press enter to continue.\n');pause;%% ========== Part 7: Training SVM with RBF Kernel (Dataset 3) ==========%  This is a different dataset that you can use to experiment with. Try%  different values of C and sigma here.% % Load from ex6data3: % You will have X, y in your environmentload('ex6data3.mat');% Try different SVM Parameters here[C, sigma] = dataset3Params(X, y, Xval, yval);% Train the SVMmodel= svmTrain(X, y, C, @(x1, x2) gaussianKernel(x1, x2, sigma));visualizeBoundary(X, y, model);fprintf('Program paused. Press enter to continue.\n');pause;

gaussianKernel.m

function sim = gaussianKernel(x1, x2, sigma)%RBFKERNEL returns a radial basis function kernel between x1 and x2%   sim = gaussianKernel(x1, x2) returns a gaussian kernel between x1 and x2%   and returns the value in sim% Ensure that x1 and x2 are column vectorsx1 = x1(:); x2 = x2(:);% You need to return the following variables correctly.sim = 0;% ====================== YOUR CODE HERE ======================% Instructions: Fill in this function to return the similarity between x1%               and x2 computed using a Gaussian kernel with bandwidth%               sigma%%sim = exp(-sum((x1 - x2).^2)/2/sigma^2);% =============================================================end

dataset3Params.m

function [C, sigma] = dataset3Params(X, y, Xval, yval)%DATASET3PARAMS returns your choice of C and sigma for Part 3 of the exercise%where you select the optimal (C, sigma) learning parameters to use for SVM%with RBF kernel%   [C, sigma] = DATASET3PARAMS(X, y, Xval, yval) returns your choice of C and %   sigma. You should complete this function to return the optimal C and %   sigma based on a cross-validation set.%% You need to return the following variables correctly.C = 1;sigma = 0.3;% ====================== YOUR CODE HERE ======================% Instructions: Fill in this function to return the optimal C and sigma%               learning parameters found using the cross validation set.%               You can use svmPredict to predict the labels on the cross%               validation set. For example, %                   predictions = svmPredict(model, Xval);%               will return the predictions on the cross validation set.%%  Note: You can compute the prediction error using %        mean(double(predictions ~= yval))%cc = [0.01, 0.03, 0.1, 0.3, 1, 3, 10, 30];ss = cc;maxx = 0;for i=1:length(cc)    for j=1:length(cc)        model = svmTrain(X, y, cc(i), @(x1, x2) gaussianKernel(x1, x2, ss(j)));        predict = svmPredict(model, Xval);        cur = mean(double(predict == yval));        if maxx < cur            maxx = cur;            C = cc(i);            sigma = ss(j);        end    endend% =========================================================================end

