Ex6 Instructions
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Programming Exercise 6:
Support Vector Machines
Machine Learning
Introduction
In this exercise, you will be using support vector machines (SVMs) to build
a spam classifier. Before starting on the programming exercise, we strongly
recommend watching the video lectures and completing the review questions
for the associated topics.
To get started with the exercise, you will need to download the starter
code and unzip its contents to the directory where you wish to complete the
exercise. If needed, use the cd command in Octave/MATLAB to change to
this directory before starting this exercise.
You can also find instructions for installing Octave/MATLAB in the “En-
vironment Setup Instructions” of the course website.
Files included in this exercise
ex6.m - Octave/MATLAB script for the first half of the exercise
ex6data1.mat - Example Dataset 1
ex6data2.mat - Example Dataset 2
ex6data3.mat - Example Dataset 3
svmTrain.m - SVM training function
svmPredict.m - SVM prediction function
plotData.m - Plot 2D data
visualizeBoundaryLinear.m - Plot linear boundary
visualizeBoundary.m - Plot non-linear boundary
linearKernel.m - Linear kernel for SVM
[?] gaussianKernel.m - Gaussian kernel for SVM
[?] dataset3Params.m - Parameters to use for Dataset 3
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ex6 spam.m - Octave/MATLAB script for the second half of the exer-
cise
spamTrain.mat - Spam training set
spamTest.mat - Spam test set
emailSample1.txt - Sample email 1
emailSample2.txt - Sample email 2
spamSample1.txt - Sample spam 1
spamSample2.txt - Sample spam 2
vocab.txt - Vocabulary list
getVocabList.m - Load vocabulary list
porterStemmer.m - Stemming function
readFile.m - Reads a file into a character string
submit.m - Submission script that sends your solutions to our servers
[?] processEmail.m - Email preprocessing
[?] emailFeatures.m - Feature extraction from emails
?indicates files you will need to complete
Throughout the exercise, you will be using the script ex6.m. These scripts
set up the dataset for the problems and make calls to functions that you will
write. You are only required to modify functions in other files, by following
the instructions in this assignment.
Where to get help
The exercises in this course use Octave1or MATLAB, a high-level program-
ming language well-suited for numerical computations. If you do not have
Octave or MATLAB installed, please refer to the installation instructions in
the “Environment Setup Instructions” of the course website.
At the Octave/MATLAB command line, typing help followed by a func-
tion name displays documentation for a built-in function. For example, help
plot will bring up help information for plotting. Further documentation for
Octave functions can be found at the Octave documentation pages. MAT-
LAB documentation can be found at the MATLAB documentation pages.
We also strongly encourage using the online Discussions to discuss ex-
ercises with other students. However, do not look at any source code written
by others or share your source code with others.
1Octave is a free alternative to MATLAB. For the programming exercises, you are free
to use either Octave or MATLAB.
2
1 Support Vector Machines
In the first half of this exercise, you will be using support vector machines
(SVMs) with various example 2D datasets. Experimenting with these datasets
will help you gain an intuition of how SVMs work and how to use a Gaussian
kernel with SVMs. In the next half of the exercise, you will be using support
vector machines to build a spam classifier.
The provided script, ex6.m, will help you step through the first half of
the exercise.
1.1 Example Dataset 1
We will begin by with a 2D example dataset which can be separated by a
linear boundary. The script ex6.m will plot the training data (Figure 1). In
this dataset, the positions of the positive examples (indicated with +) and the
negative examples (indicated with o) suggest a natural separation indicated
by the gap. However, notice that there is an outlier positive example + on
the far left at about (0.1,4.1). As part of this exercise, you will also see how
this outlier affects the SVM decision boundary.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
1.5
2
2.5
3
3.5
4
4.5
5
Figure 1: Example Dataset 1
In this part of the exercise, you will try using different values of the C
parameter with SVMs. Informally, the Cparameter is a positive value that
controls the penalty for misclassified training examples. A large Cparameter
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tells the SVM to try to classify all the examples correctly. Cplays a role
similar to 1
λ, where λis the regularization parameter that we were using
previously for logistic regression.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
1.5
2
2.5
3
3.5
4
4.5
5
Figure 2: SVM Decision Boundary with C= 1 (Example Dataset 1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
1.5
2
2.5
3
3.5
4
4.5
5
Figure 3: SVM Decision Boundary with C= 100 (Example Dataset 1)
The next part in ex6.m will run the SVM training (with C= 1) using
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SVM software that we have included with the starter code, svmTrain.m.2
When C= 1, you should find that the SVM puts the decision boundary in
the gap between the two datasets and misclassifies the data point on the far
left (Figure 2).
