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<html><head><title>d2</title></head><body bgcolor=white><h1 align=center>Description of the program: <font color=Blue>d2</font></h1><hr>This program estimates the correlation sum, the <a href="../chaospaper/node30.html">correlation dimension</a>and the <a href="../chaospaper/node34.html">correlation entropy</a> of a given, possibly multivariate, data set. It usesthe box assisted search algorithm and is quite fast as long as one isnot interested in large length scales. All length scales are computedsimultaneously and the output is written every 2 min (real time, notcpu time). It is possible to set a maximum number of pairs. If thisnumber is reached for a given length scale, the length scale will nolonger be treated for the rest of the estimate.<p>Please consult the <a href="../chaospaper/node29.html">introduction</a> paperfor initial material on dimension estimation. If you are serious, you will needto study some of the literature cited there as well.<p>In extension to what is described there, the simultaneous useof multivariate data and temporal embedding is possible using <font color=blue>d2</font>. Thus one hasto give two numbers in order to specify the embedding dimension. The first is the number of multivariate components, thesecond the number of lags in the temporal embedding. This is realized by giving two numbersto the option <font color=blue>-M</font>, seperated by a comma.For a standard scalar time series, set the first number to 1. If yourmultivariate data spans the whole phase space and no further temporalembedding is desired, set the second value to 1.In any case, the total embedding dimension in the sense of the embeddingtheorems is the product of the two numbers.<p>In order to be able to assess the convergence with increasing embeddingdimension, results are reported for several such values. The inner loop stepsthrough the number of components until the first argument of <fontcolor=blue>M</font> is reached. The outer loop increases the number of timelags until the second argument of <font color=blue>M</font> is reached.<h4>Usage of the <font color=blue>-c</font> and <font color=blue>-M</font> flags</b></h4>Suppose, the option <font color=blue>-M<em> x,y</em></font> has been specified.By default, the first <font color=blue><em> x</em></font> columns of a file are used. This behaviour canbe modified by means of the <font color=blue>-c</font> flag. It takesa series of numbers separated by commas. The numbers represent thecolomns. For instance <font color=red>-M 3,1 -c 3,5,7</font> means,use three components with no additional temporal embedding, readingcolumns 3, 5 and 7. It is not necessary to give the full number ofcomponents to the <font color=blue>-c</font> flag. If numbers aremissing, the string is filled up starting with the smallest number,larger than the largest given. For instance, <font color=red>-M 3,1 -c3 </font> would result in reading columns 3, 4 and 5.</p><hr><h2 align=center>Usage:</h2><center><font color=Red>d2 [Options]</font><p>Everything not being a valid option will be interpreted as a potential datafilename. Given no datafile at all means read stdin. Also <font color=Red>-</font>means stdin<p>Possible options are:<p><table border=2><tr><th>Option<th>Description<th>Default</tr><tr><th>-l#<td>number of data points to be used<td>whole file</tr><tr><th>-x#<td>number of lines to be ignored<td>0</tr><tr><th>-d#<td>delay for the delay vectors<td>1</tr><tr><th>-M#<td># of components,maximal embedding dimension<td>1,10</tr><tr><th>-c#<td><a href=../general.html#columns>columns to be read</a><td>1,...,# of components</tr><tr><th>-t#<td>theiler window<td>0</tr><tr><th>-R#<td>maximal length scale <td>(max data interval)</tr><tr><th>-r#<td>minimal length scale<td>(max data interval)/1000</tr><tr><th>-##<td>number of epsilon values<td>100</tr><tr><th>-N#<td>maximal number of pairs to be used <br>(0 means all possible pairs)<td>1000</tr><tr><th>-E<td>use data that is normalized to [0,1] for all components<td>not set (use natural units of the data)</tr><tr><th>-o[#]<td> <a href=../general.html#outfile>output file name</a> (without extensions)<td> 'datafile'[.c2][.d2][.h2][.stat]<br>(or if data were read from stdin: stdin[.c2][.d2][.h2][.stat])</tr><tr><th>-V#<td><a href=../general.html#verbosity>verbosity level</a><br> 0: only panic messages<br> 1: add input/output messages<br> 2: add input/output messages each time output file isopened<td>1</tr><tr><th>-h<td>show these options<td>none</tr></table></center><hr><h2 align=center>Description of the Output:</h2>The files with the extensions c2, d2 and h2 contain for each embeddingdimension and each length scaletwo columns:<br><b>first column:</b> epsilon (in units chosen)<br><b>second column:</b> the estimated quantity (correlation sum, dimension,entropy)<br>.<ul><li><b>extension .c2</b>: This file contains the correlation sums for all treated length scales and embedding dimensions.<li><b>extension .d2</b>: This file contains the local slopes of the logarithmof the correlation sum, the correlation dimension.<li><b>extension .h2</b>: This file contains the correlation entropies.<li><b>extension .stat</b>: This file shows the current status of theestimate. </ul>The output is written every two minutes (real time, not cpu time). So,you can see preliminary results even if the program is still running.Post-processing can be done in the following ways.Either of the output sequences can be <b>smoothed</b> by <ahref="av-d2.html">av-d2</a>.<b>Takens' estimator</b> can be obtained from the correlation sum (extension<b>.c2</b>) using <a href="../docs_f/c2t.html">c2t</a> and then plotted versus upper cut-off length.The <b>Gaussian kernel</b> correlation integral can be obtained from the standard correlation sum (extension<b>.c2</b>) using <a href="../docs_f/c2g.html">c2g</a>.<h2 align=center>How to get D<sub>2</sub> from the data</h2>The file with the extension <b>.d2</b> contains the skeleton toestimate the correlation dimension. As written above it contains thelocal slopes of the correlation sums for the different embeddingdimension. To get an idea of the correlation dimension it is always agood idea to plot the file using a log scale for the x-axis.<br>The figures you get could look something like one of the followingones.<img src="d2-hen.gif"></img><img src="d2-noi.gif"></img>The first figure clearly shows a plateau. This means for a wide rangeof length scales (x-axis) and for all embedding dimensions larger thana minimal one, the curves collapse and are flat. The value of theplateau gives an estimate for the dimension (1.2... in this case). Thesecond figure does not give any dimension estimate, at all. All curvesbehave different and there is no common behaviour. From this figureone can not conclude any dimension for the system.<br>Typically one finds something in between the two extremes shownabove.<hr>View the <a href="../../source_c/d2.c">C-source</a>.<hr>See also <a href="../docs_f/c2naive.html">c2naive</a>, and <ahref="../docs_f/c1.html">c1</a>. <hr><a href=../contents.html>Table of Contents</a> * <a href="../../index.html" target="_top">TISEAN home</a></body></html>⌨️ 快捷键说明
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