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))

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)

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);

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.

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).

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

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