Hardware Reference
In-Depth Information
Concerning categorical variables, the matter is not so simple. As a justification for
pragmatically treating these variables as continuous ones, there is the fact that in many
applications it is a common practice to treat categorical parameters as simple discrete
ones. So if discrete variables are treated as continuous ones, the same should apply for
categorical ones. But in this case there is a non-negligible difference: there is no order
relation between different variables values. For this reason, caution is requested: the
training database should be examined, in order to determine, from case to case, if it is
possible to treat categorical parameters as continuous ones. One criterion of decision
could be to consider if different subsets corresponding to different categories present
analogies (correlations) in response behavior. In case of positive answer, it probably
makes sense to train the RSM profitably on the full database, taking advantage of the
simple reduction to a continuous domain. If this is not the case, one possible solution
could be to treat these different categories as separate sub-problems: different RSMs
should be trained separately for each category.
As a final remark, in the design of embedded systems, many discrete input vari-
ables are power of two (e.g., memory size):
x
2
m
. In these cases, it is convenient to
perform a variable transformation, taking the exponent
m
as the actual input param-
eter, and considering
x
as a dependent (auxiliary) variable. In this way, the discrete
parameter presents two characteristics that are desirable from the point of view of any
RSM training algorithm: its values are equispaced and it is well scaled (as regards
its range of variation).
=
4.3.2
Optimal DoE
The following considerations are usually found when dealing with Radial Basis
Functions (where they have a straightforward formulation), but they can be general-
ized for all RSM algorithms. The generic application refers to multivariate scattered
training data in a continuous design space.
A generic set of scattered training points
{
=
}
is characterized by
two quantities: the
fill distance h
, and the
separation distance q
. These quantities,
defined in the followings, are shown in Fig.
4.4
.
The fill distance
h
is defined as the radius of the largest inner empty disk:
x
i
,
i
1,
...
,
n
h
=
max
x
min
≤
j
≤
n
x
−
x
j
,
(4.5)
∈
1
d
is the domain of definition of
f
(
x
). In order to achieve better
approximation quality of the RSM one should minimize the fill distance: min
h
.
The separation distance
q
is defined as the minimum distance between two training
points:
where
⊂ R
x
j
.
q
=
min
i
x
i
−
(4.6)
=
j
In order to improve the numerical stability of the RSM training algorithm, one should
maximize the separation distance: max
q
. Therefore, to improve both approximation
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