Airline Model

The airline model is a special, but often used, case of multiplicative SARIMA model.

  1. For a given seasonality length (s), the airline model is defined by four(4) parameters: \(\mu\),\(\sigma\),\(\theta\) and \(\Theta\). \[(1-L^s)(1-L)Y_t = \mu + (1-\theta L)(1-\Theta L^s)a_t\] OR \[ Z_t = (1-L^s)(1-L)Y_t = \mu + (1-\theta L)(1-\Theta L^s)a_t \] OR \[ Z_t = \mu -\theta \times a_{t-1}-\Theta \times a_{t-s} +\theta\times\Theta \times a_{t-s-1}+ a_t \] Where:
    • \(s\) is the length of seasonality.
    • \(\mu\) is the model mean
    • \(\theta\) is coefficient of first lagged innovation
    • \(\Theta\) is the coefficient of s-lagged innovation.
    • \(\left [a_t\right ] \) is the innovations time series.
  1. \(\left[Y_t\right]\) is not a stationary process, but the differenced time series \(\left[Y_t\right]\) is.
  2. After we difference \(Y_t\) (i.e. \(Z_t\), the airline model is simplified to a special MA(s) model
  3. The airline model has 5 parameters: \(\mu\,,\sigma\,,s\,,\theta\,,\Theta\)
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* Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
* Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
* D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
* Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848