JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech IV Year INFORMATION RETRIEVAL SYSTEMS Previous Question Paper (March - 2021 - R15)

The Vector Space Model (VSM) is an algebraic model used to represent documents and queries as vectors in a multi-dimensional space, where each dimension corresponds to a unique term in the collection vocabulary. By mapping text into numerical vectors, the model enables the use of geometric distance or angle-based metrics to quantify the relevance between a query and a document.

The process involves the following steps:

1. Document Representation: Each document is represented as a vector D = (w₁, w₂, ..., w_n), where w_i is the weight of term i. Weights are typically calculated using TF-IDF (Term Frequency-Inverse Document Frequency), which balances the frequency of a term within a document against its rarity across the entire corpus.
2. Query Representation: The user query is transformed into a vector Q of the same dimensionality as the document vectors, using the same indexing vocabulary.
3. Similarity Measurement: Relevance is determined by calculating the similarity between the query vector and document vectors. The most common metric is Cosine Similarity, which measures the cosine of the angle between two vectors.

In this formula, the numerator is the dot product of the query and document vectors, while the denominator is the product of their Euclidean norms. A cosine value of 1 indicates the vectors are identical in orientation, while 0 indicates they are orthogonal (no shared terms). The system then ranks documents in descending order of their similarity scores to the query vector.

The primary advantages of VSM include its ability to support partial matching, the ranking of results based on degree of relevance rather than binary retrieval, and the flexibility to incorporate term weighting schemes. However, it assumes term independence, which ignores semantic relationships like synonymy or polysemy, and can suffer from high dimensionality when dealing with large document collections.

In the Simple Term Weights method, the similarity between a query Q and a document D is typically calculated using the Inner Product (or dot product) of their respective term weight vectors. If w(i,q) is the weight of term i in the query and w(i,d) is the weight of term i in the document, the similarity is:

To calculate relevance ranking for the query Q = {gold, silver, truck}, we define four weighting schemes:

1. Binary Weighting: Weights are 1 if the term exists, 0 otherwise.
2. Term Frequency (TF): Weights equal the number of occurrences of the term in the document.
3. Collection Frequency (CF): Weights are inversely proportional to the number of documents containing the term.
4. TF-IDF: Combines TF with Inverse Document Frequency (IDF) to penalize common terms.

Using Binary Weighting (assuming Q={1,1,1}):

• Sim(Q, D1) = (1×1) + (1×0) + (1×1) = 2
• Sim(Q, D2) = (1×0) + (1×1) + (1×1) = 2
• Sim(Q, D3) = (1×1) + (1×0) + (1×1) = 2

Using TF Weighting (counting occurrences):

• Sim(Q, D1) = (1×1) + (1×0) + (1×1) = 2
• Sim(Q, D2) = (1×0) + (1×2) + (1×1) = 3
• Sim(Q, D3) = (1×1) + (1×0) + (1×1) = 2

Under CF Weighting, 'truck' is common to all, so its weight is low, while 'gold' or 'silver' might be higher. Under TF-IDF, D2 is ranked highest because it contains 'silver' twice, and 'silver' is a rare term compared to 'truck', providing higher discriminatory power for relevance.