12 Response Surface Methodology
389
12.2 Response Surface Model Construction
12.2.1 Objective
The objective of RSMs is to map the behaviour of an output variable based on the
known values of the input variables. In mathematical terms,
y = f (x),
(12.1)
where x is the vector of input variables, f is the mapping function, and y is the
output variable.
In practice, the perfect map cannot be found, except in some simple problems. A
RSM provides only an approximate map between the output and inputs:
ˆ
y = ˆ
f (x),
(12.2)
where ˆ
f is the RSM and ˆ
y is the predicted output, which usually deviates from its
real value,
y = ˆ
y + ,
(12.3)
where represents the error with respect to the real output value.
12.2.2 Classification
RSMs can be divided into two main categories: regression methods and interpolation methods.
Regression RSM (or approximation method) is typically a polynomial function
or nonlinear model with coefficients that are set to minimise the error between the
predicted surface and the sample points of the training data set. Approximation
methods are particularly useful for modelling ill-conditioned problems or in case
of low quality training data (e.g. noisy data sets).
Interpolating RSM passes through the points of the training data set. In other
words, the interpolating surface and the real response surface are equal at the
locations of the training data. Hence, interpolation is well-suited to deterministic
computer experiments where the simulation error is negligible. The choice of
inappropriate interpolation methods can result in oscillatory surfaces (e.g. when
there is a large number of model parameters). Consequently, extrapolation based
on an interpolated response surface with many model parameters is not suggested
without additional validation of the extrapolated predictions.
389
12.2 Response Surface Model Construction
12.2.1 Objective
The objective of RSMs is to map the behaviour of an output variable based on the
known values of the input variables. In mathematical terms,
y = f (x),
(12.1)
where x is the vector of input variables, f is the mapping function, and y is the
output variable.
In practice, the perfect map cannot be found, except in some simple problems. A
RSM provides only an approximate map between the output and inputs:
ˆ
y = ˆ
f (x),
(12.2)
where ˆ
f is the RSM and ˆ
y is the predicted output, which usually deviates from its
real value,
y = ˆ
y + ,
(12.3)
where represents the error with respect to the real output value.
12.2.2 Classification
RSMs can be divided into two main categories: regression methods and interpolation methods.
Regression RSM (or approximation method) is typically a polynomial function
or nonlinear model with coefficients that are set to minimise the error between the
predicted surface and the sample points of the training data set. Approximation
methods are particularly useful for modelling ill-conditioned problems or in case
of low quality training data (e.g. noisy data sets).
Interpolating RSM passes through the points of the training data set. In other
words, the interpolating surface and the real response surface are equal at the
locations of the training data. Hence, interpolation is well-suited to deterministic
computer experiments where the simulation error is negligible. The choice of
inappropriate interpolation methods can result in oscillatory surfaces (e.g. when
there is a large number of model parameters). Consequently, extrapolation based
on an interpolated response surface with many model parameters is not suggested
without additional validation of the extrapolated predictions.