From Bag-of-Words to TF-IDF

Converting text into a numerical format is essential for machine learning models. The simplest approach is often the Bag-of-Words (BoW) model. The Bag-of-Words (BoW) model represents each document as a vector, where each element corresponds to the count of a particular word from the overall vocabulary.

Imagine two simple sentences:

1. "The cat sat on the mat."
2. "The dog chased the cat."

The vocabulary is {"The", "cat", "sat", "on", "mat", "dog", "chased"}. Ignoring case and punctuation for simplicity, the BoW vectors would be:

• Sentence 1: [2, 1, 1, 1, 1, 0, 0] (corresponding to the counts of each word in the vocabulary)
• Sentence 2: [2, 1, 0, 0, 0, 1, 1]

This representation is intuitive and easy to compute. However, it has a significant limitation: it treats every word equally. Notice the word "The" has the highest count (2) in both sentences. Does "The" tell us much about the specific content or difference between these sentences? Probably not. Common words like "the", "is", "in", "a" often dominate the counts but carry less discriminative information compared to words like "sat", "mat", "dog", or "chased". BoW vectors can become heavily skewed by these frequent, yet often uninformative, words.

Furthermore, BoW only considers word counts within a single document. It doesn't account for how common or rare a word is across the entire collection of documents (the corpus). A word might appear many times in one specific document, making it seem important based on BoW alone. But if that same word also appears frequently in almost every other document in the corpus, its ability to distinguish that specific document is diminished. Conversely, a word appearing moderately frequently in one document but very rarely elsewhere might be a strong indicator of that document's unique topic.

Moving Past Simple Counts: The Need for Weighting

To address these limitations, we need a way to adjust the numerical representation to reflect the importance of a word, not just its raw frequency. We want to give higher weights to words that are frequent in a particular document but relatively rare across the entire corpus. This is precisely the motivation behind the Term Frequency-Inverse Document Frequency (TF-IDF) weighting scheme.

TF-IDF aims to quantify how relevant a word is to a document within a collection of documents. It achieves this by combining two distinct metrics:

1. Term Frequency (TF): Measures how often a term appears in a specific document.
2. Inverse Document Frequency (IDF): Measures how informative a term is by considering how common or rare it is across all documents in the corpus.

Let's look at each component.

Term Frequency (TF)

Term Frequency quantifies the prominence of a term within a single document. The basic idea is straightforward: the more times a term $t$ appears in a document $d$, the more likely it is that $t$ is relevant to the topic of $d$.

There are several ways to calculate TF:

• Raw Count: Simply the number of times term $t$ appears in document $d$. This is the BoW count. $$TF_{raw}(t, d) = \text{count of } t \text{ in } d$$
• Frequency: The raw count divided by the total number of terms in the document. This normalizes for document length, preventing longer documents from inherently having higher TF values just because they contain more words. $$TF_{freq}(t, d) = \frac{\text{count of } t \text{ in } d}{\text{total terms in } d}$$
• Logarithmic Scaling: Reduces the impact of very high term counts. A term appearing 100 times isn't necessarily 10 times more important than a term appearing 10 times. Log scaling dampens this effect. $$TF_{log}(t, d) = \log(1 + \text{count of } t \text{ in } d)$$

The choice of TF calculation can depend on the specific application and dataset characteristics. Frequency normalization is very common.

Inverse Document Frequency (IDF)

While TF tells us about a term's frequency within one document, Inverse Document Frequency tells us about its rarity or commonness across the entire corpus $D$. The intuition is that terms appearing in many documents (like "the", "a", "is") are less likely to be discriminative or informative for distinguishing between documents. Terms that appear in only a few documents are likely more specific and thus more informative.

IDF assigns a higher value to rarer terms and a lower value to common terms. It's typically calculated as follows:

$$IDF(t, D) = \log\left(\frac{N}{df_t}\right)$$

Where:

• $N$ is the total number of documents in the corpus $D$.
• $df_t$ (document frequency) is the number of documents in the corpus $D$ that contain the term $t$.

The logarithm (usually natural log or log base 10) is used to dampen the effect of IDF. Without it, very rare words could completely dominate the weighting.

Note: If a term appears in every document ($df_t = N$), the IDF becomes $\log(1) = 0$, effectively removing the term's contribution. If a term appears in no documents (e.g., a new word encountered during testing that wasn't in the training corpus), $df_t$ would be 0, leading to division by zero. To handle this, variations often add 1 to the denominator or use "smoothing":

$$IDF_{smooth}(t, D) = \log\left(\frac{N}{1 + df_t}\right) + 1$$

or

$$IDF_{smooth}(t, D) = \log\left(\frac{1 + N}{1 + df_t}\right) + 1$$

The "+ 1" added at the end ensures that terms appearing in many documents still have a small positive weight, preventing potential issues in downstream calculations.

Combining TF and IDF

The TF-IDF score for a term $t$ in a document $d$ within a corpus $D$ is simply the product of its TF score and its IDF score:

$$TFIDF(t, d, D) = TF(t, d) \times IDF(t, D)$$

This combined score achieves the desired outcome:

• It's highest for terms that appear frequently within a specific document (high TF) but rarely across the entire corpus (high IDF). These are likely the most characteristic terms for that document.
• It's lower for terms that appear infrequently in a document (low TF) or appear in many documents (low IDF).
• It's lowest (often close to zero) for terms that appear in virtually all documents (very low IDF).

Consider our simple example again: "The cat sat on the mat." and "The dog chased the cat.".

• "The": High TF in both, but appears in both (all) documents, so very low IDF. TF-IDF score will be low.
• "cat": High TF in both, appears in both (all) documents, so very low IDF. TF-IDF score will be low.
• "sat", "on", "mat": Moderate TF (1) in document 1, appears only in document 1 (rarer). IDF will be higher than for "the" or "cat". TF-IDF score will be moderate.
• "dog", "chased": Moderate TF (1) in document 2, appears only in document 2 (rarer). IDF will be higher than for "the" or "cat". TF-IDF score will be moderate.

This simple example illustrates how TF-IDF automatically down-weights common words like "the" and "cat" (in this tiny corpus) and gives more prominence to the words that differentiate the documents ("sat", "mat", "dog", "chased").

In practice, instead of manually calculating these, we use libraries like Scikit-learn, which provide efficient implementations (TfidfVectorizer). By moving from simple BoW counts to TF-IDF scores, we create a more detailed and often more effective numerical representation of text, capturing not just word presence but also a measure of word importance relative to the document and the corpus. This refined representation serves as a better foundation for subsequent steps like training machine learning models, which we will cover later. We will now look at how to calculate these scores practically and discuss other techniques like N-grams to further enhance our text features.