Chapter 1: Sampling Plans

Much of this book is made up of recipes for constructing cheap-to-evaluate `surrogate’ models that emulates the expensive response of some black
box f, given a set of samples. Excepting a few
pathological cases, the mathematical formulations of these
modelling approaches are well-posed, regardless of how the
sampling plan determines the spatial arrangement of
the observations we have built them upon. Some models do require a
minimum number n of data points but, once we have passed
this threshold, we can use them to build an unequivocally defined
surrogate.

However, a well-posed model does not necessarily generalize
well, that is, it may still be poor at predicting unseen data, and
this feature does depend on the sampling plan X.
For example, measuring the performance of a design at the extreme
values of its parameters may leave a great deal of interesting
behaviour undiscovered, say, in the centre of the design space.
Equally, spraying points liberally in certain parts of the inside
of the domain, forcing the surrogate model to make far-reaching
extrapolations elsewhere, may lead us to (false) global
conclusions based on patchy, local knowledge of the objective
landscape.

Of course, we do not always have a choice in the matter. We may be
using data obtained by someone else for some other purpose or the
available observations may come from a variety of external sources
and we may not be able to add to them. The latter situation often
occurs in conceptual design, where we wish to fit a model to
performance data relating to existing, similar products. If the
reader is only ever concerned with this type of modelling problem,
he or she may skip this chapter. However, if you
have the possibility of selecting your own objective function
sampling locations, please read on, as in what follows we discuss
a number of systematic techniques for building sampling plans that
that will enable the surrogate model to be built subsequently to
generalize well.

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