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Moreover, the search for the best regulators of the given reactions is performed by
using suitable forms of
F
-tests (based on Fisher distribution).
Let us consider the algebraic formulation of the dynamical inverse problem, given
by Eq. (3.23), for determining the vector
C
of regressor coefficients providing the
best regulators approximating a given dynamics:
T
G
×
C
×
A
= D
(3.33)
which is equivalent to (see Eq. (3.29)):
(A
⊗
G)
×
vec
(C)=
vec
(D)
.
(3.34)
Let us assume that the rank of the matrix associated to the above system is max-
imum. This happens when the stoichiometric matrix has maximum rank and the
expanded regressors are linear independent. The system above can be considered as
a multiple regression model, where
vec
is the (vector) dependent variable, while
the independent (vector) variables are the columns of matrix
(D)
. These assumptions
make it possible to deal with system 3.34 as a multiple regression model, in the
sense defined in Eq. 3.32.
For a better understanding of the following discussion we reformulate in Table 3.13
the deviations occurring in regression models in terms of the matrix
G
Δ
h
indicates variation vector of the substance of index
h
. We also extend the definition
of MSE,
R
2
and
R
2
in order to consider separately each substance of the system.
D
,where
]
−
Δ
h
[
t
i
2
SSE
h
t
(
Δ
h
[
i
i
])
d
h
=
∑
=
0
MSE
h
=
(3.35)
−
t
−
d
h
SSR
h
SST
h
=
SSE
h
SST
h
R
h
=
1
−
(3.36)
SSE
h
/
[
t
−
d
h
]
MSE
h
MST
h
.
R
h
=
1
−
=
1
−
(3.37)
SST
h
/
t
Ta b l e 3 . 1 3
The three deviations of an LGSS regression for each substance of the system. We
indicate with
h
the index of the substance, with Δ
t
h
the predicted value of Δ
t
h
by means of the
multiple regression model, and with
¯
t
h
the average of the values of Δ
t
h
.
Δ
h
−
Δ
Δ
t
h
−
¯
t
h
t
t
h
t
h
−
¯
t
h
Δ
Δ
=
Δ
+
Δ
Total
Unexplained
Explained
deviation
deviation (error)
deviation (regression)
for substance
h
for substance
h
for substance
h
¯
0
(
Δ
h
[
i
]
−
Δ
h
[
i
])
0
(
Δ
h
[
i
]
−
¯
i
h
)
2
i
2
i
h
)
2
∑
0
(
Δ
h
[
i
]
−
Δ
=
∑
+
∑
Δ
=
=
=
SST
h
SSE
h
SSR
h
Sum of Squared
Sum of Squared
Sum of Squared
Total deviations
Errors
Regressions deviations
for substance
h
for substance
h
for substance
h
