Identification Method - Climate Change: Identification and Projections - page 69

The models that we are able to identify will come from data, and solely from
climatic data.
6.2. Output error method
Following the data collection process, the identification of a process
could be reduced to a style exercise in which we determine, according to the
rules of the art, a causal mathematical relation between the input and output
signals. Nevertheless, since climatic databases are significantly noised and
relatively sparse in terms of events, it would be unrealistic to attempt to
identify a model with more than 6 or 7 parameters. This would
systematically exclude so-called non-parametric techniques, which identify
point by point frequential or temporal responses.
The most straightforward parametric identification method is that of
least-squares output error. This comes to mind naturally, without needing to
make reference to any pre-existing theory: it solely consists of minimizing,
with respect to the parameters of a model, a norm of deviations between the
output observed and the input simulated by the model. Small output errors in
the presence of sufficiently exciting inputs is often an indication of validity
for a model so that it is not necessary to go further with the analysis.
The official statistical theory of identification proposes other parametric
methods called OE, ARX, ARMAX, B&J, PEM, and so on [LJU 99].
Amongst these, OE (Output Error method) uses the least-squares method of
output error above, in a statistical context. In the present case, it so happens
that the conditions under which OE would be statistically optimal are not
met: residual output should be a white noise, but this is far from being the
case (Figure 7.6). In this event, the quest for optimality pushes us to make
the model more complex and to jointly identify statistical parameters related
to noises. Unfortunately, the results - which are not always convincing -
depend on the type of model adopted (ARX, ARMAX, etc.) and there is no
fully satisfying validation procedure. As a last resort, visual examination of
the output error remains the decisive criteria. After testing various options,
we decided to keep the OE; it has the benefit of staying away from any
arbitrary choice, except for the selection of the structure of the model itself,
without adding any further noise modeling.