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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 3 using TISEAN Nonlinear Time SeriesRoutines</title></head> <h1>Exercises using TISEAN<br><font color=blue>Part III: Time and length scales, structure,Poincare-maps, and surrogates</font></h1> </center><hr>Download and use alternatingly the data sets <a href="amplitude.dat"><b>amplitude.dat</b></a>,<a href="arch.dat"><b>arch.dat</b></a>, and <a href="whatisit.dat"</a><b>whatisit.dat</b></a><br> <br><b>Visual inspection</b><ul><li> For all data which you use for the first time:<br> look at the time series (amount of data, obviousartefacts, typical time scales, qualitative behaviour on short times), <br>compute the distribution (<a href="../docs_c/histogram.html">histogram</a>),<br> compute the auto-correlation function (<a href="../docs_c/corr.html"> corr</a>).</ul><br><b>Testing against non-stationarity</b><ul><li>Compute recurrence plots using <a href="../docs_c/recurr.html"> recurr </a> (be careful with the <font color=orange> -l </font> and the <font color=orange> -r </font> option).<br><br><li> study stationarity of <font color=blue>arch.dat</font> using the meta-recurrenceplot <a href="../docs_c/nstat_z.html"> nstat_z </a>. <br>A reasonable choice of parameters is:<br><font color=red> nstat_z arch.dat -d4 -m3 -#10 -o </font><br>Plot it in gnuplot using<br> <font color=green> set contour base</font><br><font color=green> splot 'arch.dat.nsz' with lines </font>.</ul>Without further investigation we assume that the other two datasets are stationary. We will thus extract time averaged information. Nonetheless, it is recommended to <b>always test against non-stationarity!</b><br><br><br><b>Time scales</b><ul><li> Recall typical times present in the auto-correlation functions,i.e., first zero crossing, exponential decay of an envelope.<br><br><li> Study 2-dimensional delay embeddings and determine reasonable lags.<br><br><li>Compute the space-time separation plots using <ahref="../docs_f/stp.html"> stp </a>. <br>Notice: Here you have to supply the <font color=orange> -m </font> and<font color=orange> -d </font> options to specify the embeddingspace.</br>When you want to cover a large relative time interval (e.g., <fontcolor=orange> -t500 </font>), it can be reasonable to reduce theresolution in time (e.g., <font color=orange> -#4 </font>).What is your conclusion from that about suitable Theiler-windows for each data set?</ul><a href="ex3_answer1.html"> Answers to time scales problems</a><br><br><br><b>The Poincaré surface of section</b><br>Create different map-like series of <fontcolor=blue>amplitude.dat</font>.<ul><li>Using <a href="../docs_c/poincare.html">poincare</a>, find areasonable location of the surface of section by maximizing both thevariance of the resulting data and the number of intersection points.Shift the location of the Poincaré-plane by <font color=orange> -a#</font> and flip the direction of intersection by the <fontcolor=orange> -C</font> option.<br>Example: <font color=red> poincare amplitude.dat -d8 -C1 | histogram -b0</font><br>When you have optimized the variance, store the output by use of the<font color=orange> -o </font> option and plot<br> <font color=green> plot '< delay -d8 amplitude.dat' wli,'amplitude.dat.poin' u 1:(0)</font><br> The latter allows you to control that the surface of section ismeaningful (in the plot command of the Poincaré-data, one has toreplace "0" by the number given in the <font color=orange> -a</font>option of <font color=blue>poincare</font>).<b>The number of intersections of this data set is 150</b>,for both directions of intersection.<br><br><li>Using <a href="../docs_c/extrema.html"> extrema </a>, the result isindependent of any embedding space. Use <font color=orange> -z </font>to collect minima instead of maxima and use <font color=orange> -t#</font> to suppress spurious extrema due to noise on the data.<br>Example: <font color=red>extrema amplitude.dat -z -omin.dat</font><br>You should find 152 maxima and 151 minima (Notice that there is onespurious of each due to noise. It can be identified in the 2-dembedding space of these data and by an unusually small time interval inbetween two maxima or minima, respectively. Use the <fontcolor=orange> -t#</font> option to remove them.). <br><br><br><li>Study 2-dimensional embeddings of all of the resulting timeseries.<br>Plot the resulting data as a return map: SincePoincaré data are map data, time lag 1 is the natural choice.