Fitting sparse multidimensional data with low-dimensional terms

An algorithm that fits a continuous function to sparse multidimensional data is presented. The algorithm uses a representation in terms of lower-dimensional component functions of coordinates defined in an automated way and also permits dimensionality reduction. Neural networks are used to construct the component functions.Program summaryProgram title: RS_HDMR_NNCatalogue identifier: AEEI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 19 566No. of bytes in distributed program, including test data, etc.: 327 856Distribution format: tar.gzProgramming language: MatLab R2007bComputer: any computer running MatLabOperating system: Windows XP, Windows Vista, UNIX, LinuxClassification: 4.9External routines: Neural Network Toolbox Version 5.1 (R2007b)Nature of problem:   Fitting a smooth, easily integratable and differentiatable, function to a very sparse (∼2–3∼2–3 points per dimension) multidimensional (D⩾6D⩾6) large (∼104–105∼104–105 data) dataset.Solution method: A multivariate function is represented as a sum of a small number of terms each of which is a low-dimensional function of optimised coordinates. The optimal coordinates reduce both the dimensionality and the number of the terms. Neural networks (including exponential neurons) are used to obtain a general and robust method and a functional form which is easily differentiated and integrated (in the case of exponential neurons).Running time:   Depends strongly on the dataset to be modelled and the chosen structure of the approximating function, ranges from about a minute for ∼103∼103 data in 3-D   to about a day for ∼105∼105 data in 15-D.