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% Common Information
\title{Einschub: Lokale Modellnetze}
\author[NNFS]{Vorlesung: Neuronale Netze und Fuzzy Systeme (NNFS)\\Prof. Dr.-Ing. Oliver Nelles}
\institute{Universität Siegen}
\date[09.01.2017]{09. Januar 2017}

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		\hskip10px Institut für Mechanik und Regelungstechnik - Mechatronik 
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\begin{document}

\begin{frame}
	\titlepage
\end{frame}

\section*{Überblick}
\begin{frame}
	\frametitle{Überblick}
	\tableofcontents
\end{frame}

%% Normierte RBF-Netze
\section{Normierte Radiale Basisfunktionen Netze}

\begin{frame}
  \frametitle{Normierte radiale Basis-Funktionen (NRBF-) Netze}
  \begin{columns}
    \column{.5\textwidth}
      \includegraphics[scale=0.52]{figures/nrbf_netz.pdf}
    \column{.5\textwidth}
     \begin{block}{Gleichung für Modellausgang:}
         $\hat{y} = \sum\limits_{i=1}^{M} {\color{cp-fak-color}{w_i}} \cdot \Phi_i(\vec{u}_i)$ \\
      \end{block}
      \begin{block}{\emph{Partition of Unity}:}
      		$\sum\limits_{i=1}^{M} \Phi_i(\vec{u}) = 1$
      \end{block}
      \begin{block}{Gültigkeitsfunktionen:}
      	$\Phi_i(\vec{u}) = \frac{\mu_i(\vec{u})}{\sum\limits_{j=1}^{M} \mu_j(\vec{u})}$
      \end{block}
  \end{columns}
\end{frame}

%\begin{frame}
%  \frametitle{NRBF-Netz: Interpretation als Singelton Fuzzy-System}
%  \begin{columns}
%    \column{.5\textwidth}
%      \begin{block}{NRBF-Netz}
%      	$\hat{y} = \frac{\sum\limits_{i=1}^{M} w_i \Phi_i(\vec{u}_i)}{\sum\limits_{j=1}^{M}  \Phi_j(\vec{u}_j)}$\\
%      	\begin{itemize}
%      		\item $\Phi_i$: i-te Zugehörigkeitsfunktion
%      		\item $w_i$: i-tes Gewicht
%      	\end{itemize}
%      \end{block}
%     \column{.5\textwidth}
%     \begin{block}{Singelton Fuzzy-System}
%     	$\hat{y} = \frac{\sum\limits_{i=1}^{M} s_i \mu_i(\vec{u}_i)}{\sum\limits_{j=1}^{M}  \mu_j(\vec{u}_j)}$\\
%     		\begin{itemize}
%      		\item $\mu_i$: Erfüllungsgrad der i-ten Fuzzy-Regel
%      		\item $s_i$: Singelton der i-ten Regel
%      	\end{itemize}
%     \end{block}
%  \end{columns}
%  \begin{columns}
%  	\column{1\textwidth}
%  	\vspace{0.5cm}
%  	
%  	Textform:\\
%  	if $u_1=A_{i1}$ AND $u_2=A_{i2} \cdots$ AND $u_p=A_{ip}$ THEN $w_i$ (bzw. $s_i$)
%  \end{columns}
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% Lokale Modellnetze
\section{Lokale Modellnetze}