<br><font color=green> plot '< delay max.dat','< delay min.dat','< delayamplitude.dat.poin' </font><br> In this particular case, the minima are almost identical to the pointsidentified by <font color=blue>poincare</font> with <fontcolor=orange> -C0</font>. Reason: Each sinlge revolution on theattractor is so close to harmonic, that the optimal lag of 8 is a kindof embedding of the signal and its temporal derivative. With <fontcolor=orange> -C1 -a-1</font> the resulting <fontcolor=blue>poincare</font>-data are almost identical to themaxima. For a pure sine, this should be exactly true for the optimaltime lag. All data sets have about the same variance.<br><br><li>Embed the sequences of time intervals in between intersections(second column of each of the output files).<br><font color=green> plot '< delay -c2 min.dat'</font><br>Result: also these form a reasonable attractor, since these timeintervals are also valid observables in the spirit of the embeddingtheorem. One can also use the bivariate time series of time intervalvalues and intersection points as coordinates.<br><br><li>Store the most suitable time series (best signal-to-noise ratio)for further use in <a href="ex4.html">Exercise 4</a> (our suggestion: max.dat). <br> </ul><br><b> Surrogate data</b> <br>Determine the nature of the data set <fontcolor=blue>whatisit.dat</font>! Does it represent low-dimensionalchaos? <ul><li>Find a reasonable time lag <font color=orange> d </font>from the auto-correlation function <ahref="../docs_c/corr.html">corr</a>. <br><br><li>What tells the 2-dimensional delay-embedding plot?<br><br><li>Compute the histogram in the gnuplot window,<br><font color=green> plot '< histogram whatisit.dat' w his</font><br><br><li>Compare the linear and nonlinear predictability using <a href="../docs_c/ll-ar.html">ll-ar</a> :<br><font color=red> ll-ar whatisit.dat -d10 -m3 -i2000 -s1 -o</font><br>Can you interprete the result? <a href="ex3_answer2.html">The answer</a><br><br><li>Create surrogates:<br>Create a random shuffling of the data (conserving the distribution butnot the temporal correlations) using <ahref="../docs_f/surrogates.html">surrogates</a>by<br><font color=red> surrogates whatisit.dat -i0 -o shuffle.dat</font> <br>Check: <font color=green> plot '< histogram shuffle.dat' w his,'< histogram whatisit.dat' w his </font><br>Check furthermore: The autocorrelation function of these surrogatesdecays instantaneously (<font color=green> plot '< corr shuffle.dat -D500'</font>).<br>Create phaserandomized surrogates (i.e. conserving the linear correlations butdestroying the distribution) by <br><font color=red> surrogates whatisit.dat -i0 -S -o phaseran.dat</font>. <br>(check: <font color=green> plot '< corr phaseran.dat -D500','< corrwhatisit.dat -D500'</font>and cross-check for the destruction of the distribution)<br>Compare the original data and these two surrogates by visual inspection inthe time domain and in a 2-dimensional delay embedding.<br><b>Result:</b> The data are definitely different from these two typesof surrogates!<br><br><li>Create optimal surrogates by <br><font color=red> surrogates whatisit.dat -o optimal.dat</font>,<br>and check again for distribution and auto-correlation function incomparison to the original data:<br><font color=green> plot '< corr optimal.dat -D500','< corr whatisit.dat -D500'</font><br><font color=green>plot '< histogram optimal.dat' w his,'< histogramwhatisit.dat' w his</font>.<br>If you do not find the curve for the first histogram, try<font color=green>plot '< histogram optimal.dat' w his,'< histogramwhatisit.dat' u 1:($2+.001) w his</font><br><br>Compare the two data sets in the time domain and in a two-dimensional time delayembedding.<br><br>Compare the two data sets by their predictability. (Notice that for areal surrogate data test you always have to create an ensemble ofdata sets, such as for a one-sided test with 5% confidence 19surrogates). <br>Use again <a href="../docs_c/ll-ar.html">ll-ar</a> to studyboth linear and nonlinear predictability with the same parameters as before:<br><font color=red> ll-ar optimal.dat -d10 -m3 -i2000 -s1 -o</font>,<br>and compare the results:<br><font color=green>set data style linespoints<br>plot 'whatisit.dat.ll', 'whatisit.dat.ll' u 1:3,'optimal.dat.ll','optimal.dat.ll' u 1:3</font><br> <br>Result: the results are in rather good agreementwith each other. <font color=red> Thus the hypothesis that <fontcolor=blue> whatisit.dat</font> is the nonlinear transformation of alinear stochastic process cannot be rejected.</font><br><br>In fact, the data are nonlinearly transformed AR-data.The transformation reads <font color=blue>y=atan(x/80.)</font>.<br><br><br><br><br> </body></html>⌨️ 快捷键说明
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