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delay where
x
(
t
+
τ
)
adds maximal information to the knowledge we have from
x
(
t
)
.
This is the time delay used in this chapter.
Embedding dimension:
What is the appropriate value of
d
to use as the em-
bedding dimension? The procedure used here identifies the number of
false near-
est neighbors
, points that appear to be nearest neighbors because the embedding
space is too small, of every point on the attractor associated with the orbit
y
(
n
)
,
n
N
. When the number of false nearest neighbors drops to 0, we have
unfolded or embedded the attractor in
=
1
,
2
,...,
d
,a
d
-dimensional Euclidean space.
If we are in
d
dimensions and we denote the
r
th nearest neighbor of
y
R
(
n
)
by
y
(
r
)
(
, then from Equation (16.21), the square of the Euclidean distance between
the point
y
n
)
(
n
)
and this neighbor is
d
−
1
k
=
0
[
x
(
n
+
k
τ
)
−
x
(
r
)
(
n
+
k
τ
)]
R
d
(
2
n
,
r
)=
.
(16.23)
In going from dimension
d
to dimension
d
+
1 by time-delay embedding we add
a
(
d
+
1
)
th coordinate onto each of the vectors
y
(
n
)
. This new coordinate is just
x
(
n
+
τ
d
)
. The Euclidean distance, as measured in dimension
d
+
1, between
y
(
n
)
and the same
r
th neighbor as determined in dimension
d
is given by
R
d
+
1
(
R
d
(
x
(
r
)
(
2
n
,
r
)=
n
,
r
)+[
x
(
n
+
τ
d
)
−
n
+
τ
d
)]
.
(16.24)
A natural criterion for catching embedding errors is that the increase in distance
between
y
and
y
(
r
)
(
1. The in-
crease in distance can be stated quite simply from Equations (16.23) and (16.24).
We state this criterion by designating as a false neighbor any neighbor for which
(
n
)
n
)
is large when going from dimension
d
to
d
+
R
d
+
1
(
R
d
(
n
,
r
)
−
n
,
r
)
x
(
r
)
(
=
|
x
(
n
+
τ
d
)
−
n
+
τ
d
)
|
>
R
tol
,
(16.25)
R
d
(
n
,
r
)
R
d
(
n
,
r
)
where
R
tol
is some threshold. In practical settings the number of data points is often
not large, and the following criterion handles the issue of limited data set size: If the
nearest neighbor to
y
(
n
)
is not close (
R
d
(
n
)
≈
R
A
) and it is a false neighbor, then
the distance
R
d
+
1
(
n
)
resulting from adding on a
(
d
+
1
)
th component to the data
vectors will be
R
d
+
1
(
2
R
A
[13]. That is, even distant but nearest neighbors will
be stretched to the extremities of the attractor when they are unfolded from each
other, if they are false nearest neighbors. We write this second criterion as
n
)
≈
R
d
+
1
(
)
R
A
>
n
A
tol
,
(16.26)
where
R
A
denotes the size of the attractor. Both criterions in Equations (16.25) and
(16.26) are used jointly throughout the determination of
d
E
(also see Fig. 16.1). As
a measure of
R
A
we have chosen the standard deviation
of the observed data
x
according to [12]. This source also gives us the recommended values
R
tol
=
σ
(
x
)
15
.
0
and
A
tol
=
2
.
0.
