On the use of nonlocal and nonpositive definite basis functions in radial basis function networks

It is invariably the case that when an application is developed using the radial basis function network in the neural network domain, it is constructed using Gaussian basis functions. This paper discusses the rationale for employing alternative basis functions to the prevalent Gaussian. In particular, we argue the case in support of unbounded basis functions and nonpositive definite basis functions. The use of unbounded and nonpositive basis functions, though counterintuitive in application domains such as classification and time series forecasting, have a good theoretical motivation from the domains of functional interpolation and, somewhat surprisingly, kernel-based density estimation. In addition to collating the theoretical arguments, we present a performance comparison between Gaussian and unbounded, nonpositive definite basis functions in a radial basis function network applied to a financial derivatives regression problem: estimating the price of $/DM options contracts