[simulations] first text

simulations/lecture/simulations-chapter.tex

@@ -18,12 +18,4 @@
 
 \include{simulations}
 
-\section{TODO}
-\begin{itemize}
-\item draw (and plot) random numbers
-\item draw (and plot) random functions
-\item euler forward, odeint
-\item introduce derivatives which are also needed for fitting (move box from there here)
-\end{itemize}
-
 \end{document}

simulations/lecture/simulations.tex

@@ -4,15 +4,45 @@
 \label{simulationschapter}
 \exercisechapter{Simulations}
 
-Why simulations?
-
-- Understand basic concepts (as we do in the following chapters)
-- Test your data analysis algorithms
-- Explore
-- Dynamical systems
+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 predator-prey
+interactions or the activity of neurons, evolve in time is a central
+application for simulations. Only with the availability of computers
+in the second half of the twentieth century was the exciting field of
+nonlinear dynamical systems pushed forward. Conceptually, many kinds
+of simulations are very simple and are implemented in a few lines of
+code.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Univariate data}
+The most basic simulation is to draw random numbers from a given
+distribution. This simulates repeated measurements of some quantity
+(e.g., weight of tigers or firing rate of a neuron). That is we take
+samples from a statistical population. Doing so we must specify from
+which probability distribution the data should originate from and what
+are the parameters (i.e. mean, standard deviation, ...) of that
+distribution.
+
+For drawing numbers from a normal distribution we use the
+\code{randn()} function. This function returns normally distributed
+numbers 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. For changing the mean we just add the
+desired mean to the random numbers.
+\begin{equation}
+  x_i = \mu + \sigma \xi_i
+\end{equation}
+
+
+
 draw (and plot) random numbers
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%