Information Technology Reference
In-Depth Information
The number of degrees of freedom is calculated here by considering that in Eq. (3.29)
the total number of equations for each substance is equal to
t
and the number of re-
gressors used for each substance is given by:
j
=
1
c
l
,
j
g
j
g
j
|
ϕ
l
=
d
d
h
=
(3.38)
R
o
r
l
∈
(
h
)
ϕ
l
is the regulator of the reaction
r
l
and
R
o
where
indicates the reactions of the
system consuming or producing the substance of index
h
(see Table 3.7). Following
the same reasoning, we can also extend the notion of
F
-ratio:
(
h
)
R
h
SSR
h
/
d
h
MSR
h
MSE
h
=
t
−
d
h
d
h
F
h
=
d
h
]
=
R
h
·
(3.39)
SSE
h
/
[
−
t
1
−
After having fixed a significance value
for the test, we can conclude that a linear
relationship exists if (see
F
-distribution in Sect. 7.7):
α
F
h
>
F
[
α ;
d
h
,
t
−
d
h
]
.
The evaluation of the confidence intervals for the least squares estimation of each
regressor can be done by computing, in the
t
-expansion
G
t
j
of regressor
g
j
, the right
number of degrees of freedom of the
t
-distribution. It depends on the ADA stoi-
chiometric expansion applied to the considered regressors. In fact, we recall from
the
k
-Variable Multiple Regression Model that the number of the degrees of freedom
is given by
n
1 is the number
of regressor coefficients. Now, the role of
n
is played by the number of equations
involved in the regressor
G
t
j
under consideration, which is given by (see Table 3.7):
−
(
k
+
1
)
,where
n
is the number of equations and
k
+
n
=
S
o
·|
(
r
l
)
|.
t
+
The role of
k
1, instead, is given by:
∑
h
∈
S
o
k
=
d
h
(
r
l
)
that is, the sum of all the regressors included in the model which modify at least one
substance in
S
o
. Finally, the formula that gives the confidence intervals for the
coefficients computed in Eq. (3.29), is given by:
(
r
l
)
e
β
l
,
j
=
c
l
,
j
±
t
[
α
/
2;
n
−
k
]
·
MSE
S
o
(3.40)
(
r
l
)
where
l
d
are the indexes related to reactions and re-
gressors, respectively,
e
is the element related to the regressor under examination on
the first diagonal of the matrix
(A
⊗
G)
=
1
,
2
,...,
m
and
j
=
1
,
2
,...,
×
(A
⊗
G)
−
1
T
used for the least squares
estimation of Eq. (3.30), and:
