Nonlinear autoregressive exogenous model

In time series modeling, a nonlinear autoregressive exogenous model (NARX) is a nonlinear autoregressive model which has exogenous inputs. Such a model can be stated algebraically as

$y_{t}=F(y_{t-1},y_{t-2},y_{t-3},\ldots ,u_{t-1},u_{t-2},u_{t-3},\ldots )+\varepsilon _{t}$

Here y is the variable of interest, and u is some other variable which is associated with y. In this scheme, information about u helps predict y. For example, y may be air temperature at noon, and u may be day of the year.

The function F is some nonlinear function, such as a polynomial. In some applications, F is a neural network.

References

• I.J. Leontaritis and S.A. Billings. "Input-output parametric models for non-linear systems. Part i: deterministic non-linear systems". Int'l J of Control 41:303-­328, 1985.

• I.J. Leontaritis and S.A. Billings. "Input-output parametric models for non-linear systems. Part ii: stochastic non-linear systems". Int'l J of Control 41:329-344, 1985.