Biology Reference
In-Depth Information
3.5
MODEL SELECTION
As we saw in the previous section, the model space consists of all PDSs that fit a
given data set. The problem is now to select models from the space.
There are numerous strategies for model selection. One naive approach is to simply
choose the interpolating polynomial constructed above. The problem, however, is that
predictions of the model are dependent on the form of the model. Specifically, the
terms that appear in the polynomials will “predict” the interactions among the nodes
as represented in the wiring diagramassociated to the PDS. While it is possible that the
chosen interpolation formula well characterizes the physical setting being modeled,
it is more likely that the resulting model will have little to no predictive power.
Therefore, strategies that incorporate appropriate priors are desirable (for example,
see [
27
,
28
]).
It is believed that GRNs are sparse, in the sense that they have few (less than half
of all possible) connections, and their edges follow a power-law distribution [
10
,
11
].
A model selection strategy that makes use of these assumptions is one based on
Gröbner bases. Given a set
f
j
+
I
of transition functions for a node, the method finds
a polynomial in the set that contains no terms in
I
. This is done by computing the
remainder of
f
j
upon division by the elements in
I
. As seen in Section
3.3
, polynomial
division in general is not well defined in the sense that remainders may not be unique,
but is well defined given a Gröbner basis. The strategy requires that a Gröbner basis
G
for
I
is computed. Then the process of computing the remainder of
f
j
upon division
by the elements of
I
results in a unique polynomial called the
normal form
of
f
j
with
respect to
and is denoted
NF
(
f
j
,
G
)
. Given a monomial ordering
and a Gröbner
G
basis
G
for
I
with respect to
, then
(
NF
(
f
1
,
G
),...,
NF
(
f
n
,
G
))
is a model that fits
the given data and is reduced with respect to the data.
Since Gröbner bases depend on the choice of monomial ordering, so do resulting
models.
Example 3.7.
In Example
3.6
, the graded reverse lexicographic ordering,
grevlex
,
with
x
1
, whereas the
lex
ordering with the same variable order results in models with the terms
{
x
2
x
3
will yield models involving the terms
{
1
,
x
1
,
x
2
,
x
3
}
x
3
,
x
3
}
. We can see this by doing the following in Macaulay 2 (comments
are preceded by two dashes):
1
,
x
3
,
R=ZZ/5[x1,x2,x3] --grevlex order is assumed
I1=ideal(x1-2,x2,x3) --same ideals as above
I2=ideal(x1-4,x2-3,x3-1)
I3=ideal(x1-3,x2-1,x3-4)
I=intersect{I1,I2,I3}
f1=x1ˆ2+x1+3
f2=3*x1ˆ2+x1+4
f3=2*x1ˆ2+2*x1+4
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