Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 246

Fig. 4.6. Prediction of sunspot using AR(9) model based linear regression ( a )One
step prediction ( b ) 50 step horizon prediction
4.2.1.5 Application to Sunspot Data Modeling
If we use an AR(9) model based on linear regression to the sunspot series,
omitting any data preprocessing, from a 150-step data file, we obtain a predic-
tive model. Its performances on a 50-step test set are shown on Fig. 4.6. The
difference between the real observation and the predicted value computed from
the 9 closest past observations is represented on picture (a). That prediction is
quite accurate. In picture (b), we represented the difference between the real
observation and a predicted value, which uses solely the initial 9 first values of
the data file. Of course, oscillations are damped. The damping phenomenon is
normal because the autoregressive model is stable and the predictions do not
use new measurements, but initial values only. Nevertheless, the regression
process captured the basic frequency of the phenomenon.
4.2.2 Nonlinear Identification Using Feedforward Neural Networks
4.2.2.1 Limitations of Linear Regression
When the linearity assumption on the state equation is not obeyed, linear-
regression-based modeling of controlled dynamical systems is very inaccurate
and uses very heavy models with too many parameters. That is illustrated on
Fig. 4.7.
In that example, which was described in previous section, there is no pos-
sible linear model, which exhibits both a stable equilibrium and an unstable
equilibrium. Yet, the linear regression captured the right frequency of the
oscillator.
4.2.2.2 Network with Delay (NARX Model)
The simplest example of neural identification of a controlled dynamical system
is based on regression algorithms. The model is an autoregressive model with