Extended Boolean model

In the Extended Boolean model, a document is represented as a vector (similarly to in the vector model). Each i dimension corresponds to a separate term associated with the document.

The weight of term K_x associated with document d_j is measured by its normalized Term frequency and can be defined as:

${\displaystyle w_{x,j}=f_{x,j}*{\frac {Idf_{x}}{max_{i}Idf_{i}}}}$

where Idf_x is inverse document frequency and f_x,j the term frequency for term x in document j.

The weight vector associated with document d_j can be represented as:

${\displaystyle \mathbf {v} _{d_{j}}=[w_{1,j},w_{2,j},\ldots ,w_{i,j}]}$

Figure 1: The similarities of q = (K_x ∨ K_y) with documents d_j and d_j+1.

Figure 2: The similarities of q = (K_x ∧ K_y) with documents d_j and d_j+1.

Considering the space composed of two terms K_x and K_y only, the corresponding term weights are w₁ and w₂.[2] Thus, for query q_or = (K_x ∨ K_y), we can calculate the similarity with the following formula:

${\displaystyle sim(q_{or},d)={\sqrt {\frac {w_{1}^{2}+w_{2}^{2}}{2}}}}$

For query q_and = (K_x ∧ K_y), we can use:

${\displaystyle sim(q_{and},d)=1-{\sqrt {\frac {(1-w_{1})^{2}+(1-w_{2})^{2}}{2}}}}$

We can generalize the previous 2D extended Boolean model example to higher t-dimensional space using Euclidean distances.

This can be done using P-norms which extends the notion of distance to include p-distances, where 1 ≤ p ≤ ∞ is a new parameter.[3]

• A generalized conjunctive query is given by:

${\displaystyle q_{or}=k_{1}\lor ^{p}k_{2}\lor ^{p}....\lor ^{p}k_{t}}$

• The similarity of ${\displaystyle q_{or}}$ and ${\displaystyle d_{j}}$ can be defined as:

:${\displaystyle sim(q_{or},d_{j})={\sqrt[{p}]{\frac {w_{1}^{p}+w_{2}^{p}+....+w_{t}^{p}}{t}}}}$

• A generalized disjunctive query is given by:

${\displaystyle q_{and}=k_{1}\land ^{p}k_{2}\land ^{p}....\land ^{p}k_{t}}$

• The similarity of ${\displaystyle q_{and}}$ and ${\displaystyle d_{j}}$ can be defined as:

${\displaystyle sim(q_{and},d_{j})=1-{\sqrt[{p}]{\frac {(1-w_{1})^{p}+(1-w_{2})^{p}+....+(1-w_{t})^{p}}{t}}}}$

Consider the query q = (K₁ ∧ K₂) ∨ K₃. The similarity between query q and document d can be computed using the formula:

${\displaystyle sim(q,d)={\sqrt[{p}]{\frac {(1-{\sqrt[{p}]{({\frac {(1-w_{1})^{p}+(1-w_{2})^{p}}{2}}}}))^{p}+w_{3}^{p}}{2}}}}$

Lee and Fox[4] compared the Standard and Extended Boolean models with three test collections, CISI, CACM and INSPEC. Using P-norms they obtained an average precision improvement of 79%, 106% and 210% over the Standard model, for the CISI, CACM and INSPEC collections, respectively.
The P-norm model is computationally expensive because of the number of exponentiation operations that it requires but it achieves much better results than the Standard model and even Fuzzy retrieval techniques. The Standard Boolean model is still the most efficient.

• Adaptive Feedback Methods in an Extended Boolean Model by Dr.Jongpill Choi
• Interpolation of the extended Boolean retrieval model
• Fox, E.; Betrabet, S.; Koushik, M.; Lee, W. (1992), Information Retrieval: Algorithms and Data structures; Extended Boolean model, Prentice-Hall, Inc.
• Skorkovská, Lucie; Ircing, Pavel (2009), "Experiments with Automatic Query Formulation in the Extended Boolean Model", Text, Speech and Dialogue, Lecture Notes in Computer Science, 5729, Springer Berlin / Heidelberg, pp. 371–378, doi:10.1007/978-3-642-04208-9_51, hdl:11025/16985, ISBN 978-3-642-04207-2

• Information retrieval