Nonlinear autoregressive exogenous model
Encyclopedia
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 current value of a time series which one would like to explain or predict to both:
In addition, the model contains:
which relates to the fact that knowledge of the other terms will not enable the current value of the time series to be predicted exactly.
Such a model can be stated algebraically as
Here y is the variable of interest, and u is the externally determined variable. In this scheme, information about u helps predict y, as do previous values of y itself. Here ε is the 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
. 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.
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
modeling, a nonlinear autoregressive exogenous model (NARX) is a nonlinear autoregressive model
Autoregressive model
In statistics and signal processing, an autoregressive model is a type of random process which is often used to model and predict various types of natural phenomena...
which has exogenous
Exogenous
Exogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
inputs. This means that the model relates the current value of a time series which one would like to explain or predict to 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 the other terms will not enable the current value of the time series to be predicted exactly.
Such a model can be stated algebraically as
Here y is the variable of interest, and u is the externally determined variable. In this scheme, information about u helps predict y, as do previous values of y itself. Here ε is the error
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
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
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
. F can be a neural network
Neural network
The term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...
, 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
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
can be used.