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[simulations] section on random number generators

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Jan Benda 2019-12-23 10:20:46 +01:00
parent 02e9c496a5
commit a04f0ec718
2 changed files with 42 additions and 7 deletions
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37
simulations/lecture/simulations.tex
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@ -21,6 +21,40 @@ field of nonlinear dynamical systems forward. Conceptually, many kinds
of simulations are very simple and are implemented in a few lines of
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
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Univariate data}
The most basic type of simulation is to draw random numbers from a
@ -77,8 +111,9 @@ gamma
\subsection{Random integers}
\code{randi()}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Static nonlinearities}
\section{Bivariate data and static nonlinearities}
\begin{figure}[t]
\includegraphics[width=1\textwidth]{staticnonlinearity}

12
simulations/lecture/staticnonlinearity.py
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@ -8,14 +8,14 @@ def boltzmann(x, x0, k):
return 1.0/(1.0+np.exp(-k*(x-x0)))
if __name__ == "__main__":
n = 70
n = 50
xmin = -18.0
xmax = 18.0
x0 = 2.0
k = 0.3
k = 0.25
sigma = 0.08
rng = np.random.RandomState(38281)
x = (xmax-xmin)*rng.rand(n) + xmin
rng = np.random.RandomState(15281)
x = np.linspace(xmin, xmax, n)
y = boltzmann(x, x0, k) + sigma*rng.randn(len(x))
xx = np.linspace(xmin, xmax, 200)
yy = boltzmann(xx, x0, k)
@ -30,9 +30,9 @@ if __name__ == "__main__":
ax1.set_xlabel('Hair deflection / nm')
ax1.set_ylabel('Open probability')
ax1.set_xlim(-20, 20)
ax1.set_ylim(-0.17, 1.17)
ax1.set_ylim(-0.2, 1.17)
ax1.set_xticks(np.arange(-20.0, 21.0, 10.0))
ax1.set_yticks(np.arange(0, 1.1, 0.2))
ax1.set_yticks(np.arange(-0.2, 1.1, 0.2))
ax2 = fig.add_subplot(spec[0, 1])
show_spines(ax2, 'lb')