%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Simulations} \label{simulationschapter} \exercisechapter{Simulations} 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, 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 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 from the second half of the twentieth century on pushed the exciting 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 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 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}{randomnumbers.m}{randomnumbers.out} First, read the documentation of the \varcode{rand()} function and check its output for some (small) input arguments. Generate three times the same sequence of 10 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] \includegraphics[width=1\textwidth]{randomnumbers} \titlecaption{\label{randomnumbersfig} Random numbers.} {Numerical random number generators return sequences of numbers that are as random as possible. The returned values approximate a given probability distribution. Here, for example, a uniform distribution between zero and one (top left). The serial correlation (auto-correlation) of the returned sequences is zero at any lag except for zero (bottom left). However, by setting a seed, defining the state, or spawning the random number generator, the exact same sequence can be generated (right).} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Univariate data} The most basic type of simulation is to draw random numbers from a given distribution like, for example, the normal distribution. This simulates repeated measurements of some quantity (e.g., weight of 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 have been simulated as described in the following, is described in chapter~\ref{descriptivestatisticschapter}. \subsection{Normally distributed data} For drawing numbers $x_i$ from a normal distribution we use the \code{randn()} function. This function returns normally distributed numbers $\xi_i$ 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 $\sigma$. For changing the mean we just add the desired mean $\mu$ to the random numbers: \begin{equation} x_i = \sigma \xi_i + \mu \end{equation} \begin{figure}[t] \includegraphics[width=1\textwidth]{normaldata} \titlecaption{\label{normaldatafig} Generating normally distributed data.}{With the help of a computer the weight of 300 tigers can be measured in no time using the \code{randn()} function (left). By construction we then even know the population distribution (red line, right), its mean (here 220\,kg) and standard deviation (40\,kg) from which the simulated data values were drawn (yellow histogram).} \end{figure} \begin{exercise}{normaldata.m}{normaldata.out} Write a little script that generates $n=100$ normally distributed data simulating the weight of Bengal tiger males with mean 220\,kg and standard deviation 40\,kg. Check the actual mean and standard deviation of the generated data. Do this, let's say, five times using a for-loop. Then increase $n$ to 10\,000 and run the code again. It is so simple to measure the weight of 10\,000 tigers for getting a really good estimate of their mean weight, isn't it? Finally plot from the last generated data set of tiger weights the first 1000 values as a function of their index. \end{exercise} \subsection{Other probability densities} \code{rand()} gamma \subsection{Random integers} \code{randi()} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Bivariate data and static nonlinearities} \begin{figure}[t] \includegraphics[width=1\textwidth]{staticnonlinearity} \titlecaption{\label{staticnonlinearityfig} Generating data fluctuating around a function.}{The conductance of the mechontransducer channels in hair cells of the inner ear is a saturating function of the deflection of their hairs (left, red line). Measured data will fluctuate around this function (blue dots). Ideally the residuals (yellow histogram) are normally distributed (right, red line).} \end{figure} Example: mechanotransduciton! 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 \item Add passive membrane to mechanotransduction! \item Integrate and fire \item Fitzugh-Nagumo \item Two coupled neurons? Predator-prey? \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Summary} with outook to other simulations (cellular automata, monte carlo, etc.) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printsolutions