\documentclass[11pt, a4paper, onecolumn, final]{article}

\title{}

\author{
	Hofbauer Heinz\\
	\small hhofbaue@cosy.sbg.ac.at
\and
	Berka Tobias\\
	\small tberka@cosy.sbg.ac.at
\and
	St\"utz Thomas\\
	\small tstuetz@cosy.sbg.ac.at
	}
	
\date{}

\newcommand{\sek}[1]{\section{#1}}
\newcommand{\ssek}[1]{\subsection{#1}}
\newcommand{\sssek}[1]{\subsubsection{#1}}
\newcommand{\kernel}[1]{\verb=<SNNS-DIR>/kernel/sources/#1=}

\input xy
\xyoption{all}
\usepackage[dvips]{graphics, color}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%begin%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagestyle{myheadings}
\markright{Evolution of Transfer Functions in ANN \hfill }
\Huge
\centering
\thispagestyle{empty}
\vspace*{4cm}
Evolution of Transfer Functions in Artificial Neural Networks\\

\vspace{2cm}

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

\Large
Tobias Berka\\
Heinz Hofbauer\\
Thomas St\"utz\\
\vspace{2cm}

PS Einf\"uhrung in die Bioinformatik\\
\vspace{0.5cm}
Group Number 3

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

\newpage
\vspace*{2cm}
\Huge
Introduction
\Large

\begin{itemize}
	\item NETGen
	\item - Table of functions
	\item - Cubic spline interpolation
	\item -- Smooth first and continuous second derivative
	\item Motivation : Monotonic vs Periodic Activation Functions by Sopena, Romero and Alqu\'ezar
\end{itemize}

\vspace{5cm}
\Huge 
Implementation
\Large

\begin{itemize}
	\item SNNS activation function
	\item General Function Parameter (GFP)
	\item Cubic spline
\end{itemize}

\newpage

\vspace*{2cm}
\Huge
Standard Benchmarks -- Analytical Approach\\
\Large
\vspace{2cm}

XOR\\

\begin{itemize}
	\item Net architecture
	\item Implemented sine vs interpolated sine vs adapted xor function.
\end{itemize}

\vspace{2cm}

Two Quadrants\\

\begin{itemize}
	\item Heaviside - XOR
	\item Sine - XOR
	\item Adapted heaviside
\end{itemize}

\newpage

\begin{figure}[h]
\centering
\xymatrix{
\bigcirc_{i_1} \ar[dr] & \\
 & \bigcirc_{o_1} \\
\bigcirc_{i_1} \ar[ur] & 
}
\caption{\label{xor:arc} Architecture for the XOR problem.}
\end{figure}



\begin{figure}[h]
\centering
\includegraphics{xor_sinus.ps}\\
\caption{\label{figure:xor:sine}
\small
Comparation in learing the XOR Problem with sinus and cubic spline.
Both used the same simple architecture network(figure~\ref{xor:arc}) and
the same learning pattern, only $f1$ used an interpolated sine as activation function 
in the output neuron, and $f2$
used the customized cubic spline.
}
\end{figure}


\begin{figure}[!h]
\centering
\xymatrix{
\bigcirc_{i_1} \ar[d]_<<<<<{1}& & \bigcirc_{i_2} \ar[d]^<<<<<{1} \ar@{..}[r] & input\ layer(2) \\
\bigcirc_{h_1} \ar[dr]_<<<<<{1}& & \bigcirc_{h_2} \ar[dl]^<<<<<{1} \ar@{..}[r] & hidden\ layer(2) \\
 & \bigcirc_{o} \ar@{..}[rr] &  & output\ layer(1)
}
\caption{\label{figure:quad:architecture}Architecture of the ANN for the Quadrants Problem.}
\end{figure}



\newpage

\begin{figure}[h]
\centering
\includegraphics{quad2_snns.ps}
\caption{\label{figure:quad2} Error surface for quadrants problem using adapted heaviside functions in the hidden layer.}
\end{figure}

\begin{figure}[!bh]
\centering
\includegraphics{quad_projection_col.ps}
\caption{\label{figure:projection}Actual output of the Quadrants Problem using the same architecture as in 
figure~\ref{figure:quad:architecture} with adapted heaviside as activation fuction.}
\end{figure}




\newpage

\vspace*{2cm}
\Huge
Problems
\Large

\begin{itemize}
	\item Computation time due to file access
	\item Learning nearly impossible in extrapolated part of the Cubic Spline
	\item Due to extrapolation the spline is not continuous in the terminal points
\end{itemize}

\vspace{3cm}
\Huge
Outlook
\Large
\begin{itemize}
	\item NetGen 
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%end%%%%%%%%%%%%%%%%%%%%%
\end{document}
