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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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