Introduction
Response surfaces are typically used for optimization since they provide
a visual estimate of the behavior of a function across its input space.
Their employment for this purpose implies a forward mapping in the sense
that, after data is sampled and an equation that can best approximate it
is constructed, an experimental procedure is realized until an
attractive solution -hopefully an optimum- is found. As an alternative,
in this work the idea of inverse mappings is explored towards the
optimization end, where desired output characteristics are associated to
a specific region in a function’s input space.
Consider the following hypothetical example of fitting a linear
regression to predict student weight based on their height for a sample
of size 7 (Please see Figure 1). As can be noted from the graph, there
are some points (regions) where the line is a better predictor than
others. The points in black show where data ‘behaves’ in a desirable
manner. Note that desirable behavior is defined as that region where the
data is most similar to a particular function, in this case, to the line
that minimizes the sum of squared errors. Since this phenomenon can
occur whenever using modeling to approximate datasets, we propose
extending the concept of desirable behavior - output characteristics -
to include regions in datasets where it could ‘look like’ a function
that has optimality properties; if we could identify a region in the
dataset where the it is most similar to a function with optimality
properties, we would have had found an area of potential optimality. But
how may two models generated from the same data be compared?
[CHART]
Figure 1 Hypothetical regression example, points in black show
regions where model is better predictor.
Metamodeling is when a complex model, like the ones frequently used for
simulations, is approximated by another, typically simplified, one.
Metamodels are often used for optimization purposes and require forward
mapping procedures to experimentally find optimality conditions for a
particular process. It is also common that a metamodel’s parameters are
estimated via minimization of an error function until the most
competitive fit is found. In this work, metamodels are fitted in a
different manner but also towards an optimization end; since we are
searching for the region of maximum resemblance between a data
generating model and a metamodel with optimality properties, the
parameter estimates will differ. By finding a region of maximal
similarity, we are looking to generate an inverse map in order identify
a window in the input space where optimality may be present.
Inverse mappings, when a function’s input is a specific desired
performance and its output its associated controllable variable
settings, was approached in [3]. From the intricacies the author
mentions, it was noted that the task of inverse mappings is often
reduced to finding one (or more) input parameter combinations for only
one certain output characteristic. As in the method here proposed,
solving inverse problems by the identification of the regions, instead
of points, was assessed in [4]. The Window of Maximum Similarity
(WMS) method differs from the latter in the sense that it was
constructed to be applicable to detect zones of interest in different
kinds of data and does not use probability density functions, but rather
least squares estimation and linear programming
As was first done in [6], our Optimization by Similarity method aims
to search for a region where a metamodel fits best. Their study
addresses a common problem faced in modeling polymers: the relationship
between deformation and viscosity. In contrast, we propose applying the
method to any problem, that is, any that requires modeling, abstracting
it to the mathematical space of functions. We also consider a
two-dimensional input, or ‘controllable variable’ space, as opposed to
only one. Our method entails matching a (simulated) function -one that
represents, or rather, generates random data- to another one that has
desired optimality properties-a specific form - and find their region of
maximum resemblance, through least squares estimation and optimization,
where there could exist, at least, a local optimum (Please refer to
Figure 2). The development of the method is described below, and its
applicability is tested on several common global optimization test
functions and a function created by our research group; AOG_1 .