<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt,pdftex]{exam}

\newcommand{\ptitle}{Random walk}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Grewe}{phone: 29 74588}%
{email: jan.grewe@uni-tuebingen.de}

\begin{document}

\input{../instructions.tex}


%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Random walk with memory.}

Some animals perform a random walk when searching for food. In some
cases this random walk is not completely random. In fact, sometimes
there is some memory involved. Whenever there is a positive gradient
in food gain between successive steps the animal will continue in the
very same direction as before. When the next step leads to a decrease
in food gain the animal switches back to a random walk.


\begin{questions}
  \question{} The accompanying dataset (random\_world.mat) contains a
  single variable. This is the world (10000\,m$^2$ area with
  10\,cm spatial resolution) in which there are randomly distributed
  food sources (Gaussian blotches of food).
  
  \begin{parts}
    \part{} Create a plot of the world.\\[0.5ex]
    \part{} Create a model animal that performs a pure random walk. The
    agent can walk in eight different directions (the cardinal and
    diagonal directions) with a stepsize of 10\,cm
    (approximately). Let the agent start at a random location in the
    world and count how much food it eats in 10000 steps (eaten food
    disappears from the world, of course). If the agent bumps into the
    borders of the world choose a different direction.\\[0.5ex]
    \part{} Plot a typical example walk. (You can also make an animation
    with MATLAB)\\[0.5ex]
    \part{} Same as above, but create a model animal that has some memory,
    i.e. the direction is kept constant as long as there is a positive
    gradient in the food gain. Otherwise, a random walk is performed.\\[0.5ex]
    \part{} Plot a typical example walk also for this agent.\\[0.5ex]
    \part{} Compare the performance of the two agents. Create
    appropriate plots and apply statistical methods. You will need to
    run the simulations several times to get a good estimate of the
    neumbers.
  \end{parts}
\end{questions}

\end{document}
</pre></body></html>