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非线性时间学列分析工具

If you are looking for a program that reads your signal and issues a number
that says "correlation dimension", you got yourself the wrong package.  We
think you are still better off than getting such a wrong answer. The programs
in this section carry out the calculations necessary to detect scaling and self
similarity in a fractal attractor. You will have to establish scaling and
eventually, in favourable cases, extract the dimension or entropy by careful
evaluation of the data produced by these programs.

<p>
There are two alternative implementations of the Grassbeger-Procaccia
correlation integral in this package. The program 
<a href="docs_c/d2.html">d2</a> by Rainer Hegger to our knowledge
is the fastest and most reliable implementation that is currently available.
It can also handle multivariate data and mixed embeddings.
For the very conservative, there is a slow but simple alternative
called  <a href="docs_f/c2naive.html">c2naive</a> that works on scalar data
only. Post-processing can be performed on the output in order to obtain 
Takens' estimator or the Gaussian kernel correlation integral, or just for
smoothing. 

<p>
A fixed mass algorithm for the information dimension D1 is available which also
can handle multivariate data and mixed embeddings, 
and a box-counting implementation of the order Q Renyi entropies for
multifractal studies.

<p>
You may want to consult the <a href="chaospaper/node29.html">introduction</a>
paper 
for initial material on dimension estimation. If you are serious, you will need
to study some of the literature cited there as well.

<p><table border>
<tr valign=top><td>Correlation integral C2</td>
    <td><a href="docs_c/d2.html">d2</a>, 
        <a href="docs_c/av-d2.html">av-d2</a>,
        <a href="docs_f/c2naive.html">c2naive</a>,
        <a href="docs_f/c2d.html">c2d</a></td></tr>
<tr><td>Takens estimator</td>
    <td><a href="docs_f/c2t.html">c2t</a></td></tr>
<tr><td>Gaussian kernel C2</td>
    <td><a href="docs_f/c2g.html">c2g</a></td></tr>
<tr><td>Fixed mass estiamtion of D1</td>
    <td><a href="docs_f/c1.html">c1</a></td></tr>
<tr><td>Renyi Entropies of Qth order</td>
    <td><a href="docs_c/boxcount.html">
       boxcount</a></td></tr>
</table>

<h3><a name="lyapunov">
Lyapunov exponents</a></h3>

Lyapunov exponents are an important means of quantification for unstable
systems. They are however difficult to estimate from a time series.
Unless low dimensional, high quality data is at hand, one should not attempt to
calculate the full spectrum. Try to compute the maximal exponent first.
The two implementations differ slightly.
While <a
href="docs_c/lyap_k.html">lyap_k</a> implements the formula by
<a href="chaospaper/citation.html#holger">Kantz</a>, 
<a href="docs_c/lyap_r.html">lyap_r</a> uses that by 
<a href="chaospaper/citation.html#rose">Rosenstein et al.</a> 
which differs only in the definition of the neighbourhoods.
We recommend to use the former version, 
<a href="docs_c/lyap_k.html">lyap_k</a>.
<p>
The estimation of Lyapunov exponents is also discussed in the 
<a href="chaospaper/node26.html">introduction</a> paper. A recent addition is a
programm to compute finite time exponents which are not invariant but contain
additional information. 

<p><table border>
<tr><td>Maximal exponent</td>
    <td><a href="docs_c/lyap_k.html">lyap_k</a>, 
        <a href="docs_c/lyap_r.html">lyap_r</a></td></tr>
<tr><td>Lyapunov spectrum</td>
    <td><a href="docs_c/lyap_spec.html">lyap_spec</a></tr>
<tr><td>Finite size exponents</td>
    <td><a href="docs_c/fsle.html">fsle</a></tr>
</table>

<h3><a name="surrogates">
Surrogate data</a></h3>

Before attempting any sophisticated nonlinear time series analysis, one should
try to establish that nonlinearity is indeed present. The most suitable method
for this is the approach of surrogate data. We present two schemes for the
generation of surrogate time series, one using iterative adjustments of
spectrum and distribution, and a very general framework for constrained
randomization that is based on combinatorial minimization of a cost function.
The latter approach is more like a toolbox, a starting point for your own
ideas on suitable null hypotheses etc. A few basic discriminating statistics
are also provided. 

<p>
There is a short overview page for <a
href="docs_f/test.html">nonlinearity tests</a>. There is also a section
in the <a href="chaospaper/node35.html">introduction</a> paper.

<p><table border>
<tr><td>Make surrogate data</td>
    <td><a href="docs_f/surrogates.html">surrogates</a></td></tr>
<tr><td>Determine end-to-end mismatch</td>
    <td><a href="docs_f/endtoend.html">endtoend</a></td></tr>
<tr><td>General constrained randomization</td>
    <td><a href="docs_f/randomize.html">randomize</a></td></tr>
<tr><td>Discriminating statistics</td>
    <td><a href="docs_f/timerev.html">timerev</a>,
        <a href="docs_f/predict.html">predict</a></td></tr>
</table>

<h3><a name="spike">
Spike trains</a></h3>

Sequences of times of singular events (heart beats, neuronal spikes etc.),
or sequences of intervals between such events (RR-intervals etc.) require 
specialised techniques, even for their linear analysis. Below find a list of
routines that may proove useful for this type of data. 

