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<html><body bgcolor=white><head><center><table><tr><td align=center><b>Rainer Hegger</b></td> <td width=20></td> <td align=center><b>Holger Kantz</b></td> <td width=20></td> <td align=center><b>Thomas Schreiber</b></td></tr></table><title>Exercise 2 using TISEAN Nonlinear Time SeriesRoutines</title></head> <h1>Exercises using TISEAN<br><font color=blue>Part II: Linear models and simple prediction</font></h1></center><hr>Download the data set <a href="amplitude.dat"><b>amplitude.dat</b></a> toyour local directory for use in this exercise (Press the "Shift"-keyand the left mouse button).<br> <br><b> Visual analysis of data, time scales, and correlations</b><br><ul><li> Inspect the time series visually, e.g. by gnuplot (amount of data, obviousartefacts, typical time scales, qualitative behaviour on short times)<br><br><li> Compute the autocorrelation function (<ahref="../docs_c/corr.html">corr</a>)<br><br> <li> Which is a reasonable order for an AR-model?<br>Use <a href="../docs_c/ar-model.html">ar-model</a> to fit AR-modelsto the data.<br> Study the residuals, i.e. the differences between determinsitic partof the AR-model and the next observations. Inside gnuplot:<br><font color=green> plot [0:1000]'< ar-model amplitude.dat -p10'u($0+10):1, '< ar-model amplitude.dat -p50' u($0+50):1</font><br>Plot the data also in reversed order (since one curve partly hides theother), and together with <fontcolor=blue>amplitude.dat</font>. Read the description of <ahref="../docs_c/ar-model.html">ar-model</a> to understand what yousee in the plot, and reduce and increasethe order of the model (controlled by the<font color=orange> -p</font> option) as far as your patience allowsyou to go (the computation time increases quadratically in<font color=orange> p</font>).<br><br><li> Result: the residuals have pronounced spikes at certain points of the timeseries even for very large order of the model. This demonstrates that the data do not stem from a linearstochastic process. Nonetheless, their magnitude compared to theamplitude of the signal is small. Hence, if one wants to use a linear model,<font color=orange> p=10</font> is a reasonable compromise betweenmodel complexity and performance.<br><br><li> Now use <a href="../docs_c/ar-model.html">ar-model</a> to produce a new time series:<br> <font color=red> ar-model -s5000 amplitude.dat -p10 -o</font>,the output in <font color=blue>amplitude.dat.ar</font> is now, withthe <font color=orange> -s5000</font> option, the iterated model timeseries of length 5000.<br><br><li> Compare the two time series in the time domain. Also, compute the histograms using theroutine <a href="../docs_c/histogram.html">histogram</a>:<br><font color=red>mycomputer> histogram amplitude.dat -b0 </font><br><font color=blue>Using amplitude.dat as datafile, reading column 1<br>Use 5000 lines.<br>Writing to stdout<br>#interval of data: [-1.463000e+01:1.727000e+01]<br>#average= 1.463300e-01<br>#standard deviation= 7.994755e+00<br></font>The ar-data have zero mean by construction. If you wish to superimposethe two histograms, you thus should shift the one with respect to theother by the mean value of the data:<br><font color=green> set data style histep<br>plot '< histogram amplitude.dat' u ($1-.146):2,'< histogramar.dat' </font> <br>Result: The data sets are differnt: the distribution of <fontcolor=blue> ar.dat</font> is closer to a Gaussian (and converges to aGaussian for longer time series, try <font color=green> plot '< ar-run -l100000amplitude.dat.ar | histogram' </font>).<br><br><li> Compute the auto-correlation functions and the powerspectra (by either <a href="../docs_c/mem_spec.html">mem_spec</a> or <a href="../docs_f/spectrum.html">spectrum</a>) of both of them:<br><font color=red> corr amplitude.dat -D500 -o<br>corr ar.dat -D500 -o</font><br><font color=green>set data style lines</font><br><font color=green> plot 'ar.dat.cor','amplitude.dat.cor'</font><br><font color=red>spectrum amplitude.dat -o<br>spectrum ar.dat -o</font><br><font color=green> set logscale y<br> plot 'amplitude.dat_sp','ar.dat_sp'</font><br>Result: The AR-data contain the same temporal correlations, but theydecay much faster than in <font color=blue> amplitude.dat</font>.