Implementation Note: Most SVM software packages (including
svmTrain.m) automatically add the extra feature x0= 1 for you and au-
tomatically take care of learning the intercept term θ0. So when passing
your training data to the SVM software, there is no need to add this ex-
tra feature x0= 1 yourself. In particular, in Octave/MATLAB your code
should be working with training examples x∈Rn(rather than x∈Rn+1);
for example, in the first example dataset x∈R2.
Your task is to try different values of Con this dataset. Specifically, you
should change the value of Cin the script to C= 100 and run the SVM
training again. When C= 100, you should find that the SVM now classifies
every single example correctly, but has a decision boundary that does not
appear to be a natural fit for the data (Figure 3).
1.2 SVM with Gaussian Kernels
In this part of the exercise, you will be using SVMs to do non-linear clas-
sification. In particular, you will be using SVMs with Gaussian kernels on
datasets that are not linearly separable.
1.2.1 Gaussian Kernel
To find non-linear decision boundaries with the SVM, we need to first im-
plement a Gaussian kernel. You can think of the Gaussian kernel as a sim-
ilarity function that measures the “distance” between a pair of examples,
(x(i), x(j)). The Gaussian kernel is also parameterized by a bandwidth pa-
rameter, σ, which determines how fast the similarity metric decreases (to 0)
as the examples are further apart.
You should now complete the code in gaussianKernel.m to compute
the Gaussian kernel between two examples, (x(i), x(j)). The Gaussian kernel
2In order to ensure compatibility with Octave/MATLAB, we have included this imple-
mentation of an SVM learning algorithm. However, this particular implementation was
chosen to maximize compatibility, and is not very efficient. If you are training an SVM on
a real problem, especially if you need to scale to a larger dataset, we strongly recommend
instead using a highly optimized SVM toolbox such as LIBSVM.
5
function is defined as:
Kgaussian(x(i), x(j)) = exp −kx(i)−x(j)k2
2σ2= exp
−
n
P
k=1
(x(i)
k−x(j)
k)2
2σ2
.
Once you’ve completed the function gaussianKernel.m, the script ex6.m
will test your kernel function on two provided examples and you should ex-
pect to see a value of 0.324652.
You should now submit your solutions.
1.2.2 Example Dataset 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4: Example Dataset 2
The next part in ex6.m will load and plot dataset 2 (Figure 4). From
the figure, you can obserse that there is no linear decision boundary that
separates the positive and negative examples for this dataset. However, by
using the Gaussian kernel with the SVM, you will be able to learn a non-linear
decision boundary that can perform reasonably well for the dataset.
If you have correctly implemented the Gaussian kernel function, ex6.m
will proceed to train the SVM with the Gaussian kernel on this dataset.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5: SVM (Gaussian Kernel) Decision Boundary (Example Dataset 2)
Figure 5shows the decision boundary found by the SVM with a Gaussian
kernel. The decision boundary is able to separate most of the positive and
negative examples correctly and follows the contours of the dataset well.
1.2.3 Example Dataset 3
In this part of the exercise, you will gain more practical skills on how to use
a SVM with a Gaussian kernel. The next part of ex6.m will load and display
a third dataset (Figure 6). You will be using the SVM with the Gaussian
kernel with this dataset.
In the provided dataset, ex6data3.mat, you are given the variables X,
y, Xval, yval. The provided code in ex6.m trains the SVM classifier using
the training set (X, y) using parameters loaded from dataset3Params.m.
