机器学习week7 ex6 review

这周使用支持向量机(supprot vector machine)来做一个简单的垃圾邮件分类。

Support vector machine

1.1 Example dataset 1

ex6.m首先载入ex6data1.mat中的数据绘图：

%% =============== 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 environment
load(‘ex6data1.mat‘);

% Plot training data
plotData(X, y);

fprintf(‘Program paused. Press enter to continue.\n‘);
pause;

接着用已经写好的SVM来训练，取C=1:

%% ==================== 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 environment
load(‘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;

可以看到此时左上角的一个十字点被错误地分类了。

如果改变C=100，则得到如下图形：

此时training example中的所有点都被正确分类了，但是这条线很显然并不正确。

1.2 SVM with Gaussian Kernels

1.2.1 Gaussian kernel

Gaussian kernel function:

求两点距离在MATLAB中可以直接使用pdist函数，在Octave中如果下载了Octave-forge中的statistics包也可以使用。这里是Octave下载package的方法。
故完成gaussiankernel.m代码如下：

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 vectors
x1 = 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
%
%

x = [x1‘;x2‘];      % make x1 x2 as row vectors and put together as a matrix
dis = pdist(x);     % using the pdist function in statistics package
sim = exp(-dis^2/(2*sigma^2));

% =============================================================

end

1.2.2 Example Dataset 2

根据刚刚完成的求Gaussian Kernel的函数，利用已经写好的SVM可以对ex6data2.mat中的数据进行分类。
ex6.m先对数据进行可视化如下：

经过使用了RBF kernel的SVM训练后，可以画出如下边界:

1.2.3 Example Dataset 3

需要使用交叉验证集求出最合适的C和sigma。
C和sigma的验证范围都是[0.01 0.03 0.1 0.3 1 3 10 30], 也就是说需要进行64次计算，来得到最合适的C和sigma，并用于对训练集的计算中。
完成函数dataset3Params.m如下：

function [C, sigma] = dataset3Params(X, y, Xval, yval)
%EX6PARAMS 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] = EX6PARAMS(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))
%

train_values = [0.01 0.03 0.1 0.3 1 3 10 30];
sigma_values = [0.01 0.03 0.1 0.3 1 3 10 30];

for i=1:length(train_values)
    for j=1:length(train_values)
        C = train_values(i);
        sigma = train_values(j);
        model = svmTrain(X, y, C, @(x1, x2) gaussianKernel(x1, x2, sigma));
        predictions = svmPredict(model, Xval);
        predictions_error(i,j) = mean(double(predictions ~= yval));
    end
end

mm = min(min(predictions_error));
[i j] = find(predictions_error == mm);
C = train_values(i)
sigma = train_values(j)

% Answer is C = 1 and sigma = 0.1

% =========================================================================

end

最后得到使得cross validation error最小的C和sigma分别为1和0.1。

绘制训练集散点图如下：

使用上述C和sigma，求出边界如下图：

2. Spam classification

使用SVM进行垃圾邮件分类。
表示是垃圾邮件，表示不是垃圾邮件。同时还需要将每封邮件转换成一个特征向量。
数据集来自SpamAssassin Public Corpus。这次简化的分类中，不考虑邮件的标题，只对正文进行分类。

2.1 Preprocessing Emails

需要对邮件中的内容进行Normalization:

即将正文中所有字母换成小写、去掉所有HTML的tag、把所有URL替换成“httpaddr”、所有Email地址替换成“emailaddr”，所有数字替换成“number”、所有美元符号 $ 替换成“dollar”、单词词根化、去掉非单词字符（包括标点符号，tab newline spaces均用一个空格替代）。
例如一封邮件本来含有如下内容：

经过上述处理后，则变为如下内容：

2.2 Vocabulary list

在这个简化的垃圾邮件分类中，我们只选择了最常用的单词。因为不常用的词汇只在很少一部分邮件中出现，如果增加它们作为feature，可能会导致overfitting。
在文件vocab.txt中包含了全部单词表，部分截图如下：

这些词都是在spam corpus中出现超过100次的单词，共有1899个。在实际操作中，一般使用10000-50000的单词表。

补充processEmail.m的代码，完成后如下：

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 Vocabulary
vocabList = getVocabList();

% Init return value
word_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 case
email_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 space
email_contents = regexprep(email_contents, ‘<[^<>]+>‘, ‘ ‘);

% Handle Numbers
% Look for one or more characters between 0-9
email_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 middle
email_contents = regexprep(email_contents, ‘[^\s][email protected][^\s]+‘, ‘emailaddr‘);

% Handle $ sign
email_contents = regexprep(email_contents, ‘[$]+‘, ‘dollar‘);

% ========================== Tokenize Email ===========================

% Output the email to screen as well
fprintf(‘\n==== Processed Email ====\n\n‘);

% Process file
l = 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.
    %

    index = find(strcmp(vocabList,str) == 1);    % find the index of str in vocablist
    word_indices = [word_indices; index];         % add index to  word_indices

    % =============================================================

    % 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 footer
fprintf(‘\n\n=========================\n‘);

end

经过处理的样例如下所示：

Extracting features from Emails

将上述经过处理的邮件表示成特征向量 , 其中 , 分别代表单词表中的号单词是否出现在了邮件中。即我们需要把一封邮件转化为如下的向量形式：

完成emailFeatures.m如下：

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 dictionary
n = 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 ..];
%
%

x(word_indices) = 1; 

% =========================================================================

end

2.3 Training SVM for spam classification

训练集中提供了4000封非垃圾邮件和4000封垃圾邮件。而测试集中则有1000封邮件用于测试。
在ex6_spam.m中使用SVM进行训练，并进行测试，分别输出训练集和测试集分类的准确率：

%% =========== 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 environment
load(‘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 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));
end

fprintf(‘\n\n‘);
fprintf(‘\nProgram paused. Press enter to continue.\n‘);
pause;

结果如上图，正确率分别高达99.8%和98.9%。