[simulations] some improvements

simulations/lecture/simulations.tex

@@ -7,12 +7,12 @@
 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
+the ones we introduce in the following chapters and many more, 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
+possible outcomes of our 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
@@ -23,31 +23,41 @@ code.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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
+At the heart of many simulations are random numbers that we get from
+\enterm[random number generator]{random number generators}.  These are
+numerical algorithms that return sequences of numbers that appear to
+be as random as possible. If we draw random numbers using, for
+example, the \code{rand()} function, then these numbers are indeed
+uniformly distributed and have a mean of one half. Subsequent numbers
+are 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 of numbers. This differentiates them from truely random
+numbers and hence they are called \enterm[random number
+generator!pseudo]{pseudo random number generators}. In rare cases this
+periodicity can induce problems in simulations whenever more random
+numbers than the period of the random number generator are
+used. Luckily, nowadays the periods of random nunmber generators are
+very large, $2^{64}$, $2^{128}$, or even larger.
+
+The pseudo randomness of numerical random number generators also has
+an advantage.  They allow to repeat exactly the same sequence of
+random numbers. After defining the state of the generator or setting a
+\enterm{seed} with the \code{rng()} function, a particular 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.
+generator. This way we can not only precisly define the statistics of
+noise in our simulated data, but we can repeat an experiment with
+exactly the same sequence of noise values.  This is useful for plots
+that involve some random numbers but should look the same whenever the
+script is run.
 
 \begin{exercise}{}{}
   Generate three times the same sequence of 20 uniformly distributed
   numbers using the \code{rand()} and \code{rng()} functions.
+
+  Generate 10\,000 uniformly distributed random numbers and compute
+  the correlation coefficient between each number and the next one in
+  the sequence. This is the serial correlation at lag one.
 \end{exercise}
 
 \begin{figure}[t]
@@ -72,8 +82,8 @@ 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
+matter whether they have been actually measured or whether they have
+been simulated as described in the following, is described in
 chapter~\ref{descriptivestatisticschapter}.
 
 \subsection{Normally distributed data}
@@ -142,6 +152,7 @@ draw (and plot) random functions (in statistics chapter?)
 \section{Dynamical systems}
 
 \begin{itemize}
+\item iterated maps
 \item euler forward, odeint
 \item introduce derivatives which are also needed for fitting (move box from there here)
 \item Passive membrane

simulations/lecture/staticnonlinearity.py

@@ -22,7 +22,7 @@ if __name__ == "__main__":
 
     fig = plt.figure(figsize=cm_size(16.0, 6.0))
     spec = gridspec.GridSpec(nrows=1, ncols=2,
-                             left=0.10, bottom=0.23, right=0.97, top=0.96, wspace=0.4)
+                             left=0.12, bottom=0.23, right=0.97, top=0.96, wspace=0.4)
     ax1 = fig.add_subplot(spec[0, 0])
     show_spines(ax1, 'lb')
     ax1.plot(xx, yy, colors['red'], lw=2)