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Property 1.6. For any two point processes s i ,
s j ∈S ( T )
the triangle inequality holds.
That is,
λ s i + λ s j λ s i + λ s j .
As before, the proof is given in the appendix.
1.5.2 Estimation
As previous stated, spike trains are realizations of underlying point processes, but
the memoryless cross-intensity kernel as presented so far is a deterministic operator
on the point processes rather than on the observed spike trains. Using a well-known
methodology for the estimation of the intensity function we now derive an estimator
for the memoryless cross-intensity kernel. One of the advantages of this route is that
the conceptual construction of spike train kernel is dissociated from the problem
of estimation from data. Put differently, in this way it is possible to have a clear
statistical interpretation while later approaching the problem from a practical point
of view. The connection between the mCI kernel and
will now become obvious.
A well-known method for intensity estimation from a single spike train is kernel
smoothing [5, 26]. Accordingly, given a spike train s i comprising of spike times
{
κ
t i m ∈T
: m
=
1
,...,
N i }
the estimated intensity function is
N i
m = 1 h ( t t i m ) ,
ˆ
λ s i (
t
)=
(1.17)
where h is the smoothing function. This function must be nonnegative and inte-
grate to one over the real line (just like a probability distribution function (pdf)).
Commonly used smoothing functions are the Gaussian, Laplacian, and
α
-functions,
among others.
From a filtering perspective, Equation (1.17) can be seen as a linear convolution
between the filter impulse response given by h
and the spike train given as a sum
of Dirac functionals centered at the spike times. In particular, binning is nothing
but a special case of this procedure in which the spike times are first quantized
according to the binsize and h is a rectangular window [5]. Moreover, compared with
pdf estimation with Parzen windows [21], we immediately observe that intensity
estimation as shown above is directly related to the problem of pdf estimation except
for a normalization term, a connection made clear by Diggle and Marron [6].
Consider spike trains s i
(
t
)
ˆ
,
s j
∈S ( T )
with estimated intensity functions
λ
(
t
)
and
s i
ˆ
λ s j (
according to Equation (1.17). Substituting the estimated intensity functions
in the definition of the mCI kernel (Equation (1.12)) yields
t
)
N j
n = 1 κ ( t i m t n ) ,
N i
m = 1
I
(
s i ,
s j )=
(1.18)
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