Short introduction to Vector Space Model (VSM)

In information retrieval or text mining, the term frequency – inverse document frequency also called tf-idf, is a well know method to evaluate how important is a word in a document. tf-idf are also a very interesting way to convert the textual representation of information into a Vector Space Model (VSM), or into sparse features, we’ll discuss more about it later, but first, let’s try to understand what is tf-idf and the VSM.

VSM has a very confusing past, see for example the paper The most influential paper Gerard Salton Never Wrote that explains the history behind the ghost cited paper which in fact never existed; in sum, VSM is an algebraic model representing textual information as a vector, the components of this vector could represent the importance of a term (tf–idf) or even the absence or presence (Bag of Words) of it in a document; it is important to note that the classical VSM proposed by Salton incorporates local and global parameters/information (in a sense that it uses both the isolated term being analyzed as well the entire collection of documents). VSM, interpreted in a lato sensu, is a space where text is represented as a vector of numbers instead of its original string textual representation; the VSM represents the features extracted from the document.

Let’s try to mathematically define the VSM and tf-idf together with concrete examples, for the concrete examples I’ll be using Python (as well the amazing scikits.learn Python module).

Going to the vector space

The first step in modeling the document into a vector space is to create a dictionary of terms present in documents. To do that, you can simple select all terms from the document and convert it to a dimension in the vector space, but we know that there are some kind of words (stop words) that are present in almost all documents, and what we’re doing is extracting important features from documents, features do identify them among other similar documents, so using terms like “the, is, at, on”, etc.. isn’t going to help us, so in the information extraction, we’ll just ignore them.

Let’s take the documents below to define our (stupid) document space:

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Train Document Set:

d1: The sky is blue.
d2: The sun is bright.

Test Document Set:

d3: The sun in the sky is bright.
d4: We can see the shining sun, the bright sun.

Now, what we have to do is to create a index vocabulary (dictionary) of the words of the train document set, using the documents d1 and d2 from the document set, we’ll have the following index vocabulary denoted as \mathrm{E}(t) where the t is the term:

   \mathrm{E}(t) =   \begin{cases}   1, & \mbox{if } t\mbox{ is ``blue''} \\   2, & \mbox{if } t\mbox{ is ``sun''} \\   3, & \mbox{if } t\mbox{ is ``bright''} \\   4, & \mbox{if } t\mbox{ is ``sky''} \\   \end{cases}

Note that the terms like “is” and “the” were ignored as cited before. Now that we have an index vocabulary, we can convert the test document set into a vector space where each term of the vector is indexed as our index vocabulary, so the first term of the vector represents the “blue” term of our vocabulary, the second represents “sun” and so on. Now, we’re going to use the term-frequency to represent each term in our vector space; the term-frequency is nothing more than a measure of how many times the terms present in our vocabulary \mathrm{E}(t) are present in the documents d3 or d4, we define the term-frequency as a couting function:

   \mathrm{tf}(t,d) = \sum\limits_{x\in d} \mathrm{fr}(x, t)

where the \mathrm{fr}(x, t) is a simple function defined as:

   \mathrm{fr}(x,t) =   \begin{cases}   1, & \mbox{if } x = t \\   0, & \mbox{otherwise} \\   \end{cases}

So, what the tf(t,d) returns is how many times is the term t is present in the document d. An example of this, could be  tf(``sun'', d4) = 2 since we have only two occurrences of the term “sun” in the document d4. Now you understood how the term-frequency works, we can go on into the creation of the document vector, which is represented by:

   \displaystyle \vec{v_{d_n}} =(\mathrm{tf}(t_1,d_n), \mathrm{tf}(t_2,d_n), \mathrm{tf}(t_3,d_n), \ldots, \mathrm{tf}(t_n,d_n))

Each dimension of the document vector is represented by the term of the vocabulary, for example, the \mathrm{tf}(t_1,d_2) represents the frequency-term of the term 1 or t_1 (which is our “blue” term of the vocabulary) in the document d_2.

Let’s now show a concrete example of how the documents d_3 and d_4 are represented as vectors:

   \vec{v_{d_3}} = (\mathrm{tf}(t_1,d_3), \mathrm{tf}(t_2,d_3), \mathrm{tf}(t_3,d_3), \ldots, \mathrm{tf}(t_n,d_3)) \\   \vec{v_{d_4}} = (\mathrm{tf}(t_1,d_4), \mathrm{tf}(t_2,d_4), \mathrm{tf}(t_3,d_4), \ldots, \mathrm{tf}(t_n,d_4))

which evaluates to:

   \vec{v_{d_3}} = (0, 1, 1, 1) \\   \vec{v_{d_4}} = (0, 2, 1, 0)

As you can see, since the documents d_3 and d_4 are:

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d3: The sun in the sky is bright.
d4: We can see the shining sun, the bright sun.

