Two methods for identifying Wiener cascades having noninvertible static nonlinearities

Two methods are proposed for identifying the component elements of a Wiener cascade that is comprised of a dynamic linear element (L) followed by a static nonlinearity (N). Both methods avoid potential problems of instability in a procedure presented by Paulin [M. G. Paulin, Biol. Cybern. 69: 67-76, 1993], which itself is a modification of a method described earlier by Hunter and Korenberg [I. W. Hunter and M. J. Korenberg, Biol. Cybern. 55: 135-144, 1996]. The latter method is a rapidly convergent iterative procedure that produces accurate estimates of the L and N elements from short data records, provided that the static nonlinearity N is invertible. Subsequently, Paulin introduced a modification that removed this limitation and enabled identification of Wiener cascades with nonmonotonic static nonlinearities. However, Paulin presented his modification employing an autoregressive moving average (ARMA) model representation for the dynamic linear element. To remove the possibility that the estimated ARMA model could be unstable, we recast the procedure by utilizing instead a rapid method for finding an impulse response representation for the dynamic linear element. However, in this form the procedure did not have good convergence properties, so we introduced two key ideas, both of which provide effective alternatives for identifying Wiener cascades whether or not the static nonlinearities therein are invertible. The new procedures are illustrated on challenging examples involving high-degree polynomial static nonlinearities, of odd or even symmetry, a high-pass linear element, and output noise corruption of 50%.