Digital Signal Processing Reference
In-Depth Information
typically considered to be binary operators, so that the theory can be readily
applied to nonlinear filtering in the context of binary representation, as in the
case of stack filters. In particular, here we consider binary-valued functions
defined on
d
-dimensional Euclidean space.
For filter class
C
and classifier
ψ
∈
ε
ψ
C
, the
empirical error
, ˆ
n
[
], on the sample
data is the fraction of times that
in the sample. It is the
error rate on the sample. The empirical-error classifier,
ˆ
ψ(
X
i
)
=
Y
i
for
(
X
i
,Y
i
)
ψ
n, C
, minimizes the
empirical error. We denote its mean-absolute error by ˆ
ε
n, C
. The design error is
ˆ
n, C
=
ε
n, C
−
ε
C
. In the decomposition of Equation 13.3,
E
[
ˆ
ε
n, C
] and
E
[
n, C
]
ε
n, C
] and
E
[
ˆ
are replaced by
E
[ˆ
n, C
], respectively.
Associated with the filter class
C
is its
Vapnik-Chervonenkis
(
VC
)
dimension
,
V
C
, whose details we leave to the literature.
88
This dimension can either be
finite or infinite. If it is finite, then
C
is called a
VC class
. Constraining the
filter class lowers the VC dimension. As a consequence of the fundamental
VC theorem,
89
,
90
for VC classes with dimension exceeding 2, the expected
value of the empirical-error design cost is given by
4
V
C
log
n
+
4
E
[
ˆ
n, C
]
≤
.
2
n
The VC dimension provides a bound in the design cost.
This bound can be applied to the design of nonlinear filters. Consider an
increasing filter
d
. Such a filter possesses a minimal morpholog-
ical representation in terms of erosions.
91
,
92
The class of all increasing filters
of this kind possessing a minimal representation with
m
erosions has VC
dimension bounded by
d
m
78
. Hence,
ψ
:
→{
0
,
1
}
4
d
m
/
2
log
n
+
4
E
[
ˆ
n, C
]
≤
.
2
n
The kind of issues studied in pattern recognition theory apply to genetic
regulatory networks, in particular to Boolean networks.
An important role in the inference of multivariate relationships between
genes was played by the
coefficient of determination
(COD), introduced by
Dougherty et al.
71-73
in the context of optimal nonlinear filter design. The
COD was subsequently proposed for inference of probabilistic Boolean net-
works as models of genetic regulatory networks.
74
The COD measures the
degree to which the expression levels of an observed gene set can be used to
improve the prediction of the expression of a target gene relative to the best
possible prediction in the absence of observations. The method allows incor-
poration of knowledge of other conditions relevant to the prediction, such
as the application of particular stimuli, or the presence of inactivating gene
mutations, as predictive elements affecting the expression level of a given
gene. Using the COD, one can find sets of genes related multivariately to a
given target gene.
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