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assumed to be linear combinations of polynomial regressors, it is very common to
meet multicollinearity problems.
In order to overcome the problem, the stepwise algorithm implemented in LGSS
was extended by filtering the expanded regressors, during the forward selection
phase, in order to avoid the insertion of a regressor which is highly correlated with
others already inserted in the regression model. To this end, LGSS computes the
variance inflation factor
(VIF) for each regressor, which gives an idea of the degree
of multicollinearity it introduces with respect to the other regressors in the regres-
sion equation. To calculate the VIF for a regressor
g
, we need to run a multiple
regression by considering
g
as the dependent variable and the set of already inserted
regressors as the set of independent variables. The variance inflation factor associ-
ated with
g
is:
1
VIF
(
g
)=
(3.46)
1
−
R
g
where
R
g
is the coefficient of determination for the multiple regression of
g
with
respect to the other regressors.
A VIF of 6, for example, means that the variance of the regression coefficient es-
timator for the considered regressor is 6 times what it should be when no collinear-
ity exists. In LGSS the user can select a threshold value for the variance inflation
factor in order to avoid the insertion of collinear regressors. This solution, how-
ever, may affect the performance of the algorithm since it requires many additional
computations.
Of course, a way to overcome the problem of multicollinearity is to drop collinear
variables before launching LGSS. We extended LGSS by including a procedure,
based on a hierarchical clustering technique [86, 104], which allows us to cluster
the time series of the regressors associated to the same reaction and to select those
which are less correlated and that best satisfy the log-gain principle. In many cases
this allows us to significantly reduce the set of regressors and provides a better
approximation.
3.4.4
Models of Mitosis
LGSS represents a solution, in terms of MP systems, of the inverse dynamics prob-
lem, that is, of the identification of (discrete) mathematical models exhibiting an
observed dynamics and satisfying all the constraints required by the specific knowl-
edge about the modeled phenomenon.
In Table 3.14 a short list of models obtained by LGSS is presented. In this section
we outline an application of LGSS to Golbeter's oscillator given in Fig. 3.8 [81, 82,
83]. LGSS provided 700 different models of this oscillator, which, for the most
part, provide the same order of approximation of Golbeter's model. Moreover, by
considering the phenomenon at different timescales, we obtained different models
and in many cases the analytical form of these models is simpler than Golbeter's
model [101].
