Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.2. NONLINEAR MODELING OF TIME SERIES 37
Symmetry arguments in equations (3.19, 3.20 and 3.21) are also valid for the back-
ward prediction. The number of coefficients for the cubic part is reduced from r 3
to r 2 + r!/(r − 3)!3! and the total number of prediction coefficients is reduced form
p + q 2 + r 3 to p + (q (q + 3)/2 − q) + r 2 + r!/(r − 3)!3!.
and
The form of the forward and backward prediction equations is now given by:
x f n = a f
1xn−1 + a f
2xn−2 · · · + a f pxn−p
+ b f
11x 2 n−1 + 2b f
12xn−1xn−2 + 2b f
13xn−1xn−3 + · · · + b f qqx 2 n−q
+ c f
111x 3 n−1 + 3c f
112x 2 n−1xn−2 + · · · + 6c f
123xn−1xn−2xn−3 + · · · + c f rrrx 3 n−r
+ εn , (3.22)
x b n = a b 1xn+1 + a b 2xn+2 · · · + a b pxn+p
+ b b 11x 2 n+1 + 2b b 12xn+1xn+2 + 2b b 31xn+1xn+3 + · · · + b b qqx 2 n+q
+ c b 111x 3 n+1 + 3c b 112x 2 n+1xn+2 + · · · + 6c b 123xn+1xn+2xn+3 + · · · + c b rrrx 3 n+r
+ εn . (3.23)
As in the linear prediction problem, I will assume that the Volterra coefficients
(linear, quadratic, and cubic) are obtained using actual observations. For example,
assume p = 1, q = 2, c = 3, and a time series of N = 7 points: