Neural network modeling of data with gaps: method of principal curves, Carleman's formula, and other

Abstract

A method of modeling data with gaps by a sequence of curves has been developed. The new method is a generalization of iterative construction of singular expansion of matrices with gaps. Under discussion are three versions of the method featuring clear physical interpretation: linear - modeling the data by a sequence of linear manifolds of small dimension; quasilinear - constructing "principal curves: (or "principal surfaces"), univalently projected on the linear principal components; essentially non-linear - based on constructing "principal curves": (principal strings and beams) employing the variation principle; the iteration implementation of this method is close to Kohonen self-organizing maps. The derived dependencies are extrapolated by Carleman's formulas. The method is interpreted as a construction of neural network conveyor designed to solve the following problems: to fill gaps in data; to repair data - to correct initial data values in such a way as to make the constructed models work best; to construct a calculator to fill gaps in the data line fed to the input.