# Changes

## TISEAN package

, 03:52, 10 June 2019
m
Remove redundant Category:Packages.
== Porting TISEAN ==
This section will focus which focuses on demonstrating how the capabilities of the TISEAN package. The previous information about is to be ported and what is the porting procedure has been moved current state of that process is located in [[TISEAN_package:Procedure|here]].
== Tutorials ==
* [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/tutorial/amplitude.dat amplitude.dat]
=== False Nearest Neighbors ===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|Analyzing false nearest neighbors|<syntaxhighlight lang="octave" style="font-size:13px"># Create mapsikd = ikeda (10000);hen = henon (10000);hen_noisy = hen + std (hen) * 0.02 .* (-6 + sum (rand ([size(hen), 12]), 3));# Create and plot false nearest neighbors[dim_ikd, frac_ikd] = false_nearest (ikd(:,1));[dim_hen, frac_hen] = false_nearest (hen(:,1));[dim_hen_noisy, frac_hen_noisy] = false_nearest (hen_noisy(:,1));plot (dim_ikd, frac_ikd, '-b*;Ikeda;',... dim_hen, frac_hen, '-r+;Henon;',... dim_hen_noisy, frac_hen_noisy, '-gx;Henon Noisy;');</syntaxhighlight>}}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}}.[[File:tisean_false_neigh.png|400px|center]] === Finding Unstable Periodic Orbits ===Here I will demonstrate how to find unstable periodic orbits. This section is based on the TISEAN documentation chapter [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node19.html#SECTION00053000000000000000 Finding unstable periodic orbits]. We will start by finding these orbits using function {{Codeline|upo}}.{{Code|Finding unstable periodic orbits|<syntaxhighlight lang="octave" style="font-size:13px"># Create mapshen = henon (1000);hen = hen + std (hen) * 0.02 .* (-6 + sum (rand ([size(hen), 12]), 3));# Find orbits[lengths, data] = upo(hen(:,1), 2, 'p',6,'v',0.1, 'n', 100);</syntaxhighlight>}}The vector {{Codeline|lengths}} contains the size of each of the found orbits and {{Codeline|data}} contains their coordinates. To obtain delay coordinates and plot these orbits onto the original data we will use {{Codeline|upoembed}}.{{Code|Plotting unstable periodic orbits|<syntaxhighlight lang="octave" style="font-size:13px"># Create delay coordinates for the orbits and dataup = upoembed (lengths, data, 1);delay_hen = delay (hen(:,1));# Plot all of the dataplot (delay_hen(:,1), delay_hen(:,2), 'r.;Noisy Henon;','markersize',2,... up{4}(:,1), up{4}(:,2),'gx;Fixed Point;','markersize',20,'linewidth',1, ... up{3}(:,1), up{3}(:,2),'b+;Period 2;','markersize',20,'linewidth',1, ... up{2}(:,1), up{2}(:,2),'ms;Period 6;','markersize',20,'linewidth',1, ... up{1}(:,1), up{1}(:,2),'cs;Period 6;','markersize',20,'linewidth',1);</syntaxhighlight>}}[[File:Upo.png|400px|center]]The plotting options are passed to make the orbits more visible. === Nonlinear Prediction ===In this section we will demonstrate some functions from the 'Nonlinear Prediction' chapter of the TISEAN documentation (located [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node16.html#SECTION00050000000000000000 here]). For now this section will only demonstrate functions that are connected to the [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node18.html#SECTION00052000000000000000 Simple Nonlinear Prediction] section. <br/>There are three functions in this section: {{Codeline|lzo_test}}, {{Codeline|lzo_gm}} and {{Codeline|lzo_run}}. The first is used to estimate the forecast error for a set of chosen parameters, the second gives some global information about the fit and the third produces predicted points. Let us start with the first one (before starting this example remember to download 'amplitude.dat' from above and start Octave in the directory that contains it). The pairs of parameters {{Codeline|(m,d)}} where chosen after the TISEAN documentation.