Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 251

Fig. 4.10. Behavior of the empirical mean estimate. ( a ) Original signal ( b ) estimate
of the parameter using a constant-gain filter ( c ) Estimate of the parameter using
decreasing gains
compare a first order filter and a recursive estimation of the mean, the behav-
ior of such an estimator has been shown on Fig. 4.10. The task is to track the
quasi-periodic variation of the deterministic component of a random signal
with a signal-to-noise ratio of 1/5. The original signal is shown on picture
(a). On picture (b), the results for various values of the gain are compared.
On picture (c), the performances of slowly decreasing gain estimates are com-
pared. It is shown that the ability of the estimate to track the slow variations
of the parameter in that case are poor.
One can notice that the empirical mean estimation is based on the
minimization of a quadratic cost function using gradient descent. Actually,
in the case of the stationary model, the data are a sample of the prob-
ability distribution of a random variable X . The quadratic cost function