<br>The spectra have to be compared with both linear and logarithmicy-scale. The frequency around 0.03 is dominant in both data sets, theharmonics of that visible in <font color=blue>amplitude.dat_sp</font>are suppressed in <font color=blue>ar.dat_sp</font>. This reflectsthat the AR-model contains the relevant time scales, but has shortcomingsin a quantitative comparison. However, these are not too dramatic whenonly viewed with second order statistics. The differences will be moreevident in the higher order correlations and other nonlinear concepts.<br><br><li>Repeat the exercise starting from the ar-data you generated (filear.dat). You should observe that fitting an ar-model to ar-data willyield residuals with a gaussian distribution, and that the histograms, auto-correlationfunctions and power spectra of the model data are identical to thoseof the input data, if the order of the fit (<fontcolor=orange> -p</font>) is not smaller than the order of the model bywhich the data were produced.<br><br></ul><b>Embedding and time lags</b><br><ul><li> Visualize both amplitude.dat and ar.dat in a delay embedding (donot forget to reset the gnuplot, e.g., <font color=green> set nologs</font>), using <a href="../docs_f/delay.html">delay </a>:<br>Start with <font color=orange>-d1</font> and increase it, at least upto 50. What is optimal by a) visual impression, and what should beoptimal when b) considering the auto-correlation function?<br><b> Answers:</b><br> <font color=blue> amplitude.dat</font>: a)About 8, when unfolding is good but overlap is still small. b) about 8: the first zero of the autocorrelation functionwould be optimal for a harmonic, periodic signal embedded in 2dimensions. <br><font color=blue>ar.dat</font>: a) for delay 8, the shape of the blob of lines comes close tocircular, hence indicating sufficient decorrelation of the componentsof the delay vectors. b) The auto-correlation function yields aboutthe same as for <font color=blue> amplitude.dat</font>.</ul><br><b>Determinism and predictability</b><br><ul><li> compute the false nearest neighbour statistics (<a href="../docs_c/false_nearest.html">false_nearest</a>):<br><font color=red> false_nearest amplitude.dat -M8 -d8 -o-t200 -f5 </font><br>Study the output, <font color=blue>amplitude.dat.fnn</font>, andobserve the invariance of the result (namely that the embedding dimension3 is insufficient but 4 is o.k.) under change of the time lag. <br><br><li> Use the zeorth-order predictor (<a href="../docs_c/zeroth.html">zeroth</a>)on amplitude.dat and on ar.dat.<br><font color=red> zeroth amplitude.dat -m1,4 -d8 -o-s250<br>zeroth ar.dat -m1,4 -d8 -o-s250</font><br><font color=green>plot[][0:1.5] 'amplitude.dat.zer','ar.dat.zer',.05*exp(.02*x) </font></ul><br>You should be able to verify the following observations:<br>For increasing prediction horizion, the prediction errors of amplitude.datshow two regimes: Exponential increase of the error due to chaos(the regime of nonlinear deterministic dynamics), slow linearincrease due to loss of phase locking (the regime of linearcorrelations due to the rather constant period of the oscillations), constantwhen the predictions lose all correlations to the actualvalues (limit of unpredictability for a large prediction horizon of moretime steps than can be computed with this data set, the relative prediction error saturates at 1. In order toarrive a prediction horizons larger than one half of the data set,you must switch off the causality window by the <fontcolor=orange>-C0</font> option in <a href="../docs_c/zeroth.html">zeroth</a>). <br> <br>No succesful prediction for ar.dat beyond the linear correlations.Since ar.dat is a linear stochastic data set,it does not contain phase space information.<br><br><br></body></html>⌨️ 快捷键说明
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