[CT4100]: Add Week 2 lecture notes
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\pagestyle{fancy}
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\usepackage{microtype} % Slightly tweak font spacing for aesthetics
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\usepackage{amsmath}
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\usepackage[english]{babel} % Language hyphenation and typographical rules
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\usepackage{xcolor}
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\definecolor{linkblue}{RGB}{0, 64, 128}
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@ -140,6 +141,206 @@ web search engines, digital libraries, \& recommender systems.
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It is finding material (usually documents) of an unstructured nature that satisfies an information need within large
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collections (usually stored on computers).
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\section{Information Retrieval Models}
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\subsection{Introduction to Information Retrieval Models}
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\textbf{Data collections} are well-structured collections of related items; items are usually atomic with a
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well-defined interpretation.
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Data retrieval involves the selection of a fixed set of data based on a well-defined query (e.g., SQL, OQL).
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\\\\
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\textbf{Information collections} are usually semi-structured or unstructured.
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Information Retrieval (IR) involves the retrieval of documents of natural language which is typically not
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structured and may be semantically ambiguous.
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\subsubsection{Information Retrieval vs Information Filtering}
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The main differences between information retrieval \& information filtering are:
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\begin{itemize}
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\item The nature of the information need.
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\item The nature of the document set.
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\end{itemize}
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Other than these two differences, the same models are used.
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Documents \& queries are represented using the same set of techniques and similar comparison algorithms are also
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used.
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\subsubsection{User Role}
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In traditional IR, the user role was reasonably well-defined in that a user:
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\begin{itemize}
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\item Formulated a query.
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\item Viewed the results.
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\item Potentially offered feedback.
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\item Potentially reformulated their query and repeated steps.
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\end{itemize}
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In more recent systems, with the increasing popularity of the hypertext paradigm, users usually intersperse
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browsing with the traditional querying.
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This raises many new difficulties \& challenges.
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\subsection{Pre-Processing}
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\textbf{Document pre-processing} is the application of a set of well-known techniques to the documents \& queries
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prior to any comparison.
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This includes, among others:
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\begin{itemize}
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\item \textbf{Stemming:} the reduction of words to a potentially common root.
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The most common stemming algorithms are Lovin's \& Porter's algorithms.
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E.g. \textit{computerisation},
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\textit{computing}, \textit{computers} could all be stemmed to the common form \textit{comput}.
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\item \textbf{Stop-word removal:} the removal of very frequent terms from documents, which add little to the
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semantics of meaning of the document.
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\item \textbf{Thesaurus construction:} the manual or automatic creation of thesauri used to try to identify
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synonyms within the documents.
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\end{itemize}
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\textbf{Representation} \& comparison technique depends on the information retrieval model chosen.
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The choice of feedback techniques is also dependent on the model chosen.
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\subsection{Models}
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Retrieval models can be broadly categorised as:
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\begin{itemize}
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\item Boolean:
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\begin{itemize}
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\item Classical Boolean.
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\item Fuzzy Set approach.
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\item Extended Boolean.
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\end{itemize}
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\item Vector:
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\begin{itemize}
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\item Vector Space approach.
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\item Latent Semantic indexing.
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\item Neural Networks.
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\end{itemize}
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\item Probabilistic:
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\begin{itemize}
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\item Inference Network.
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\item Belief Network.
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\end{itemize}
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\end{itemize}
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We can view any IR model as being comprised of:
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\begin{itemize}
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\item $D$ is the set of logical representations within the documents.
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\item $Q$ is the set of logical representations of the user information needs (queries).
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\item $F$ is a framework for modelling representations ($D$ \& $Q$) and the relationship between $D$ \& $Q$.
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\item $R$ is a ranking function which defines an ordering among the documents with regard to any query $q$.
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\end{itemize}
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We have a set of index terms:
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$$
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t_1, \dots , t_n
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$$
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A \textbf{weight} $w_{i,j}$ is assigned to each term $t_i$ occurring in the $d_j$.
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We can view a document or query as a vector of weights:
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$$
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\vec{d_j} = (w_1, w_2, w_3, \dots)
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$$
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\subsection{Boolean Model}
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The \textbf{Boolean model} of information retrieval is based on set theory \& Boolean algebra.
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A query is viewed as a Boolean expression.
