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following shifting operation:
T
A
−
m
=[
. Usually, two memory
levels are enough for defining MP approximations of very complex functions. MP
grammars
M
1
and
M
2
[100], having the form given in Table 3.12, which approximate
(
0
,...,
0
,
t
A
,
1
,...,
t
A
,
n
−
m
]
3
3
(
)
,
(
)) = (
(
)
,
(
)
)
(
(
)
,
(
)) = (
(
.
)
,
(
.
))
f
1
x
g
1
x
cos
x
sin
x
and
f
2
x
g
2
x
cos
0
99
x
sin
0
99
x
10
−
3
,
τ
=
×
respectively (initial values 3
2, shift of each substance 2,
2
π
and
ν
=
μ
=
1).
M
1
provides approximation order 10
−
6
and uses two memory levels with:
c
1
=
0
.
0133055979719,
c
2
,
0
=
38
.
2585289672269,
c
2
,
1
=
−
76
.
6872395145061,
c
2
,
2
=
38
.
4397714886722,
c
3
=
0
.
0053099516581,
c
4
,
0
=
35
.
6377416209758,
c
4
,
1
=
−
70
.
8226212812310,
c
4
,
2
=
35
.
1955107358582,
c
5
,
0
=
14
.
5027497660384,
c
5
,
1
=
−
29
.
4732582998089,
c
5
,
2
=
14
.
9749362227182,
c
6
,
0
=
57
.
3945207457749,
c
6
,
1
=
−
115
.
0336102507111,
c
6
,
2
=
57
.
6568032453146.
M
2
does not use memories, and provides approximation order 10
−
14
with:
c
1
=
0
.
000196018399012,
c
2
,
0
=
0
.
009850833684540,
c
3
=
−
0
.
019701667369079,
c
4
,
0
=
0
.
009899838284292,
c
5
,
0
=
0
.
009899838284293,
c
6
,
0
=
0
.
009948842884046.
Ta b l e 3 . 1 2
The Goniometricus MP grammar generalized with two memory levels
r
1
:0
→
A
ϕ
1
=
c
1
+
c
2
,
0
A
+
c
2
,
1
A
−
1
+
c
2
,
2
A
−
2
r
2
:
A
→
B
ϕ
2
=
c
3
+
c
4
,
0
B
+
c
4
,
1
B
−
1
+
c
4
,
2
B
−
2
+
c
5
,
0
A
+
c
5
,
1
A
−
1
+
c
5
,
2
A
−
2
r
3
:
B
→
0
ϕ
3
=
c
6
,
0
B
+
c
6
,
1
B
−
1
+
c
6
,
2
B
−
2
3.4
Stoichiometric Expansion and Stepwise Regression
The method which we presented in the previous section, for approximating real
functions, is based on grammars with very simple rules. In this case, we assumed
regulators having linear forms. In general, an MP grammar generating a given dy-
namics can be very complex, since we cannot
apriori
restrict the forms of reg-
ulators. For this reason we need to extend our method for solving the Regulation
Discovery Problem within MP grammars.
Let us suppose we know some time series of states
2
, that is, the vector sequence
(
s
[
i
]
|
i
∈
N)
,
2
Very often the time series from which the inverse dynamical problem starts are not at
regular time intervals. In this case a preprocessing phase is appropriate for determining an
interpolation curve fitting the observed values along the observation points. After this, we
will consider the time series given by the values of the interpolation curve at steps with the
same time interval.
