Neural Model Identification

Abstract

Suppose that the sample ${D_n} = \left\{ {({x_{i,1}},{x_{i,2}},...,{x_{i,m}},{y_i})} \right\}_{i = 1}^n = \left\{ {\left( {{x_{i,}}{y_i}} \right)} \right\}_{i = 1}^n$ comprises n independent observations on m explanatory variables x _j , j = 1,…, m and one dependent variable y and that each observation can be regarded as a realization of an (m + 1)-dimensional distribution function Ξ(x, y) =Ψ(y|x)Ω(x) (the operating model) which will sometimes also be denoted by F for simplicity. We view the observations as being generated by an unknown function ø(x) with the addition of a stochastic component, commonly taken to be independently and identically distributed (i.i.d.) with zero mean and constant variance σ², i.e.

$$ {y_i} = \phi \left( {{x_i}} \right) + {\varepsilon _i} $$

(2.1)