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The model also assumes terms are present or absent, hence term weights $w_{i,j}$ are binary \& discrete, i.e.,
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$w_{i,j}$ is an element of $\{0, 1\}$.
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\\\\
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Advantages of the Boolean model include:
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\begin{itemize}
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\item Clean formalism.
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\item Widespread \& popular.
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\item Relatively simple
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\end{itemize}
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Disadvantages of the Boolean model include:
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\begin{itemize}
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\item People often have difficulty formulating expressions, harbours some difficulty in use.
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\item Documents are considered either relevant or irrelevant; no partial matching allowed.
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\item Poor performance.
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\item Suffers badly from natural language effects of synonymy etc.
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\item No ranking of results.
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\item Terms in a document are considered independent of each other.
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\end{itemize}
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\subsubsection{Example}
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$$
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q = t_1 \land (t_2 \lor (\neg t_3))
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$$
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\begin{minted}[linenos, breaklines, frame=single]{sql}
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q = t1 AND (t2 OR (NOT t3))
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\end{minted}
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This can be mapped to what is termed \textbf{disjunctive normal form}, where we have a series of disjunctions
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(or logical ORs) of conjunctions.
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$$
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q = 100 \lor 110 \lor 111
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$$
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If a document satisfies any of the components, the document is deemed relevant and returned.
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\subsection{Vector Space Model}
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The \textbf{vector space model} attempts to improve upon the Boolean model by removing the limitation of binary
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weights for index terms.
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Terms can have non-binary weights in both queries \& documents.
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Hence, we can represent the documents \& the query as $n$-dimensional vectors.
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$$
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\vec{d_j} = (w_{1,j}, w_{2,j}, \dots, w_{n,j})
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$$
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$$
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\vec{q} = (w_{1,q}, w_{2,q}, \dots, w_{n,q})
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$$
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We can calculate the similarity between a document \& a query by calculating the similarity between the vector
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representations of the document \& query by measuring the cosine of the angle between the two vectors.
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$$
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\vec{a} \cdot \vec{b} = \mid \vec{a} \mid \mid \vec{b} \mid \cos (\vec{a}, \vec{b})
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$$
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$$
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\Rightarrow \cos (\vec{a}, \vec{b}) = \frac{\vec{a} \cdot \vec{b}}{\mid \vec{a} \mid \mid \vec{b} \mid}
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$$
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We can therefore calculate the similarity between a document and a query as:
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$$
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\text{sim}(q,d) = \cos (\vec{q}, \vec{d}) = \frac{\vec{q} \cdot \vec{d}}{\mid \vec{q} \mid \mid \vec{d} \mid}
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$$
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Considering term weights on the query and documents, we can calculate similarity between the document \& query as:
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$$
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\text{sim}(q,d) =
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\frac
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{\sum^N_{i=1} (w_{i,q} \times w_{i,d})}
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{\sqrt{\sum^N_{i=1} (w_{i,q})^2} \times \sqrt{\sum^N_{i=1} (w_{i,d})^2} }
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$$
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Advantages of the vector space model over the Boolean model include:
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\begin{itemize}
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\item Improved performance due to weighting schemes.
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\item Partial matching is allowed which gives a natural ranking.
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\end{itemize}
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The primary disadvantage of the vector space model is that terms are considered to be mutually independent.
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\subsubsection{Weighting Schemes}
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We need a means to calculate the term weights in the document and query vector representations.
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A term's frequency within a document quantifies how well a term describes a document;
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the more frequently a term occurs in a document, the better it is at describing that document and vice-versa.
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This frequency is known as the \textbf{term frequency} or \textbf{tf factor}.
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\\\\
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If a term occurs frequently across all the documents, that term does little to distinguish one document from another.
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This factor is known as the \textbf{inverse document frequency} or \textbf{idf-frequency}.
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Traditionally, the most commonly-used weighting schemes are know as \textbf{tf-idf} weighting schemes.
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\\\\
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For all terms in a document, the weight assigned can be calculated as:
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$$
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w_{i,j} = f_{i,j} \times \log \left( \frac{N}{N_i} \right)
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$$
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where
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\begin{itemize}
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\item $f_{i,j}$ is the (possibly normalised) frequency of term $t_i$ in document $d_j$.
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\item $N$ is the number of documents in the collection.
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\item $N_i$ is the number of documents that contain term $t_i$.
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\end{itemize}
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