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25_Outline/sections/20_foundations.tex
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\section{Definitions}
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Set of \emph{variables}:
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\begin{equation}
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V = \{v_1, \dotsc, v_m\}
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\end{equation}
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A corresponding \emph{domain mapping function} that maps a variable to its possible values:
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\begin{equation}
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\mathfrak{D} : V \to D; x \mapsto \mathfrak{D}(x) \qquad where \ D = \{d_1, \dotsc, d_o\}
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\end{equation}
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Set of \emph{users}:
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\begin{equation}
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U = \{1, \dotsc, n\}
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\end{equation}
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A users \emph{utility function} for a domain value of a variable. This is the utility that a user has from a certain selected domain value. It is a function that only the user himself knows:
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\begin{equation}
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\begin{split}
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u_i(d_j), \qquad \text{where}\ & d_j \in \mathfrak{D}(j),\\
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& 1 <= j <= m, \\
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& 1 <= i <= n
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\end{split}
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\end{equation}
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\emph{User preferences} that are entered into the system by the user according to his utility function:
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\begin{gather}
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P = \{ P_1, \dotsc, P_n\},\ \text{where} \\
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P_i = \{(d,\ u_i(d)) \ | \ \forall d \in \mathfrak{D}(i),\ i=1,\dotsc,m \} \notag
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\end{gather}
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A \emph{configuration} has a state it is defined by a tuple of variables and their corresponding domain value. Essentially it is a set of variables and assigned values:
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\begin{equation}
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S = \{ (v_i,\ d) \ |\ v_i \in V \ \land \ d \in \mathfrak{D}(i),\ i=1,\dotsc,m \}
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\end{equation}
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A \emph{finished configuration} is a configuration that contains all variables:
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\begin{equation}
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\begin{split}
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S_F \subset S,\ where \ & \forall v_i \in V (\exists (v_i, d) \in S_F : d \in \mathfrak{D}(i)) \\
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& \land is\_valid(S_F)
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\end{split}
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\end{equation}
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% TODO: define valid configuration state
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\emph{Group configuration scoring function} using preferences and current configuration state. This function gives a score for a finished configuration (while using the current configuration state and all user preferences):
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\begin{equation}
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score_{group}: S \times P \times S_F \to \mathbb{R}
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\end{equation}
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An example group configuration scoring function is $score_{group}$ with
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\begin{equation}
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\notag \alpha \in \mathbb{R}, \qquad changed(d,\overline{s}, s) =
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\begin{cases}
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1, & d \in \overline{s} \land d \notin s \\
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0, & \text{otherwise}
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\end{cases}
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\end{equation}
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\begin{equation}
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\begin{split}
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score_{group}(\overline{s},\ \overline{p},\ s)
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& = score(\overline{p},\ s) - penalty(\overline{s},\ s) \\
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& = score(\overline{p},\ s) - \sum_{d \in \overline{s}} changed(d,\overline{s}, s) \cdot \alpha
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\end{split}
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\end{equation}
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\begin{mdframed}[frametitle={Forest Example}]
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In this example we have two users. The use case is a piece of forest and variables are for example harvesting activity, which trees to grow and accessibility for people.
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\begin{align}
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\begin{split}
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V = \{ & \textit{Heimisch}, \textit{Klimaresilient}, \textit{Verwertbar}, \textit{Ernteaufwand}, \\
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& \textit{Menge}, \textit{Preis}, \textit{Walderfahrung} \},
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\end{split} \notag \\
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\mathfrak{D}(\textit{Heimisch}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag \\
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\mathfrak{D}(\textit{Klimaresilient}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag \\
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\mathfrak{D}(\textit{Verwertbar}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag \\
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\mathfrak{D}(\textit{Ernteaufwand}) = \{ & \text{Motormanuel}, \text{Harvester}, \text{Vollautomatisch}\}, \notag \\
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\mathfrak{D}(\textit{Menge}) = \{ & \text{Keine}, \text{Gering}, \text{Hoch}\}, \notag \\
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\mathfrak{D}(\textit{Preis}) = \{ & \text{Gering}, \text{Mittel}, \text{Hoch}\}, \notag\\
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\mathfrak{D}(\textit{Walderfahrung}) = \{ & \text{Gering}, \text{Mittel}, \text{Intensiv}\},\notag \\
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U = \{ & 1,2\} \notag\\
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P = \{ & P_1, P_2\} \notag\\
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\begin{split}
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P_1 = \{ & (\text{Motormanuel}, 0.5), (\text{Harvester}, -0.3) \} \\
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& \cup \{ (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V,\ i \notin \{ \text{Motormanuel}, \text{Harvester}\} \ \} \
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\end{split} \notag \\
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P_2 = \{ & (d,0)\ |\ d \in \mathfrak{D}(i),\ i \in V \} \notag \\
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S = \{ & (\textit{Heimisch}, \text{Gering}), (\textit{Menge}, \text{Gering}) \} \notag \\
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\begin{split}
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S_F = \{ & (\textit{Heimisch}, \text{Gering}), (\textit{Klimaresilient}, \text{Gering}), (\textit{Verwertbar}, \text{Gering}), \\
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& (\textit{Ernteaufwand}, \text{Motormanuel}),
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(\textit{Menge}, \text{Keine}), (\textit{Preis}, \text{Hoch}),\\
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& (\textit{Walderfahrung}, \text{Gering}) \}
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\end{split} \notag
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\end{align}
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\end{mdframed}
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\section{Recommender Systems}
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\subsection{Advantages over Collaborative Filtering}
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\begin{itemize}
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\item No cold start problem for items
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\item No grey sheep problem as not dependent on similar groups having existed before.
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\item Domain knowledge is existent
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\item No issues with data sparsity as item description is given by product structure
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\item No reliance on preferences that would result in a comparison space that is too large
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\item No dependence of historic group preference accuracy
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\end{itemize}
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\subsection{Advantages over Constrained-Based Recommendation}
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\begin{itemize}
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\item Configuration state does not cause absence of recommendations
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\item Expendable to also support constraints
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\item No need to handle inconsistencies explicitly
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\end{itemize}
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\begin{table}
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\begin{center}
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\begin{tabularx}{\columnwidth}{X|X|X}
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\hline
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Collaborative Filtering
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& \begin{itemize}
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\item Serendipity of results
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\item Automatic learning of market segments
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\item Grey sheep problem
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\item No domain knowledge required
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\end{itemize}
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& \begin{itemize}
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\item Cold start problem for users and items
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\item Grey sheep problem
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\item Quality based on rating quality
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\item Data sparsity
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\item Privacy not guaranteed
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\end{itemize} \\
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\hline
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Content-Based Filtering
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& \begin{itemize}
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\item No community required
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\item User independent
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\item Transparent
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\item No item cold start
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\item Simplicity
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\item Robust
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\item Stable to constant influx of new users
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\item Possible to have profitability metric
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\end{itemize}
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& \begin{itemize}
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\item Overspecialisation
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\item No serendipity
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\item User cold start problem
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\item Requires domain knowledge
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\end{itemize} \\
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\hline
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Constraint-Based Recommendation
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& \begin{itemize}
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\item Transparent
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\item Good for non discrete values
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\end{itemize}
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& \begin{itemize}
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\item Inconsistent constraints
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\item No results
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\end{itemize} \\
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\end{tabularx}
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\caption{A description of the advantages and disadvantages of common recommendation techniques}
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\label{tab:RecommenderComparison}
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\end{center}
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\end{table}
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\FloatBarrier
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