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for
j
=1
,
2
,...,W
. It is an approximation of the local patterns
v
(
w
l
)
of
i
the original delay matrix
X
(
w
l
)
, for window size
w
l
=
w
0
W
l
.
Consider
v0
(100
,
1)
in our example. The first
k
=2outof
kW
=4num-
1
bers in
v
(100
,
1)
approximate the patterns among the 2-dimensional vec-
1
tors
p
(100
,
0)
(
j
)
, which in turn capture patterns among the 100-dimensional
vectors
x
(100
,
0)
(
i
)
of the original time-delay matrix. Thus, but forming the
appropriate linear combination of the 100-dimensional patterns
v
(100
,
0)
≡
i
v0
(100
,
0)
(i.e., the columns of
V
(100
,
0)
≡
V0
(100
,
0)
), weighted according
to
v
(100
,
1
1
[1 : 2], we can construct the first half of the 200-dimensional
linear combination of the columns of
V
(100
,
0)
i
≡
V0
(100
,
0)
weighted ac-
cording to
v
(100
,
1)
1
[3 : 4] gives us the second half of the 200-dimensional
level
l
= 2 we similarly combine the columns of
V0
(100
,
1)
according to
v
(100
,
2)
[1 : 2] (for the first half,
v0
(100
,
2)
[1 : 200]) and to
v
(100
,
2)
[
3 : 4] (for
1
1
the second half,
v0
(100
,
2)
[201 : 400]) and so on, for the higher levels.
1
Lemma 5.12 (Orthonormality of
v0
(
w
0
,l
)
)
We have
i
=
j
,
v0
(
w
0
,l
)
T
v0
(
w
0
,l
)
=0
,where
i,j
=
v0
(
w
0
,l
)
=1
and, for
i
i
i
j
1
,
2
,...,k
.
Proof.
For level
l
= 0 they are orthonormal since they coincide with
the original patterns
v
(
w
0
i
which are by construction orthonormal. We
proceed by induction on the level
l
1. Without loss of generality,
assume that
k
= 2 and, for brevity, let
B
≡
V0
(
w
0
,l−
1)
and
b
i,
1
≡
v
(
w
0
,l
)
≥
[1 :
k
],
b
i,
2
≡
v
(
w
0
,l
)
[
k
+1:
k
], so that
v
(
w
0
,l
)
=[
b
i,
1
,
b
i,
2
]. Then
i
i
i
v0
(
w
0
,l
)
2
=[
Bb
i,
1
Bb
i,
2
]
2
=
Bb
i,
1
2
+
Bb
i,
2
2
i
v
(
w
0
,l
)
2
+
2
=
2
=1
,
=
b
i,
1
b
i,
2
i
and
v0
(
w
0
,l
)
T
v0
(
w
0
,l
)
=[
Bb
i,
1
Bb
i,
2
]
T
[
Bb
j,
1
Bb
j,
2
]
=
b
i,
1
B
T
Bb
j,
1
+
b
i,
2
B
T
Bb
j,
2
=
b
i,
1
b
j,
1
+
b
i,
2
b
j,
2
=
v
(
w
0
,l
)
i
j
T
v
(
w
0
,l
)
=0
,
i
j
