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. This means that the model relates the presentcurrent value of thea time series to both:

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In [[time series]] modeling, a '''nonlinear autoregressive exogenous model''' (NARX) is a [[nonlinear]] [[autoregressive]] model which has [[exogenous]] inputs. This means that the model relates the present value of the time series to both:

* past values of the same series; and

* presentcurrent 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 the other terms will not enable the presentcurrent value of the time series to be predicted exactly.

In addition, the model contains:

* an "error" or "residual" term

which relates to the fact that knowledge of the other terms will not enable the present 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}, u_{t-1}, u_{t-2}, u_{t-3}, \ldots) + \varepsilon_t </math>

Here ''y'' is the variable of interest, and ''u'' is somethe otherexternally variable which isdetermined associated with ''y''variable. In this scheme, information about ''u'' helps predict ''y'', as do previous values of ''y'' itself. Here <math>\varepsilon</math>''ε'' is the error[[errors orand residualresiduals 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'' iscan 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.

== 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 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.

* 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:TimeNonlinear time series analysis]]