2.2垃圾邮件分类器
ex6_spam.m

%% Machine Learning Online Class%  Exercise 6 | Spam Classification with SVMs%%  Instructions%  ------------% %  This file contains code that helps you get started on the%  exercise. You will need to complete the following functions:%%     gaussianKernel.m%     dataset3Params.m%     processEmail.m%     emailFeatures.m%%  For this exercise, you will not need to change any code in this file,%  or any other files other than those mentioned above.%%% Initializationclear ; close all; clc%% ==================== Part 1: Email Preprocessing ====================%  To use an SVM to classify emails into Spam v.s. Non-Spam, you first need%  to convert each email into a vector of features. In this part, you will%  implement the preprocessing steps for each email. You should%  complete the code in processEmail.m to produce a word indices vector%  for a given email.fprintf('\nPreprocessing sample email (emailSample1.txt)\n');% Extract Featuresfile_contents = readFile('emailSample1.txt');word_indices  = processEmail(file_contents);% Print Statsfprintf('Word Indices: \n');fprintf(' %d', word_indices);fprintf('\n\n');fprintf('Program paused. Press enter to continue.\n');pause;%% ==================== Part 2: Feature Extraction ====================%  Now, you will convert each email into a vector of features in R^n. %  You should complete the code in emailFeatures.m to produce a feature%  vector for a given email.fprintf('\nExtracting features from sample email (emailSample1.txt)\n');% Extract Featuresfile_contents = readFile('emailSample1.txt');word_indices  = processEmail(file_contents);features      = emailFeatures(word_indices);% Print Statsfprintf('Length of feature vector: %d\n', length(features));fprintf('Number of non-zero entries: %d\n', sum(features > 0));fprintf('Program paused. Press enter to continue.\n');pause;%% =========== Part 3: Train Linear SVM for Spam Classification ========%  In this section, you will train a linear classifier to determine if an%  email is Spam or Not-Spam.% Load the Spam Email dataset% You will have X, y in your environmentload('spamTrain.mat');fprintf('\nTraining Linear SVM (Spam Classification)\n')fprintf('(this may take 1 to 2 minutes) ...\n')C = 0.1;model = svmTrain(X, y, C, @linearKernel);p = svmPredict(model, X);fprintf('Training Accuracy: %f\n', mean(double(p == y)) * 100);%% =================== Part 4: Test Spam Classification ================%  After training the classifier, we can evaluate it on a test set. We have%  included a test set in spamTest.mat% Load the test dataset% You will have Xtest, ytest in your environmentload('spamTest.mat');fprintf('\nEvaluating the trained Linear SVM on a test set ...\n')p = svmPredict(model, Xtest);fprintf('Test Accuracy: %f\n', mean(double(p == ytest)) * 100);pause;%% ================= Part 5: Top Predictors of Spam ====================%  Since the model we are training is a linear SVM, we can inspect the%  weights learned by the model to understand better how it is determining%  whether an email is spam or not. The following code finds the words with%  the highest weights in the classifier. Informally, the classifier%  'thinks' that these words are the most likely indicators of spam.%% Sort the weights and obtin the vocabulary list[weight, idx] = sort(model.w, 'descend');vocabList = getVocabList();fprintf('\nTop predictors of spam: \n');for i = 1:15    fprintf(' %-15s (%f) \n', vocabList{idx(i)}, weight(i));endfprintf('\n\n');fprintf('\nProgram paused. Press enter to continue.\n');pause;%% =================== Part 6: Try Your Own Emails =====================%  Now that you've trained the spam classifier, you can use it on your own%  emails! In the starter code, we have included spamSample1.txt,%  spamSample2.txt, emailSample1.txt and emailSample2.txt as examples. %  The following code reads in one of these emails and then uses your %  learned SVM classifier to determine whether the email is Spam or %  Not Spam% Set the file to be read in (change this to spamSample2.txt,% emailSample1.txt or emailSample2.txt to see different predictions on% different emails types). Try your own emails as well!filename = 'spamSample1.txt';% Read and predictfile_contents = readFile(filename);word_indices  = processEmail(file_contents);x             = emailFeatures(word_indices);p = svmPredict(model, x);fprintf('\nProcessed %s\n\nSpam Classification: %d\n', filename, p);fprintf('(1 indicates spam, 0 indicates not spam)\n\n');

