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Fig. 2.32.
Input-output representation, output noise assumption
Fig. 2.33.
The ideal model for an input-output representation with output noise
assumption
where
w
is a vector of parameters, and where function
ϕ
NN
is computed by
a feedforward neural network. Assume that the network has been trained so
that
ϕ
NN
is exactly equal to
ϕ
. Moreover, assume that the prediction error
is equal to the noise at the first
n
time steps:
y
p
(
k
)
−
g
(
k
)=
b
(
k
)for
k
=0to
n
g
(
k
)=
b
(
k
) for all
k
. Thus, the prediction error of
the model is equal to the noise: the model is therefore ideal, since it accounts
for all that is deterministic in the representation, and does not model noise.
If the initial condition is not obeyed, but nevertheless
ϕ
RN
=
ϕ
,andif
the model is stable irrespective of the initial conditions, the modeling error
vanishes as
k
increases.
Note that, in that case, the ideal model is
recurrent
.
−
1. Then one has
y
p
(
k
)
−
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