📄 ex4.html
字号:
<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 4 using TISEAN Nonlinear Time SeriesRoutines</title></head> <h1>Exercises using TISEAN<br><font color=blue>Part IV: Dimensions, Lyapunov exponents and entropies</font></h1></center><hr>Use again the data set <a href="amplitude.dat"><b>amplitude.dat</b></a>.<br> <br><ul><li> Recall reasonable embedding parameters from the results of thelast exercises and their Theiler-windows.<br><br><li>Compute the correlation sums of the sets using <a href="../docs_c/d2.html"> d2 </a>.<br><font color=red> d2 amplitude.dat -d8 -t100 -o</font><br>Study carefully the output files <font color=blue>amplitude.dat.c2</font> and<font color=blue>amplitude.dat.d2</font>.<br> <br> Can you give reasonable dimension estimates?<br>The best result will be obtained by fitting by eye a power law to the<font color=blue>amplitude.dat.c2</font>-curves:<br><font color=green>set logscale<br>plot 'amplitude.dat.c2',.001*x**2.1<br></font>Errors of the <font color=blue>ampitude.dat.d2</font>-curves areanti-correlated! Use <a href="../docs_c/av-d2.html">av-d2</a> withthe option <font color=orange> -a#</font> to smoothen them. Observe theedge effects and folding effects in the m=1-curve, on larger scales also the m>1-curvescannot show the right dimension. Anyway, although the data represent achaotic flow where theory tells us that the dimension must be largerthan 2, a value of D<sub>2</sub>>2 can hardly bederived from these curves!<br>Give an estimate of the noise level.<br><br><li>Entropies: plot the file <font color=blue>amplitude.dat.h2</font> toestimate the entropies (on a log-linear scale). Write dimension and entropy estimates and thenoise level into a table (on paper). Do not forget to divide theh<sub>2</sub>-estimate by the time lag.<br><br><li>Redo the computation with a smaller and a larger time lag, e.g., <fontcolor=orange> -d1</font> and <font color=orange> -d24</font>. You shouldbe able to confirm the values found above, but the scaling rangeseither shrink or larger <font color=blue>m</font>-values are neededfor convergence. For smaller lag you find more edge effects, forlarger lags more folding effects. Entropy estimates might somehowprofit from larger lags.<br><br> <li> Maximal Lyapunov exponent: Use <a href="../docs_c/lyap_k.html">lyap_k </a> and/or <a href="../docs_c/lyap_r.html">lyap_r</a>, e.g.,.<br><font color=red>lyap_k amplitude.dat -M6 -m3 -d8 -t100 -s500 -r.1-o</font><br> If necessary, use the option <font color=orange> -n1000</font> to keep the numerical effortlimited. Try to understand on which time interval you expect theexponential increase. Admittedly, flow data are not really suited forsuch kinds of algorithms.<br><br><li> Compute the Lyapunov exponent forthe Poincaré-map data of this flow which you produced in thelast exercise, e.g. with the following parameters:<br><font color=red> lyap_k max.dat -M5 -m2 -s25 -o</font><br>Recall that for map data, the optimal settings are <fontcolor=orange>-d1 -t0</font> which are the defaults, and that there isno use to consider time horizons which are larger than the data set(hence, <font color=orange> -s25</font>).<br>The estimated exponent (slope of the straight line) carries the units of 1/time and thus has to bedivided by the average time in between two intersections of the flowwith the Poincaré surface of section in order to be compared to thevalue obtained from the flow data. This time is computed by (length of the flow data set)/(#number of intersections).<br><br><li> Use the Poincaré-map data <font color=blue> max.dat</font> or <font color=blue>amplitude.dat.poin</font> generatedwith <font color=orange> -C1</font> as a data base for <ahref="../docs_c/nstep.html">nstep</a> to create a much longersynthetic trajectory of length 5000:<br><font color=red> nstep -k10 -L5000 max.dat -o</font><br>Since the data of <font color=blue> max.dat</font> form an almost1-dimensional graph of a map, <font color=orange> -m1,2</font>, i.e.,the default values, are fine. Due to the low number of data points,<font color=orange> -k10</font> is a reasonable number of neighbours;to require more would extend the neighbourhoods to outside the linearregime. Verify the validity of the forecasts by acomparison of the attractors in thedelay embedding space. Evidently, there is some fake additionalstructure whose existence cannot be supported by the originaldata. Other choices of parameters/Poincaré data of the flow mightyield better agreements. In any case, the data base (152 points) is rather poor.<br><br><li> Compute the correlation dimension for these numerical data.