Your task is to use the cross validation set Xval, yval to determine the
best Cand σparameter to use. You should write any additional code nec-
essary to help you search over the parameters Cand σ. For both Cand σ, we
suggest trying values in multiplicative steps (e.g., 0.01,0.03,0.1,0.3,1,3,10,30).
Note that you should try all possible pairs of values for Cand σ(e.g., C= 0.3
and σ= 0.1). For example, if you try each of the 8 values listed above for C
and for σ2, you would end up training and evaluating (on the cross validation
set) a total of 82= 64 different models.
After you have determined the best Cand σparameters to use, you
should modify the code in dataset3Params.m, filling in the best parameters
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−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Figure 6: Example Dataset 3
−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Figure 7: SVM (Gaussian Kernel) Decision Boundary (Example Dataset 3)
you found. For our best parameters, the SVM returned a decision boundary
shown in Figure 7.
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Implementation Tip: When implementing cross validation to select
the best Cand σparameter to use, you need to evaluate the error on
the cross validation set. Recall that for classification, the error is de-
fined as the fraction of the cross validation examples that were classi-
fied incorrectly. In Octave/MATLAB, you can compute this error using
mean(double(predictions ~= yval)), where predictions is a vector
containing all the predictions from the SVM, and yval are the true labels
from the cross validation set. You can use the svmPredict function to
generate the predictions for the cross validation set.
You should now submit your solutions.
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2 Spam Classification
Many email services today provide spam filters that are able to classify emails
into spam and non-spam email with high accuracy. In this part of the exer-
cise, you will use SVMs to build your own spam filter.
You will be training a classifier to classify whether a given email, x, is
spam (y= 1) or non-spam (y= 0). In particular, you need to convert each
email into a feature vector x∈Rn. The following parts of the exercise will
walk you through how such a feature vector can be constructed from an
email.
Throughout the rest of this exercise, you will be using the the script
ex6 spam.m. The dataset included for this exercise is based on a a subset of
the SpamAssassin Public Corpus.3For the purpose of this exercise, you will
only be using the body of the email (excluding the email headers).
2.1 Preprocessing Emails
> Anyone knows how much it costs to host a web portal ?
>
Well, it depends on how many visitors youre expecting. This can be
anywhere from less than 10 bucks a month to a couple of $100. You
should checkout http://www.rackspace.com/ or perhaps Amazon EC2 if
youre running something big..
To unsubscribe yourself from this mailing list, send an email to:
groupname-unsubscribe@egroups.com
Figure 8: Sample Email
Before starting on a machine learning task, it is usually insightful to
take a look at examples from the dataset. Figure 8shows a sample email
that contains a URL, an email address (at the end), numbers, and dollar
amounts. While many emails would contain similar types of entities (e.g.,
numbers, other URLs, or other email addresses), the specific entities (e.g.,
the specific URL or specific dollar amount) will be different in almost every
email. Therefore, one method often employed in processing emails is to
“normalize” these values, so that all URLs are treated the same, all numbers
are treated the same, etc. For example, we could replace each URL in the
email with the unique string “httpaddr” to indicate that a URL was present.
3http://spamassassin.apache.org/old/publiccorpus/
10
This has the effect of letting the spam classifier make a classification decision
based on whether any URL was present, rather than whether a specific URL
was present. This typically improves the performance of a spam classifier,
since spammers often randomize the URLs, and thus the odds of seeing any
particular URL again in a new piece of spam is very small.
In processEmail.m, we have implemented the following email prepro-
cessing and normalization steps:
•Lower-casing: The entire email is converted into lower case, so
that captialization is ignored (e.g., IndIcaTE is treated the same as
Indicate).
•Stripping HTML: All HTML tags are removed from the emails.
Many emails often come with HTML formatting; we remove all the
HTML tags, so that only the content remains.
•Normalizing URLs: All URLs are replaced with the text “httpaddr”.
•Normalizing Email Addresses: All email addresses are replaced
with the text “emailaddr”.
