time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

modeling, a nonlinear autoregressive exogenous model (NARX) is a

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...

autoregressive model In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...

which has

exogenous In a variety of contexts, exogeny or exogeneity () is the fact of an action or object originating externally. It contrasts with endogeneity or endogeny, the fact of being influenced within a system. Economics In an economic model, an exogeno ...

inputs. This means that the model relates the current value of a time series 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 other terms will not enable the current value of the time series to be predicted exactly. Such a model can be stated algebraically as :

y_t = F(y_, y_, y_, \ldots, u_, u_, u_, u_, \ldots) + \varepsilon_t

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 An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistics ...

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 mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

. ''F'' can be a

neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

, a

wavelet network A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...

, a

sigmoid network Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include: * Sigmoid function, a mathematical function * Sigmoid colon, part of the ...

and so on. To test for non-linearity in a time series, the

BDS test BDS may refer to: Companies and organizations * Bergenske Dampskibsselskab, a Norwegian shipping service * Black Dragon Society, a Japanese paramilitary organization * Blessed Damien Society, a leprosy charity * Blue Diamond Society, a Nepales ...

(Brock-Dechert-Scheinkman test) developed for

econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

can be used.

References

* S. A. Billings. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains, Wiley, , 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

Open-source implementation of the NARX model using neural networks
Time series models Nonlinear time series analysis