Nonparametric nonlinear regression using polynomial and neural approximators: a numerical comparison

The solution of nonparametric regression problems is addressed via polynomial approximators and one-hidden-layer feedforward neural approximators. Such families of approximating functions are compared as to both complexity and experimental performances in finding a nonparametric mapping that interpolates a finite set of samples according to the empirical risk minimization approach. The theoretical background that is necessary to interpret the numerical results is presented. Two simulation case studies are analyzed to fully understand the practical issues that may arise in solving such problems. The issues depend on both the approximation capabilities of the approximating functions and the effectiveness of the methodologies that are available to select the tuning parameters, i.e., the coefficients of the polynomials and the weights of the neural networks. The simulation results show that the neural approximators perform better than the polynomial ones with the same number of parameters. However, this superiority can be jeopardized by the presence of local minima, which affects the neural networks but does not regard the polynomial approach.