•Normalizing Numbers: All numbers are replaced with the text
“number”.
•Normalizing Dollars: All dollar signs ($) are replaced with the text
“dollar”.
•Word Stemming: Words are reduced to their stemmed form. For ex-
ample, “discount”, “discounts”, “discounted” and “discounting” are all
replaced with “discount”. Sometimes, the Stemmer actually strips off
additional characters from the end, so “include”, “includes”, “included”,
and “including” are all replaced with “includ”.
•Removal of non-words: Non-words and punctuation have been re-
moved. All white spaces (tabs, newlines, spaces) have all been trimmed
to a single space character.
The result of these preprocessing steps is shown in Figure 9. While pre-
processing has left word fragments and non-words, this form turns out to be
much easier to work with for performing feature extraction.
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anyon know how much it cost to host a web portal well it depend on how
mani visitor your expect thi can be anywher from less than number buck
a month to a coupl of dollarnumb you should checkout httpaddr or perhap
amazon ecnumb if your run someth big to unsubscrib yourself from thi
mail list send an email to emailaddr
Figure 9: Preprocessed Sample Email
1 aa
2 ab
3 abil
...
86 anyon
...
916 know
...
1898 zero
1899 zip
Figure 10: Vocabulary List
86 916 794 1077 883
370 1699 790 1822
1831 883 431 1171
794 1002 1893 1364
592 1676 238 162 89
688 945 1663 1120
1062 1699 375 1162
479 1893 1510 799
1182 1237 810 1895
1440 1547 181 1699
1758 1896 688 1676
992 961 1477 71 530
1699 531
Figure 11: Word Indices for Sample Email
2.1.1 Vocabulary List
After preprocessing the emails, we have a list of words (e.g., Figure 9) for
each email. The next step is to choose which words we would like to use in
our classifier and which we would want to leave out.
For this exercise, we have chosen only the most frequently occuring words
as our set of words considered (the vocabulary list). Since words that occur
rarely in the training set are only in a few emails, they might cause the
model to overfit our training set. The complete vocabulary list is in the file
vocab.txt and also shown in Figure 10. Our vocabulary list was selected
by choosing all words which occur at least a 100 times in the spam corpus,
resulting in a list of 1899 words. In practice, a vocabulary list with about
10,000 to 50,000 words is often used.
Given the vocabulary list, we can now map each word in the preprocessed
emails (e.g., Figure 9) into a list of word indices that contains the index
of the word in the vocabulary list. Figure 11 shows the mapping for the
sample email. Specifically, in the sample email, the word “anyone” was first
normalized to “anyon” and then mapped onto the index 86 in the vocabulary
list.
Your task now is to complete the code in processEmail.m to perform
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this mapping. In the code, you are given a string str which is a single word
from the processed email. You should look up the word in the vocabulary
list vocabList and find if the word exists in the vocabulary list. If the word
exists, you should add the index of the word into the word indices variable.
If the word does not exist, and is therefore not in the vocabulary, you can
skip the word.
Once you have implemented processEmail.m, the script ex6 spam.m will
run your code on the email sample and you should see an output similar to
Figures 9&11.
Octave/MATLAB Tip: In Octave/MATLAB, you can compare two
strings with the strcmp function. For example, strcmp(str1, str2) will
return 1 only when both strings are equal. In the provided starter code,
vocabList is a “cell-array” containing the words in the vocabulary. In
Octave/MATLAB, a cell-array is just like a normal array (i.e., a vector),
except that its elements can also be strings (which they can’t in a normal
Octave/MATLAB matrix/vector), and you index into them using curly
braces instead of square brackets. Specifically, to get the word at index
i, you can use vocabList{i}. You can also use length(vocabList) to
get the number of words in the vocabulary.
You should now submit your solutions.
2.2 Extracting Features from Emails
You will now implement the feature extraction that converts each email into
a vector in Rn. For this exercise, you will be using n= # words in vocabulary
list. Specifically, the feature xi∈ {0,1}for an email corresponds to whether
the i-th word in the dictionary occurs in the email. That is, xi= 1 if the i-th
word is in the email and xi= 0 if the i-th word is not present in the email.
