Definition:ARIMA Model/Linguistic Note
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Linguistic Note on ARIMA Model
The I in the acronym ARIMA stands for integrated.
This arises because the relationship which is the inverse of the backward difference operator:
- $w_t = \map {\nabla^d} {z_t}$
is:
- $z_t = S^d w_T$
where $S$ is the summation operator, defined as:
- $S = \nabla^{-1} = \paren {1 - B}^{-1}$
so that:
- $\map S {w_t} = \ds \sum_{j \mathop = 0}^\infty w_{t - j} = w_t + w_{t - 1} + w_{t - 2} + \dotsb$
Hence the autoregressive integrated moving average (ARIMA) process can be generated by summing or integrating the ARMA process a total of $d$ times.
Sources
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- $1$: Introduction:
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Nonstationary models
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: