Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.3. 1-D SYNTHETIC AND REAL DATA EXAMPLES 41
clear that quadratic and cubic terms are needed to capture the strong variability
observed in the time series.
In Figure 3.5(c) I portray the contribution due to the linear terms of a third-
order Volterra series to predicted the data in Figure 3.5(b) due to the linear terms of
third order Volterra series. Note that this contribution is negligible and shows that
linear terms can not model the highly nonstationary part of signal. Figures 3.5(d)
and 3.5(e) illustrate the parts of the prediction associated with quadratic (q = 8)
and cubic (r = 8) terms in the third-order Volterra series. Figure 3.5(f) shows the
contribution of nonlinear terms associated with quadratic and cubic terms (q = 8
and r = 8). It is clear that nonlinear quadratic and cubic terms are required to
properly model the full aperture.
A real data example corresponding to the so called Arctic oscillation time series
(AO)-a time series from 1950 to 1999 of sea level pressures-is used to characterize
the long term variability of nonseasonal sea level oscillations (Thomson, 2004).
Figures 3.6(a) and 3.7(a) show the nonlinear AO data for the period from 1950
to 1999. The data consist of 104 samples (3 observations per year-January, February
- March). Figures 3.6(b) and 3.6(c) illustrate predicted AO values for a Volterra
system consisting of linear and nonlinear terms (p = 10, q = 10, and r = 10) and
associated error, respectively. Figures 3.6(d) and 3.6(e) represent our attempt to
model the data with a linear prediction filter (p = 10) and corresponding error.
Again, it is clear that the dynamics of the time series is better captured by the
third-order Volterra system.
Figure 3.7(b) is the prediction using a third-order Volterra series with p = 10,
q = 10, and r = 10. In Figure 3.7(c) I portray the part of the prediction attributed