246 8 Filters and Predictors
means that we must solve a prediction problem because the knowledge of
the system dynamics is equivalent to the knowledge of the future evolution
and vice versa. Since we have no information about the real system dynamics
and we obtain also in future no more information as the continuation of the
observation records, it is no longer necessary to estimate the complete state
evolution of the system. The present situation allows us not to see more than
the observations, i.e., neither it can be proven a certain assumption about
the intrinsic dynamics nor this assumption can be disproved. From this point
of view, the treatment of such black box systems is an application of the
principle of Occam’s razor [47, 48]. This idea is attributed to the 14th-century
Franciscan monk William of Occam, which states that entities should not be
multiplied unnecessarily. The most useful statement of this principle is that
the better theory of two competing theories which make exactly the same
predictions is the simpler one. Occam’s razor is used to cut away unprovable
concepts.
In principle, each forecasting concept about the observations belonging to
a system with hidden intrinsic dynamics defines also a more or less suitable
model connecting the current and historical observations with an estimation
of the future evolution. In so far, this models represent a substitute system
from which we may obtain substitute evolution equations which are the neces-
sary constraints for a successful control. The uncertainties of such models are
considered in appropriable noise terms. Thus, if we have estimated the evolu-
tion of the underlying system, we come back to the classical stochastic control
problems. The system output, the observations, is automatically the input of
the control function while all functions defining the control law are obtainable
from the estimated evolution equations, i.e., the forecasting equations.
In the subsequent sections we will give some few ideas which may be helpful
for the characterization and application of several prediction methods. Since
these techniques do not belong to the central topics of the control theory, we
restrict our carrying out to a brief discussion of the main features.
8.6.4 Regression and Autoregression
For simplicity, we consider again time discrete processes. At the beginning
of the last century, standard predictions were undertaken by simply extrap-
olating a given time series through a global fit procedure. The principle is
very simple. Suppose we have a time series of observations {Y
1
,Y
2
,...,Y
L
}
with the corresponding points in time {t
1
,t
2
,...,t
L
} and Y
n
vectors of the
p-dimensional observation space. Then we can determine a regression function
f in such a way that the distance between the observations Y
n
and the cor-
responding values f (t
n
) becomes sufficiently small. There are two problems.
The first one is the choice of a suitable parametrized regression function. This
is usually an empirical step which depends often on the amount of experience.
The second problem is the definition of a suitable measure for the distance.