<br>Result: The file <font color=blue>max.dat.cast.c2</font> supports adimension of D=1.2, but under consideration of what we said beforeabout the fine structure, this number cannot be trusted. Whencomparing the curves to <font color=blue>max.dat.c2</font>, thedimension estimates on <font color=blue>max.dat</font> itself, thereis some compatibility, but in this case everything is speculation. <br><font color=red><b>This illustrates our standard warning: One can apply the routinesto and exctract numbers from many data sets, but whether the resultsare meaningful or not remains in the interpretation of the applicant!</b></font><br><br><li> Compute the maximal Lyapunov exponent for the synthetic Poicare-map data. Are the results compatible with what you found for <fontcolor=blue>max.dat</font> ?<br><br><li> Can you expect more than one positive exponent in the map data?<br>Use <a href="../docs_c/lyap_spec.html">lyap_spec</a> to compute thefull Lyapunov spectrum of the synthetic data. <br>The resulting set of exponents contains spurious ones. For theiridentification, run the algorithm with modified neighbourhood size,larger embedding dimension, or a different time lag. How many positiveexponents do you find?<br>Our result is 0.53 for the positive and -1. for the negativeexponent. In a 2-dimensional embedding, there are no spurious exponents!Enter the maximal exponent in your table, the sum of all positive asthe estimate of the KS-entropy, and compute the Kaplan-Yorke-dimensionfrom the non-spurious exponents.<br><br><li>How good is the mutual agreement of the different methods?<br>This is our result:<br><table><tr><td align=center><font color=darkgreen><b>data type</b></font></td> <td width=20></td> <td align=center><font color=darkgreen><b>dimension</b></font></td> <td width=20></td> <td align=center><font color=darkgreen><b>max. Lyapunov</b></font></td> <td width=20></td> <td align=center><font color=darkgreen><b>entropy</b></font></td> <td width=20></td> <td align=center><font color=darkgreen><b>noise level</b></font></td></tr><tr> <td align=center>flow data</td> <td width=20></td> <td align=center>D<sub>2</sub>=2.1±0.05</td> <td width=20></td> <td align=center>lyap_k=0.015±0.002</td> <td width=20></td> <td align=center>h<sub>2</sub>=0.02±0.005</td> <td width=20></td> <td align=center> 0.5 units </td></tr><tr> <td align=center>map data max.dat</td> <td width=20></td> <td align=center>D<sub>2</sub>=1 + 1.2±0.2</td> <td width=20></td> <td align=center>lyap_k=0.015</td> <td width=20></td> <td align=center>h<sub>2</sub>=0.45/32.9=0.0135</td> <td width=20></td> <td align=center>0.5 units </td></tr><tr> <td align=center>synthetic data max.dat.cast</td> <td></td> <td align=center>D<sub>2</sub>=1+ 1.2±.1</td> <td width=20></td> <td align=center>lyap_k: 0.5/32.9 </td> <td width=20></td> <td align=center>h<sub>2</sub>=0.4/32.9 </td> <td width=20></td> <td align=center></td></tr><tr> <td align=center>synthetic data max.dat.cast</td> <td></td> <td align=center>D<sub>KY</sub>=1+1.5</td> <td width=20></td> <td align=center> lyap_spec: 0.53/32.9= 0.016 </td> <td width=20></td> <td align=center> </td> <td width=20></td> <td align=center></td></tr></table></font><br><br><font color=red><b>Comments</b></font><br>The <b>flow data</b> are numerically generated data with additivenoise, and in so far optimal (no drifts, no interactive noise, noartifacts). However, they represent only about 150 revolutioins of thetrajectory on its attractor, and thus are rather few data. <br>Thenumerically exact Lyapunov exponent (obtained during the integrationof the system) is 0.0173, not so far from the time series value. <br>The entropy should be bounded by this exponent, but the times seriesvalue is also close enough. In fact, the convergence of the entropyestimates as a function of the embedding dimension is from above,i.e., higher embeddings might yield better estimates.However, this fact yields a mismatch of theentropy estimate and the maximal Lypunov exponent obtained from thetime series data.<br>The dimension estimate is reasonable.<br><br>The <b>Poincaré map data</b> are very few, but since the Poincaré mapis essentially one-dimensional, they contain still meaningfulinformation. Nonetheless, the agreement between the results on the mapdata and the flow data is not excellent.<br><br>The invariant characteristics of the <b> synthetic map data</b> are inagreement with the Poincaré map data, but as we have seen, theattractor contains structure which cannot be supported by theobservations. <br>The computation of the Lyapunov spectrum yields an overestimation ofthe dimension by the Kaplan-Yorke formula, and it can be suspectedthat the negative Lyapunov exponent is underestimated. <br><br><br><br><br></body></html>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -