Information Technology Reference
In-Depth Information
Fig. 3.2
Principal
components:
c
1
and
c
2
z
c
2
c
1
y
x
i
=
1
λ
i
i
=
1
λ
i
≥
v
(3.17)
being
v
the degree of desired information, or variability index (1 value, means that we
use all the eigenvalues). The PCA works in a vector space. If we work in a space of
functions, the analysis will be the FPCA. Let
f
1
(
x
),
f
1
(
x
),...,
f
n
(
x
)
be functions
in separable Hilbert space endowed with inner product
X
L
2
f
i
|
f
j
=
f
i
(
x
)
f
j
(
x
)
dx
∀
f
i
,
j
∈
[
0
,
X
]
(3.18)
0
If each function
f
i
(
x
)
may be decomposed in
L
c
i
T
f
i
(
x
)
=
c
il
θ
l
(
x
)
=
(
x
)
(3.19)
l
=
1
The mean and covariance functions of
f
i
, will be
f
c
T
(
x
)
=
E
(
f
(
x
))
= ¯
(
x
)
(3.20)
T
cov
[
(
),
(
)
]=
(
)
(
)(
)
Cov
f
x
f
s
x
C
s
(3.21)
where
C
={
c
il
,
i
=
1
,...,
n
,
l
=
1
,...,
L
}
. We define the covariance operator as:
X
L
2
C
(
f
(
x
))
=
Cov
[
f
(
x
),
f
(
s
)
]
f
(
s
)
ds
,
∀
f
∈
[
0
,
X
]
,
∀
x
,
s
∈[
0
,
X
]
0
(3.22)
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