Thus, for a typical email, this feature would look like:
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x=
0
.
.
.
1
0
.
.
.
1
0
.
.
.
0
∈Rn.
You should now complete the code in emailFeatures.m to generate a
feature vector for an email, given the word indices.
Once you have implemented emailFeatures.m, the next part of ex6 spam.m
will run your code on the email sample. You should see that the feature vec-
tor had length 1899 and 45 non-zero entries.
You should now submit your solutions.
2.3 Training SVM for Spam Classification
After you have completed the feature extraction functions, the next step of
ex6 spam.m will load a preprocessed training dataset that will be used to train
a SVM classifier. spamTrain.mat contains 4000 training examples of spam
and non-spam email, while spamTest.mat contains 1000 test examples. Each
original email was processed using the processEmail and emailFeatures
functions and converted into a vector x(i)∈R1899.
After loading the dataset, ex6 spam.m will proceed to train a SVM to
classify between spam (y= 1) and non-spam (y= 0) emails. Once the
training completes, you should see that the classifier gets a training accuracy
of about 99.8% and a test accuracy of about 98.5%.
2.4 Top Predictors for Spam
our click remov guarante visit basenumb dollar will price pleas nbsp
most lo ga dollarnumb
Figure 12: Top predictors for spam email
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To better understand how the spam classifier works, we can inspect the
parameters to see which words the classifier thinks are the most predictive
of spam. The next step of ex6 spam.m finds the parameters with the largest
positive values in the classifier and displays the corresponding words (Figure
12). Thus, if an email contains words such as “guarantee”, “remove”, “dol-
lar”, and “price” (the top predictors shown in Figure 12), it is likely to be
classified as spam.
2.5 Optional (ungraded) exercise: Try your own emails
Now that you have trained a spam classifier, you can start trying it out on
your own emails. In the starter code, we have included two email exam-
ples (emailSample1.txt and emailSample2.txt) and two spam examples
(spamSample1.txt and spamSample2.txt). The last part of ex6 spam.m
runs the spam classifier over the first spam example and classifies it using
the learned SVM. You should now try the other examples we have provided
and see if the classifier gets them right. You can also try your own emails by
replacing the examples (plain text files) with your own emails.
You do not need to submit any solutions for this optional (ungraded)
exercise.
2.6 Optional (ungraded) exercise: Build your own dataset
In this exercise, we provided a preprocessed training set and test set. These
datasets were created using the same functions (processEmail.m and emailFeatures.m)
that you now have completed. For this optional (ungraded) exercise, you will
build your own dataset using the original emails from the SpamAssassin Pub-
lic Corpus.
Your task in this optional (ungraded) exercise is to download the original
files from the public corpus and extract them. After extracting them, you
should run the processEmail4and emailFeatures functions on each email
to extract a feature vector from each email. This will allow you to build a
dataset X, y of examples. You should then randomly divide up the dataset
into a training set, a cross validation set and a test set.
While you are building your own dataset, we also encourage you to try
building your own vocabulary list (by selecting the high frequency words
4The original emails will have email headers that you might wish to leave out. We have
included code in processEmail that will help you remove these headers.
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that occur in the dataset) and adding any additional features that you think
might be useful.
Finally, we also suggest trying to use highly optimized SVM toolboxes
such as LIBSVM.
You do not need to submit any solutions for this optional (ungraded)
exercise.
Submission and Grading
After completing various parts of the assignment, be sure to use the submit
function system to submit your solutions to our servers. The following is a
breakdown of how each part of this exercise is scored.
Part Submitted File Points
Gaussian Kernel gaussianKernel.m 25 points
Parameters (C,σ) for Dataset 3 dataset3Params.m 25 points
Email Preprocessing processEmail.m 25 points
Email Feature Extraction emailFeatures.m 25 points
Total Points 100 points
You are allowed to submit your solutions multiple times, and we will take
only the highest score into consideration.
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