Q Consider the following documents.
D1: I went to the park to play D2: park is nearby to play
D3: going to the park is fun Q: park nearby play
Apply the vector model to rank the above documents.
Ans:
Step 1: Represent Documents and Query by the index terms
• D1: I, went, to, the, park, to, play
• D2: park, is, nearby, to, play
• D3: going, to, the, park, is, fun
• Q: park, nearby, play
Step 2: Create the Vocabulary
Combine unique terms from all documents and the query.
Vocabulary:
{ I, went, to, the, park, play, is, nearby, going, fun }
Step 3: Construct Term Frequency (TF) Matrix
Count the frequency of each term from the vocabulary in each document and query.
Term D1 (fij) D2 (fij) D3 (fij) Q (fij)
i 1 0 0 0
went 1 0 0 0
to 2 1 1 0
the 1 0 1 0
park 1 1 1 1
play 1 1 0 1
is 0 1 1 0
nearby 0 1 0 1
going 0 0 1 0
fun 0 0 1 0
Step 4: Compute Inverse Document Frequency (IDF)
IDF is calculated as:
• N: Total number of documents (N=3).
• n: Number of documents containing the term.
Term ni (Documents containing term) IDF = log(3/ni)
i 1 log(3/1) = log(3) ≈ 0.477
went 1 log(3/1) = log(3) ≈ 0.477
to 3 log(3/3) = log(1) = 0.000
the 2 log(3/2) = log(1.5) ≈ 0.176
park 3 log(3/3) = log(1) = 0.000
play 2 log(3/2) = log(1.5) ≈ 0.176
is 2 log(3/2) = log(1.5) ≈ 0.176
nearby 1 log(3/1) = log(3) ≈ 0.477
going 1 log(3/1) = log(3) ≈ 0.477
fun 1 log(3/1) = log(3) ≈ 0.477
Step 5: Calculate TF-IDF Weight (wij):
Term D1 (wij) D2 (wij) D3 (wij) Q (wij)
(1 + log 1) * 0.477 =
i 1 * 0.477 = 0.477 0 0 0
(1 + log 1) * 0.477 =
went 1 * 0.477 = 0.477 0 0 0
(1 + log 2) * 0.000 = (1 + log 1) * 0.000 = (1 + log 1) * 0.000 =
to 1.301 * 0 = 0 1*0=0 1*0=0 0
(1 + log 1) * 0.176 = (1 + log 1) * 0.176 =
the 1 * 0.176 = 0.176 0 1 * 0.176 = 0.176 0
(1 + log 1) * 0.000 = (1 + log 1) * 0.000 = (1 + log 1) * 0.000 = (1 + log 1) * 0.000 =
park 1*0=0 1*0=0 1*0=0 1*0=0
(1 + log 1) * 0.176 = (1 + log 1) * 0.176 = (1 + log 1) * 0.176 =
play 1 * 0.176 = 0.176 1 * 0.176 = 0.176 0 1 * 0.176 = 0.176
(1 + log 1) * 0.176 = (1 + log 1) * 0.176 =
is 0 1 * 0.176 = 0.176 1 * 0.176 = 0.176 0
(1 + log 1) * 0.477 = (1 + log 1) * 0.477 =
nearby 0 1 * 0.477 = 0.477 0 1 * 0.477 = 0.477
(1 + log 1) * 0.477 =
going 0 0 1 * 0.477 = 0.477 0
(1 + log 1) * 0.477 =
fun 0 0 1 * 0.477 = 0.477 0
Step 6: Represent Documents and Query as Vectors
Using the TF-IDF weights, we form vectors for each document and the query, in the order of
our vocabulary: {i, went, to, the, park, play, is, nearby, going, fun}.
• D1 Vector: [0.477, 0.477, 0, 0.176, 0, 0.176, 0, 0, 0, 0]
• D2 Vector: [0, 0, 0, 0, 0, 0.176, 0.176, 0.477, 0, 0]
• D3 Vector: [0, 0, 0, 0.176, 0, 0, 0.176, 0, 0.477, 0.477]
• Q Vector: [0, 0, 0, 0, 0, 0.176, 0, 0.477, 0, 0]
Step 7: Calculate Cosine Similarity:
Cosine similarity is given by:
Calculate Dot Products (Q. D):
• Q. D1 = (0 * 0.477) + (0 * 0.477) + (0 * 0) + (0 * 0.176) + (0 * 0) + (0.176 * 0.176) +
(0 * 0) + (0.477 * 0) + (0 * 0) + (0 * 0) = 0.030976
• Q. D2 = (0 * 0) + (0 * 0) + (0 * 0) + (0 * 0) + (0 * 0) + (0.176 * 0.176) + (0 * 0.176) +
(0.477 * 0.477) + (0 * 0) + (0 * 0) = 0.030976 + 0.227529 = 0.258505
• Q. D3 = (0 * 0) + (0 * 0) + (0 * 0) + (0 * 0.176) + (0 * 0) + (0.176 * 0) + (0 * 0.176) +
(0.477 * 0) + (0 * 0.477) + (0 * 0.477) = 0
Calculate Magnitudes:
• ||D1|| = sqrt(0.477^2 + 0.477^2 + 0.176^2 + 0.176^2) = sqrt(0.227529 + 0.227529 +
0.030976 + 0.030976) = sqrt(0.51701) ≈ 0.7190
• ||D2|| = sqrt(0.176^2 + 0.176^2 + 0.477^2) = sqrt(0.030976 + 0.030976 + 0.227529) =
sqrt(0.289481) ≈ 0.5380
• ||D3|| = sqrt(0.176^2 + 0.176^2 + 0.477^2 + 0.477^2) = sqrt(0.030976 + 0.030976 +
0.227529 + 0.227529) = sqrt(0.51701) ≈ 0.7190
• ||Q|| = sqrt(0.176^2 + 0.477^2) = sqrt(0.030976 + 0.227529) = sqrt(0.258505) ≈ 0.5084
Calculate Cosine Similarities:
• Sim(Q, D1) = 0.030976 / (0.5084 * 0.7190) = 0.030976 / 0.3655316 ≈ 0.0847
• Sim(Q, D2) = 0.258505 / (0.5084 * 0.5380) = 0.258505 / 0.2735272 ≈ 0.9451
• Sim(Q, D3) = 0 / (0.5084 * 0.7190) = 0
Step 8: Rank Documents
Based on the cosine similarity scores (higher score means more similar):
1. D2 (Cosine Similarity ≈ 0.9451)
2. D1 (Cosine Similarity ≈ 0.0847)
3. D3 (Cosine Similarity = 0)
