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Fig. 4.13.
Neural network training using the extended Kalman filter
network). The input-output pair sequence of the neural network is the mea-
surement process that provides information on the evolution of the configura-
tion. Thus, that adaptive algorithm is well suited to tracking slow variations
of the environment, which is a typical task for an adaptive algorithm.
In that context, training amounts to estimating the state. Operating the
network amounts to measuring the current state. Therefore, the innovation
is the classical error for supervised learning i.e., the difference between the
desired network output and the computed network output when an input is
presented to the network.
The linear state equation
X
(
k
+1) =
AX
(
k
)+
Bu
(
k
)+
V
(
k
+1) becomes
the following expression:
W
(
k
+1)=
W
(
k
)+
V
(
k
+1)
,
where
W
(
k
) is the configuration vector of the network (weights+biases) at
time
k
.
The nonlinear measurement equation
Y
(
k
)=
h
[
X
(
k
)] +
W
(
k
) becomes
y
(
k
)=
g
[
x
(
k
)
,
w
(
k
)] +
W
(
k
)
.
−
g
[
x
(
k
+1)
,
w
(
k
)]. It is indeed the expression of the learning error that was
Therefore, the innovation error of the model is:
ϑ
(
k
+1) =
y
(
k
+1)
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