Recurrence Plot & Quantification

Simple MATLAB functions for calculating recurrence plots and recurrence quantification.

Functions

EMBED

Creates embedding vector using time delay embedding.

Syntax

Y=EMBED(X,M,T) creates the embedding vector Y from the time series X using a time delay embedding with dimension M and delay T. The resulting embedding vector has length N-T*(M-1), where N is the length of the original time series.

Reference

• Packard, N. H., Crutchfield, J. P., Farmer, J. D., Shaw, R. S. (1980). Geometry from a time series. Physical Review Letters 45, 712-716.

Example

N = 300; % length of time series
x = .9*sin((1:N)*2*pi/70); % exemplary time series
y = embed(x,2,17); % embed into 2 dimensions using delay 17
plot(y(:,1),y(:,2))

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RP

Calculates a recurrence plot.

Syntax

R=RP(X,E,THRESH,NORM,ALG) calculates the recurrence plot R from an embedding vector X and using the threshold E. X is a N-by-M matrix corresponding to N time points and M embedding dimensions.

[R,D]=RP(...) outputs the recurrence plot R and the underlying distance matrix D.

Optional arguments:

NORM - is a string setting the norm for distance calculation in phasespace. Can be 'euc' for euclidian norm (default) or 'max' for maximum norm.

ALG - is a string specifying the algorithm of calculating the distance matrix. Can be 'loops', 'vector' (default), or 'matlabvector'.

THRESH - is a string that specifies how the threshold epsilon will be calculated. With 'fix' (default) the RP is computed with a fixed threshold epsilon specified by the input parameter E. With 'var' the RP is computed with a fixed threshold epsilon, which corresponds to the lower 'E'-quantile (specified by E) of the distance distribution of all points in phasespace. With 'fan' the RP is computed with a variable threshold resulting in a fixed amount of nearest neighbours in phasespace, specified % by the fraction E of recurrence points

Reference

• Marwan, N., Romano, M. C., Thiel, M., Kurths, J. (2007). Recurrence plots for the analysis of complex systems. Physics Reports, 438, 237-329.
• Kraemer, K. H., Donner, R. V., Heitzig, J., & Marwan, N. (2018). Recurrence threshold selection for obtaining robust recurrence characteristics in different embedding dimensions. Chaos, 28, 085720.

Example

N = 300; % length of time series
x = .9*sin((1:N)*2*pi/70); % exemplary time series
xVec = embed(x,2,17); % embed into 2 dimensions using delay 17
R = rp(xVec,.1,'fix','max'); % calculate RP using maximum norm and fixed threshold
imagesc(R)

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RQA

Calculates recurrence quantification analysis.

Syntax

Q=RQA(R,L,T) calculates measures of recurrence quantification analysis for recurrence plot R using minimal line length L and a Theiler window `T.

Output:

• Y(1) = RR (recurrence rate)
• Y(2) = DET (determinism)
• Y(3) = <L> (mean diagonal line length)
• Y(4) = Lmax (maximal diagonal line length)
• Y(5) = ENTR (entropy of the diagonal line lengths)
• Y(6) = LAM (laminarity)
• Y(7) = TT (trapping time)
• Y(8) = Vmax (maximal vertical line length)
• Y(9) = RTmax (maximal white vertical line length)
• Y(10) = T2 (recurrence time of 2nd type)
• Y(11) = RTE (recurrence time entropy, i.e., RPDE)
• Y(12) = Clust (clustering coefficient)
• Y(13) = Trans (transitivity)

Reference

• Marwan, N., Romano, M. C., Thiel, M., Kurths, J. (2007). Recurrence plots for the analysis of complex systems. Physics Reports, 438, 237-329.
• Marwan, N., Donges, J. F., Zou, Y., Donner, R. V., Kurths, J. (2009). Complex network approach for recurrence analysis of time series. Physics Letters A, 373, 4246-4254.

Example

N = 300; % length of time series
x = .9*sin((1:N)*2*pi/70); % exemplary time series
xVec = embed(x,2,17);
R = rp(xVec,.1);
Y = rqa(R);

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RP_ISO

Calculates the isodirectional recurrence plot

Syntax

R=RP_ISO(X,E,W) calculates the isodirectional recurrence plot R from an embedding vector X and using the threshold E for the vector distances and threshold W for the angle to be considered as isodirectional.

R=RP_ISO(X,E,W,TAU) estimates tangential vector using time delay TAU.

Example

[t x] = ode45('lorenz',[0 100],[-6.2 -10 14]);
[R1, SP, R0] = rp_iso(x(3000:5000,:),10,.2);

nexttile
imagesc(R0) % regular RP
axis square

nexttile
imagesc(R1) % isodirectional RP
axis square

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RP_PERP

Calculates the perpendicular recurrence plot

Syntax

R=RP_PERP(X,E,W) calculates the perpendicular recurrence plot R from an embedding vector X and using the threshold E for the vector distances and threshold W for the angle to be considered as perpendicular.

R=RP_PERP(X,E,W,TAU) estimates tangential vector using time delay TAU (works only if condition in line 95 is set to 0).

Example

[t x] = ode45('lorenz',[0 100],[-6.2 -10 14]);
[R1, SP, R0] = rp_perp(x(3000:5000,:),10,.25);

nexttile
imagesc(R0) % regular RP
axis square

nexttile
imagesc(R1) % perpendicular RP
axis square

Application

Part of this code was used in the study

• M. H. Trauth, A. Asrat, W. Duesing, V. Foerster, K. H. Kraemer, N. Marwan, M. A. Maslin, F. Schaebitz: Classifying past climate change in the Chew Bahir basin, southern Ethiopia, using recurrence quantification analysis, Climate Dynamics, 53(5), 2557–2572 (2019). DOI:10.1007/s00382-019-04641-3

How to cite

• Use the citation provided by Zenodo https://zenodo.org/record/6325324

Background

read more about recurrence plot analysis at http://www.recurrence-plot.tk/

License

(see LICENSE file)

Copyright 2016-2020, Potsdam Institute for Climate Impact Research (PIK), Institute of Geosciences, University of Potsdam, K. Hauke Kraemer, Norbert Marwan, Martin H. Trauth