Prediction Domain in Nonlinear Models

Abstract

Most of the difficulties arising in the interpretation of linear regression are due to collinearity, which is inherent to the structure of the design points (the X space) in the classical linear model

$${y_{n \times 1}} = {X_{n \times p}}{\Theta _{p \times 1}} + {e_{n \times 1}},$$

where the subscripts indicate the dimensions of the vectors and matrices. The structure of the X space has to be analysed as a warning to limit a correct use of a regression model for prediction purposes; the portion of space where prediction is good was introduced as the ‘effective prediction domain’ or EPD by Mandel (1985). This notion may be extended when a linear model contains x variables that are nonlinear functions of one or more of the other variables, such as x ²_j or x_jx_k.

In this paper we extend the notion of EPD to nonlinear models, which have the general form

$${y_{n \times 1}} = \eta ({X_{n \times p}},{\Theta _{p \times 1}}) + {e_{n \times 1}}$$

.