Term Frequency › Worked Example

Let's work through all five variants on a small example to build concrete intuition. Consider two documents:

• Doc A: "the cat sat on the mat the cat" (8 tokens)
• Doc B: "the dog chased the cat across the yard the dog" (10 tokens)

The vocabulary is: {across, cat, chased, dog, mat, on, sat, the, yard}

Raw counts:

Applying each weighting scheme to the term "the":

• Raw: Doc A = 3, Doc B = 4
• Log: Doc A = $1 + \ln(3) \approx 2.10$, Doc B = $1 + \ln(4) \approx 2.39$
• Boolean: Doc A = 1, Doc B = 1 (both contain "the")
• Augmented ($K=0.5$): In Doc A, the maximum count is 3 (for "the"), so $\text{tf}_{\text{aug}} = 0.5 + 0.5 \cdot (3/3) = 1.0$. In Doc B, the maximum count is 4 (also "the"), so $\text{tf}_{\text{aug}} = 0.5 + 0.5 \cdot (4/4) = 1.0$. Both get weight 1.0 since "the" is the dominant term in both.
• L2 normalization: Divide each raw count vector by its Euclidean norm before comparison.

Notice how augmented TF reveals that "the" dominates both documents equally (weight 1.0 in both), while "cat" gets weight $0.5 + 0.5 \cdot (2/3) \approx 0.833$ in Doc A and $0.5 + 0.5 \cdot (1/4) = 0.625$ in Doc B, correctly showing that "cat" is proportionally more important in Doc A.

Now let's compute the L2 norms for these two documents:

$$\|\mathbf{v}_A\|_2 = \sqrt{0^2 + 2^2 + 0^2 + 0^2 + 1^2 + 1^2 + 1^2 + 3^2 + 0^2} = \sqrt{4 + 1 + 1 + 1 + 9} = \sqrt{16} = 4$$ $$\|\mathbf{v}_B\|_2 = \sqrt{1^2 + 1^2 + 1^2 + 2^2 + 0^2 + 0^2 + 0^2 + 4^2 + 1^2} = \sqrt{1+1+1+4+16+1} = \sqrt{24} \approx 4.90$$

After normalization, the "cat" entry in Doc A becomes $2/4 = 0.5$, and in Doc B it becomes $1/4.90 \approx 0.204$. So the L2 representation also correctly reflects that "cat" plays a larger role in Doc A than in Doc B, and does so without requiring any free parameter like $K$.

This worked example illustrates a key insight: all five variants agree qualitatively on the relative importance of terms within each document, but they differ in how much weight they assign to high-count terms like "the" versus low-count terms like "sat." The choice of variant affects the geometry of the document space, which in turn affects the behavior of any downstream classifier or retrieval system.