Nonlinear autoregressive exogenous model: Difference between revisions

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In [[time series]] modeling, a '''nonlinear autoregressive exogenous model''' (NARX) is a [[nonlinear]] [[autoregressive model]] model which has [[exogenous]] inputs. SuchThis ameans that the model canrelates bethe current value of a statedtime algebraicallyseries asto both:

* past values of the same series; and

* current and past values of the driving (exogenous) series — that is, of the externally determined series that influences the series of interest.

In addition, the model contains an error term which relates to the fact that knowledge of other terms will not enable the current value of the time series to be predicted exactly.

Such a model can be stated algebraically as

:<math> 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 </math>

: <math> y_t = F(y_{t-1}, y_{t-2}, y_{t-3}, \ldots, u_{t}, u_{t-1}, u_{t-2}, u_{t-3}, \ldots) + \varepsilon_t </math>

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.

Here ''y'' is the variable of interest, and ''u'' is somethe otherexternally variabledetermined which is associated with ''y''variable. In this scheme, information about ''u'' helps predict ''y'', as do previous values of ''y'' itself. Here ''ε'' is the [[errors and residuals in statistics|error]] term (sometimes called noise). For example, ''y'' may be air temperature at noon, and ''u'' may be the day of the year (day-number within year).

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

[[Image:narx.jpg|frame|NARX with two input memory taps and three output memory taps]]

The function ''F'' is some nonlinear function, such as a [[polynomial]]. ''F'' can be a [[neural network]], a [[wavelet network]], a [[sigmoid network]] and so on. To test for non-linearity in a time series, the [[BDS test]] (Brock-Dechert-Scheinkman test) developed for [[econometrics]] can be used.

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== References ==

* S. A. Billings. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains, Wiley, {{ISBN|978-1-1199-4359-4}}, 2013.

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

* O. Nelles. "Nonlinear System Identification". Springer Berlin, {{ISBN|3-540-67369-5}}, 2000.

* W.A. Brock, J.A. Scheinkman, W.D. Dechert and B. LeBaron. "A Test for Independence based on the Correlation Dimension". ''Econometric Reviews'' 15:197-235, 1996.

==External links==

* 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.

* [https://sourceforge.net/projects/narxsim Open-source implementation of the NARX model using neural networks]

[[Category:StochasticTime processesseries models]]

[[Category:Nonlinear time series analysis]]