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It is finding material (usually documents) of an unstructured nature that satisfies an information need within large collections (usually stored on computers). +\section{Information Retrieval Models} +\subsection{Introduction to Information Retrieval Models} +\textbf{Data collections} are well-structured collections of related items; items are usually atomic with a +well-defined interpretation. +Data retrieval involves the selection of a fixed set of data based on a well-defined query (e.g., SQL, OQL). +\\\\ +\textbf{Information collections} are usually semi-structured or unstructured. +Information Retrieval (IR) involves the retrieval of documents of natural language which is typically not +structured and may be semantically ambiguous. + +\subsubsection{Information Retrieval vs Information Filtering} +The main differences between information retrieval \& information filtering are: +\begin{itemize} + \item The nature of the information need. + \item The nature of the document set. +\end{itemize} + +Other than these two differences, the same models are used. +Documents \& queries are represented using the same set of techniques and similar comparison algorithms are also +used. + +\subsubsection{User Role} +In traditional IR, the user role was reasonably well-defined in that a user: +\begin{itemize} + \item Formulated a query. + \item Viewed the results. + \item Potentially offered feedback. + \item Potentially reformulated their query and repeated steps. +\end{itemize} + +In more recent systems, with the increasing popularity of the hypertext paradigm, users usually intersperse +browsing with the traditional querying. +This raises many new difficulties \& challenges. + +\subsection{Pre-Processing} +\textbf{Document pre-processing} is the application of a set of well-known techniques to the documents \& queries +prior to any comparison. +This includes, among others: +\begin{itemize} + \item \textbf{Stemming:} the reduction of words to a potentially common root. + The most common stemming algorithms are Lovin's \& Porter's algorithms. + E.g. \textit{computerisation}, + \textit{computing}, \textit{computers} could all be stemmed to the common form \textit{comput}. + \item \textbf{Stop-word removal:} the removal of very frequent terms from documents, which add little to the + semantics of meaning of the document. + \item \textbf{Thesaurus construction:} the manual or automatic creation of thesauri used to try to identify + synonyms within the documents. +\end{itemize} + +\textbf{Representation} \& comparison technique depends on the information retrieval model chosen. +The choice of feedback techniques is also dependent on the model chosen. + +\subsection{Models} +Retrieval models can be broadly categorised as: +\begin{itemize} + \item Boolean: + \begin{itemize} + \item Classical Boolean. + \item Fuzzy Set approach. + \item Extended Boolean. + \end{itemize} + + \item Vector: + \begin{itemize} + \item Vector Space approach. + \item Latent Semantic indexing. + \item Neural Networks. + \end{itemize} + + \item Probabilistic: + \begin{itemize} + \item Inference Network. + \item Belief Network. + \end{itemize} +\end{itemize} + +We can view any IR model as being comprised of: +\begin{itemize} + \item $D$ is the set of logical representations within the documents. + \item $Q$ is the set of logical representations of the user information needs (queries). + \item $F$ is a framework for modelling representations ($D$ \& $Q$) and the relationship between $D$ \& $Q$. + \item $R$ is a ranking function which defines an ordering among the documents with regard to any query $q$. +\end{itemize} + +We have a set of index terms: +$$ +t_1, \dots , t_n +$$ + +A \textbf{weight} $w_{i,j}$ is assigned to each term $t_i$ occurring in the $d_j$. +We can view a document or query as a vector of weights: +$$ +\vec{d_j} = (w_1, w_2, w_3, \dots) +$$ + +\subsection{Boolean Model} +The \textbf{Boolean model} of information retrieval is based on set theory \& Boolean algebra. +A