APPROXIMATION

The fulfilment of the last two equations are only a necessary cond:.
tion for a minimum. However, it is possible to show that we have obtained
a minimum.
It is still possible to write down expressions for maximal error
bounds similar to eq. (2.34), now using both the norm in the reproducing
kernel Hilbert space, and the norms implicitly introduced in the n-dimen-
sional space of measurements by D and the m-dimensional space of parame-
ters by P, see APG (p.128). For the mean square error of prediction we
find with
that
and
2.5 Consequences of the mathematical model
In order to use LSC we must select an appropriate RKHS. This is done
by chosing a reproducing kernel as discussed in the next chapter. But what
are the consequences of these choices in case we for example need to or
want to use spherical or planar approximation? Something which look nume-
rically or computationally nice may become a failure in practice.
We must also be able to compute the quantities I K(Li,L.) l for all
J
kinds of linear functionals, which correspond to the data types we en-
counter in practice as well as the vectors AL. Aiso the noise variance-
covariance matrix must be available. However in practice we generally on-
ly know the variance8 for groups of measurements and no information is
available for the covariances.
Also systems of linear equations must be solved, with dimension as
large as the number of observations plus the number of parameters. How we