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As requested by the method, we build the input vectors of the regression
x
(
k
)=[
x
(
k
);
x
(
k
1.
The computation of regression coe
cients yields the following numerical
results:
−
1);
u
(
k
);
u
(
k
−
1)] for
k
varying from 2 to
N
−
a
1
=1
.
29
, a
2
=
−
0
.
83
, c
1
=0
.
49
, c
2
=
−
0
.
51
.
Assume now that we ignore the input data; then the regression input
vectors are two-dimensional
x
(
k
)=[
x
(
k
);
x
(
k
1)]. We have a simple AR
model, and the computation gives the following, inaccurate estimations:
−
a
1
=1
.
17
, a
2
=
−
0
.
71
.
The model was not relevant: since the input trajectory was white noise,
an AR model was used to process data that were actually generated by an
ARMA model with vector noise (
u
k
,v
k
).
Now, assume that a measurement noise is introduced into the simulator,
so that our observation of the state is inaccurate while the process dynamics
is unaffected (that point will be developed further in the section devoted to
filtering). Then the data-generating process is the following:
x
(
k
+1)=
a
1
x
(
k
)+
a
2
x
(
k
−
1) +
c
1
u
(
k
)+
c
2
u
(
k
−
1)
.
y
(
k
)=
x
(
k
)+
b
0
w
(
k
)
In that case we have poor results if we use the ARX regression, even when
the input trajectory is taken into account. We get
a
1
=0
.
61
, a
2
=
−
0
.
36
, c
1
=0
.
49
, c
2
=
−
0
.
11
.
That numerical simulated example shows how it is important to get a
relevant model for the noises to achieve linear regression. This problem was
already addressed in Chap. 2 in the framework of dynamical neural modeling.
We shall give a more detailed account further in this chapter: the occurrence
of measurement noise creates a new problem, namely the filtering problem.
4.2.1.3 Statistical Background
Statistical analysis of time series is well known and will not be detailed here-
after. One can consult [Chatfield 1994] for a classical and practical handbook
and [Gourieroux 1995; Azencott 1984] for more mathematical details. We shall
just outline here the explanation of the least-squares methodology in the sim-
plest case of a stable autoregressive model with a gaussian centered noise.
The variables are written with capital letters because they are considered as
random variables.
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