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5.6 Nonlinear Time-Series Analysis (by N. Marwan) 111

The reconstruction of the phase space portrait using only the fi rst state and
a delay of six

tau = 6;
plot3(x(1:end-2*tau,1),x(1+tau:end-tau,1),x(1+2*tau:end,1))
xlabel('x_1'), ylabel('x_2'), zlabel('x_3')
grid, view([100 60])
reveals a similar phase portrait with the two typical ears (Fig. 5.13). The
characteristic properties of chaotic systems are also seen in this reconstruc-
tion.
The time delay and embedding dimension has to be chosen with a pre-
ceding analysis of the data. The delay can be estimated with the help of the
autocovariance or autocorrelation function. For our example of a periodic
oscillation,

x = 0 : pi/10 : 3*pi;
y1 = sin(x);

we compute and plot the autocorrelation function

for i = 1 : length(y1) - 2
r = corrcoef(y1(1 : end-i), y1(1 + i : end));
C(i) = r(1,2);
end

plot(C)
xlabel('Delay'), ylabel('Autocorrelation')
grid on

Now we choose such a delay at which the autocorrelation function equals
zero for the fi rst time. In our case this is 5, which is the value that we have
already used in our example of phase space reconstruction. The appropriate
embedding dimension can be estimated by using the false nearest neigh-
bours method or, simpler, recurrence plots, which are introduced in the next
chapter. Tthe embedding dimension is gradually increased until the majority
of the diagonal lines are parallel to the line of identity.
The phase space trajectory or its reconstruction is the base of several mea-
sures defined in nonlinear data analysis, like Lyapunov exponents, Rényi
entropies or dimensions. The book on nonlinear data analysis by Kantz and
Schreiber (1997) is recommended for more detailed information on these
methods. Phase space trajectories or their reconstructions are also the neces-
sary for constructing recurrence plots.