Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 67

m is defined. For the deterministic system we
can describe the dynamics by an explicit system of m first-order ordinary differential
equations :

m , in which a state x

vector space R ∈ R

dx ( t )

m

= f ( t , x ( t )) , x ∈ R (2.126)

dt

If the time is treated as a discrete variable, the representation can take a form of
an m -dimensional map:

x n + 1 = F ( x n ) , n ∈ Z (2.127)

solving the above equations is called a trajectory of
the dynamical system. Typical trajectories can run away to infinity or can be confined
to the certain space, depending on F (or f ). An attractor is a geometrical object to
which a dynamical system evolves after a long enough time. Attractor can be a point,
a curve, a manifold or a complicated object called a strange attractor . Attractor is
considered strange if it has non-integer dimension. Non-linear chaotic systems are
described by strange attractors.
Now we have to face the problem that, what we observe is not a phase space
object but a time series and, moreover, we don't know the equations describing the
process that generates them. The delay embedding theorem of Takens [Takens, 1981]
provides the conditions under which a smooth attractor can be reconstructed from
a sequence of observations of the state of a dynamical system. The reconstruction
preserves the properties of the dynamical system that do not change under smooth
coordinate changes. A reconstruction in d dimensions can be obtained by means of
the retarded vectors:

A sequence of points x n or x ( t )

ξ ( t )= x ( t i ) , x ( t i + τ ) ,..., x ( t i +( m − 1 ) τ ) (2.128)

Number m is called the embedding dimension, τ is called delay time or lag.
According to the above formula almost any scalar observation (e.g., time series),
is sufficient to learn about the evolution of a very complex high-dimensional deter-
ministic evolving system. However, we don't know in advance how long the retarded
vector must be and we don't know the delay τ.
Choosing a too small value of τ would give a trivial result and too large τ would
hamper the information about the original system. Usually the time coordinate of the
first minimum of the autocorrelation function is taken as τ. The phase portrait of an
ECG obtained by embedding in three-dimensional space is shown in Figure 2.22.
Please note (picture b) the distortion of the phase portrait for too large τ. Finding m
is even more complex. The method of false neighbors [Kantz and Schreiber, 2000]
is difficult to apply and doesn't give unequivocal results. Usually the embedding
dimension is found by increasing the dimension step by step.

2.5.1 Lyapunov exponent
Lyapunov exponents describe the rates at which nearby trajectories in phase space
converge or diverge; they provide estimates of how long the behavior of a mechanical