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\title{Simulation of \cite{Grainger245} with Rescorla Wagner equations}
\shorttitle{Grainger et al. (2012) simulation with RW equations}
\author{Robert Geirhos (3827808), Klara Grethen (3899962), \\David-Elias Künstle (3822829), Felicia Saar (3818590)}
\affiliation{Linguistics for Cognitive Science Course, University of Tübingen}

\abstract{TODO TODO TODO our abstract goes here TODO TODO TODO}

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\begin{document}
\maketitle

%\cite{}
\section{Introduction}
\todo{statement of the problem}

\section{Simulations}
\subsection{Experimental Code}
The simulation code is split in three parts, the creation of the trials, the learning of the monkey and the analysis of the learning results, implemented in the \emph{R Programming Language} \parencite{Rcore}.

\subsubsection{Trial creation}
The algorithm follows in general the structure defined in the reference paper and supplemental materials and described above.
The word-nonword corpus is the one used by the monkey DAN in \cite{Grainger245}.

The lack of information leaded to our own design decision in some edge cases.
Trials will always be created in blocks of 100.
To ensure this constraint the new word block part can be replaced by learned words if there is no new word left in the corpus and vice versa if there's no word learned the learned word part will be filled by the new word.
The new words, learned words and nonwords get picked randomly out of their pool with repetition allowed.

\subsubsection{Monkey learning}
After a block the presented new word can be marked as learned by the definition in \cite{Grainger245}. The rescorla wagner learner therefore has to learn a block, return the guesses and then continue learning with the next block.
This is not easily possible with \emph{ndl} \parencite{Rndl} where for we implemented a rescorla wagner learner ourself.

Since preliminary experiments showed that the monkeys performed with very high accuracies (>90\%), we decided to introduce a random parameter $ r $ in the experiment, defined as the fraction of times the monkey would make a random guess instead of an experience-based prediction.

\subsubsection{Data analysis}
To compare the accuracy with different learning rates we used not only standard tools like linear regression models \emph{(lm)} and \emph{anova} \parencite{Rcore} but also more advanced non linear general additive models \emph{(GAM)} provided by the package \emph{mgcv} \parencite{Rmgcv} compared and visualized with \emph{itsadug} \parencite{Ritsadug}

\subsection{Choice of Parameters}
\subsubsection{Number of Trials}
The six monkeys in the original experiment participated in a different number of trials (min: 43.041, max: 61.142, mean: 52.812). For the sake of simplicity, we presented exactly 50.000 trials in each of our experiments.
\subsubsection{Random Parameter} The random parameter $ r $ was set to 0.65, which proved to be reasonable value. That means, in 65\% of the cases the monkey would guess for either word or nonword with equal probabilities. Therefore, the maximum possible performance $ p_{max} $ is:
$$ p_{max} = 1 - \frac{r}{2} = 0.675$$
In other words, the maximum possible performance is no longer 1.0 (for a very intelligent monkey) but rather restricted by $ r $. If a monkey's performance is slightly better than $ p_{max} $, this is assured to be due to chance.
\subsubsection{Alpha and Beta} Both $ \alpha $ and $ \beta $ were our independent variables which we manipulated over the course of the experiments. We gathered data for every possible combination of $ \alpha $ and $ \beta $ values within an equally spaced range from 0.0 to 0.3. A total of 15 values for each $ \alpha $ and $ \beta $ were combined to $ 15*15 = 225 $ possible combinations. Since $ \alpha $ and $ \beta $ were internally multiplied to a single value, we expected the outcome to be symmetrical due to the commutativity of the multiplication operation and therefore calculated each combination of $ \alpha $ and $ \beta $ only once, which we used as a trick to improve the overall runtime. Therefore, $\sum_{i=1}^{15}i = 120$ combinations  remained to be explored.

\subsubsection{Lambda}
The independent variable $\lambda$ represents the maximum activation in the Rescorla-Wagner model and therefore limits the learning.
It makes it possible to modulate saliency of a stimulus. A more salient stimulus could not only have higher learning rates but also a higher maximum activation. In the original experiment the stimulus were same colored words and nonwords with four letters on a equally colored background. We assume the single words and nonwords are equally salient and keep therefore $\lambda$ constant (1). 

\subsection{Running Parallelized Experiments}
Running an experiment with a single combination of $ \alpha $ and $ \beta $ on a normal desktop computer took about 75 minutes. Therefore, the parameter space one could explore within a reasonable amount of time was quite restricted. We decided to write a parallelized version of the code to reduce the overall runtime. Using the R packages foreach \parencite{Rforeach}, parallel \parencite{Rparallel} and doParallel \parencite{RdoParallel}, restructured the experiment. Since conflicts can easily occur when more than one core is trying to access a shared data structure at the same time, we implemented a parallelized version that is able to run without even containing critical sections. Instead, each thread has its own data structure, a .txt file, and in the end the results are harvested and combined. This version of the experiment ran on a cluster with 15 cores, each performing a total amount of eight experiments. Altogether, 120 combinations of $ \alpha $ and $ \beta $ were explored overnight, which would have taken about 150 hours in a non-parallelized version.

\section{Results}
\todo{results}

\begin{figure*}[ht]
  \centering
  \includegraphics[width=0.9\textwidth]{../plots/plot_accuracy}
  \caption{
    Top row shows model output  accuracies in dependence of modulated alpha and beta.
    Second row visualizes corresponding nonlinear regressions (GAM).
    Accuracy seem to approximate a maximal accuracy with growing alpha, beta parameter.
    In the GAM you see the small influence of one of the parameter.
    Therefore this resoults could be approximated with just one nonlinear parameter.
  }
  \label{fig:accuracy}
\end{figure*}

\begin{figure*}[ht]
  \centering
  \includegraphics[width=0.9\textwidth]{../plots/plot_numwords}
  \caption{
    The left plot shows the raw num words learned of the model with modulated parameter (alpha, beta).
    The model performs always quite well, just several parameter value result in lower perfomance. The corresponding nonlinear regression plot (middle) doesn't mirror a first hipothesis of growing words learned with growing parameter values.
    This is not necessarily a consequense of a wrong hypothesis but of a wrong regression model because of the weak data with very high frequency of around 305 learned words but almost no other number of learned words (right plot).
}
  \label{fig:numwords}
\end{figure*}


\section{Discussion}
\todo{"your conclusions about what is most likely to underlie the different success rates of the baboons" In our simulation the influence of the random factor was a lot higher than of the learning rates. So the motivation of really trying to make a correct guess could result in different success rates.}

\printbibliography{}

\appendix

\onecolumn

\section{Complete Results}
Here are the complete results of our experiments. The abbreviations used are:
\begin{APAitemize}
\item \#Trials: Number of trials
\item \#LearnedW: Number of learned words
\item \#PresW: Number of presented words
\item GenAcc: General accuracy
\item WAcc: Word accuracy
\item NWAcc: Nonword accuracy
\end{APAitemize}

\input{result_tables.tex}

\lstinputlisting[language=R]{../baboonSimulation.R}

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