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s
j
)=
Λ
s
i
,
Λ
s
j
I
(
s
i
,
H
I
.
(1.22)
It must be remarked that
H
I
is in fact a functional space. More specifically, that
points in
H
I
are functions of spike trains defined on
S
(
T
)
. This is a key difference
between the space of intensity functions
L
2
(
T
)
H
I
,
in that the latter allows for statistics of the transformed spike trains to be estimated
as functions of spike trains
. The usefulness of an RKHS for optimization and general
computation with spike trains can be appreciated, for example, in the derivation of
principal component analysis in Section 1.7.
explained above and the RKHS
1.6.3 mCI Kernel and the RKHS Induced by
κ
The mCI kernel estimator in Equation (1.18) shows the evaluation written in terms
of elementary kernel operations on the spike times. This fact alone provides a dif-
ferent perspective on how the mCI kernel uses the statistics of the spike times. To
see this more clearly, if
is chosen according to Section 1.3 as symmetric positive
definite, then it can be substituted by its inner product (Equation (1.4)) in the mCI
kernel estimator, yielding
κ
N
j
N
i
m
=
1
n
=
1
Φ
n
I
i
m
(
s
i
,
s
j
)=
,
Φ
H
κ
N
i
m
=
1
Φ
(1.23)
N
j
n
=
1
Φ
i
m
j
n
=
,
.
H
κ
When the number of samples approaches infinity (so that the intensity functions and,
consequently the mCI kernel, can be estimated exactly) the mean of the transformed
spike times approaches the expectation. Hence, Equation (1.23) results in
N
i
N
j
E
Φ
i
,
E
Φ
j
I
(
s
i
,
s
j
)=
H
κ
,
(1.24)
where
E
Φ
i
,
E
Φ
j
denotes the expectation of the transformed spike times and
N
i
,
N
j
are the expected number of spikes in spike trains
s
i
and
s
j
, respectively.
Equation (1.23) explicitly shows that the mCI kernel can be computed as an inner
product of the expectation of the transformed spike times in the RKHS
H
κ
induced
G
H
κ
and
H
by
κ
. In other words, there is a congruence
between
I
in this
ca
se given
N
i
E
i
,
explicitly by the expectation of the transformed spike times,
G
(
Λ
)=
Φ
s
i
such that
Λ
s
i
,
Λ
s
j
H
I
=
G
(
Λ
s
i
)
,G
(
Λ
s
j
)
H
κ
=
N
i
E
Φ
i
,
N
j
E
Φ
j
H
κ
.
(1.25)
Recall that the transformed spike times form a manifold (the subset of a hyper-
sphere) and, since these points have constant norm, the kernel inner product depends
