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to create a multidimensional reconstruction (embedding) space. If the embedding
space is generated properly, the behavior of trajectories in this embedding space
will have the same geometric and dynamical properties that characterize the actual
trajectories in the full multidimensional phase space of the system. The method of
delays was suggested by Packard et al. [17] in 1980 and was put on a firm theoretical
basis by Takens [32] in 1981.
From the set of observations
x
(
t
0
+
n
Δ
t
)=
x
(
n
)
, multivariate vectors in
d
-
dimensional space
y
(
n
)=(
x
(
n
)
,
x
(
n
+
τ
)
,...,
x
(
n
+(
d
−
1
)
τ
))
(16.21)
are used to trace out the orbit of the system. The observations,
x
, are a projection
of the multivariate phase space of the system onto the 1D axis of the
x
(
n
)
's. The
purpose of time-delay embedding technique is to unfold the projection back to a
multivariate phase space that is representative of the original system. In practice,
the natural questions of what time delay
(
n
)
and what embedding dimension
d
to use
in this reconstruction have had a variety of answers. The following sections present
the methods used in this chapter for determining
τ
τ
and
d
.
:
The choice of time delay is not a straightforward
problem. If it is taken too small, there is almost no difference between the different
elements of the delay vectors. If on the other hand
The time-delay parameter
τ
is very large, the different co-
ordinates may be almost uncorrelated. In this case the reconstructed attractor may
become very complicated, even if the true underlying attractor is simple. This is
typical of chaotic systems, where the autocorrelation function decays fast. Unfor-
tunately, since
τ
has no relevance in the mathematical framework, there exists no
rigorous way of determining its optimal value. At least a dozen different methods
have been suggested for the estimation of
τ
, and since all these methods yield val-
ues of similar magnitude, we should estimate
τ
just by a single preferred tool and
work with this estimate [11]. Past studies have made use of the autocorrelation func-
tion, but a quite reasonable objection to this procedure is that it is based on linear
statistics, not taking into account nonlinear dynamical correlations. Therefore, it is
sometimes recommended that one look for the first minimum of the time-delayed
mutual information
. This is the information we already possess about the value of
x
τ
.
On the interval explored by the data, we create a histogram for the probability
distribution of the data. We denote by
p
i
the probability that the signal assumes a
value inside the
i
th bin of the histogram, and let
p
ij
(
τ
)
(
t
+
τ
)
if we know
x
(
t
)
be the probability that
x
(
t
)
is
in bin
i
and
x
(
t
+
τ
)
is in bin
j
. Then the mutual information for time delay
τ
reads
(
τ
)=
i
,
j
p
ij
(
τ
)
ln
p
ij
(
τ
)
−
2
∑
i
I
p
i
ln
p
i
.
(16.22)
The value of the mutual information is independent of the particular choice of
histogram, as long as it is fine enough.
3
The first minimum of
I
(
τ
)
marks the time
3
Throughout this chapter 512 bins have been used.
