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where
is the 'kernel' obtained by the autocorrelation of the smoothing function
h
. Notice that ultimately the obtained estimator linearly combines and weights the
contribution of a kernel operating on a pair of event coordinates. Moreover, this
estimator operates directly on the event coordinates of the whole realization without
loss of resolution and in a computationally efficient manner since it takes advantage
of the, typically, sparse occurrence of events.
If the kernel
κ
is chosen such that it satisfies the requirements in Section 1.3, then
the mCI kernel corresponds to a summation of all pairwise inner products between
spike times of the spike trains, evaluated by kernel on the spike time differences.
Put in this way, we can now clearly see how the mCI inner product on spike trains
builds upon the inner product on spike times denoted by
κ
and the connection to
Equation (1.10). The later approach, however, clearly states the underlying point
process model.
κ
1.6 Induced RKHS and Congruent Spaces
H
Some considerations about the RKHS space
I
induced by the mCI kernel and
congruent spaces are made in this section. The relationship between
I
and its con-
gruent spaces provides alternative perspectives and a better understanding of the
mCI kernel. Figure 1.1 provides a diagram of the relationships among the various
spaces discussed next.
H
Fig. 1.1: Relation between the original space of spike trains
and the vari-
ous Hilbert spaces. The
double-line
bi-directional connections denote congruence
between spaces.
S
(
T
)