{{Code|Analyzing forecast errors for various parameters|<syntaxhighlight lang="octave" style="font-size:13px"># Load dataload amplitude.dat# Create different forecast error resultssteps = 200;res1 = lzo_test (amplitude(1:end-200), 'm', 2, 'd', 6, 's', steps);res2 = lzo_test (amplitude(1:end-200), 'm', 3, 'd', 6, 's', steps);res3 = lzo_test (amplitude(1:end-200), 'm', 4, 'd', 1, 's', steps);res4 = lzo_test (amplitude(1:end-200), 'm', 4, 'd', 6, 's', steps);plot (res1(:,1), res1(:,2), 'r;m = 2, d = 6;', ... res2(:,1), res2(:,2), 'g;m = 3, d = 6;',... res3(:,1), res3(:,2), 'b;m = 4, d = 1;',... res4(:,1), res4(:,2), 'm;m = 4, d = 6;');</syntaxhighlight>}}[[File:tisean_nl_prediction.png|400px|center]] It seems that the last pair {{Codeline|(m &#61; 4, d &#61; 6)}} is suitable. We will use it to determine the the best neighborhood to use when generating future points.{{Code|Determining the best neighborhood size using lzo_gm|<syntaxhighlight lang="octave" style="font-size:13px">gm = lzo_gm (amplitude(1:end-200), 'm', 4, 'd', 6, 's', steps, 'rhigh', 20);</syntaxhighlight>}}After analyzing {{Codeline|gm}} it is easy to observe that the least error is for the first neighborhood size of {{Codeline|gm}} which is equal to 0.49148. We will use it to produce the prediction vectors. Only the first 4800 datapoints from {{Codeline|amplitude}} are used in order to compare the predicted values with the actual ones for the last 200 points.{{Code|Creating forecast points|<syntaxhighlight lang="octave" style="font-size:13px">steps = 200;forecast = lzo_run (amplitude(1:end-200), 'm', 4, 'd', 6, 'r', 0.49148, 'l', steps);forecast_noisy = lzo_run (amplitude(1:end-200), 'm', 4, 'd', 6, 'r', 0.49148, ... 'dnoise', 10, 'l', steps);plot (amplitude(end-199:end), 'g;Actual Data;', ... forecast, 'r.;Forecast Data;',... forecast_noisy, 'bo;Forecast Data with 10% Dynamic Noise;');</syntaxhighlight>}}[[File:tisean_nl_prediction_2.png|400px|center]] The difference between finding the proper {{Codeline|r}} and not finding it can be very small. <br/>The produced data is the best local zeroth order fit on the {{Codeline|amplitude.dat}} for {{Codeline|(m &#61; 4, d &#61; 6)}}. === Nonlinear Noise Reduction ===This tutorial show different methods of the 'Nonlinear Noise Reduction' section of the TISEAN documentation (located [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node22.html#SECTION00060000000000000000 here]). It shows the use of simple nonlinear noise reduction (function {{Codeline|lazy}}) and locally projective nonlinear noise reduction (function {{Codeline|ghkss}}). To start let's create noisy data to work with.
{{Code|Creating a noisy henon map|<syntaxhighlight lang="octave" style="font-size:13px">
hen = henon (10000);
delay_noisy = delay (hen_noisy);
# Plot both on one chart
plot (delay_noisy(:,1), delay_noisy(:,2), 'b.;Noisy Data;','markersize',3,... delay_clean(:,1), delay_clean(:,2), 'r.;Clean Data;','markersize',3)
</syntaxhighlight>}}
[[File:tisean_nl_noisereduction.png|400px|center]]

On the chart created the red dots represent cleaned up data. It is much closer to the original than the noisy blue set.<br/>
Now we will do the same with {{Codeline|ghkss}}.
