Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.2. NONLINEAR MODELING OF TIME SERIES 36
and
x b n = a b 1xn+1 + a b 2xn+2 · · · + a b pxn+p
+ b b 11x 2 n+1 + b b 12xn+1xn+2 + b b 21xn+2xn+1 + b b 22xn+2xn+2 + · · · + b b qqx 2 n+q
+ c b 111x 3 n+1 + c b 112x 2 n+1xn+2 + · · · + c b 123xn+1xn+2xn+3 + · · · + c b rrrx 3 n+r
+ εn , (3.18)
where I, II, and III are the linear, quadratic nonlinear, and cubic nonlinear contri-
butions, respectively. I can see from equations (3.17) and (3.18) that the quadratic
(Powers et al., 1990) and cubic coefficients of the Volterra series must obey sym-
metry properties as mentioned in the subsection 3.1.3. For instance,
b f
12xn−1xn−2 = b f
21xn−2xn−1. (3.19)
It is clear that the number of coefficients to be computed is reduced from q 2 to (q (q+
3)/2−q). The cubic part of the Volterra series has also similar symmetry properties,
however, the symmetry relations are more complicated than the quadratic part. For
example,
c f
112xn−1xn−1xn−2 = c f
121xn−1xn−2xn−1 = c f
211xn−2xn−1xn−1 , (3.20)
c f
123xn−1xn−2xn−3 = c f
132xn−1xn−3xn−2 = c f
213xn−2xn−1xn−3 =
c f
231xn−2xn−3xn−1 = c f
312xn−3xn−1xn−2 = c f
321xn−3xn−2xn−1 . (3.21)