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In-Depth Information
the next
w
0
= 100 points
x
101
,x
102
,...,x
200
alsointotwonumbers,
p
(100)
and
p
(100)
2
,
2
, and so on. These projections, by construction, ap-
proximate the original series well. Therefore, we can represent the first
row
x
(200)
(1)
2
,
1
≡
[
x
1
,...,x
200
]
T
200
of
X
(200)
with just four numbers,
∈
R
≡
p
(100)
2
,
2
T
x
(100
,
1)
(1)
1
,
1
,p
(100)
1
,
2
,p
(100)
2
,
1
,p
(100)
4
. Doing the same for the
other rows of
X
(200)
, we construct a matrix
X
(100
,
1)
with just
n
=4
columns, which is a very good approximation of
X
(200)
.Consequently,
we compute the local patterns using
X
(100
,
1)
instead of
X
(200)
. Repeat-
ing this process recursively, we can find the local patterns for a window
w
(100
,
2)
= 100
∈
R
2
2
= 400 and so on.
Definition 5.9 (Level-(
w
0
,l
) window)
The level-
(
w
0
,l
)
window cor-
responds to an original window size (or
scale
)
w
l
:=
w
0
·
·
W
l
. Patterns at
each level
l
are found recursively, using patterns from the previous level
l
1
.
In the above example, we have
w
0
= 100 and
l
=0
,
1. Since
w
0
and
W
are fixed for a particular sequence of scales
w
l
, we will simply refer
to level-
l
windows and patterns. The recursive construction is based on
the level-
l
delay matrix and corresponding patterns.
Definition 5.10 (Level-
l
delay matrix
X
(
w
0
,l
)
)
Given a starting win-
dow
w
0
and a scale factor
W
, the level-
l
delay matrix is simply
X
(
w
0
,
0)
:=
X
(
w
0
)
for
l
=0
and for
l
=1
,
2
,...
it is recursively defined by
X
(
w
0
,l
)
:= Delay
P
(
w
0
,l−
1)
,W
,
where
P
(
w
0
,l
)
:=
X
(
w
0
,l
)
V
(
w
0
,l
)
is the projection onto the level-
l
patterns
V
(
w
0
,l
)
which are found based on
X
(
w
0
,l
)
. The level-
l
delay matrix is an
approximation of the delay matrix
X
(
w
l
)
for window size
w
l
=
w
0
W
l
.
In our example, the patterns extracted from
X
(100
,
1)
are four-dimensional
vectors,
v
(100
,
1)
−
4
, whereas the patterns for
X
(200)
would be 200-
dimensional vectors
v
(200)
∈
R
i
200
. However, we can appropriately com-
∈
R
i
bine
v
(100
,
1)
i
to estimate
v
(200
i
.
Definition 5.11 (Level-
l
local pattern
v0
(
w
0
,l
)
and
v
(100
,
0)
≡
v
(100)
i
i
i
)
The level-
l
pat-
tern
v0
(
w
0
,l
i
, for all
i
=1
,
2
,...,k
, corresponding to a window of
w
l
=
w
0
W
l
is simply
v0
(
w
0
,
0)
:=
v
(
w
0
)
for
l
=0
and for
l
=1
,
2
,...
it is
i
i
defined recursively by
v0
(
w
0
,l
)
[(
j
−
1)
w
l−
1
+1:
jw
l−
1
]:=
V0
(
w
0
,l−
1)
v
(
w
0
,l
)
i
1)
k
+1:
jk
]
,
(5.7)
[(
j
−
i
