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?
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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 .