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[[File:lyap_k.png|400px|center]]
 
[[File:lyap_k.png|400px|center]]
  
=== Dimensions and Entropies ===
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=== Dimensions and entropies ===
This section is discussed on the [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node29.html#SECTION00080000000000000000 TISEAN documentation page]. One of the functions discussed is {{Codeline|d2}}. It is used to estimate the correlation sum, correlation dimension and correlation entropy of a time series. The time series used here will be the Henon map.
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This function uses a method to determine the minimum sufficient embedding dimension. It is based on the [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node9.html#SECTION00032200000000000000 False Nearest Neighbors] section of the TISEAN documentation. As a demonstration we will create a plot that contains an Ikeda Map, a Henon Map and a Henon Map corrupted by 10% of Gaussian noise.
{{Code|Calculation correlation sum, dimension and entropy|<syntaxhighlight lang="octave" style="font-size:13px">
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{{Code|Analyzing false nearest neighbors|<syntaxhighlight lang="octave" style="font-size:13px">
 
# Create maps
 
# Create maps
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ikd      = ikeda (10000);
 
hen      = henon (10000);
 
hen      = henon (10000);
# Calculate the correlation sum, dimension and entropy
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hen_noisy = hen + std (hen) * 0.02 .* (-6 + sum (rand ([size(hen), 12]), 3));
vals = d2 (hen, 'd', 1, 'm', 5, 't',50);
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# Create and plot false nearest neighbors
# Plot correlation sum
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[dim_ikd, frac_ikd] = false_nearest (ikd(:,1));
subplot (2,3,1)
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[dim_hen, frac_hen] = false_nearest (hen(:,1));
do_plot_corr  = @(x) loglog (x{1}(:,1),x{1}(:,2),'b');
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[dim_hen_noisy, frac_hen_noisy] = false_nearest (hen_noisy(:,1));
hold on
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plot (dim_ikd, frac_ikd, '-b*;Ikeda;',...
arrayfun (do_plot_corr, {vals.c2});
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      dim_hen, frac_hen, '-r+;Henon;',...
hold off
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      dim_hen_noisy, frac_hen_noisy, '-gx;Henon Noisy;');
xlabel ("Epsilon")
 
ylabel ("Correlation sums")
 
title ("c2");
 
# Plot correlation entropy
 
subplot (2,3,4)
 
do_plot_entrop  = @(x) semilogx (x{1}(:,1),x{1}(:,2),'g');
 
hold on
 
arrayfun (do_plot_entrop, {vals.h2});
 
hold off
 
xlabel ("Epsilon")
 
ylabel ("Correlation entropies");
 
title ("h2")
 
# Plot correlation dimension
 
subplot (2,3,[2 3 5 6])
 
do_plot_slope = @(x) semilogx (x{1}(:,1),x{1}(:,2),'r');
 
hold on
 
arrayfun (do_plot_slope, {vals.d2});
 
hold off
 
xlabel ("Epsilon")
 
ylabel ("Local slopes")
 
title ("d2");
 
 
</syntaxhighlight>}}
 
</syntaxhighlight>}}
[[File:d2_out.png|400px|center]]
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The {{Codeline|dim_*}} variables hold the dimension (so here 1:5), and {{Codeline|frac_*}} contain the fraction of false nearest neighbors. From this chart we can obtain the sufficient embedding dimension for each system. For a Henon Map {{Codeline|m &#61; 2}} is sufficient, but for an Ikeda map it is better to use {{Codeline|m &#61; 3}}.
The output of {{Codeline|d2}} can be further processed using the following functions: {{Codeline|av_d2}}, {{Codeline|c2t}}, {{Codeline|c2g}}. This tutorial will show how to use {{Codeline|av_d2}} which smooths the output of {{Codeline|d2}} (usually used to smooth the "{{Codeline|d2}}" field of the output).
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[[File:tisean_false_neigh.png|400px|center]]
{{Code|Smooth output of d2|<syntaxhighlight lang="octave" style="font-size:13px">
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# Smooth d2 output
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figure 2
 
smooth = av_d2 (vals,'a',2);
 
# Plot the smoothed output
 
do_plot_slope = @(x) semilogx (x{1}(:,1),x{1}(:,2),'b');
 
hold on
 
arrayfun (do_plot_slope, {smooth.d2});
 
hold off
 
xlabel ("Epsilon")
 
ylabel ("Local slopes")
 
title ("Smooth");
 
</syntaxhighlight>}}
 
[[File:tisean_av_d2_out.png|400px|center]]
 
Optionally the line "{{Codeline|figure 2}}" can be omitted, which will cause the smoothed version to be superimposed on the "raw" version that came straight from {{Codeline|d2}}.
 
  
 
=== Testing for Nonlinearity ===
 
=== Testing for Nonlinearity ===

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