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+\renewcommand\thesubsection{\arabic{subsection}.} +\renewcommand\thesubsubsection{\thesubsection\roman{subsubsection}.} +\makeatother +%NEW QUOTING ENVIORNMENT WITH AUTHOR AT THE END +\newenvironment{aquote}[1]{% + \pushQED{#1}% + \begin{quote}}{ + \par\nointerlineskip\noindent\hfill(\popQED)% + \end{quote}% +} + + + +% Mit den neuen commands CA kann man schneller im Schreibefluss zitieren +\newcommand\CA[1]{\citeauthor{#1} \autocite*{#1}} +\newcommand\CAs[1]{\citeauthor{#1}'s \autocite*{#1}} + +%Declare your custom MathOperators +\DeclareMathOperator*{\E}{\mathbb{E}} %Expected Value +\DeclareMathOperator*{\Var}{var} %Variance +\DeclareMathOperator*{\Cov}{cov} +\DeclareMathOperator*{\plim}{plim} %plim convergence +\DeclareMathOperator*{\bias}{bias} %bias +\DeclareMathOperator*{\se}{s.e.} %standard error +\DeclareMathOperator*{\power}{power} %power +\DeclareMathOperator*{\trace}{trace} %Trace + + +% In order to let the tags of equations appear on the left use \leqn, +% for right \reqn within the Latex document. Here, they are defined. +\makeatletter +\def\leqn{\tagsleft@true} +\def\reqn{\tagsleft@false} +\def\fleq{\@fleqntrue \let\mathindent\@mathmargin \@mathmargin=\leftmargini} +\def\cneq{\@fleqnfalse} +%\g@addto@macro{\endsubequations}{\addtocounter{equation}{-1}} +\makeatother + +% Vermeide Hurenkinder und Schusterjungen +\widowpenalty=5000 +\clubpenalty = 5000 + +%Make new column types for tables +\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}%% centered +\newcolumntype{R}[1]{>{\raggedleft\arraybackslash}m{#1}} %% right aligned +\newcolumntype{L}[1]{>{\raggedright\arraybackslash}m{#1}} %% right aligned + +\parskip 6pt +%Package to include Code +\usepackage{listings} +\usepackage{color} +\usepackage{textcomp} +\definecolor{listinggray}{gray}{0.7} +\definecolor{lbcolor}{rgb}{0.9,0.9,0.9} +\lstset{ + backgroundcolor=\color{lbcolor}, + tabsize=4, + rulecolor=, + language=SAS, + basicstyle=\scriptsize, + linewidth=0.9\linewidth, + upquote=true, + aboveskip={0.5\baselineskip}, + columns=fixed, + showstringspaces=false, + extendedchars=true, + breaklines=true, + prebreak = \raisebox{0ex}[0ex][0ex]{}, + frame=single, + showtabs=false, + showspaces=false, + showstringspaces=false, + identifierstyle=\ttfamily, + keywordstyle=\color[rgb]{0,0,1}, + commentstyle=\color[rgb]{0.133,0.545,0.133}, + stringstyle=\color[rgb]{0.627,0.126,0.941}, +} + +\usepackage{fancyhdr} + +% \footskip 60pt +\pagestyle{fancy} %eigener Seitenstil +\fancyhf{} %alle Kopf- und Fu�zeilenfelder bereinigen +\fancyhead[L]{\textbf{\large Introduction to Applied Data Science}\\ +\small Prof.\ Dr.\ Thomas Dimpfl, Dr.\ Johannes Bleher, \& Sophia Koch\\ +Department of Business Mathematics \\ +and Data Science} +\fancyhead[R]{\includegraphics[width=3.7cm]{KeyVisual_ohne_Text.pdf}} +%Kopfzeile links \fancyhead[C]{} %zentrierte Kopfzeile +% \fancyhead[R]{Name} %Kopfzeile rechts +\renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie +\fancyfoot[C]{ + \thepage +} %Seitennummer + +\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie + + + +\date{} + +\begin{document} +\begin{center} + {\LARGE \bfseries Exercise Sheet 8\\ + --\\ +$k$-Nearest Neighbors from scratch \\ +and the \texttt{caret}-package} +\end{center} + + +\section*{Requirements} +For this assignment you will need the R packages \texttt{caret} and \texttt{data.table} installed on your system. + + +\section{Clone a git repository} +\begin{enumerate}[label={\alph*.}] + \item Clone the \texttt{git} repository \href{https://aidaho-edu.uni-hohenheim.de/gitlab/jbleher/AIDAHO_IADS_AS8.git}{\texttt{AIDAHO\_IADS\_AS8}} . + \item Change the remote repository address to an address within your own user space on the \href{https://aidaho-edu.uni-hohenheim.de/gitlab}{AIDAHO-Gitlab} +\end{enumerate} + +In the git repository you will find a dataset taken from \href{https://www.coursera.org/projects/data-science-coding-challenge-loan-default-prediction}{Cousera's Loan Prediction Challenge} available for download on \href{https://www.kaggle.com/datasets/nikhil1e9/loan-default?resource=download}{Kaggle}. +The variables contained in the dataset are listed in Table~\ref{tab:loan_data_features}. +\begin{table}[h!] +\begin{center} +\caption{Summary of Loan Dataset} +\label{tab:loan_data_features} +\begin{minipage}{0.9\linewidth} +\footnotesize + The table lists the variables contained in the loan dataset. Each loan is identified by the variable \texttt{LoanID}. The variable + \texttt{default} is the target variable. All other variables are features that may be helpful in predicting the default variable.