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equations in which there are more equations than unknowns. “Least-squares” means
that the overall solution minimizes the sum of the squares of the errors made in
solving the equation (see Sect. 7.7 for a mathematical explanation of the method).
Let us explain the procedure of metabolic approximation, in the particular case
of a function from
2
R
R
(
[
]
,
[
])
=
,...,
t
,by
means of the following MP grammar ( 0 is the empty multiset) in which the values
of
c
1
,
c
2
,
to
providing the time series
A
i
B
i
with
i
0
...
,
c
6
are unknowns:
0
→
A
ϕ
1
(
A
,
B
)=
c
1
A
+
c
2
AB
2
A
→
B
ϕ
2
(
B
)=
c
3
B
+
c
4
B
→
0
ϕ
3
(
A
)=
c
5
B
+
c
6
.
The stoichiometric matrix of the system is
1
−
20
01
A =
,
−
1
therefore the approximation system of equations will be, for
i
=
0
,
1
,...,
t
−
1,
Δ
A
[
i
]=
A
[
i
+
1
]
−
A
[
i
]=
c
1
A
[
i
]+
c
2
A
[
i
]
B
[
i
]
−
2
(
c
3
B
[
i
]+
c
4
)
(3.14)
Δ
B
[
i
]=
B
[
i
+
1
]
−
B
[
i
]=
c
3
B
[
i
]+
c
4
−
(
c
5
B
[
i
]+
c
6
)
.
If we call
M
the matrix of coefficients presented on the right-hand side of Eq. (3.14),
the least-squares approximation of the coefficient vector
T
,ex-
pressed as a transposed row (exponent
T
denotes matrix transposition), for which
our MP system exhibits a dynamics close to the time series we started from, is given
by (see Sect. 7.7 and [225]):
(
c
1
,
c
2
,
c
3
,
c
4
,
c
5
,
c
6
)
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
Δ
A
[
0
]
c
1
Δ
B
[
0
]
c
2
Δ
A
[
1
]
=
M
T
M
−
1
c
3
M
T
]
......
Δ
A
[
[
×
×
×
Δ
1
B
c
4
c
5
]
i
c
6
Δ
[
i
]
B
assuming that
M
has maximum rank
1
(i.e., equal to the number of its columns). At
this point we can apply EMA to the MP system which uses
c
1
,
c
2
,...,
c
6
as values
for
c
1
,
c
2
,...,
c
6
. If the system provides a dynamics close enough to the time se-
ries
t
, then the method stops. If the approximation error
is already too big, the method tries to modify the initial MP grammar by intro-
ducing some rules deduced by a suitable correlation analysis of the approximation
error.
1
(
A
[
i
]
,
B
[
i
])
with
i
=
0
,...,
If
M
does not have a maximum rank, then the matrix given by the product
M
T
×
M
is not
invertible.
