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\chapter{Simulations}
\label{simulationschapter}
\exercisechapter{Simulations}

The real power of computers for data analysis is the possibility to
run simulations. Experimental data of almost unlimited sample sizes
can be simulated in no time. This allows to explore basic concepts,
like the ones we introduce in the following chapters, with well
controlled data sets that are free of confounding pecularities of real
data sets. With simulated data we can also test our own analysis
functions. More importantly, by means of simulations we can explore
possible outcomes of our planned experiments before we even started
the experiment or we can explore possible results for regimes that we
cannot test experimentally. How dynamical systems, like for example
predator-prey interactions or the activity of neurons, evolve in time
is a central application for simulations. Computers becoming available
from the second half of the twentieth century on pushed the exciting
field of nonlinear dynamical systems forward. Conceptually, many kinds
of simulations are very simple and are implemented in a few lines of
code.

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\section{Random numbers}
At the heart of many simulations are random numbers. Pseudo random
number generator XXX.  These are numerical algorithms that return
sequences of numbers that appear to be as random as possible. If we
draw random number using, for example, the \code{rand()} function,
then these numbers are indeed uniformly distributed and have a mean of
one half. Subsequent numbers are also independent of each other,
i.e. the autocorrelation function is zero everywhere except at lag
zero. However, numerical random number generators have a period, after
which they repeat the exact same sequence. This differentiates them
from truely random numbers and hence they are called \enterm{pseudo
  random number generators}. In rare cases this periodicity can induce
problems in your simulations. Luckily, nowadays the periods of random
nunmber generators very large, $2^{64}$, $2^{128}$, or even larger.

An advantage of pseudo random numbers is that they can be exactly
repeated given a defined state or seed of the random number
generator. After defining the state of the generator or setting a
\term{seed} with the \code{rng()} function, the exact same sequence of
random numbers is generated by subsequent calls of the random number
generator. This is in particular useful for plots that involve some
random numbers but should look the same whenever the script is run.

Figure XXX: three sequences - initial one, second different one with
seed, third with same seed. Fourth panel with autocorrelation
function.

\begin{exercise}{}{}
  Generate three times the same sequence of 20 uniformly distributed
  numbers using the \code{rand()} and \code{rng()} functions.
\end{exercise}


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\section{Univariate data}
The most basic type of simulation is to draw random numbers from a
given distribution like, for example, the normal distribution. This
simulates repeated measurements of some quantity (e.g., weight of
tigers or firing rate of neurons). Doing so we must specify from which
probability distribution the data should originate from and what are
the parameters (mean, standard deviation, shape parameters, etc.)
that distribution. How to illuastrate and quantify univariate data, no
matter whether they have been actually measured or whether they are
simulated as described in the following, is described in
chapter~\ref{descriptivestatisticschapter}.

\subsection{Normally distributed data}
For drawing numbers $x_i$ from a normal distribution we use the
\code{randn()} function. This function returns normally distributed
numbers $\xi_i$ with zero mean and unit standard deviation. For
changing the standard deviation we need to multiply the returned data
values with the required standard deviation $\sigma$. For changing the
mean we just add the desired mean $\mu$ to the random numbers:
\begin{equation}
  x_i = \sigma \xi_i + \mu
\end{equation}

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{normaldata}
  \titlecaption{\label{normaldatafig} Generating normally distributed
    data.}{With the help of a computer the weight of 300 tigers can be
    measured in no time using the \code{randn()} function (left).  By
    construction we then even know the population distribution (red
    line, right), its mean (here 220\,kg) and standard deviation
    (40\,kg) from which the simulated data values were drawn (yellow
    histogram).}
\end{figure}

\begin{exercise}{normaldata.m}{normaldata.out}
  First, read the documentation of the \varcode{randn()} function and
  check its output for some (small) input arguments.  Write a little
  script that generates $n=100$ normally distributed data simulating
  the weight of Bengal tiger males with mean 220\,kg and standard
  deviation 40\,kg. Check the actual mean and standard deviation of
  the generated data. Do this, let's say, five times using a
  for-loop. Then increase $n$ to 10\,000 and run the code again. It is
  so simple to measure the weight of 10\,000 tigers for getting a
  really good estimate of their mean weight, isn't it?  Finally plot
  from the last generated data set of tiger weights the first 1000
  values as a function of their index.
\end{exercise}

\subsection{Other probability densities}
\code{rand()}
gamma

\subsection{Random integers}
\code{randi()}


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\section{Bivariate data and static nonlinearities}

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{staticnonlinearity}
  \titlecaption{\label{staticnonlinearityfig} Generating data
    fluctuating around a function.}{The open probability of the
    mechontransducer channel in hair cells of the inner ear is a
    saturating function of the deflection of hairs (left, red line).
    Measured data will fluctuate around this function (blue dots).
    Ideally the residuals (yellow histogram) are normally distributed
    (right, red line).}
\end{figure}

Example: mechanotransduciton!

draw (and plot) random functions (in statistics chapter?)

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\section{Dynamical systems}

\begin{itemize}
\item euler forward, odeint
\item introduce derivatives which are also needed for fitting (move box from there here)
\item Passive membrane
\item Add passive membrane to mechanotransduction!
\item Integrate and fire
\item Fitzugh-Nagumo
\item Two coupled neurons? Predator-prey?
\end{itemize}

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\section{Summary}
with outook to other simulations (cellular automata, monte carlo, etc.)



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\printsolutions