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821 bytes added ,  14:05, 6 July 2015
=== Dimensions and entropies Entropies ===This function uses a method to determine the minimum sufficient embedding dimension. It section is based discussed on the [ False Nearest NeighborsSECTION00080000000000000000 TISEAN documentation page] section . One of the TISEAN documentationfunctions discussed is {{Codeline|d2}}. As a demonstration we will create a plot that contains an Ikeda MapIt is used to estimate the correlation sum, a Henon Map correlation dimension and correlation entropy of a time series. The time series used here will be the Henon Map corrupted by 10% of Gaussian noisemap.{{Code|Analyzing false nearest neighborsCalculation correlation sum, dimension and entropy|<syntaxhighlight lang="octave" style="font-size:13px">
# Create maps
ikd = ikeda (10000);
hen = henon (10000);
hen_noisy # Calculate the correlation sum, dimension and entropyvals = d2 (hen + std , 'd', 1, 'm', 5, 't',50);# Plot correlation sumsubplot (hen2,3,1) * 0.02 .* do_plot_corr = @(-6 + sum x) loglog (rand x{1}([size:,1),x{1}(hen:,2), 12]'b');hold onarrayfun (do_plot_corr, 3{vals.c2});hold offxlabel ("Epsilon")ylabel ("Correlation sums")title ("c2");# Create and plot false nearest neighborsPlot correlation entropy[dim_ikdsubplot (2,3, frac_ikd] 4)do_plot_entrop = false_nearest @(x) semilogx (ikdx{1}(:,1),x{1}(:,2),'g');[dim_henhold onarrayfun (do_plot_entrop, frac_hen] = false_nearest {vals.h2});hold offxlabel (hen"Epsilon")ylabel (:,1"Correlation entropies");title ("h2");# Plot correlation dimensionsubplot (2,3,[dim_hen_noisy, frac_hen_noisy2 3 5 6] )do_plot_slope = false_nearest @(x) semilogx (hen_noisyx{1}(:,1));plot ,x{1}(dim_ikd:, frac_ikd2), '-b*r');Ikeda;'hold onarrayfun (do_plot_slope,{vals...d2});hold offxlabel ("Epsilon") dim_hen, frac_hen, '-r+;Henon;',...ylabel ("Local slopes") dim_hen_noisy, frac_hen_noisy, '-gx;Henon Noisy;'title ("d2");
[[File:d2_out.png|400px|center]]The output of {{Codeline|dim_*d2}} variables hold can be further processed using the dimension (so here 1following functions:5){{Codeline|av_d2}}, {{Codeline|c2t}}, and {{Codeline|frac_*c2g}} contain . This tutorial will show how to use {{Codeline|av_d2}} which smooths the fraction output of false nearest neighbors. From this chart we can obtain the sufficient embedding dimension for each system. For a Henon Map {{Codeline|m &#61; 2d2}} is sufficient, but for an Ikeda map it is better (usually used to use smooth the "{{Codeline|m &d2}}" field of the output).{{Code|Smooth output of d2|<syntaxhighlight lang="octave" style="font-size:13px">#61Smooth d2 outputfigure 2smooth = av_d2 (vals,'a',2); 3# Plot the smoothed outputdo_plot_slope = @(x) semilogx (x{1}(:,1),x{1}(:,2),'b');hold onarrayfun (do_plot_slope, {smooth.d2});hold offxlabel ("Epsilon")ylabel ("Local slopes")title ("Smooth");</syntaxhighlight>}}[[File:tisean_false_neightisean_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}}.


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