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This principle is a discrete formulation [92, 93] of a general principle holding very
often in biological systems, in order to keep a global equilibrium among all its com-
ponents. The term log-gain is related to the fact that the continuous version of the
log-gain of a real derivable function
f
, is the derivative of its logarithm, express-
ing its relative (infinitesimal) variation
d
log
f
(
x
)
dt
. In general terms, we can
assume that in a biological system dynamics obeys a global equilibrium among vari-
ables. Namely, the relative variation of a variable, depending on certain independent
variables, has to be a linear combination of its independent variables. In the influ-
ential topic [66], the biological significance and relevance of this principle, in many
specific contexts, is widely discussed.
For evaluating to which extent a regressor verifies the log-gain principle, we com-
pute for each of them a log-gain score, by using a least- squares approximation,
where the log-gain of the regressor, along the observation points, is equal to a linear
combination of the log-gain of its tuners. For example, if we want to test the good-
ness of the regressor
AB
, given by the product of two time series (
A
and
B
), then we
need to calculate the least- squares approximation for
c
1
,
df
(
x
)
dt
/
x
=
c
2
such that
Lg
(
A
[
i
]
B
[
i
]) =
c
1
Lg
(
A
[
i
])+
c
2
Lg
([
B
[
i
])
≤
<
where 0
t
. Then the obtained approximation error determines a criterion for
establishing a
log-gain ordering
among regressors by assigning greater log-gain
scores to the ones which provide smaller approximation errors.
i
3.4.2
The Stepwise Regression LGSS
Once regressors are ranked according to their log-gain scores, their choice is based
on a regression procedure which develops the ideas introduced in Sect. 3.3, by ex-
tending the classical method of stepwise regression [216, 76, 85]. Before presenting
LGSS regression, we will recall the classical
k-variable multiple regression
.The
reader can find more details and statistical motivations in Chapter 7 (Sect. 7.7) and
in Aczel and Sounderpandian's topic [216], from which we adopt the notation.
The following equation is the general form of a linear regression. Statistics pro-
vides methods for finding the right coefficients, possibly null, if they exist, ex-
pressing the kind of relationship between a
dependent variable Y
and one or more
independent variables X
1
,
X
2
, ...,
X
k
(if
k
>
1, then the regression equation above
is called a
multiple regression model
):
Y
=
β
0
+
β
1
X
1
+
β
2
X
2
+
...
+
β
k
X
k
+
ε
.
(3.32)
The LGSS algorithm, given in the gray box at the end of this section, is a pro-
cedure extending the one defined in Sect. 3.3, where MP systems were applied to
the problem of approximating real functions. Also, LGSS is based on least squares
estimation, but differently from standard regression, metabolic approximation in
LGSS has the form of a grammar imposed by the phenomenon under investigation.
