Data-based mathematical modeling: Development and application

This research study presents the mathematical basis for building the MC-HARP data-processing environment. The MC-HARP strategy determines the functional structure and parameters of a mathematical model simultaneously. A Monte Carlo (MC) strategy combined with the concept of Hierarchical Adaptive Random Partitioning (HARP) and fuzzy subdomains determines the multivariate parallel distributed mappings. The constructed mapping can be modeled as a neural network. The HARP algorithm is based on a divide-and-conquer strategy that partitions the input space into measurable connected subdomains and builds a local approximation for the mapping task. Fuzziness promotes continuity of the mapping constructed by HARP and smooths the mismatching of the local approximations in the neighboring subdomains. The Monte Carlo superposition of a sample of random partitions, reduces the localized disturbances among the fuzzy subdomains, controls the global smoothness of the mean average mapping, and improves the generalization of the constructed mapping.The tree structure of the HARP modules and the independence of both the subdomain approximations and the random partitions enable the MC-HARP environment to quickly converge to a series of equally plausible solutions without user interaction. The MC-HARP environment enjoys a large-scale granularity produced by the Monte Carlo parallelism and the geometric parallelism achieved by partitioning the input space. Therefore this environment can exhibit good performance on parallel computers for large and complex scientific databases.The developed MC-HARP philosophy for building data-based approximate mappings leads to a novel model selection criterion and an original framework for classifying data-fitting problems. The MC-HARP environment not only can build approximate multivariate mappings with self-organization capability, noise and fault tolerance, adaptivity, generalization, highly plastic and stable learning characteristics with respect to the addition of new data points, and parallel structure but also can answer fundamental questions in data-based mathematical modeling. These questions include: (1) What is the confidence level for each predicted output of the constructed model? (2) What is the approximation confidence measure for the constructed model? (3) How does the functional complexity of the actual multivariate mapping change over the input space? (4) What is the suitable structural complexity for a data-based model using noisy data? (5) What is the level of noise in the data? (6) Is the amount of training data adequate? If not, which regions of the input space need more data? (7) Is the selected parametric model suitable? (8) What is the conditioning of a data-fitting problem? (9) Is data-based mathematical modeling promising for the given task?The developed MC-HARP environment can support the diverse needs of the scientific and engineering community. It has the versatility to develop and verify parametric and nonparametric mathematical models and also global and local approximate mappings. Furthermore, It establishes an environment for unifying existing mathematical modeling techniques in statistics, approximation theory, information theory, system identification, and neural networks.U of I OnlyETDs are only available to UIUC Users without author permissio