\\ +\end{minipage} +\begin{tabular}{llp{9cm}} +\toprule +Variable Name& Data Type& Description \\ +\midrule +LoanID & string & A unique identifier for each loan. \\ +Age & integer & Age of the borrower. \\ +Income & integer & Annual income of the borrower. \\ +LoanAmount & integer & Amount of money being borrowed. \\ +CreditScore & integer & Credit score of borrower. \\ +MonthsEmployed & integer & Number of months the borrower has been employed. \\ +NumCreditLines & integer & Number of credit lines the borrower has open. \\ +InterestRate & float & Interest rate for the loan. \\ +LoanTerm & integer & Term length of the loan in months. \\ +DTIRatio & float & Debt-to-Income ratio. \\ +Education & string & Highest level of education attained by the borrower. \\ +EmploymentType & string & Employment status of the borrower \\ +MaritalStatus & string & Marital status of the borrower \\ +HasMortgage & string & Whether the borrower has a mortgage. \\ +HasDependents & string & Whether the borrower has dependents. \\ +LoanPurpose & string & Purpose of loan. \\ +HasCoSigner & string & Whether the loan has a co-signer. \\ +Default & integer & Whether the loan defaulted (1) or not (0). \\ +\bottomrule +\end{tabular} +\end{center} +\end{table} + +\section{$k$-Nearest Neighbors from scratch} +\subsection*{Getting to know and preparing the data} +\begin{enumerate}[label=\thesectionenumi\arabic*.,resume=tasks] + \item Read in the data and check whether the column types concur with the types listed in table~\ref{tab:loan_data_features}. Look at the first lines to get a feeling for the data. \\ + \underline{Hint:} Use the function \texttt{sapply} and \texttt{typeof}. + \item Change all character variables (except the variable \texttt{LoanID} to factors.\\ + \underline{Hint:} Use the function \texttt{lapply} and the fact that a \texttt{data.frame} is similar to a list. + \item For each level of the factor variables create a dummy variable. \\ + \underline{Hint:} Use the function \texttt{dummyVars} from the \texttt{caret}-package and the corresponding \texttt{predict} method to generate the full data frame called \texttt{XX}. +\end{enumerate} + +\subsection*{Distance function and determining the $k$ nearest neighbors} +In order to calculate the $k$ nearest neighbors, we need to determine what the term \texttt{near} means. For this assignment, we will use the Euclidean distance. Figure~\ref{fig:euclidian_distance} depicts the distance. + +\begin{figure}[h!] + \caption{Visualization of the Euclidean distance} +\label{fig:euclidian_distance} +\begin{center} +\begin{minipage}{0.9\linewidth} +\footnotesize + The figure illustrates how the Euclidian distance between two points $p$ and $q$ in a two dimensional Euclidean space is calculated. The Euclidian distance is the length of the solid black line in the plot. To determine the length, the Pythagorean theorem and the presence of rectangular triangle is used. The rectangle is indicated with the square. + \\ +\end{minipage} +\begin{tikzpicture}[scale=1.5, dot/.style={circle,inner sep=1pt,fill,name=#1}, + extended line/.style={shorten >=-#1,shorten <=-#1}, + extended line/.default=1cm] + +% Define points +\node[dot=p,label=below left:{$p$}] at (1,2) {}; +\node[dot=p1,label=below right:{$p_1$}] at (1,0) {}; +\node[dot=p2,label=below left:{$p_2$}] at (0,2) {}; + +\node[dot=q,label=below right:{$q$}] at (4,3) {}; +\node[dot=q1,label=below right:{$q_1$}] at (4,0) {}; +\node[dot=q2,label=below left:{$q_2$}] at (0,3) {}; + +% Draw lines +\draw[thin, ->] (0,0) -- (5,0) node[anchor=north west] {$x$}; +\draw[thin, ->] (0,0) -- (0,4) node[anchor=south east] {$y$}; +\draw[dashed] (p2) -- (p) -- node[below] {$q_1 - p_1$} (q |- p2) --node[left] {$q_2 - p_2$} (q) -- (q2); +\draw[dashed] (p1) -- (p); +\draw[dashed] (q1) -- (q |- p2); +\draw[solid] (p) -- (q); +\draw (2.5,4) node {$d(p,q)^2 = (q_1 - p_1)^2 + (q_2 - p_2)^2$}; +% Draw right angle symbol +\draw (q |- p) rectangle ++(-0.2,0.2); +\end{tikzpicture} +\end{center} +\end{figure} + +\begin{enumerate}[label=\thesectionenumi\arabic*.,resume=tasks] + \item Write a function that calculates the Euclidean distance for two vectors $p$ and $q$. Call the function \texttt{eclidean\_distance} The function should also check whether the two vectors are of the same length. if they are not of the same length, an error message should be issued. + \item \label{enum:knn_function}Write a function that given a dataset \texttt{X} determines the $k$ nearest neighbors based on a distance function for each observation in a different dataset \texttt{Y}. The function should also be handed corresponding outcomes to \texttt{X} in the vector \texttt{outcomes\_X}. \\ + \underline{Hints and Remarks:} + \begin{itemize} + \item Make sure that the function issues an error \ldots + \begin{itemize} + \item[\ldots] if \texttt{X} and \texttt{Y} do not have the same number of columns. + \item[\ldots] if \texttt{X} and \texttt{outcomes\_X} do not have the same number of rows. + \end{itemize} + \item The function should take the argument \texttt{X}, \texttt{outcomes\_X}, \texttt{Y}, \texttt{k} and \texttt{DISTFUN} where \texttt{DISTFUN} is the function used to calculate the distance between two vectors. + \item All distances can be calculated using the \texttt{apply} function in one short line of code. + \item Use the order command to determine the indices of the $k$ nearest neighboring points in \texttt{X}. + \item The function should return a list that contains the following three matrices: + \begin{enumerate} + \item the indices, + \item the distances, + \item and the outcomes + \end{enumerate} + of the $k$ nearest neighboring points in \texttt{X} for each observation in \texttt{Y}. + \item Test your function with the following call + \end{itemize} + \begin{R} + knn.obj <- nearest_neighbors(X = XX[1:100,], + Y = XX[4001:4003,], + k = 3, + DISTFUN = euclidean_distance, + outcome = thedata$Default) + \end{R} + +\item \label{enumi:majority_vote} Write a function called \texttt{majority\_vote} which, given a vector \texttt{x} returns the most frequent value as a string. +\item Write a function \texttt{knn\_prediction} that takes the resulting list of task \ref{enum:knn_function} and an evaluation function \texttt{EVALFUN} to predict the outcome based on kNN. The evaluation function could be the \texttt{mean} function or the \texttt{majority\_vote} functionn of task \ref{enumi:majority_vote}. +\item Is the scale of the variables an issue for the prediction? +\end{enumerate} + + +\section{kNN with the \texttt{caret}-package} +\subsection*{Background: The \texttt{caret} package} +The \href{https://topepo.github.io/caret/index.html}{\texttt{caret}-package} provides a useful conglomeration of tools for statistical/machine learning applications. The documentation of the package lists the capabilities of the package: +\begin{quote} + \itshape +The caret package (short for Classification And REgression Training) is a set of functions that attempt to streamline the process for creating predictive models. The package contains tools for: +\begin{itemize}[label=] + \item data splitting + \item pre-processing + \item feature selection + \item model tuning using resampling + \item variable importance estimation +\end{itemize} +as well as other functionality. +\end{quote} + +\subsection*{Splitting the data set based on the outcome variable} +\begin{enumerate}[label=\thesectionenumi\arabic*.,resume=tasks2] + \item Have a look at the documentation of the \texttt{createDataPartition} from the \texttt{caret} package. What does it do? What other functions are there? + \item \label{split:step1} Use the function \texttt{createDataPartition} to create the indices that split the dataset into two parts (80\%/20\%) based on the \texttt{Default} variable. The output should not be a list. + \item \label{split:step2} Create a training data set and a test data set. Both sets should not contain the column \texttt{LoanID}. + \item \label{split:freq} Have a look at the frequencies of the outcome variable. How many times does \texttt{Default = 1} and \texttt{Default = 0} occur in the output dataset? + \item Change the column type of \texttt{Default} to \texttt{factor}. Repeat the splitting in tasks \ref{split:step1} and \ref{split:step2}. Do the frequencies change in \ref{split:freq}? +\end{enumerate} +\subsection*{Model training} +\begin{enumerate}[label=\thesectionenumi\arabic*.,resume=tasks2] + \item \label{step:kNN} Use the following code to train a \texttt{knn}-prediction model on the \texttt{trainData} + \begin{R} + ctrl <- trainControl( method="repeatedcv", + repeats = 3, + number=10) + knnFit <- train(as.factor(Default) ~ ., + data = trainData, + method = "knn", + trControl = ctrl, + preProcess = c("center","scale"), + tuneLength = 10) + \end{R} + + \item Have a look at the resulting object \texttt{knnFit}. What is \texttt{Accuracy}, how is it calculated? +\end{enumerate} +\subsection*{Model evaluation} +\begin{enumerate}[label=\thesectionenumi\arabic*.,resume=tasks2] + \item Use the method predict on the \texttt{knnFit} object from step \ref{step:kNN} with the argument \texttt{newdata=testData}. + \item Compare the prediction with the actual values in the \texttt{Default} column of the the \texttt{testData} data frame. What is the Accuracy? + \end{enumerate} +\end{document}