{{Code|Locally projective nonlinear noise reduction|<syntaxhighlight lang="octave" style="font-size:13px">
clean = ghkss (hen,'m',7,'q',2,'r',0.05,'k',20,'i',2);
</syntaxhighlight>}}The rest of the code is the same as the code used in the {{Codeline|lazy}} example. <br/>Once both results are compared it is quite obvious that for this particular example {{Codeline|ghkss}} is superior to {{Codeline|lazy}}. The TISEAN documentation points out that this is not always the case.[[File:tisean_nl_noisereduction_2.png|400px|center]] === Lyapunov Exponents ===Here I will demonstrate how to use the function {{Codeline|lyap_k}}. It estimates the maximal Lyapunov exponent from a time series (more information available from the TISEAN documentation located [http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/node27.html here]). In this tutorial we will estimate the maximal Lyapunov exponent for various embedding dimensions and then plot them. {{Code|Creating Lyapunov exponents|<syntaxhighlight lang="octave" style="font-size:13px"># Create delay vectors time seriesin = sin((1:2500).'./360) + cos((1:2500).'./180);# Estimate Lyapunov exponentsmmax_val = 20lyap_exp = lyap_k (in, 'mmin',2,'mmax',mmax_val,'d',6,'s',400,'t',500);</syntaxhighlight>}}In this function the output ({{Codeline|lyap_exp}} is a {{Codeline|5 x 20}} struct array. We will only use one row for both the clean plot.{{Code|Plotting Lyapunov exponents|<syntaxhighlight lang="octave" style="font-size:13px">cla resethold onfor j=2:mmax_val plot (lyap_exp(1,j-1).exp(:,1),lyap_exp(1,j-1).exp(:,2),'r');endforxlabel ("t [flow samples]");ylabel ("S(eps, embed, t)");hold off</syntaxhighlight>}}[[File:lyap_k.png|400px|center]] === 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 noisy datacorrelation entropy of a time series. The time series used here will be the Henon map.{{Code|Calculation correlation sum, dimension and entropy|<syntaxhighlight lang="octave" style="font-size:13px"># Create mapsdelay_clean hen = delay henon (clean10000);delay_noisy # Calculate the correlation sum, dimension and entropyvals = delay d2 (hen_noisyhen, 'd', 1, 'm', 5, 't',50);# Plot both on one chartcorrelation sumsubplot (2,3,1)plot do_plot_corr = @(delay_noisyx) loglog (x{1}(:,1), delay_noisyx{1}(:,2), 'b');hold onarrayfun (do_plot_corr, {vals.c2});Noisy Datahold offxlabel ("Epsilon")ylabel ("Correlation sums")title ("c2");# Plot correlation entropysubplot (2,3,4)do_plot_entrop = @(x) semilogx (x{1}(:,1),x{1}(:,2),'g');hold onarrayfun (do_plot_entrop,'markersize{vals.h2});hold offxlabel ("Epsilon")ylabel ("Correlation entropies");title ("h2")# Plot correlation dimensionsubplot (2,3,[2 3 5 6])do_plot_slope = @(x) semilogx (x{1}(:,1),x{1}(:,2),'r');hold onarrayfun (do_plot_slope, {vals.d2});hold offxlabel ("Epsilon")ylabel ("Local slopes")title ("d2");</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). delay_clean{{Code|Smooth output of d2|<syntaxhighlight lang="octave" style="font-size:13px"># Smooth d2 outputfigure 2smooth = av_d2 (vals,'a',2);# Plot the smoothed outputdo_plot_slope = @(x) semilogx (x{1}(:,1), delay_cleanx{1}(:,2), 'rb');hold onarrayfun (do_plot_slope, {smooth.d2});hold offxlabel ("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 ===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) &#61; xn^3}}. We then run it through {{Codeline|surrogates}} and plot the data.{{Code|Creating data from Gaussian process|<syntaxhighlight lang="octave" style="font-size:13px"># Create Gaussian process datag = zeros (2000,1);for i = 2:2000 g(i) = 0.7 * g(i-1) + (-6 + sum (rand ([size(1), 12]), 3));endfor# Create a measurement of itspike = g.^3;Clean Data# Create the surrogatesur = surrogates (spike);# Plot the datasubplot (2,1,1)plot (spike,'g');title ("spike")subplot (2,1,2)plot (sur,'markersize,3b');title ("surrogate")
</syntaxhighlight>}}
[[CategoryFile:Octave-Forgesurrogate_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.