Natural Language Processing · M2 · formulas

Unigrams, bigrams and trigrams: construct them correctly

N-grams & probability

Formulas explained through full worked examples

Definition. An n-gram is a contiguous sequence of n items. For word n-grams, the items are consecutive word tokens. A unigram contains one word; a bigram contains two; a trigram contains three. Character n-grams use characters instead, so always identify the unit being counted.

Tokens: we | study | natural | language

Move a window one token at a time

Bigrams: we study / study natural / natural language

For m tokens without padding, the number of n-gram positions is max(0, m−n+1). Repeated positions may contain the same n-gram type. A bag-of-n-grams representation counts each n-gram type and loses the overall sentence order beyond the local order inside each n-gram.

\text{Number of unpadded n-gram positions}=\max(0,m-n+1)

Question. List all word unigrams, bigrams and trigrams for “the prime minister of our country”. Do not add boundary tokens.

Given

Six tokens: the, prime, minister, of, our, country. n = 1, 2 and 3.

To find

All three lists and their counts.

Formula and why it applies

\#\text{positions}=m-n+1

Each window must contain consecutive words and move one position. No wraparound or invented final word is allowed.

Solution

Step 1. Take one word at a time

Unigrams: (the), (prime), (minister), (of), (our), (country). Count = 6−1+1 = 6.

Step 2. Take two consecutive words

Bigrams: (the, prime), (prime, minister), (minister, of), (of, our), (our, country). Count = 6−2+1 = 5.

Step 3. Take three consecutive words

Trigrams: (the, prime, minister), (prime, minister, of), (minister, of, our), (of, our, country). Count = 6−3+1 = 4.

Final answer. 6 unigrams, 5 bigrams and 4 trigrams.

Question. Construct bigrams for “I like tea” using one start and one end marker.

Given

Original tokens: I, like, tea. Add <s> and </s> once each.

To find

The boundary-aware bigram sequence.

Formula and why it applies

\#\text{bigrams}=(m+2)-1=m+1

The start context predicts the first word, and the last word predicts that the sentence ends.

Solution

Step 1. Write the padded sequence

Step 2. Move a two-token window

(<s>, I), (I, like), (like, tea), (tea, </s>).

Step 3. Count the windows

3+1=4

Final answer. There are four boundary-aware bigrams.

Question. Construct trigrams for “I like tea” using two start markers and one end marker.

Given

Original sentence has three words. For a trigram model each predicted token needs two prior context tokens.

To find

All trigram prediction events.

Formula and why it applies

P(w_i\mid w_{i-2},w_{i-1})

Two start markers provide a complete context even for the first word.

Solution

Step 1. Pad with two starts

Step 2. Slide a three-token window

(<s>, <s>, I), (<s>, I, like), (I, like, tea), (like, tea, </s>).

Step 3. Count predicted events

The model predicts I, like, tea and the end marker: four predictions.

Final answer. Four trigrams. Start markers are context; they are not ordinary predicted words.

Question. Give the bag-of-bigrams counts for “a b a b” without padding.

Given

Four tokens in their given order.

To find

Bigram positions, distinct bigram types and their counts.

Formula and why it applies

\#\text{positions}=4-2+1=3

A bag stores frequencies; repeated bigrams must not be discarded.

Solution

Step 1. Enumerate all positions

(a,b), (b,a), (a,b).

Step 2. Group identical pairs

Bigram type	Count
a b	2
b a	1

Final answer. Three bigram positions but two distinct bigram types.

Yaad rakho: n-gram mein n words; context mein n−1 words.