Polynomial Theory of Complex Systems

A complex multidimensional decision hypersurface can be approximated by a set of polynomials in the input signals (properties) which contain information about the hypersurface of interest. The hypersurface is usually described by a number of experimental (vector) points and simple functions of their coordinates. The approach taken in this paper to approximating the decision hypersurface, and hence the input-output relationship of a complex system, is to fit a high-degree multinomial to the input properties using a multilayered perceptronlike network structure. Thresholds are employed at each layer in the network to identify those polynomials which best fit into the desired hypersurface. Only the best combinations of the input properties are allowed to pass to succeeding layers, where more complex combinations are formed. Each element in each layer in the network implements a nonlinear function of two inputs. The coefficients of each element are determined by a regression technique which enables each element to approximate the true outputs with minimum mean-square error. The experimental data base is divided into a training and testing set. The training set is used to obtain the element coefficients, and the testing set is used to determine the utility of a given element in the network and to control overfitting of the experimental data. This latter feature is termed "decision regularization.