Information Technology Reference
In-Depth Information
Proj
ξ
k
(
Λ
s
)=
Λ
s
,
ξ
k
H
I
i
=
1
b
ki
Λ
s
,
N
s
i
˜
=
Λ
(1.38)
i
=
1
b
ki
I
(
s
,
s
i
)
−
N
N
j
=
1
I
(
s
,
s
j
)
1
N
=
.
1.7.2 Optimization in the Space Spanned by the Intensity
Functions
As before, let
denote the set of spike trains for which we
wish to determine the principal components, and
{
s
i
∈S
(
T
)
,
i
=
1
,...,
N
}
{
λ
s
i
(
t
)
,
t
∈T,
i
=
1
,...,
N
}
the
corresponding intensity functions. The mean intensity function is
N
i
=
1
λ
s
i
(
t
)
,
1
N
¯
λ
(
)=
t
(1.39)
and, therefore, the centered intensity functions are
˜
¯
(
)=
λ
(
)
−
λ
(
)
.
λ
t
t
t
(1.40)
s
i
s
i
Again, the problem of finding the principal components of a set of data can be
stated as the problem of finding the eigenfunctions of unit norm such that the pro-
jections have maximum variance. This can be formulated in terms of the following
optimization problem. A function
ζ
(
t
)
∈
L
2
(
λ
s
i
(
t
)
,
t
∈T
)
is a principal component
if it maximizes the cost function
i
=
1
Proj
ζ
(
λ
s
i
)
2
1
N
2
J
(
ζ
)=
−
γ
ζ
−
(1.41)
i
=
1
λ
s
i
,
ζ
2
1
N
2
=
L
2
−
γ
ζ
−
,
where
γ
is the Lagrange multiplier constraining
ζ
to have unit norm. It can be
˜
shown that
ζ
(
t
)
lies in the subspace spanned by the intensity functions
{
λ
s
i
(
t
)
,
i
=
1
,...,
N
}
. Therefore, there exist coefficients
b
1
,...,
b
N
∈
R
such that
N
j
=
1
b
j
λ
s
j
(
t
)=
b
T
˜
ζ
(
t
)=
r
(
t
)
.
(1.42)
˜
T
. Substituting in Equation (1.31)
˜
T
with
b
=[
b
1
,...,
b
N
]
and ˜
r
(
t
)=
λ
s
1
(
t
)
,...,
λ
s
N
(
t
)
yields
