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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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