68 lines
2.8 KiB
TeX
68 lines
2.8 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 predator-prey
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interactions or the activity of neurons, evolve in time is a central
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application for simulations. Only with the availability of computers
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in the second half of the twentieth century was the exciting field of
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nonlinear dynamical systems pushed 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 simulation is to draw random numbers from a given
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distribution. This simulates repeated measurements of some quantity
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(e.g., weight of tigers or firing rate of a neuron). That is we take
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samples from a statistical population. Doing so we must specify from
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which probability distribution the data should originate from and what
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are the parameters (i.e. mean, standard deviation, ...) of that
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distribution.
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For drawing numbers from a normal distribution we use the
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\code{randn()} function. This function returns normally distributed
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numbers with zero mean and unit standard deviation. For changing the
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standard deviation we need to multiply the returned data values with
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the required standard deviation. For changing the mean we just add the
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desired mean to the random numbers.
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\begin{equation}
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x_i = \mu + \sigma \xi_i
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\end{equation}
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draw (and plot) random numbers
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\section{Static nonlinearities}
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draw (and plot) random functions
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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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\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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