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Figure 4. Phase space representation of a popular song (left) and a classical piece (right)
(a) pop music
(b) classic
Figure 5. Intervals found in the index dimension are summarized.
(a) Intervals in index dimension
(b) Smoothing of intervals
Definition 6
Vectors within
phase
space
are con-
structed, where the components
are parts of the
original series:
separating classic from the more percussive pop
music as shown in Figure 4.
Basis transformations most often are revers-
ible, because only the basis, not the position of the
elements is changed. In contrast, if intervals in the
index dimension are used, the transformation is
not reversible. If, for instance, we summarize the
original series by some time intervals and assign
a value to each interval, the transformed series
has still the same number of elements but fewer
different values (see Figure 5 for illustration).
We will discuss some possible ways to detect
intervals in different dimensions of a value series
in Section 3.3.
p
=
(
x
,
x
,
x
,...,
x
)
i
i
i
+
d
i
+
2
d
i
+
(
m
−
1
d
where d is the delay, and m the dimension of the
phase space. The set
P
=
{
p
|
i
=
1
,...,
n
−
(
m
−
1
d
}
d
,
m
i
is the phase space representation of the original
series
(
xi
)
i
∈ {1
,…,n
}
.
Within the phase space, several features can
be extracted, for example, the angles between
vectors. Small variances of angles indicate smooth
changes of the state variables, large variances
harsh changes. This is a dominant feature when
filters
Filters transform elements of a series to another
location within the same space. Moving average
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