\begin{frame}
  \frametitle{Aufbau Lokaler Modellnetze (LMN)}
  \begin{columns}
    \column{.5\textwidth}
      \includegraphics[scale=0.52]{figures/lmn.pdf}
    \column{.5\textwidth}
     \begin{block}{Gleichung für Modellausgang:}
         $\hat{y} = \sum\limits_{i=1}^{M} {\color{cp-fak-color}{\hat{y}_i}} \cdot \Phi_i(\vec{u})$
      \end{block}
      \begin{block}{Partition of Unity:}
         $\sum\limits_{i=1}^{M} \Phi_i(\vec{u}) = 1$
      \end{block}
      \begin{block}{Gültigkeitsfunktionen:}
      	$\Phi_i(\vec{u}) = \frac{\mu_i(\vec{u})}{\sum\limits_{j=1}^{M} \mu_j(\vec{u})}$
      \end{block}
  \end{columns}
%    \begin{columns}
%  	\column{1\textwidth}
%  	\vspace{0.2cm}
%  	
%  	Textform:\\
%  	if $u_1=A_{i1}$ AND $u_2=A_{i2} \cdots$ AND $u_p=A_{ip}$ THEN $\hat{y}_i = f(u_1, u_2, \cdots , u_p)$
%  \end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%% NRBF vs. LMN
\section{Gegenüberstellung: NRBF vs. LMN Modelle}

\begin{frame}
	\frametitle{NRBF Netz mit 1 Eingang und 2 Basisfunktionen}
	\begin{columns}
		\column{0.5\textwidth}
		\begin{block}{NRBF Modellausgang:}
			$\hat{y} = \sum\limits_{i=1}^{2} {\color{myorange}{w_i}} \cdot {\color{cp-blue}{\Phi_i}}(\vec{u}_i)$
		\end{block}
		\column{0.5\textwidth}
		\begin{block}{Interpretation als Regeln:}
			\textcolor{cp-blue}{if} $u_1$ is small \textcolor{myorange}{THEN} $\hat{y}_i = w_1$\\
			\textcolor{cp-blue}{if} $u_1$ is big \textcolor{myorange}{THEN} $\hat{y}_i = w_2$
		\end{block}
		%\vspace{-5px}
	\end{columns}
	\vspace{20px}
	\begin{columns}
		\column{1\textwidth}
		 \includegraphics[scale=0.5]{figures/validity_functions}
		 \includegraphics[scale=0.5]{figures/local_models_nrbf}
		 \includegraphics[scale=0.5]{figures/global_model_nrbf}
	\end{columns}
	
\end{frame}

\begin{frame}
	\frametitle{LMN mit 1 Eingang und 2 Lokalen Modellen}
	\begin{columns}
		\column{0.5\textwidth}
		\begin{block}{LMN Modellausgang:}
			$\hat{y} = \sum\limits_{i=1}^{2} {\color{myorange}{\hat{y}_i}} \cdot {\color{cp-blue}{\Phi_i}}(\vec{u})$
		\end{block}
		\column{0.5\textwidth}
		\begin{block}{Interpretation als Regeln:}
			\textcolor{cp-blue}{if} $u_1$ is small \textcolor{myorange}{THEN} $\hat{y}_1 = u_1$\\
			\textcolor{cp-blue}{if} $u_1$ is big \textcolor{myorange}{THEN} $\hat{y}_2 = 2 u_1 - 0.2$
		\end{block}
		%\vspace{-5px}
	\end{columns}
	\vspace{20px}
	\begin{columns}
		\column{1\textwidth}
		\includegraphics[scale=0.5]{figures/validity_functions}
		\includegraphics[scale=0.5]{figures/local_models_lmn}
		\includegraphics[scale=0.5]{figures/global_model_lmn}
	\end{columns}
	
\end{frame}

\begin{frame}
	\frametitle{LMN mit 2 Eingängen und 3 Lokalen Modellen}
	\begin{columns}
		\column{0.4\textwidth}
		\begin{block}{LMN Modellausgang:}
			$\hat{y} = \sum\limits_{i=1}^{3} {\color{myorange}{\hat{y}_i}} \cdot {\color{cp-blue}{\Phi_i}}(\vec{u})$
		\end{block}
		\column{0.6\textwidth}
		\begin{block}{Interpretation als Regeln:}
			\textcolor{cp-blue}{if} $u_1$ is big \textcolor{myorange}{THEN} $\hat{y}_1$\\
			\textcolor{cp-blue}{if} $u_1$ is small AND $u_2$ is small \textcolor{myorange}{THEN} $\hat{y}_2$\\
			\textcolor{cp-blue}{if} $u_1$ is small AND $u_2$ is big \textcolor{myorange}{THEN} $\hat{y}_3$
		\end{block}
		%\vspace{-5px}
	\end{columns}
	\vspace{10px}
	\begin{columns}
		\column{1\textwidth}
		\includegraphics[scale=0.5]{figures/partitioning_lolimot_3LM}
		\includegraphics[scale=0.55]{figures/validityFunctions_lolimot}
		\includegraphics[scale=0.55]{figures/process_with_localModels_lolimot}
	\end{columns}
	
