Biology Reference
In-Depth Information
Draw the state space and wiring diagram of this stochastic PDS labeling the edges
with the corresponding probabilities.
Project 3.6.
Consider the four time series in Section 4.3 of [
30
]. Find the ideal
of polynomials that vanish on the series and using the software package Gfan [
31
],
compute its Gröbner fan.
3.6
DISCRETIZATION
For reasons explained at the beginning of Section
3.2
, we have been assuming that
the experimental data we use for reverse engineering have already been discretized
into a (small) finite number of states. Typically, however, experimental measurements
come to us represented by computer floating point numbers and consequently data
discretization is in fact part of the modeling process and can be viewed as a prepro-
cessing step. We will use the definition of discretization presented in [
35
].
Definition 3.11.
A
discretization
of a real-valued vector
v
=
(v
1
,...,v
N
)
is an
=
(
d
1
,...,
d
N
)
integer-valued vector
d
with the following properties:
1.
Each element of
d
is in the set 0
1 for some (usually small) positive
integer
D
, called the degree of the discretization.
2.
For all 1
,
1
,...,
D
−
≤
,
≤
≤
v
i
≤
v
j
.
i
j
N
,wehave
d
i
d
j
if and only if
,v
i
v
j
.
Spanning
discretizations of degree
D
satisfy the additional property that the smallest
element of
d
is equal to 0 and that the largest element of
d
is equal to
D
Without loss of generality, assume that
v
is sorted, i.e., for all
i
<
j
1.
There is no universal way for data discretization that works for all data sets and
all purposes. Sometimes discretization is a straightforward process. For example, if a
gene expression time series has a sigmoidal shape, e.g.,
−
(
.
,
.
,
,
.
,
)
0
1
1
2
2
23
04
26
,itis
reasonable to discretize it as
. More complicated expression profiles may
be easy to discretize too and it is often true that the human eye is the best discretization
“tool” whose abilities to discern patterns cannot be reproduced by any software.
Large data sets, on the other hand, do require some level of automatization in the
discretization process. Regardless of the particular situation, it is good practice to look
at the data first and explore for any patterns that may helpwith the discretization before
inputting the data into any discretization algorithm. Afterwards, the way you choose
to discretize your data, which includes selecting the number of discrete states, should
depend on the type and amount of data and the specific reason for discretization. Below
we present several possible approaches which by no means comprise a complete list.
Binary discretizations
are the simplest way of discretizing data, used, for instance,
for the construction of Boolean network models for gene regulatory networks [
36
,
37
].
The expression data are discretized into only two qualitative states as either present
or absent. An obvious drawback of binary discretization is that labeling real-valued
data according to a present/absent scheme may cause the loss of large amounts of
information.
(
0
,
0
,
0
,
1
,
1
)
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