Obtaining Volterra Kernels from Neural Networks

Abstract

Both neural networks (NN) and Volterra series (VS) are widely used in nonlinear dynamic system identification. In VS approach, the system is modeled using a set of kernel functions that correspond to different order convolutions. Kernels in VS are typically estimated using an orthogonal expansion technique. In this study, we discuss the method of obtaining VS representation of nonlinear systems from their NN models as an alternative approach and compare its modeling performances against the popular Laguerre basis expansion (LBE) technique. In LBE approach, the critical issues are to select a suitable pole parameter and number of basis functions to be used in the expansions, so that the kernels can be accurately represented. We devised novel approaches to address both issues, the pole parameter is selected using a systematic optimization approach and the number of basis functions is decided using the minimum description length criterion. Our preliminary results on synthetic data indicate that when used with these provisions, LBE yields more accurate kernels estimation results than the NN approach. However, LBE is typically used without these provisions in literature. We demonstrate that with its typical use, kernels estimated using the LBE approach can be quite misleading even though the estimation error may seem to be reasonable. Therefore, we suggest the use NN approach as a reference method to confirm the morphology of the kernels estimated via other approaches, including LBE.