Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.3. 1-D SYNTHETIC AND REAL DATA EXAMPLES 40
It is clear that the problem is linear in the coefficients. One possible solution vector
is given by the regularized least squares solution (damped least squares). Adopt-
ing the least squares method with zero-order quadratic regularization leads to the
solution of the filter coefficients:
m = (A T A) + µI) −1 A T d . (3.29)
For small systems (small order of parameters p, q, and r) one can use direct in-
version methods to solve equation (3.29). For systems that involve long operators,
I suggest the use of semi-iterative solvers like the method of conjugate gradients
(Wang and Treitel, 1973). In our examples, however, I have adopted direct inver-
sion methods. In general, f − x algorithms do not require the inversion of large
systems of equations. In many cases this is a consequence of working with small
spatial windows.
3.3 1-D Synthetic and Real Data Examples
I have developed an algorithm to invert the coefficients of a third-order Volterra
series. I focus on a 1-D synthetic time series which is generated with a real second-
order Volterra system. Figures 3.4(a) and 3.5(a) show a 1-D input data for 100
samples.
Figures 3.4(b) and 3.4(c) represent the predicted series modeled via the third-
order Volterra series (with parameters p = 8, q = 8, and r = 8) and the associated
modeling error, respectively. Figures 3.4(d) and 3.4(e) portray the predicted data
using linear prediction theory and modeling error (p = 8, q = 0, and r = 0). It is