query is viewed as a Boolean expression. +The model also assumes terms are present or absent, hence term weights $w_{i,j}$ are binary \& discrete, i.e., +$w_{i,j}$ is an element of $\{0, 1\}$. +\\\\ +Advantages of the Boolean model include: +\begin{itemize} + \item Clean formalism. + \item Widespread \& popular. + \item Relatively simple +\end{itemize} + +Disadvantages of the Boolean model include: +\begin{itemize} + \item People often have difficulty formulating expressions, harbours some difficulty in use. + \item Documents are considered either relevant or irrelevant; no partial matching allowed. + \item Poor performance. + \item Suffers badly from natural language effects of synonymy etc. + \item No ranking of results. + \item Terms in a document are considered independent of each other. +\end{itemize} + +\subsubsection{Example} +$$ +q = t_1 \land (t_2 \lor (\neg t_3)) +$$ + +\begin{minted}[linenos, breaklines, frame=single]{sql} +q = t1 AND (t2 OR (NOT t3)) +\end{minted} + +This can be mapped to what is termed \textbf{disjunctive normal form}, where we have a series of disjunctions +(or logical ORs) of conjunctions. + +$$ +q = 100 \lor 110 \lor 111 +$$ + +If a document satisfies any of the components, the document is deemed relevant and returned. + +\subsection{Vector Space Model} +The \textbf{vector space model} attempts to improve upon the Boolean model by removing the limitation of binary +weights for index terms. +Terms can have non-binary weights in both queries \& documents. +Hence, we can represent the documents \& the query as $n$-dimensional vectors. + +$$ +\vec{d_j} = (w_{1,j}, w_{2,j}, \dots, w_{n,j}) +$$ +$$ +\vec{q} = (w_{1,q}, w_{2,q}, \dots, w_{n,q}) +$$ + +We can calculate the similarity between a document \& a query by calculating the similarity between the vector +representations of the document \& query by measuring the cosine of the angle between the two vectors. +$$ +\vec{a} \cdot \vec{b} = \mid \vec{a} \mid \mid \vec{b} \mid \cos (\vec{a}, \vec{b}) +$$ +$$ +\Rightarrow \cos (\vec{a}, \vec{b}) = \frac{\vec{a} \cdot \vec{b}}{\mid \vec{a} \mid \mid \vec{b} \mid} +$$ + +We can therefore calculate the similarity between a document and a query as: +$$ +\text{sim}(q,d) = \cos (\vec{q}, \vec{d}) = \frac{\vec{q} \cdot \vec{d}}{\mid \vec{q} \mid \mid \vec{d} \mid} +$$ + +Considering term weights on the query and documents, we can calculate similarity between the document \& query as: +$$ +\text{sim}(q,d) = +\frac +{\sum^N_{i=1} (w_{i,q} \times w_{i,d})} +{\sqrt{\sum^N_{i=1} (w_{i,q})^2} \times \sqrt{\sum^N_{i=1} (w_{i,d})^2} } +$$ + +Advantages of the vector space model over the Boolean model include: +\begin{itemize} + \item Improved performance due to weighting schemes. + \item Partial matching is allowed which gives a natural ranking. +\end{itemize} + +The primary disadvantage of the vector space model is that terms are considered to be mutually independent. + +\subsubsection{Weighting Schemes} +We need a means to calculate the term weights in the document and query vector representations. +A term's frequency within a document quantifies how well a term describes a document; +the more frequently a term occurs in a document, the better it is at describing that document and vice-versa. +This frequency is known as the \textbf{term frequency} or \textbf{tf factor}. +\\\\ +If a term occurs frequently across all the documents, that term does little to distinguish one document from another. +This factor is known as the \textbf{inverse document frequency} or \textbf{idf-frequency}. +Traditionally, the most commonly-used weighting schemes are know as \textbf{tf-idf} weighting schemes. +\\\\ +For all terms in a document, the weight assigned can be calculated as: +$$ +w_{i,j} = f_{i,j} \times \log \left( \frac{N}{N_i} \right) +$$ +where +\begin{itemize} + \item $f_{i,j}$ is the (possibly normalised) frequency of term $t_i$ in document $d_j$. + \item $N$ is the number of documents in the collection. + \item $N_i$ is the number of documents that contain term $t_i$. +\end{itemize} +