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only on the angle between points. This is typically not true for the average of these
points, however. Observe that the circular variance [14] of the transformed spike
times of spike trains
s
i
is
E
Φ
m
−
E
Φ
i
,
E
Φ
i
H
κ
i
i
m
i
var
(
Φ
)=
,
Φ
H
κ
(1.26)
)
−
E
Φ
i
2
H
κ
.
=
κ
(
0
So, the norm of the mean transformed spike times is inversely proportional to the
variance of the elements in
H
κ
. This means that the inner product between two spike
trains depends also on the dispersion of these average points. This fact is important
because data reduction techniques rely heavily on optimization with the data vari-
ance. For instance, kernel principal component analysis [30] directly maximizes the
variance expressed by Equation (1.26) [19].
1.6.4 mCI Kernel as a Covariance Kernel
In Section 1.5.1 it was shown that the mCI kernel is indeed a symmetric positive
definite kernel. As mentioned in Section 1.2, Parzen [22] showed that any symmetric
and positive definite kernel is also a covariance function of a random process defined
in the original space of the kernel (see also Wahba [38, Chapter 1]). In the case of
the mCI kernel, this means the random processes are defined on
S
(
T
)
.
Let
X
denote this random process. Then, for any
s
i
∈S
(
T
)
,
X
(
s
i
)
is a random
variable on a probability space
with measure
P
. As proved by Parzen, this
random process is Gaussian distributed with zero mean and covariance function
(
Ω
,B,
P
)
E
ω
X
s
j
)
.
I
(
s
i
,
s
j
)=
(
s
i
)
X
(
(1.27)
Notice that the expectation is over
ω
∈
Ω
since
X
(
s
i
)
is a random variable defined
on
.
This means that
X
is actually a doubly stochastic random process. An intrigu-
ing perspective is that, for any given
Ω
, a situation which can be written explicitly as
X
(
s
i
,
ω
)
,
s
i
∈S
(
T
)
,
ω
∈
Ω
ω
,
X
(
s
i
,
ω
)
is an ordered and almost surely
nonuniform sampling of
X
(
·,
ω
)
. The space spanned by these random variables is
L
2
(
X
(
s
i
)
,
s
i
∈S
(
T
))
since
X
is obviously square integrable (that is,
X
has finite
covariance).
The RKHS
H
I
induced by the mCI kernel and the space of random functions
L
2
are clearly congruent. This fact is a consequence of the ba-
sic congruence theorem [22] since the two spaces have the same dimension or, al-
ternatively, by verifying that the congruence mapping between the two spaces ex-
ists. For this reason we may consider the mCI kernel also as a covariance measure
of random variables directly dependent on the spike trains with well-defined sta-
tistical properties. Allied to our familiarity and intuitive knowledge of the use of
covariance (which is nothing but cross-correlation between centered random vari-
ables) this concept can be of great importance in optimization and design of optimal
(
X
(
s
i
)
,
s
i
∈S
(
T
))
