Hardware Reference
In-Depth Information
4.4
Algorithms Description
This section describes the fundamental concepts of the RSMs used in the Multicube
design flow. All the presented RSMs have been integrated either in the Multicube
Explorer open source tool and/or in the modeFRONTIER design tool. The set of
RSMs consists of the following models:
Linear regression (Regression-based).
Splines (Regression-based).
RBF (Interpolation-based).
Neural Networks (Regression-based).
Kriging (Interpolation/Regression-based).
Evolutionary Design (Regression-based).
In the following paragraphs we will describe each model in detail, while some results
of the application of RSM techniques to some industrial case studies will be reported
in Chap. 8.
4.4.1
Linear Regression
Linear regression is a technique for building and tuning an analytic model
ρ
(
x
)asa
linear combination of
x
's parameters in order to minimize the prediction residual
ε
.
We apply regression by taking into account also the interaction between the pa-
rameters and the quadratic behavior with respect to a single parameter. We thus
consider the following general model:
n
n
n
n
a
j
x
j
+
(
x
,
w
=
[
a
,
b
,
c
])
=
a
0
+
b
j
,
k
x
l
x
j
+
c
j
x
j
(4.8)
j
=
1
l
=
1
j
=
1,
j
=
l
j
=
1
where
x
j
is the
level
(numerical representation) associated with the
j
-th parameter
of the system configuration, while
a
,
b
and
c
are a decomposition of the RSM family
parameters
w
and
n
is the number of parameters of the design space.
Least squares analysis can be used to determine a suitable approximation of
w
. The
least squares technique determines the values of unknown quantities in a statistical
model by minimizing the sum of the squared residuals (the difference between the
approximated and observed values).
A measure of the
quality of fit
associated with the resulting model is called
coefficient of determination
and defined as follows:
k
(
y
k
−
y
)
2
R
2
=
k
(
ρ
k
−¯
y
)
2
.
(4.9)
where
y
k
is the
k
-th observation,
y
is the average of the observations, and
ρ
k
is the
prediction for the
y
k
observation.
R
2
¯
corresponds to the ratio of variability in a data
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