390
P. Z. Korondi et al.
12.2.3 Construction Stages
The construction of a response surface model can be divided into four stages: data
preparation, algorithm choice, parameter tuning, and model validation. Each stage
consists in multiple steps described in more detail in the following sections. The
descriptions are not immutable. For example, the first stage can be skipped if a
sufficient amount of data is available. The stages do not necessarily have to be
performed sequentially. In numerous applications these stages are done iteratively.
The training data can be refined in regions where the model validation phase
indicates lower accuracy and models can be recalculated after the introduction of
new sample instances.
12.2.3.1 Data Preparation
The first step is to identify the relevant design variables and responses based on
available knowledge or sensitivity analysis [5]. The region of interest (i.e. the ranges
of the design variables) should also be defined.
The second step consists in sampling the design space. The number of possible
sample points is constrained by the cost of the experiments. Different techniques
have been developed to obtain the maximum amount of information using the
minimal amount of resources. These techniques are called Design of Experiments
(DOE) [6]. Many DOE techniques solve an optimisation problem to achieve a welldistributed sample data set. It should not contain data points that are too close to
each other or large areas not covered by any points. Two basic quality measures are
introduced below and shown in Fig. 12.1: the Separation Distance index and Fill
Distance index.
The Separation Distance q is the distance between the two closest training points
and is defined as,
q = min
i =j
||x i − x j ||,
(12.4)
where x i and x j is any pair of sample points and the norm || . . . || is the Euclidean
distance in a multidimensional space. Training points that are too close to each
other can lead to numerical instabilities in the generation of the RSM, hence, the
Separation Distance should be maximised.
The Fill Distance index h is a metric to quantify how well the training points
cover the investigated domain Ω of the variables. In case of a two-variable domain,
the Fill Distance gives the radius of the biggest possible circle that can be drawn in
the domain without having any training point inside the circle.
h = max
xΩ
min
1≤j ≤n
||x − x j ||.
(12.5)
P. Z. Korondi et al.
12.2.3 Construction Stages
The construction of a response surface model can be divided into four stages: data
preparation, algorithm choice, parameter tuning, and model validation. Each stage
consists in multiple steps described in more detail in the following sections. The
descriptions are not immutable. For example, the first stage can be skipped if a
sufficient amount of data is available. The stages do not necessarily have to be
performed sequentially. In numerous applications these stages are done iteratively.
The training data can be refined in regions where the model validation phase
indicates lower accuracy and models can be recalculated after the introduction of
new sample instances.
12.2.3.1 Data Preparation
The first step is to identify the relevant design variables and responses based on
available knowledge or sensitivity analysis [5]. The region of interest (i.e. the ranges
of the design variables) should also be defined.
The second step consists in sampling the design space. The number of possible
sample points is constrained by the cost of the experiments. Different techniques
have been developed to obtain the maximum amount of information using the
minimal amount of resources. These techniques are called Design of Experiments
(DOE) [6]. Many DOE techniques solve an optimisation problem to achieve a welldistributed sample data set. It should not contain data points that are too close to
each other or large areas not covered by any points. Two basic quality measures are
introduced below and shown in Fig. 12.1: the Separation Distance index and Fill
Distance index.
The Separation Distance q is the distance between the two closest training points
and is defined as,
q = min
i =j
||x i − x j ||,
(12.4)
where x i and x j is any pair of sample points and the norm || . . . || is the Euclidean
distance in a multidimensional space. Training points that are too close to each
other can lead to numerical instabilities in the generation of the RSM, hence, the
Separation Distance should be maximised.
The Fill Distance index h is a metric to quantify how well the training points
cover the investigated domain Ω of the variables. In case of a two-variable domain,
the Fill Distance gives the radius of the biggest possible circle that can be drawn in
the domain without having any training point inside the circle.
h = max
xΩ
min
1≤j ≤n
||x − x j ||.
(12.5)