Functional estimation problem

B. Neural Network Solution to the Functional Estimation Problem.449... 

In this study the problem of estimating an unknown function from its examples is revisited. Its mathematical description is attempted to map as closely as possible the practical problem that the potential NN user has to face. The objective of the chapter is twofold (1) to draw the framework in which NN solutions to the problem can be developed and studied, and (2) to show how careful considerations on the fundamental issues naturally lead to the Wave-Net solution. The analysis will not only attempt to justify the development of the Wave-Net, but will also refine its operational characteristics. The motivation for studying the functional estimation problem is the derivation of a modeling framework suitable for process control. The applicability of the derived solution, however, is not limited to control implementations. 

Fig. 1. Example of a functional estimation problem evolution of the model (solid line) and comparison with the real function (dashed line) as more data (asterisks) become available.

For an introduction to NNs and their functionality, the reader is referred to the rich literature on the subject (e.g., Rumelhart et al, 1986 Barron and Barron, 1988). For our purposes it suffices to say that NNs represent nonlinear mappings formulated inductively from the data. In doing so, they offer potential solutions to the functional estimation problem and will be studied as such. 

The parameter identification problem associated with the conventional permeability experiments is within the first class (with m= 1). By contrast, the problems we consider here are within the second and third classes these areJunctional estimation problems. Ultimately, however, these are solved with finite-dimensional representations, although an essential aspect of the solution of these infinite-dimensional (function) estimation problems is the selection of the appropriate representations. 

The magnitude of /emp(g) will be referred as the empirical error. All regression algorithms, by minimizing the empirical risk /emp(g), produce an estimate, g(x), which is the solution to the functional estimation problem. 

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