<p><table border>
<tr><td>Event/intervcal conversion</td>
    <td><a href="docs_f/intervals.html">intervals</a></td></tr>
<tr><td>Interval/event conversion</td>
    <td><a href="docs_f/events.html">events</a></td></tr>
<tr><td>Autocorrelation function of event times</td>
    <td><a href="docs_f/spikeauto.html">spikeauto</a></td></tr>
<tr><td>Power spectrum of event times</td>
    <td><a href="docs_f/spikespec.html">spikespec</a></td></tr>
<tr><td>Surrogate data preserving event time autocorrelations</td>
    <td><a href="docs_f/randomize_spike.html">randomize_spikeauto_exp_random</a></td></tr>
<tr><td>Surrogate data preserving event time power spectrum</td>
    <td><a href="docs_f/randomize_spike.html">randomize_spikespec_exp_event</a></td></tr>
</table>

<h3><a name="multivariate">
Multivariate time series</a></h3>

TISEAN's multivariate section is still growing - like the research body in
this area. As a convention, program names for ``cross-'' quantities
computed between two signals start with <font color=blue><tt>x</tt></font>. 
Those programs that can handle scalar as
well as multivariate data don't follow any name convention.
<p>
Currently, three (cross-) similarity measures are available, a linear and
two nonlinear ones. Further, there is a zeroth order predictor using
multivariate embeddings.
A few general routines that can handle multivariate data are also mentioned
below. Note in particular the multivariate surrogate data generator, the
Grassberger-Procaccia correlation sum, and the fixed mass D1 estimator.
<p>
The general convention for column selection is as follows.
Columns can be given as a comma seperated list with the option
<font color=blue><tt> -c#[,#]  </tt></font>. Wherever the number of columns is
selectable, this can be done with the option <font color=blue><tt> -m#
</tt></font> or the first number in its argument. The precedence of these
settings are as follows:
<ul><dt>
  <font color=blue><tt> -m </tt></font> overrides 
  <font color=blue><tt> -c </tt></font>
   <dt>
  <font color=blue><tt> -c </tt></font> overrides the default 
    <font color=red>only</font> if <font color=red>more</font> columns are
    specified. 
</ul>
Exceptions to this general convention are possible and mentioned in the
specific program descriptions.

<p><table border>
<tr><td>Multivariate linear model</td>
   <td><a href="docs_c/ar-model.html">ar-model</a></td></tr>
<tr><td>Multivariate noise generation</td>
   <td><a href="docs_c/makenoise.html">makenoise</a></td></tr>
<tr><td>Linear cross-correlations</td>
   <td><a href="docs_c/xcor.html">xcor</a></td></tr>
<tr><td>Extrema of a multivariate signal</td>
   <td><a href="docs_c/extrema.html">extrema</a></td></tr>
<tr><td>Savitzky-Golay filter</td>
   <td><a href="docs_c/sav_gol.html">sav_gol</a></td></tr>
<tr><td>Recurrence plot</td>
   <td><a href="docs_c/recurr.html">recurr</a></td></tr>
<tr><td>Nonlinear cross-prediction</td>
   <td><a href="docs_c/xzero.html">xzero</a></td></tr>
<tr><td>Cross-correlation integral</td>
   <td><a href="docs_c/xc2.html">xc2</a></td></tr>
<tr><td>Correlation integral, also for multivariate data</td>
   <td><a href="docs_c/d2.html">d2</a></td></tr>
<tr><td>Fixed mass approach to D1, also for multivariate data</td>
   <td><a href="docs_f/c1.html">c1</a></td></tr>
<tr><td>Lyapunov spectra</td>
   <td><a href="docs_c/lyap_spec.html">lyap_spec</a></td></tr>
<tr><td>Renyi entropies</td>
   <td><a href="docs_c/boxcount.html">boxcount</a></td></tr>
<tr><td>Zeroth order prediction on multivariate time series</td>
   <td><a href="docs_c/zeroth.html">zeroth</a></td></tr>
<tr><td>Locally linear prediction on multivariate time series</td>
   <td><a href="docs_c/nstep.html">nstep</a></td></tr>
<tr><td>Compare two signals</td>
   <td><a href="docs_f/compare.html">compare</a></td></tr>
<tr><td>Choose sub-sequence or columns</td>
    <td><a href="docs_f/choose.html">choose</a></td></tr>
<tr><td>Make surrogate data, also multivariate</td>
    <td><a href="docs_f/surrogates.html">surrogates</a></td></tr>
<tr><td>Determine end-to-end mismatch, also multivariate</td>
    <td><a href="docs_f/endtoend.html">endtoend</a></td></tr>
</table>

<hr>
<em>Copyright &#169; Rainer Hegger, Holger Kantz, Thomas Schreiber (1999)</em>
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