\end{frame}

%\begin{frame}
%  \frametitle{Ansatzfunktionen für gewählte Modellklasse}
%	\begin{columns}
%		\column{0.3\textwidth}
%	\includegraphics[scale=0.25]{figures/validityFunctions.eps}
%		\column{0.7\textwidth}
%     \begin{block}{Affiner Ansatz für lokale Modelle:}
%        $\hat{y}_i(\vec{u}) = \sum\limits_{i=1}^{M} (w_{i0} + w_{i1}u_1 + w_{i2}u_2 + \hdots + w_{ip}u_p)$
%      \end{block}
%	\vspace{-5px}
%	\begin{block}{Gültigkeitsfunktionen:}
%		$\Phi_i(\vec{u}) = \frac{\mu_i(\vec{u})}{\sum\limits_{j=1}^{M} \mu_j(\vec{u})}$
%	\end{block}
%	\vspace{-5px}
%	\begin{block}{Zugehörigkeitsfunktionen:}
%		$\mu_i(\vec{u}) = e^{-\frac{1}{2}\left(\frac{(u_1 - c_{i1})^2}{\sigma_{i1}^2}+\hdots + \frac{(u_p - c_{ip})^2}{\sigma_{ip}^2}\right)}$
%	\end{block}
%	\end{columns}
%
%\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%% LOLIMOT Trainingsalgorithmus
\section{LOLIMOT Trainingsalgorithmus}

%\begin{frame}
%  \frametitle{Ansatzfunktionen für gewählte Modellklasse}
%	\begin{columns}
%		\column{0.3\textwidth}
%	\includegraphics[scale=0.25]{figures/validityFunctions.eps}
%		\column{0.7\textwidth}
%     \begin{block}{Affiner Ansatz für lokale Modelle:}
%        $\hat{y}_i(\vec{u}) = \sum\limits_{i=1}^{M} (w_{i0} + w_{i1}u_1 + w_{i2}u_2 + \hdots + w_{ip}u_p)$
%      \end{block}
%	\vspace{-5px}
%	\begin{block}{Gültigkeitsfunktionen:}
%		$\Phi_i(\vec{u}) = \frac{\mu_i(\vec{u})}{\sum\limits_{j=1}^{M} \mu_j(\vec{u})}$
%	\end{block}
%	\vspace{-5px}
%	\begin{block}{Zugehörigkeitsfunktionen:}
%		$\mu_i(\vec{u}) = e^{-\frac{1}{2}\left(\frac{(u_1 - c_{i1})^2}{\sigma_{i1}^2}+\hdots + \frac{(u_p - c_{ip})^2}{\sigma_{ip}^2}\right)}$
%	\end{block}
%	\end{columns}
%
%\end{frame}

\begin{frame}
  \frametitle{Local Linear Model Tree (LOLIMOT)}
  \begin{columns}
    \begin{column}{.45\textwidth}
      	\includegraphics[scale=0.4]{figures/lolimot_baum.pdf}
	\end{column}
    \begin{column}{.55\textwidth}
     \begin{block}{(Haupt-)Problematik:}
        Bestimmung der Lage und Standardabweichungen der Zugehörigkeitsfunktionen.
      \end{block}
	\begin{block}{Lösungsansatz:}
		Sukzessive Unterteilung des Eingangsraumes (heuristisch, iterativ).
%		\begin{itemize}
%			\item Heuristisch
%			\item Iterativ
%			\item Sukzessive Unterteilung des Eingangsraumes
%		\end{itemize}
	\end{block}
	\end{column}
  \end{columns}

