# Text classification

The goal is to take a new text document (an article from a website, perhaps) and choose 1, 2, maybe more categories that describe the document. These documents and categories relate to artificial intelligence; the documents are obtained from searching the web with terms like "artificial intelligence," "robot," and "Turing test."

The 19 categories are: AIOverview, Agents, Applications, CognitiveScience, Education, Ethics, Games, History, Interfaces, MachineLearning, NaturalLanguage, Philosophy, Reasoning, Representation, Robots, ScienceFiction, Speech, Systems, Vision. These links go to the AITopics website, which provides a news service called AI in the News that must solve the problem of categorizing news articles into one or more of these 19 categories.

These notes describe how to do the categorization; obtaining documents and their content is beyond our scope. Thus, we will utilize a prepared collection of documents. The zip files linked here provide about 3000 documents with human-chosen categories. We'll take a portion of these, say 90%, and train our categorization tools. The remaining 10% will be used for testing purposes. Which 90% and 10% are training and testing sets should be set randomly (and differently in perhaps 5 or 10 different experiments to ensure a categorizer's performance is robust).

Zip file of all AI articles — these documents have the raw original text.

Zip file of all AI articles (stemmed documents) — these documents are the same as above, but every word in the text has been "stemmed" (see below). This is the document set we will be working with.

## Word stemming

In linguistic morphology and information retrieval, stemming is the process for reducing inflected (or sometimes derived) words to their stem, base or root form—generally a written word form. The stem need not be identical to the morphological root of the word; it is usually sufficient that related words map to the same stem, even if this stem is not in itself a valid root.

[…]

A stemmer for English, for example, should identify the string "cats" (and possibly "catlike", "catty" etc.) as based on the root "cat", and "stemmer", "stemming", "stemmed" as based on "stem". A stemming algorithm reduces the words "fishing", "fished", "fish", and "fisher" to the root word, "fish". (Wikipedia)

## Feature vectors

A document $$j$$ is represented as:

$$X_j = (x_{j1}, x_{j2}, \dots, x_{jc}),$$

where $$c$$ is the count of unique words across all documents in the database. Thus, all documents have vectors of the same number of dimensions ($$c$$ dimensions).

### Binary features

$$x_{ji} = \cases{1 & \text{if $$j$$ is in i} \cr 0 & \text{otherwise}}$$

### Frequency count features

$$x_{ji} = tf_{ji},$$

where $$tf_{ji}$$ is the frequency (count) of the term in the document.

### Normalized frequency count features

$$x_{ji} = \frac{tf_{ji}}{\sqrt{\sum_{k} tf_{ki}^2}}$$

## tf-idf

The document vectors described above all suffer from over-scoring common words. A common word like "the" does not give any evidence that one document is closely related to another one just because they both contain many instances of that word. So in addition to recording how often a term appears in a document, we need to divide out some measure of how common is the term across different documents. A term appearing in nearly every document is not as relevant as a term appearing in few. Additionally, if a rare term appears more often in a document, it is even more indicative of the content of that document.

### Motivation

Suppose we have a set of English text documents and wish to determine which document is most relevant to the query "the brown cow." A simple way to start out is by eliminating documents that do not contain all three words "the", "brown", and "cow", but this still leaves many documents. To further distinguish them, we might count the number of times each term occurs in each document and sum them all together; the number of times a term occurs in a document is called its term frequency.

However, because the term "the" is so common, this will tend to incorrectly emphasize documents which happen to use the word "the" more frequently, without giving enough weight to the more meaningful terms "brown" and "cow". The term "the" is not a good keyword to distinguish relevant and non-relevant documents and terms, unlike the less common words "brown" and "cow". Hence an inverse document frequency factor is incorporated which diminishes the weight of terms that occur very frequently in the collection and increases the weight of terms that occur rarely. (Wikipedia)

### The math

Every dimension (word) in the document vector has the following value:

$$x_{ji} = \log(tf_{ji} + 1) \log(n/df_j + 1),$$

where $$j$$ is the index of a word, $$i$$ is the index of a document, $$x_{ji}$$ is the value in position $$j$$ (for word $$j$$) in the vector representing document $$i$$, $$tf_{ji}$$ is the frequency (count) of the term $$j$$ in the document $$i$$, $$n$$ is the number of documents in the database, and $$df_j$$ is the number of documents in the database that contain at least one occurrence of the term $$j$$.

This graph shows the tf-idf value for a particular term in a document that comes from a database of 1,010 documents. The bottom axis is $$df$$ (document frequency for that term); the top numbered axis (the "depth" axis) is $$tf$$ or (term frequency; the number of times that term appears in the document under consideration); the right axis is the tf-idf value.

We see in the graph that an uncommon term ($$df$$ is low) produces a higher tf-idf value. However, due to the use of logarithms, the tf-idf value grows little as the term appears more often in the document. In all, a term that's appears several times in the document and is somewhat unique to the document is a strong indicator (of category membership, of relevance to a query containing that term, etc.).

