Constrained topological mapping for nonparametric regression analysis

For multivariate problems the proposed neural approach may alleviate “the curse of dimensionality,” that is, reduce the size of the training set required for evaluation of the unknown function (of many variables), by increasing the number of units (knots) in the topological map.

Abstract

The idea of using Kohonen's self-organizing maps is applied to the problem of nonparametric regression analysis, that is, evaluation (approximation) of the unknown function of N-1 variables given a number of data points (possibly corrupted by random noise) in N-dimensional input space. Simple examples show that the original Kohonen's algorithm performs poorly for regression problems of even low dimensionality, due to the fact that topologically correct ordering of units in N-dimensional space may violate the natural topological ordering of projections of those units onto (N-1)-dimensional subspace of independent variables. A modification of the original algorithm called the constrained topological mapping algorithm is proposed for regression analysis applications. Given a number of data points in N-dimensional input space, the proposed algorithm performs correct topological mapping of units (as the original algorithm) and at the same time preserves topological ordering of projections of these units onto (N-1)-dimensional subspace of independent coordinates. Simulation examples illustrate good performance (i.e., accuracy, convergence) of the proposed algorithm for approximating 2- and 3-variable functions. Moreover, for multivariate problems the proposed neural approach may alleviate “the curse of dimensionality,” that is, reduce the size of the training set required for evaluation of the unknown function (of many variables), by increasing the number of units (knots) in the topological map.