\end{frame}

%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT1}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT2}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT3}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT4}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT5}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT6}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT7}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT8}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT9}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT10}
%		\end{column}
%	\end{columns}
%	
%\end{frame}
%
%\begin{frame}
%	\frametitle{LOLIMOT Trainingsbeispiel}
%	\begin{columns}
%		\begin{column}{1\textwidth}
%			\centering
%			\includegraphics[scale=0.4]{figures/LOLIMOT_IT11}
%		\end{column}
%	\end{columns}
%	
%\end{frame}

\begin{frame}
	\frametitle{Schätzung der lokalen Modellparameter - Global}
	\begin{columns}
		\begin{column}{1\textwidth}
			\vspace{-18px}
			\begin{block}{LMN mit 2 Eingängen und 2 lokal affinen Modellen}
				$\hat{y} = \sum\limits_{i=1}^{2} {\color{myorange}{\underbrace{\left(w_{0,i} + w_{1,i}u_1 + w_{2,i}u_2\right)}_{\hat{y}_i}}} \cdot {\color{cp-blue}{\Phi_i}}(\vec{u})$
			\end{block}
		\end{column}
	\end{columns}
	\begin{columns}
		\begin{column}{1\textwidth}
			\vspace{-10px}
			\begin{block}{Globale Schätzung}
			\scalebox{0.8}{
				$\vec{X} = \left[\begin{array}{ccc|ccc}
				\Phi_1(\vec{u}) & u_1(1) \Phi_1(\vec{u}) & u_2(1) \Phi_1(\vec{u}) & \Phi_2(\vec{u}) & u_1(1) \Phi_2(\vec{u}) & u_2(1) \Phi_2(\vec{u}) \\ 
				\Phi_1(\vec{u}) & u_1(2) \Phi_1(\vec{u}) & u_2(2) \Phi_1(\vec{u}) & \Phi_2(\vec{u}) & u_1(2) \Phi_2(\vec{u}) & u_2(2) \Phi_2(\vec{u}) \\ 
				\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 
				\undermat{\text{Lokales Modell 1 (} \vec{X}_{\text{LM1}} \text{)}}{\Phi_1(\vec{u}) & u_1(N) \Phi_1(\vec{u})  & u_2(N) \Phi_1(\vec{u})} & \undermat{\text{Lokales Modell 2 (} \vec{X}_{\text{LM2}} \text{)}}{\Phi_2(\vec{u}) & u_1(N) \Phi_2(\vec{u}) & u_2(N) \Phi_2(\vec{u})}
			\end{array}\right] $
%			\newline
%			$\vec{w} = \left[\begin{array}{cccccc}
%			w_{0,1} & w_{1,1} & w_{2,1} & w_{0,2} & w_{1,2} & w_{2,2}
%			\end{array} \right]$
			}
			\ \newline
			\  \newline
			\scalebox{0.8}{$
				\vec{w} = \left[
					\begin{array}{ccc|ccc}
						%w_{0,1} & w_{1,1} & w_{2,1} & w_{0,2} & w_{1,2} & w_{2,2} \\
						\undermat{\vec{w}_{\text{LM1}}}{w_{0,1} & w_{1,1} & w_{2,1}} & \undermat{\vec{w}_{\text{LM2}}}{w_{0,2} & w_{1,2} & w_{2,2}}
					\end{array}
					\right]
					\Longrightarrow
					\hat{\vec{w}} = \left( \vec{X}^T \vec{X} \right)^{-1} \vec{X}^T \vec{y} 
				$}
			\end{block}
		\end{column}
	\end{columns}
\end{frame}