An example of binary, frequency, and tfidf document vectors can be found in the doc-vectors.xls spreadsheet.

## k-nearest neighbor

Since we have different ways of representing the document vectors, we have different "distance" calculations. Below, take $$\hat{X} = (\hat{x}_0, \dots, \hat{x}_m)$$ to be the document we want to classify, and $$X_j = (x_{j0}, \dots, x_{jn})$$ to be a document from the database ($$j$$ will range so we check the distance to every document in the database).

### With binary features

$$d = \frac{\sum_{i} \hat{x}_{i} x_{ji}}{\sum_{i} \hat{x}_{i}}$$

The numerator of the fraction gives the number of shared words; the denominator gives the number of words in the document to be classified. Thus, $$0 \leq d \leq 1$$.

### With frequency count features

$$d = \sqrt{\sum_{i} (\hat{x}_{i} - x_{ji})^2}$$

This is just Euclidean distance between the two document vectors. Thus, $$0 \leq d < \infty$$.

### With tf-idf

$$d = \sum_{i} \hat{x}_{i} x_{ji},$$

assuming the vectors $$\hat{X}$$ (the document we are trying to classify) and $$X_j$$ (each document in the database) have been normalized so that each of their Euclidean distances to the origin vector (all zeros) is equal to 1.0.

The above is a dot-product of two vectors. Because these vectors have been normalized, the dot-product can also be called the cosine similarity:

$$\cos \theta = \hat{X} \cdot X_j,$$

where $$0 \leq \theta \leq \pi/2$$ and $$\theta$$ is the angle (from the origin) between the two document vectors in the higher-dimensional space. More similar documents have a smaller angle between them, meaning their cosine similarity is closer to 1.0. Documents with absolutely no terms in common have a dot-product equal to 0.0, or equivalently, a 90-degree angle between them.

### Take-away lessons

• Good choices for what I am calling the "algorithm" (tf-idf, count with normalization, etc.) depends on the dataset. However, tf-idf is robust and works best on all datasets represented here. This is a significant result.
• Using a weighted voting technique is generally a good idea.
• Increasing $$k$$ does not always increase performance.
• The best value for $$k$$ depends on the the dataset and the algorithm.

## Category centroids

To construct the "centroid" (average, mean) of a category, we gather up all the documents that are members of the category, and compute the mean of their document vectors:

$$C_c = \frac{\sum_j X_j^c}{n_c},$$

where $$C_c$$ is the centroid for category $$c$$, $$X_j^c$$ is a document vector from the category $$c$$, and $$n_c$$ is the number of documents in the category.

When we have a new test document that we want to classify, we compute its distance from each category centroid. Then, we can either choose to take the "closest" or "most similar" category, or the best $$n$$ categories, or all categories that have a distance less than some threshold.

This is a plot (after applying multidimensional scaling) of how "different" each category centroid is from each other. It seems that Ethics, AIOverview, and History are quite similar, for example.

Data from 10 experiments are shown in these box plots. Performance is not great on any dataset.

The AI dataset often has multiple category membership (the other datasets always have one category per document), so we have to choose how many categories to select. We can use a threshold (choose all categories where the similarity of the document with the category centroid is greater than a threshold), or we can choose the best $$n$$ categories. The plot above shows performance for this second approach, using various values for $$n$$. For all such values, performance is quite poor.

### Benefits of category centroids

• The centroids are computed ahead of time, and when we get a new document, we only need to compute its distance to a handful of category centroids (rather than compute its distance to every document in the database).

### Drawbacks of category centroids

• An averaged document vector may be distorted due to the presence of outliers (very bizarre documents that are members of the category). Perhaps some category outlier detector can remove these documents from the calculation of the mean.
• When using a dot-product technique, say $$A \cdot B$$, if a vector $$B$$ has more non-zero terms than another vector $$B'$$ then the dot-product $$A \cdot B$$ has a good chance of being greater than $$A \cdot B'$$. This means that categories with more documents will produce centroids with more non-zero terms (because the documents in the category span more words than a category with fewer documents), and therefore, everything else being equal, larger dot-products. So the category with the most members will be chosen most often. (N.B., this is all conjecture; I have not verified this suspicion.)

## Support vector machines

We will not be learning about support vector machines in this class; the topic is too advanced. However, it should be noted that they perform very well:

Dataset (with tfidf) Average F-score
ai 0.925
la1 0.950
oh10 0.957
re0 0.974
tr21 0.953
wap 0.981

However, support vector machine training is very slow: training the "la1" dataset (only 2883 documents and 6 categories) took 2.5 hours on a quad-core machine (using all cores). Training with the naïve Bayes method (described in Naïve Bayesian classification notes) takes only a few seconds.