Discovery of function forms in three variables

Abstract

To find a mathematical description of a certain class of events is the goal of mathematical modeling. Traditionally, it is the task of mathematicians and engineer-scientists. The goal of function form discovery is to develop machine intelligence systems to tackle this problem. Though the machine intelligence approach is still in its infancy, it has been demonstrated that systems based on such approach are able to give more compact and meaningful forms that describe the input data than the traditional numerical methods.

This thesis presents a function form discovery system known as FFD-II which is a significant extension of the FFD system. The adoption and extension of the data transformation mechanism of FFD allows FFD-II to discover significantly wider variety of functional forms from numerical data than its predecessors. FFD was developed initially for finding real-valued function forms of one independent variable. It could also be used to find families of functions in an indirect way. FFD-II is able to discover function forms of two independent variables directly from numeric data for it can make use of three dimensional information that cannot be used by the indirect methods which, for example, have to rely on "cross-effects" in the discovery. Hence, FFD-II not only exhibits better performance in handling the discovery problem, but is also more flexible for future extensions. Another significant characteristics of FFD-II is its new adaptive error control. It identifies the noise patterns according to the smoothness of an observed functional image and monitors the magnitude of propagated errors according to the theoretical error analysis results. In FFD-II special treatments are also added to reduce the effects of noise. Hence, the new system has a greater tolerance to both the computational error as well as the noise of the input than FFD.

Other new contributions of FFD-II include: 1) the construction and analysis of a three dimensional based function form description language; 2) the design of special purpose numeric methods which can recognize primitive functional patterns, conduct factorization and handle partial differential transformations of three dimensional data; 3) the quantified measurements of the qualitative characteristics of a functional image and 4) the implementation of a new heuristic search process.