119 lines
5.2 KiB
TeX
119 lines
5.2 KiB
TeX
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\chapter{Simulations}
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\label{simulationschapter}
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\exercisechapter{Simulations}
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The real power of computers for data analysis is the possibility to
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run simulations. Experimental data of almost unlimited sample sizes
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can be simulated in no time. This allows to explore basic concepts,
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like the ones we introduce in the following chapters, with well
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controlled data sets that are free of confounding pecularities of real
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data sets. With simulated data we can also test our own analysis
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functions. More importantly, by means of simulations we can explore
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possible outcomes of our planned experiments before we even started
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the experiment or we can explore possible results for regimes that we
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cannot test experimentally. How dynamical systems, like for example
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predator-prey interactions or the activity of neurons, evolve in time
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is a central application for simulations. Computers becoming available
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from the second half of the twentieth century on pushed the exciting
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field of nonlinear dynamical systems forward. Conceptually, many kinds
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of simulations are very simple and are implemented in a few lines of
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code.
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\section{Univariate data}
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The most basic type of simulation is to draw random numbers from a
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given distribution like, for example, the normal distribution. This
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simulates repeated measurements of some quantity (e.g., weight of
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tigers or firing rate of neurons). Doing so we must specify from which
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probability distribution the data should originate from and what are
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the parameters (mean, standard deviation, shape parameters, etc.)
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that distribution. How to illuastrate and quantify univariate data, no
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matter whether they have been actually measured or whether they are
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simulated as described in the following, is described in
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chapter~\ref{descriptivestatisticschapter}.
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\subsection{Normally distributed data}
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For drawing numbers $x_i$ from a normal distribution we use the
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\code{randn()} function. This function returns normally distributed
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numbers $\xi_i$ with zero mean and unit standard deviation. For
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changing the standard deviation we need to multiply the returned data
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values with the required standard deviation $\sigma$. For changing the
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mean we just add the desired mean $\mu$ to the random numbers:
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\begin{equation}
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x_i = \sigma \xi_i + \mu
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\end{equation}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{normaldata}
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\titlecaption{\label{normaldatafig} Generating normally distributed
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data.}{With the help of a computer the weight of 300 tigers can be
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measured in no time using the \code{randn()} function (left). By
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construction we then even know the population distribution (red
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line, right), its mean (here 220\,kg) and standard deviation
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(40\,kg) from which the simulated data values were drawn (yellow
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histogram).}
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\end{figure}
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\begin{exercise}{normaldata.m}{normaldata.out}
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First, read the documentation of the \varcode{randn()} function and
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check its output for some (small) input arguments. Write a little
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script that generates $n=100$ normally distributed data simulating
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the weight of Bengal tiger males with mean 220\,kg and standard
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deviation 40\,kg. Check the actual mean and standard deviation of
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the generated data. Do this, let's say, five times using a
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for-loop. Then increase $n$ to 10\,000 and run the code again. It is
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so simple to measure the weight of 10\,000 tigers for getting a
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really good estimate of their mean weight, isn't it? Finally plot
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from the last generated data set of tiger weights the first 1000
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values as a function of their index.
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\end{exercise}
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\subsection{Other probability densities}
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\code{rand()}
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gamma
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\subsection{Random integers}
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\code{randi()}
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\section{Static nonlinearities}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{staticnonlinearity}
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\titlecaption{\label{staticnonlinearityfig} Generating data
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fluctuating around a function.}{The open probability of the
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mechontransducer channel in hair cells of the inner ear is a
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saturating function of the deflection of hairs (left, red line).
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Measured data will fluctuate around this function (blue dots).
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Ideally the residuals (yellow histogram) are normally distributed
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(right, red line).}
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\end{figure}
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Example: mechanotransduciton!
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draw (and plot) random functions (in statistics chapter?)
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\section{Dynamical systems}
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\begin{itemize}
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\item euler forward, odeint
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\item introduce derivatives which are also needed for fitting (move box from there here)
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\item Passive membrane
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\item Add passive membrane to mechanotransduction!
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\item Integrate and fire
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\item Fitzugh-Nagumo
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\item Two coupled neurons? Predator-prey?
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\end{itemize}
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\section{Summary}
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with outook to other simulations (cellular automata, monte carlo, etc.)
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\printsolutions
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