Contents I. Introduction
Function approximation methods and theories are the most important foundation for various complicated systems modeling. Starting from Taylor expansion, polynomial approximation and Fourier series, various function approximation methods and schemes have been proposed such as splines approximation [6], neural networks (NNs)[1], [5], fuzzy systems[7], [8], [10]–[13], wavelets [3], kernel and support vector based approximation, and hybrid schemes which combines different approximation methods. From a methodology point of view, the main feature of all these approximation schemes is using some simple functions as the basic construction element and compose (or combine) these simple functions to form the final approximation schemes. For example, for polynomial approximation, the basic functions are simple power functions such as 1, , … and the linear combination of them to form the final approximation scheme; For fuzzy systems, the basic functions are the membership functions and the weighting combination of them to form the final fuzzy systems; For forward NNs, the basic functions are linear, threshold or sigmoid functions and the composition of them to form the final multilayer NNs. From the application point of view, the main feature of the existing approximation schemes is a forward thinking to answer “If -What” type of questions. For example, if a student is “x=clever” and “y=work reasonable hard”, then what performance he/she will be: “z=f(x,y)=good”; If a student is “x=reasonable clever” and “y=work hard”, then what performance he/she will be: “z=f(x,y)=good”. It can be seen from the given examples here that, in spite of the same output “good performance”, each input case is treated based on its local feature (i.e., it is around ““x=clever” and “y=work reasonable hard” or around “x=reasonable clever” and “y=work hard”) and its corresponding function value is interpolated or predicted separately. Therefore it is a local approach. In other words, from the application point of view, the main feature of the existing function approximation schemes is a forward and local based thinking to answer “If -What” type of questions.
