Data-Based Mathematical Modeling: Development and Application, or How to Build a Mapping Neural Network

Abstract

This report presents a general method for developing data-based mathematical models for complex problems with large data bases. The method uses the Monte Carlo method in conjunction with a hierarchical adaptive random partitioning scheme with fuzzy sub domains (Me-HARP). Me-HARP provides an environment for simultaneously building and training a mapping neural network. The method is self-organizing and can operate with minimal external adjustment. It can interactively accept knowledge and provide guidance for efficiently improving the model and the data base. The Me-HARP environment enjoys a large-scale granularity produced by the Monte Carlo parallelism and the geometric parallelism achieved by partitioning the input space. We study the performance of the Me-HARP method by applying it to an experimental data base on pavement performance under a variety of environmental and traffic conditions. Numerical simulations are used throughout the report to demonstrate that the method is able to deal with high-dimensional, noisy, non-homogeneous data. The Me-HARP method leads to a novel model selection criterion and an original framework for classifying data-fitting problems, and can be used to answer fundamental questions in data-based mathematical modeling. These questions include: What is the confidence level in the constructed model and the data base? What is the optimal functional structure of the model for noisy data? How appropriate is a particular parametric model for the given data? The Me-HARP method established an environment for unifying existing mathematical modeling techniques in statistics, approximation theory, information theory, system identification, and neural networks.