# Q7b · Co-occurrence, dot products and plot

Subject: Natural Language Processing

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/nlp/#tee/papers/legacy-f24/q7b

Canonical: https://deep-learning-study-room.vercel.app/read/nlp/tee/papers/legacy-f24--q7b/

Previous examination · complete worked answers

**Question. Define target words, context window, context words and co-occurrence matrix. For the printed boy/ball/dog/sun corpus below, use one whole sentence as the window; contexts={kicked,field,tree,ran}. Find boy·dog and boy·sun, plot on kicked/field dimensions and comment.** [10 marks]

### Answer

**Target** is the word whose row we represent. **Context** is an observed neighbouring word whose column we count. A **window** defines which tokens count as neighbours; here all other tokens in the same sentence. A **co-occurrence matrix** records those counts.

### Given

The boy kicked the ball into the field. The dog barked at the boy. The sun was shining over the field. The boy ran after the dog. The ball rolled to the tree.

### To find

Matrix, two dot products and the requested 2D coordinates.

### Formula: which one and why



$$
u\cdot v=\sum_i u_iv_i
$$



Use lowercase token-pair counts within each sentence; do not cross sentence boundaries.

### Step 1. List contributions and build the matrix

| Sentence | Target occurrences | Context occurrences |
| --- | --- | --- |
| 1 | boy, ball | kicked, field |
| 2 | dog, boy |  |
| 3 | sun | field |
| 4 | boy, dog | ran |
| 5 | ball | tree |

| Target / context | kicked | field | tree | ran |
| --- | --- | --- | --- | --- |
| boy | 1 | 1 | 0 | 1 |
| ball | 1 | 1 | 1 | 0 |
| dog | 0 | 0 | 0 | 1 |
| sun | 0 | 1 | 0 | 0 |

### Step 2. Dot boy and dog

[1, 1, 0, 1] · [0, 0, 0, 1] = 1×0 + 1×0 + 0×0 + 1×1 = 1

### Step 3. Dot boy and sun

[1, 1, 0, 1] · [0, 1, 0, 0] = 1×0 + 1×1 + 0×0 + 1×0 = 1

### Step 4. Plot coordinates: kicked on x, field on y

| Target | (x,y) |
| --- | --- |
| boy | (1, 1) |
| ball | (1, 1) |
| dog | (0, 0) |
| sun | (0, 1) |

kickedfielddog (0,0)sun (0,1)boy, ball (1,1)

Two coordinates omit tree and ran, so boy and ball overlap here despite different full vectors.

### Final answer

**boy·dog=1; boy·sun=1.** Both share one counted context contribution. Dot-product similarity reflects this corpus and magnitude, not a claim that boy is equally semantically related to dog and sun in general.