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TeX/main.tex

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+\documentclass[a4paper, 12pt]{article}
+
+\usepackage[margin=1in]{geometry} 
+\usepackage[english]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage{libertine}
+\usepackage{amsmath}
+%\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{amsfonts}
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\usepackage{enumerate}
+\usepackage{sectsty}
+\usepackage{setspace}
+\usepackage{booktabs}
+\usepackage{caption}
+\usepackage[justification=RaggedRight, singlelinecheck=false]{caption} 
+%\usepackage{natbib} %[numbers]
+\usepackage{multirow}
+\usepackage{3parttable}
+\usepackage{mathtools}
+\usepackage{tabu}
+\usepackage{makecell}
+\usepackage{dsfont}
+\usepackage {tikz}
+\usetikzlibrary{positioning}
+\usepackage{pgfplots}
+\usepgfplotslibrary{colorbrewer}
+\usepackage{subcaption}
+\usetikzlibrary{arrows,decorations.markings}
+\pgfplotsset{compat = 1.16}
+\usepackage[export]{adjustbox}
+
+
+
+\usepackage[style=authoryear, backend=bibtex]{biblatex}
+\addbibresource{Literaturverzeichnis.bib}
+\urlstyle{same}
+\bibliography{Literaturverzeichnis.bib}
+\numberwithin{figure}{section}
+\numberwithin{table}{section}
+\numberwithin{equation}{section}
+
+\sectionfont{\centering}
+ \input{insbox}
+
+\parindent0in
+\pagestyle{plain}
+\thispagestyle{plain}
+\newtheorem{Theorem}{Theorem}[section]
+\newtheorem{Definition}[Theorem]{Definition}
+\newtheorem{Algorithm}[Theorem]{Algorithm}
+\newtheorem{Example}[Theorem]{Example}
+
+
+\DeclareMathOperator*{\plim}{\mathbb{P}\text{-}\lim}
+\begin{document}
+
+        
+\newcommand{\plimn}[0]{\plim\limits_{n \to \infty}}
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+
+
+\newcommand{\abs}[1]{\ensuremath{\left\vert#1\right\vert}}
+%\newcommand{\myrightarrow}[1]{\xrightarrow{\makebox[2em][c]{$\scriptstyle#1$}}}‌
+
+%Arndt Tobias \hfill 21.12.2017\newline
+%\textbf{Seminar Machine--Learning: Unsupervised %Learning} \newline
+%Institut für Mathematik der Universität %Augsburg\\
+%Lehrstuhl für Rechnerorientierte Statistik und %Datenanalyse\\
+\smallskip\hrule\bigskip
+
+\begin{center}
+	{\huge{Electricity Price Forecasting based on Regression Tree Models}}
+\end{center} 
+\hrulefill
+\pagenumbering{gobble}
+\newpage
+%\setcounter{tocdepth}{4}
+\tableofcontents 
+\newpage
+
+% Theorem 3.8
+\input{theo_3_8.tex}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:

TeX/theo_3_8.tex

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+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "main"
+%%% End:
+
+With the following Theorem we will have an explicit desrctiption for the
+limes of RN as the amount of nodes is increased.
+
+\begin{Theorem}[Ridge weight penaltiy corresponds to adapted spline]
+  \label{theo:main1}
+  For arbitrary training data \(\left(x_i^{train}, y_i^{train}\right)\) it holds
+  \[
+    \plimn \norm{\mathcal{RN^{*, \tilde{\lambda}}} - f^{*,
+        \tilde{\lambda}}_{g, \pm}}_{W^{1,\infty}(K)} = 0.
+  \]
+
+  With
+  \begin{align*}
+    \label{eq:1}
+    \tilde{\lambda} &\coloneqq \lambda n g(0), \\
+    g(x) &\coloneqq
+           g_{\xi}(x)\mathbb{E}\left[ v_k^2 \vert \xi_k = x \right], \forall x
+           \in \mathbb{R}
+  \end{align*}
+  and \(RN^{*, \tilde{\lambda}}}\), \(f^{*,\tilde{\lambda}}_{g, \pm}\)
+as defined in ??? and ??? respectively. 
+\end{Theorem}
+In order to proof Theo~\ref{theo:main1} we need to proof a number of
+auxilary Lemmata first.