The resulting vector \vec{v_{d_3}} shows that we have, in order, 0 occurrences of the term “blue”, 1 occurrence of the term “sun”, and so on. In the \vec{v_{d_3}}, we have 0 occurences of the term “blue”, 2 occurrences of the term “sun”, etc.

But wait, since we have a collection of documents, now represented by vectors, we can represent them as a matrix with |D| \times F shape, where |D| is the cardinality of the document space, or how many documents we have and the F is the number of features, in our case represented by the vocabulary size. An example of the matrix representation of the vectors described above is:

   M_{|D| \times F} =   \begin{bmatrix}   0 & 1 & 1 & 1\\   0 & 2 & 1 & 0   \end{bmatrix}

As you may have noted, these matrices representing the term frequencies tend to be very sparse (with majority of terms zeroed), and that’s why you’ll see a common representation of these matrix as sparse matrices.

Python practice

Environment Used: Python v.2.7.2, Numpy 1.6.1, Scipy v.0.9.0, Sklearn (Scikits.learn) v.0.9.

Since we know the  theory behind the term frequency and the vector space conversion, let’s show how easy is to do that using the amazing scikit.learn Python module.

Scikit.learn comes with lots of examples as well real-life interesting datasets you can use and also some helper functions to download 18k newsgroups posts for instance.

Since we already defined our small train/test dataset before, let’s use them to define the dataset in a way that scikit.learn can use:

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train_set = ("The sky is blue.", "The sun is bright.")
test_set = ("The sun in the sky is bright.",
"We can see the shining sun, the bright sun.")

In scikit.learn, what we have presented as the term-frequency, is called CountVectorizer, so we need to import it and create a news instance:

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from sklearn.feature_extraction.text import CountVectorizer
vectorizer = CountVectorizer()

The CountVectorizer already uses as default “analyzer” called WordNGramAnalyzer, which is responsible to convert the text to lowercase, accents removal, token extraction, filter stop words, etc… you can see more information by printing the class information:

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print vectorizer
CountVectorizer(analyzer__min_n=1,
analyzer__stop_words=set(['all', 'six', 'less', 'being', 'indeed', 'over', 'move', 'anyway', 'four', 'not', 'own', 'through', 'yourselves', (...)

Let’s create now the vocabulary index:

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vectorizer.fit_transform(train_set)
print vectorizer.vocabulary
{'blue': 0, 'sun': 1, 'bright': 2, 'sky': 3}

See that the vocabulary created is the same as E(t) (except because it is zero-indexed).

Let’s use the same vectorizer now to create the sparse matrix of our test_set documents:

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smatrix = vectorizer.transform(test_set)

print smatrix

(0, 1)        1
(0, 2)        1
(0, 3)        1
(1, 1)        2
(1, 2)        1

Note that the sparse matrix created called smatrix is a Scipy sparse matrix with elements stored in a Coordinate format. But you can convert it into a dense format:

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smatrix.todense()

matrix([[0, 1, 1, 1],
........[0, 2, 1, 0]], dtype=int64)

Note that the sparse matrix created is the same matrix M_{|D| \times F} we cited earlier in this post, which represents the two document vectors \vec{v_{d_3}} and \vec{v_{d_4}}.

We’ll see in the next post how we define the idf (inverse document frequency) instead of the simple term-frequency, as well how logarithmic scale is used to adjust the measurement of term frequencies according to its importance, and how we can use it to classify documents using some of the well-know machine learning approaches.

I hope you liked this post, and if you really liked, leave a comment so I’ll able to know if there are enough people interested in these series of posts in Machine Learning topics.

As promised, here is the second part of this tutorial series.

References

The classic Vector Space Model

The most influential paper Gerard Salton never wrote

Wikipedia: tf-idf

Wikipedia: Vector space model

Scikits.learn Examples

Updates

21 Sep 11 - fixed some typos and the vector notation
22 Sep 11 – fixed import of sklearn according to the new 0.9 release and added the environment section
02 Oct 11 – fixed Latex math typos
18 Oct 11 – added link to the second part of the tutorial series

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