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ARIMA

noun/uh-REE-muh/Also: autoregressive integrated moving average, ARMA, SARIMA, ARIMAX, Box–Jenkins method

In plain wordsA classic statistical recipe for forecasting a series of numbers over time: remove any steady trend, then predict the next value from recent values and recent prediction errors.

  1. A class of linear statistical models for time series forecastingtime series forecasting, n.: Predicting future values of something measured over time, such as sales, electricity demand, prices or rainfall, from its past values and related information. ('autoregressive integrated moving average'), written ARIMA(p, d, q). The series is first differenced d times to remove trends and make it stationary, meaning its statistical properties no longer drift over time; the differenced series is then modeled as a linear combination of its own p previous values (the autoregressive part) and of the q previous random shocks, or forecast errors (the moving-average part). Seasonal ARIMA (SARIMA) adds the same terms at seasonal lags, and ARIMAX adds external regressors. The Box–Jenkins methodology fits models in three repeated stages: identification (choosing p, d and q from autocorrelation plots and unit-root tests), estimation (usually by maximum likelihood) and diagnostic checking of the residuals; automatic order selection with information criteria is now common. ARIMA models are interpretable and yield prediction intervals, but they assume linear dynamics and model one series at a time.

    A seasonal ARIMA(1,1,1)(0,1,1)₁₂ is still our baseline for monthly passenger numbers.

(1−∑i=1pϕiLi)(1−L)d yt=c+(1+∑j=1qθjLj)εt\Big(1 - \sum_{i=1}^{p} \phi_i L^i\Big)(1 - L)^d \, y_t = c + \Big(1 + \sum_{j=1}^{q} \theta_j L^j\Big)\varepsilon_t

Origin

Builds on George Udny Yule's autoregressive model (1927), Eugen Slutsky's moving-average processes (1927) and Peter Whittle's theory of ARMA models (1951). George Box and Gwilym Jenkins assembled the integrated and seasonal model class, with its identify–estimate–check methodology, in their 1970 book Time Series Analysis: Forecasting and Control, making ARIMA a standard tool in economics, operations and engineering.

Usage note

In machine learning, ARIMA and seasonal naive forecasts serve as standard baselinesbaseline, n.: A simple reference method that a new model must beat to show it is worth anything, such as always guessing the most common answer or reusing yesterday's value.; a complex model that cannot beat them adds no value. In the formula, L is the lag operator (L yₜ = yₜ₋₁).

Related

See also
  • time series forecastingtime series forecasting, n.: Predicting future values of something measured over time, such as sales, electricity demand, prices or rainfall, from its past values and related information.
  • exponential smoothingexponential smoothing, n.: A simple forecasting method that predicts the next value as a weighted average of past values, giving the most weight to recent ones and steadily less to older ones.
  • autoregressive modelautoregressive model, n.: A model that produces a sequence one piece at a time, each new piece depending on the pieces already produced — like writing a sentence word by word without going back.
  • Akaike information criterionAkaike information criterion, n.: A score for comparing statistical models that rewards fitting the data well but charges a penalty for every extra adjustable number, to avoid picking needlessly complicated models. Lower is better.
  • baselinebaseline, n.: A simple reference method that a new model must beat to show it is worth anything, such as always guessing the most common answer or reusing yesterday's value.
  • regressionregression, n.: Predicting a number rather than a category — a house's price, tomorrow's temperature, a patient's blood pressure — by learning from past examples where the answer is known.

Sources

  1. Box, G. E. P. & Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.
  2. Whittle, P. (1951). Hypothesis Testing in Time Series Analysis. Uppsala: Almqvist & Wiksells.
  3. Hyndman, R. J. & Khandakar, Y. (2008). Automatic Time Series Forecasting: The forecast Package for R. Journal of Statistical Software 27(3).

Mentioned in

autoregressive modelautoregressive model, n.: A model that produces a sequence one piece at a time, each new piece depending on the pieces already produced — like writing a sentence word by word without going back., time series forecastingtime series forecasting, n.: Predicting future values of something measured over time, such as sales, electricity demand, prices or rainfall, from its past values and related information., Akaike information criterionAkaike information criterion, n.: A score for comparing statistical models that rewards fitting the data well but charges a penalty for every extra adjustable number, to avoid picking needlessly complicated models. Lower is better., Bayesian information criterionBayesian information criterion, n.: A score for comparing statistical models that, like AIC, rewards good fit and penalizes extra adjustable numbers, but penalizes them more as the dataset grows, favoring simpler models. Lower is better., exponential smoothingexponential smoothing, n.: A simple forecasting method that predicts the next value as a weighted average of past values, giving the most weight to recent ones and steadily less to older ones., time series foundation modeltime series foundation model, n.: A large AI model pretrained on huge numbers of time series, such as sales, traffic and sensor readings, that can forecast a new series it has never seen without being trained on it.