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remove word 'we' from thesis

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30_Thesis/sections/10_foundations.tex
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@@ -10,7 +10,7 @@ Formally a configuration problem can be specified as a \emph{constraint satisfac
\begin{equation} \label{eq:Foundations:ProductConfiguration:ConstraintSatisfactionProblem}
CSP(V,\mathfrak{D},C),
\end{equation}
where we have a set of \emph{variables} $V$ (which in this thesis will also be referred to as \emph{features}) with
where $V$ is a set of \emph{variables} (which in this thesis will also be referred to as \emph{features}) with
\begin{equation} \label{eq:Foundations:ProductConfiguration:Variables}
V = \{v_1, \dotsc, v_m\},
\end{equation}
@@ -25,14 +25,14 @@ and \emph{constraints} $C$ that limit the solution space with
\subsection{Configuration State}
We will define a \emph{configuration} $S$ as a tuple of variables (\autoref{eq:Foundations:ProductConfiguration:Variables}) and their corresponding domain value with
A \emph{configuration} $S$ will be defined as a tuple of variables (\autoref{eq:Foundations:ProductConfiguration:Variables}) and their corresponding domain value with
\begin{equation} \label{eq:Foundations:ProductConfiguration:ConfigurationState}
S = \{ (v_i,\ d) \ |\ v_i \in V \ \land \ d \in \mathfrak{D}(i),\ i=1,\dotsc,m \}.
\end{equation}
Essentially it is a set of variables and assigned values.
\subsection{Finished Configuration}
To define a \emph{finished configuration} we first need to define what a valid configuration is. Therefore we define $is\_valid$ as
To define what a \emph{finished configuration} is, it is required to first define what it means for a configuration to be valid. Therefore $is\_valid$ is defined as
\begin{equation} \label{eq:Foundations:ProductConfiguration:IsValid}
is\_valid : S \to \{true, false\}; x \mapsto
\begin{cases}
@@ -44,13 +44,13 @@ with $solution\_space$ being the solution space of the corresponding constraint
\begin{equation} \label{eq:Foundations:ProductConfiguration:FinishedConfiguration}
S_F \subset S,\ where \ \forall v_i \in V (\exists (v_i, d) \in S_F : d \in \mathfrak{D}(i)) \land is\_valid(S_F).
\end{equation}
In practice a finished configuration of a product (or solution) is something that is ready to be produced. For example if we are configuring a car, this means that the car could be produced in the specified way that is given by the finished configuration.
In practice a finished configuration of a product (or solution) is something that is ready to be produced. For example if a care is being configured, this means that the car could be produced in the specified way that is given by the finished configuration.
\section{Group-Based Product Configuration}
\label{sec:Foundations:GroupBasedProductConfiguration}
Instead of a single person configuring a product, a group of people is configuring one product which can be useful in multi-stakeholder decisions. This setting needs mechanisms for describing the preferences of multiple people. Therefore we will add to our definitions, a set of users $U$ with
Instead of a single person configuring a product, a group of people is configuring one product which can be useful in multi-stakeholder decisions. This setting needs mechanisms for describing the preferences of multiple people. Therefore to the definitions there will be added a set of users $U$ with
\begin{equation}\label{eq:Foundations:ProductConfiguration:Users}
U = \{1, \dotsc, n\},
\end{equation}
@@ -91,7 +91,7 @@ In collaborative filtering a users rating for unknown items is predicted by find
Collaborative Filtering can not only be done using users, it can also be item-based. Hereby the similarity between items is used for a recommendation and not similar users \cite{ricciRecommenderSystemsHandbook2015}. In the context of configuration the similarity to other historic configurations can be used which makes it an item based approach.
\autoref{tab:Foundations:RecommenderSystem:MoviePreferences} shows an example rating matrix. A simple user-based way to calculate a rating would be to use a k-nearest neighbour (kNN) algorithm and then take the average of those ratings. Using this method with $k := 2$ and euclidean distance our closest neighbours are \textit{Lucy} and \textit{Diane} therefore giving us a predicted rating of $4$. If we use an item based approach instead, we will try to find similar items based on the users rating. An example of similar items here would be \textit{Forest Gump} and \textit{Wall-E} as John and Lucy each have given them the sane rating and Eric's rating is off by one. Using again kNN with $k := 2$ we find that \textit{Forest Gump} and \textit{Wall-E} are the most similar to \textit{Titanic} thereby having a predicted rating of $4.5$.
\autoref{tab:Foundations:RecommenderSystem:MoviePreferences} shows an example rating matrix. A simple user-based way to calculate a rating would be to use a k-nearest neighbour (kNN) algorithm and then take the average of those ratings. Using this method with $k := 2$ and euclidean distance our closest neighbours are \textit{Lucy} and \textit{Diane} therefore giving us a predicted rating of $4$. If an item-based approach is used instead, it will be tried to find similar items based on the user's rating. An example of similar items here would be \textit{Forest Gump} and \textit{Wall-E} as John and Lucy each have given them the sane rating and Eric's rating is off by one. Using again kNN with $k := 2$ it is found that \textit{Forest Gump} and \textit{Wall-E} are the most similar to \textit{Titanic} thereby having a predicted rating of $4.5$.
However this simple similarity and prediction function does not take into account different distances. For example Lucy's ratings are more similar compared to Eric's than Diane's but Diane's and Lucy's rating is valued the same amount.
\todo[inline]{
@@ -109,7 +109,7 @@ Our movie example (see \autoref{tab:Foundations:RecommenderSystem:MoviePreferenc
\subsection{Content-Based Filtering}
For the content-based filtering approach, items and users are assigned to categories. Based on consumption and rating of items a user will have implicit ratings for categories. Predictions are now made based on a categories of the new item \cite[~ pp. 10, 11]{felfernigDecisionTasksBasic2018}.
Using our example from \autoref{tab:Foundations:RecommenderSystem:MoviePreferences} and using an additional category matrix (see \autoref{tab:Foundations:RecommenderSystem:ContentBasedFilteringCategories}) we can derive a rating matrix per category (using the average rating of the user of each movie contained in this category). The result can be seen in \autoref{tab:Foundations:RecommenderSystem:ContentBasedFilteringProfiles}. To predict Eric's rating of Titanic we now can use the categories of \textit{Titanic} and average out Eric's implicit rating per category. Titanic is only in the category romance and as Eric's rating of \textit{Forest Gump} is $5$ the prediction is a rating of $5$. Categories don't have to be the genre, they could be any kind of data about a movie.
Using the example from \autoref{tab:Foundations:RecommenderSystem:MoviePreferences} and using an additional category matrix (see \autoref{tab:Foundations:RecommenderSystem:ContentBasedFilteringCategories}) it a rating matrix per category can be derived (using the average rating of the user of each movie contained in this category). The result can be seen in \autoref{tab:Foundations:RecommenderSystem:ContentBasedFilteringProfiles}. To predict Eric's rating of Titanic, the categories of \textit{Titanic} and averages of Eric's implicit rating per category are used. Titanic is only in the category romance and as Eric's rating of \textit{Forest Gump} is $5$ the prediction is a rating of $5$. Categories don't have to be the genre, they could be any kind of data about a movie.
\begin{table}
\centering