Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

3.2. NONLINEAR MODELING OF TIME SERIES 33
Equation (2.9) (linear prediction model) is a time-invariant linear system where
I have replaced the output by the one-step ahead prediction of the input. Similarly,
I can consider equation (3.12) and construct a time-variant nonlinear prediction
operator with a nonlinear Volterra model of the form (order of p, q, r):
xn =
+
+
p
ai xn−i
i=1
q q
bjk xn−jxn−k
j=1 k=1
r r r
clms xn−lxn−mxn−s
l=1 m=1 s=1
+ εn , (3.13)
where ai, bjk, and clms are the linear, the nonlinear quadratic, and the nonlinear
cubic impulse responses of the nonlinear system, respectively, also known as Volterra
kernels. The first term on the right hand side of equation (3.13) represents the
classical linear prediction problem with a prediction operator of order p. The second
and third terms of equation (3.13) represent the expansion of the signal in terms
of quadratic and cubic nonlinearities. The modeling error is given by εn. Note
that equation (3.13) is the nonlinear AR formulation as an extension of linear AR
modeling.
The last expression can also be written in prediction error form
εn = xn −
−
−
p
ai xn−i
i=1
q q
bjk xn−jxn−k
j=1 k=1
r r r
clms xn−lxn−mxn−s . (3.14)
l=1 m=1 s=1