\begin{frame}
	\frametitle{Schätzung der lokalen Modellparameter - Lokal}
	\begin{columns}
		\begin{column}{1\textwidth}
			%\vspace{-18px}
			\begin{block}{Lokale Schätzung}
				
				\begin{itemize}
					\item $M$ x lokale Least-Squares Schätzung: Geringerer Rechenaufwand\\ $\Rightarrow$ $\mathcal O\left(M (p + 1)^3\right)$ statt $\mathcal O\left(M^3 (p + 1)^3\right)$ 
					\fontsize{9.5}{7.2}{
						\begin{align*}
						\hat{\vec{w}}_{\text{LM}i} = &\left( \vec{X}_{\text{LM}i}^T \vec{X}_{\text{LM}i} \right)^{-1} \vec{X}_{\text{LM}i}^T \vec{y}_{LMi} \\
						&\text{mit } i = 1, 2, \ldots, M
						\end{align*}}
					\item Interaktion zwischen lokalen Modellen wird ignoriert\\ $\Rightarrow$ Erhöhung des Bias-Fehlers
					\item Einschränkung der Flexibilität $\Rightarrow$ Regularisierender Effekt!
				\end{itemize}
%				\fontsize{8.5}{7.2}{
%				\begin{align*}
%					\hat{\vec{w}}_{\text{LM1}} &= \left( \vec{X}_{\text{LM1}}^T \vec{X}_{\text{LM1}} \right)^{-1} \vec{X}_{\text{LM1}}^T \vec{y} \\
%					\hat{\vec{w}}_{\text{LM2}} &= \left( \vec{X}_{\text{LM2}}^T \vec{X}_{\text{LM2}} \right)^{-1} \vec{X}_{\text{LM2}}^T \vec{y} \\
%					\vdots \\
%					\hat{\vec{w}}_{\text{LM,M}} &= \left( \vec{X}_{\text{LM,M}}^T \vec{X}_{\text{LM,M}} \right)^{-1} \vec{X}_{\text{LM,M}}^T \vec{y} 
%				\end{align*}}
%				\fontsize{8.5}{7.2}{
%				\begin{align*}
%				\hat{\vec{w}}_{\text{LMi}} &= \left( \vec{X}_{\text{LMi}}^T \vec{X}_{\text{LMi}} \right)^{-1} \vec{X}_{\text{LMi}}^T \vec{y} 
%				\end{align*}}
			\end{block}
		\end{column}
	\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% Beispiele
\section{MATLAB Beispiele}

\begin{frame}
  \frametitle{Demonstrationsbeispiele in MATLAB}
\begin{columns}
    \column{.3\textwidth}
      \includegraphics[scale=0.18]{figures/process1d.eps} \newline
	\newline 
      \includegraphics[scale=0.18]{figures/process2d.eps}
    \column{.7\textwidth}
     \begin{block}{Hyperbel (1D):}
         $y(u_1) = \frac{0.05}{0.05 + (1-u_1)}$
      \end{block}
      \begin{block}{Hyperbel 30°/60° (2D):}
         $y(u_1,u_2) = \frac{0.55}{0.05 + (1-0.5u_1) + (1-u_2)}$
      \end{block}
  \end{columns}

\end{frame}

\begin{frame}
  \frametitle{Demonstrationsbeispiele in MATLAB}
\begin{columns}
    \column{.3\textwidth}
      \includegraphics[scale=0.18]{figures/process2d45.eps} \newline
	\newline 
      \includegraphics[scale=0.18]{figures/hyperbelPlusEbene.eps}
    \column{.7\textwidth}
     \begin{block}{Hyperbel 45°/45° (2D):}
         $y(u_1,u_2) = \frac{0.05}{0.05 + (1-u_1) + (1-u_2)}$
      \end{block}
      \begin{block}{Hyperbel+Ebene (2D):}
         $y(u_1,u_2) = \frac{0.05}{0.05 + (1-u_1)} + 0.8 \cdot u_2$
      \end{block}
  \end{columns}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
\end{document}