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spike trains clarifies exactly from basic principles which kernels can be used and
what are the general properties of the mCI kernel defined. Even though it may be
argued that kernel methods can be applied directly for spike trains data given a ker-
nel, the true meaning of using such a kernel cannot be well determined. This is
one of the strengths of the explicit construction followed. In this way, the general
structure of the RKHS space induced is well understood allowing for methods to be
derived from their basic ideas. Additionally, we were able to establish a close math-
ematical relationship to several congruent spaces where the derived methods can be
thoroughly comprehended. Still, it must be remarked that the mCI kernel presented
here will likely not be the most appropriate for a number of problems. This was
not the goal of this chapter. Instead one of our aims was to show how other kernels
that operate with spike trains may be easily formulated. Depending on a specific
application other kernels may be defined which lead to simpler solutions and/or are
computationally simpler.
It is noteworthy that the mCI kernel is not restricted to applications with spike
trains but rather can be applied to processing with any Poisson point processes. In
fact, the mCI kernel can be applied for even more general point processes. Natu-
rally, it might not be the optimum inner product for point processes other than Pois-
son processes since the intensity function does not fully characterizes the process
but, in a sense, this is similar to the use of cross-correlation in continuous random
processes, which is only sensitive to second-order statistics.
Acknowledgments A. R. C. Paiva was supported by Funda¸ ˜aoparaaCiencia e a Tecnologia
(FCT), Portugal, under grant SRFH/BD/18217/2004. This work was partially supported by NSF
grants ECS-0422718 and CISE-0541241.
Appendix: Proofs
This section presents the proofs for Properties 1.2, 1.5, and 1.6 in Section 1.5.1.
Proof (Property 1.2). The symmetry of the matrix results immediately from Prop-
erty 1.1.
By definition, a matrix is nonnegative definite if and only if
T
a
Va
0, for any
T
a
=[
a 1 ,...,
a n ]
with a i R
. So, we have that
n
i = 1
n
j = 1 a i a j I ( s i , s j ) ,
T
a
Va =
(1.44)
which, making use of the mCI kernel definition (Equation (1.12)), yields
n
i = 1 a i λ s i ( t )
n
j = 1 a j λ s j ( t )
dt
T
a
Va =
T
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