# Q4a · Jaccard, cosine and Euclidean without preprocessing

Subject: Natural Language Processing

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/nlp/#tee/papers/legacy-r23/q4a

Canonical: https://deep-learning-study-room.vercel.app/read/nlp/tee/papers/legacy-r23--q4a/

Previous examination · complete worked answers

**Question. Explain Jaccard, cosine and Euclidean measures. Calculate them without preprocessing for D₁=“Data is the new oil of the digital economy” and D₂=“Data is a new oil”.** [10 marks]

### Answer

### Given

Use exactly the given case and words; do not remove stopwords.

### To find

Three measures in explicitly stated representations.

### Formula: which one and why



$$
J(A,B)=\frac{|A\cap B|}{|A\cup B|},\quad d_E(u,v)=\sqrt{\sum_i(u_i-v_i)^2}
$$



Jaccard uses sets of distinct tokens. Cosine and Euclidean use raw-count BoW vectors in one shared order; do not secretly turn repeated the into a single count.

### Step 1. Build the common vocabulary and counts

| Document | Data | is | the | new | oil | of | digital | economy | a |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| D1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 0 |
| D2 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |

### Step 2. Jaccard sets

Intersection={Data,is,new,oil}, size4; union size9. J=4/9=0.444444; Jaccard distance=1−J=5/9.

### Step 3. Cosine

Dot product=4. D1 squared length=1+1+4+1+1+1+1+1=11; D2 squared length=5. Cosine=4/√55=0.539359890.

### Step 4. Euclidean

Coordinate differences: [0, 0, 2, 0, 0, 1, 1, 1, -1]. Sum of squares=8. Distance=√8=2.828427125. Smaller distance means greater proximity; this is a distance, not a probability.

### Final answer

**Jaccard=0.444444; cosine=0.539360; Euclidean distance=2.828427.**