processEmail.m

function word_indices = processEmail(email_contents)%PROCESSEMAIL preprocesses a the body of an email and%returns a list of word_indices %   word_indices = PROCESSEMAIL(email_contents) preprocesses %   the body of an email and returns a list of indices of the %   words contained in the email. %% Load VocabularyvocabList = getVocabList();% Init return valueword_indices = [];% ========================== Preprocess Email ===========================% Find the Headers ( \n\n and remove )% Uncomment the following lines if you are working with raw emails with the% full headers% hdrstart = strfind(email_contents, ([char(10) char(10)]));% email_contents = email_contents(hdrstart(1):end);% Lower caseemail_contents = lower(email_contents);% Strip all HTML% Looks for any expression that starts with < and ends with > and replace% and does not have any < or > in the tag it with a spaceemail_contents = regexprep(email_contents, '<[^<>]+>', ' ');% Handle Numbers% Look for one or more characters between 0-9email_contents = regexprep(email_contents, '[0-9]+', 'number');% Handle URLS% Look for strings starting with http:// or https://email_contents = regexprep(email_contents, ...                           '(http|https)://[^\s]*', 'httpaddr');% Handle Email Addresses% Look for strings with @ in the middleemail_contents = regexprep(email_contents, '[^\s]+@[^\s]+', 'emailaddr');% Handle $ signemail_contents = regexprep(email_contents, '[$]+', 'dollar');% ========================== Tokenize Email ===========================% Output the email to screen as wellfprintf('\n==== Processed Email ====\n\n');% Process filel = 0;while ~isempty(email_contents)    % Tokenize and also get rid of any punctuation    [str, email_contents] = ...       strtok(email_contents, ...              [' @$/#.-:&*+=[]?!(){},''">_<;%' char(10) char(13)]);    % Remove any non alphanumeric characters    str = regexprep(str, '[^a-zA-Z0-9]', '');    % Stem the word     % (the porterStemmer sometimes has issues, so we use a try catch block)    try str = porterStemmer(strtrim(str));     catch str = ''; continue;    end;    % Skip the word if it is too short    if length(str) < 1       continue;    end    % Look up the word in the dictionary and add to word_indices if    % found    % ====================== YOUR CODE HERE ======================    % Instructions: Fill in this function to add the index of str to    %               word_indices if it is in the vocabulary. At this point    %               of the code, you have a stemmed word from the email in    %               the variable str. You should look up str in the    %               vocabulary list (vocabList). If a match exists, you    %               should add the index of the word to the word_indices    %               vector. Concretely, if str = 'action', then you should    %               look up the vocabulary list to find where in vocabList    %               'action' appears. For example, if vocabList{18} =    %               'action', then, you should add 18 to the word_indices     %               vector (e.g., word_indices = [word_indices ; 18]; ).    %     % Note: vocabList{idx} returns a the word with index idx in the    %       vocabulary list.    %     % Note: You can use strcmp(str1, str2) to compare two strings (str1 and    %       str2). It will return 1 only if the two strings are equivalent.    %    for i=1:length(vocabList)        if strcmp(vocabList{i}, str)            word_indices = [word_indices; i];            break;        end    end    % =============================================================    % Print to screen, ensuring that the output lines are not too long    if (l + length(str) + 1) > 78        fprintf('\n');        l = 0;    end    fprintf('%s ', str);    l = l + length(str) + 1;end% Print footerfprintf('\n\n=========================\n');end

emailFeatures.m

function x = emailFeatures(word_indices)%EMAILFEATURES takes in a word_indices vector and produces a feature vector%from the word indices%   x = EMAILFEATURES(word_indices) takes in a word_indices vector and %   produces a feature vector from the word indices. % Total number of words in the dictionaryn = 1899;% You need to return the following variables correctly.x = zeros(n, 1);% ====================== YOUR CODE HERE ======================% Instructions: Fill in this function to return a feature vector for the%               given email (word_indices). To help make it easier to %               process the emails, we have have already pre-processed each%               email and converted each word in the email into an index in%               a fixed dictionary (of 1899 words). The variable%               word_indices contains the list of indices of the words%               which occur in one email.% %               Concretely, if an email has the text:%%                  The quick brown fox jumped over the lazy dog.%%               Then, the word_indices vector for this text might look %               like:%               %                   60  100   33   44   10     53  60  58   5%%               where, we have mapped each word onto a number, for example:%%                   the   -- 60%                   quick -- 100%                   ...%%              (note: the above numbers are just an example and are not the%               actual mappings).%%              Your task is take one such word_indices vector and construct%              a binary feature vector that indicates whether a particular%              word occurs in the email. That is, x(i) = 1 when word i%              is present in the email. Concretely, if the word 'the' (say,%              index 60) appears in the email, then x(60) = 1. The feature%              vector should look like:%%              x = [ 0 0 0 0 1 0 0 0 ... 0 0 0 0 1 ... 0 0 0 1 0 ..];%%for i=1:length(word_indices)    x(word_indices(i)) = 1;end% =========================================================================end