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5.6 Nonlinear Time-Series Analysis (by N. Marwan) 115

t = 0.01 : 0.05 : 50;
y = x(1:5:5000,1);
m = 3; tau = 2;

N = length(y);
N2 = N - tau*(m - 1);
The original data series has a length of 5000, after resampling 1000 data points
or 50 sec, but because of the time delay method, the reconstructed phase space
trajectory has the length 996. Now we create the phase space trajectory with
for mi = 1:m
xe(:,mi) = y([1:N2] + tau*(mi-1));
end
We can accelerate the pair-wise test between each points on the trajectory
with a fully vectorized algorithm supported by MATLAB. For that we need
to transfer the trajectory vector into two test vectors, whose component-wise
test will provide the pair-wise test of the trajectory vector:

x1 = repmat(xe,N2,1);
x2 = reshape(repmat(xe(:),1,N2)',N2*N2,m);
Using these vectors we calculate the recurrence plot using the Euclidean
norm without any FOR loop.

S = sqrt(sum((x1 - x2).^ 2,2 ));
S = reshape(S,N2,N2);
imagesc(t(1:N2),t(1:N2),S<10)
colormap([1 1 1;0 0 0])
xlabel('Time'), ylabel('Time')

This recurrence plot reveals many short diagonal lines (Fig. 5.16). These
lines represent epochs, where the phase space trajectory runs parallel to for-
mer or later sequences of this trajectory, i.e., the states and the dynamics are
similar at these times. The distances between these diagonal lines, represent-
ing the periods of the cycles, differ and are not constant – just as they are in
a harmonic oscillation.
The structure of recurrence plots can also be described by a suite of quan-
titative measures. Several measures are based on the distribution of the
lengths of diagonal or vertical lines. These parameters can be used to trace
hidden transitions in a process. Bivariate and multivariate extensions of re-
currence plots furthermore offer nonlinear correlation tests and synchroni-
zation analysis. A detailed introduction to recurrence plot based methods
can be found at the web site
http://www.recurrence-plot.tk