Autoregressive integrated moving average

Autoregressive integrated moving average

In statistics, an autoregressive integrated moving average (ARIMA) model is a generalisation of an autoregressive moving average or (ARMA) model. These models are fitted to time series data either to better understand the data or to predict future points in the series. The model is generally referred to as an ARIMA("p","d","q") model where "p", "d", and "q" are integers greater than or equal to zero and refer to the order of the autoregressive, integrated, and moving average parts of the model respectively.

Given a time series of data X_t where t is an integer index and the X_t are real numbers, then an ARMA("p","q") model is given by:left(1 - sum_{i=1}^p phi_i L^i ight) X_t = left(1 + sum_{i=1}^q heta_i L^i ight) varepsilon_t,where L is the lag operator, the phi_i are the parameters of the autoregressive part of the model, the heta_i are the parameters of the moving average part and the varepsilon_t are error terms. The error terms varepsilon_t are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.

An ARIMA("p","d","q") process is obtained by integrating an ARMA("p","q") process. That is,:left(1 - sum_{i=1}^p phi_i L^i ight) (1-L)^d X_t = left(1 + sum_{i=1}^q heta_i L^i ight) varepsilon_t,where "d" is a positive integer that controls the level of differencing (or, if d=0, this model is equivalent to an ARMA model). Conversely, applying term-by-term differencing "d" times to an ARMA("p","q") process gives an ARIMA("p","d","q") process. Note that it is only necessary to difference the AR side of the ARMA representation, because the MA component is always I(0).

It should be noted that not all choices of parameters produce well-behaved models. In particular, if the model is required to be stationary then conditions on these parameters must be met.

Some well-known special cases arise naturally. For example, an ARIMA(0,1,0) model is given by::X_t = X_{t-1} + varepsilon_twhich is simply a random walk.

A number of variations on the ARIMA model are commonly used. For example, if multiple time series are used then the X_t can be thought of as vectors and a VARIMA model may be appropriate. Sometimes a seasonal effect is suspected in the model. For example, consider a model of daily road traffic volumes. Weekends clearly exhibit different behaviour from weekdays. In this case it is often considered better to use a SARIMA (seasonal ARIMA) model than to increase the order of the AR or MA parts of the model. If the time-series is suspected to exhibit long-range dependence then the d parameter may be replaced by certain non-integer values in a Fractional ARIMA (FARIMA also sometimes called ARFIMA) model.

ee also

*Partial autocorrelation
*Radial basis function


*Mills, Terence C. "Time Series Techniques for Economists." Cambridge University Press, 1990.
*Percival, Donald B. and Andrew T. Walden. "Spectral Analysis for Physical Applications." Cambridge University Press, 1993.

External links

* [ The US Census Bureau uses ARIMA for "seasonally adjusted" data (programs, docs, and papers here)]

Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Autoregressive Integrated Moving Average - ARIMA — A statistical analysis model that uses time series data to predict future trends. It is a form of regression analysis that seeks to predict future movements along the seemingly random walk taken by stocks and the financial market by examining the …   Investment dictionary

  • Autoregressive fractionally integrated moving average — In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA ( autoregressive integrated moving average ) models by allowing non integer values of the differencing parameter and are… …   Wikipedia

  • Autoregressive moving average model — In statistics, autoregressive moving average (ARMA) models, sometimes called Box Jenkins models after the iterative Box Jenkins methodology usually used to estimate them, are typically applied to time series data.Given a time series of data X t …   Wikipedia

  • Autoregressive conditional heteroskedasticity — ARCH redirects here. For the children s rights organization, see Action on Rights for Children. In econometrics, AutoRegressive Conditional Heteroskedasticity (ARCH) models are used to characterize and model observed time series. They are used… …   Wikipedia

  • ARIMA — autoregressive integrated moving average …   Medical dictionary

  • ARIMA — • autoregressive integrated moving average …   Dictionary of medical acronyms & abbreviations

  • Time series — Time series: random data plus trend, with best fit line and different smoothings In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at …   Wikipedia

  • Exponential smoothing — is a technique that can be applied to time series data, either to produce smoothed data for presentation, or to make forecasts. The time series data themselves are a sequence of observations. The observed phenomenon may be an essentially random… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”