Digital Signal Processing Reference
In-Depth Information
Let us briefly discuss the COD in the context of Boolean networks. Let
x
i
be a
target
gene that we wish to predict by observing some other genes
x
i
1
,x
i
2
,
...
,x
i
k
. Also, suppose
f
(
x
i
1
,x
i
2
,
...
,x
i
k
)
is an optimal predictor of
x
i
ε
relative to some error measure
. For example, in the case of mean-square
error (MSE) estimation, the optimal predictor is the conditional expectation
of
x
i
given
x
i
1
,x
i
2
,
opt
be the optimal error achieved by
f
. Then,
the COD for
x
i
relative to
x
i
1
,x
i
2
,
...
,x
i
k
. Let
ε
,x
i
k
is defined as
θ
=
ε
i
−
ε
opt
ε
i
...
,
(13.5)
where
i
is the error of the best (constant) estimate of
x
i
in the absence of
any conditional variables. It is easily seen that the COD must be between 0
and 1 and measures the relative decrease in error from estimating
x
i
via
f
rather than by just the best constant estimate. In practice, the COD must be
estimated from training data with designed approximations used in place of
f
. Those sets of (predictive) genes that yield the highest COD, compared to all
other sets of genes, are the ones used to construct the optimal predictor of the
target gene. Given limited amounts of training data, it is prudent to constrain
the complexity of the predictor by limiting the number of possible predictive
genes that can be used. This corresponds to limiting the connectivity
K
of
the Boolean network. Finally, the above procedure is applied to all target
genes, thus estimating all the functions in a Boolean network. The method
is computationally intensive and massively parallel architectures have been
employed to handle large gene sets.
75
ε
13.4 Concluding Remarks
The modeling and analysis of genetic regulatory networks represent an im-
portant and rapidly developing area in computational genomics research,
requiring a multidisciplinary approach. In this chapter, by using Boolean
networks as an illustrative example, we have discussed several intimate con-
nections with nonlinear signal processing theories related to root signals and
optimal design of nonlinear digital filters. Many methods and algorithms
in nonlinear signal processing, such as the design of nonlinear filters with a
specified set of root signals, can be carried over to Boolean networks and their
generalizations. For example, ideas and results from mathematical morphol-
ogy were recently used to characterize important mappings between so-called
probabilistic Boolean networks,
76
which have been proposed as models for ge-
netic regulatory networks. It is our belief that researchers with a background
in nonlinear signal processing have the potential to make significant contri-
butions and bring their unique perspectives to this exciting and important
field.
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