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=== Testing for Nonlinearity ===
=== Testing for Nonlinearity ===
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.
This section is discussed on the [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node35.html#SECTION00090000000000000000 TISEAN documentation page]. The focus of this section will be the function {{Codeline|surrogates}}. It uses surrogate data to determine weather data is nonlinear. Let us first create the input data which will be a stationary Gaussian linear stochastic process. It is measured by {{Codeline|s(xn) = xn^3}}. We then run it through {{Codeline|surrogates}} and plot the data.
{{Code|Calculation correlation sum, dimension and entropy|<syntaxhighlight lang="octave" style="font-size:13px">
{{Code|Creating data from Gaussian process|<syntaxhighlight lang="octave" style="font-size:13px">
# Create maps
# Create Gaussian process data
hen      = henon (10000);
g = zeros (2000,1);
# Calculate the correlation sum, dimension and entropy
for i = 2:2000
vals = d2 (hen, 'd', 1, 'm', 5, 't',50);
  g(i) = 0.7 * g(i-1) + (-6 + sum (rand ([size(1), 12]), 3));
# Plot correlation sum
endfor
subplot (2,3,1)
# Create a measurement of it
do_plot_corr = @(x) loglog (x{1}(:,1),x{1}(:,2),'b');
spike = g.^3;
hold on
# Create the surrogate
arrayfun (do_plot_corr, {vals.c2});
sur  = surrogates (spike);
hold off
# Plot the data
xlabel ("Epsilon")
subplot (2,1,1)
ylabel ("Correlation sums")
plot (spike,'g');
title ("c2");
title ("spike")
# Plot correlation entropy
subplot (2,1,2)
subplot (2,3,4)
plot (sur,'b');
do_plot_entrop  = @(x) semilogx (x{1}(:,1),x{1}(:,2),'g');
title ("surrogate")
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]]
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).
{{Code|Smooth output of d2|<syntaxhighlight lang="octave" style="font-size:13px">
# Smooth d2 output
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}}.


[[Category:Octave-Forge]]
[[File:surrogate_tutorial.png|400px|center]]
It is crucial that the length of the input to surrogates is factorizable by only 2,3 and 5. Therefore, if it is not the excess of data is truncated accordingly. Padding with zeros is not allowed. To solve this problem one can use {{Codeline|endtoend}}, and choose the best subset of the input data to be used to generate a surrogate.


== External links ==
== External links ==
* [https://bitbucket.org/josiah425/tisean Bitbucket repository ] where the porting is taking place.
* [https://bitbucket.org/josiah425/tisean Bitbucket repository ] where the porting is taking place.
* [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/ TISEAN package website] where the package is described along with references to literature, tutorials and manuals.
* [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/ TISEAN package website] where the package is described along with references to literature, tutorials and manuals.
[[Category:Octave Forge]]

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