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In-Depth Information
∑
SSE
S
o
r
l
)
=
SSE
h
(3.41)
(
S
o
h
∈
(
r
l
)
SSE
S
o
(
r
l
)
n
−
MSE
S
o
(
r
l
)
=
k
.
(3.42)
The same observations made for computing the degrees of freedom in Eq. (3.40)
must to be applied to the formula of the partial
F
-test. When we want to test the
statistical significance of an expanded regressor in LGSS, we apply the following
partial
F
-test:
SSE
S
o
(
r
l
)
−
SSE
S
o
(
r
l
)
MSE
S
o
F
[
1
,
n
−
k
]
=
(3.43)
(
r
l
)
where
r
l
is the rule to which the regressor
g
j
is added (
G
t
j
is its expansion),
S
o
(
r
l
)
indicates the set of substances produced or consumed by
r
l
,
SSR
S
o
(
r
l
)
is the sum of
squares for error of the reduced model (i.e. the model without the regressor under
examination),
SSE
S
o
is the sum of squares for error of the full model (i.e. the
model with the regressor under examination), and
MSE
S
o
(
r
l
)
(
r
l
)
is the mean square
error of the full model.
After fixing a significance value
for the test, we can conclude that the full
model is statistically better than the reduced one (i.e. that the regressor under ex-
amination should be included in the model) when the value of the partial
F
-test is
greater than the threshold value:
α
F
[
α ;1
,
n
−
k
]
.
(3.44)
The partial
F
-statistics given in Eq. (3.43) makes possible the application of the
stepwise regression in LGSS, because it allows us to select the right set of regressors
that should be used in ADA expansion. We will test only a subset of the expanded
regressors, that is, the ones which have log-gain scores higher than a fixed theshold
(see Sect. 3.4.1). When the stepwise algorithm stops, LGSS saves the computed
multiple regression model and then tries to modify it by running again the stepwise
regression algorithm with a larger set of expanded regressors. We repeat this phase
until there are no regressors that can be considered. The final regression model will
be the one, among those saved, which will provide the best approximation. The
estimation of the approximation of models is calculated by means of Eqs. (3.35),
(3.36), (3.37), (3.39), and (3.40).
The least squares estimation used in LGSS gives the best approximations with
respect to the “observed steps”, while when the dynamics is produced by means of
the regulators, substance variations are computed by means of the previous “com-
puted step”. For this reason the dynamics generated by the MP system provided by
LGSS can be improved by a
tuning process
which systematically searches in small
neighborhoods of the estimated coefficients, inside the range of the confidence in-
tervals computed by means of Eq. (3.40). In this way, usually, values providing an
MP dynamics have a significant improvement in the approximation of the observed
dynamics.
