Definition:Summation Operator
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Definition
Let $f: \R \to \R$ be a real function.
The summation operator $S$ on $f$ is defined as:
- $\map S f = \ds \sum_{j \mathop = 0}^\infty \map f {x - j} = \map f x + \map f {x - 1} + \map f {x - 2} + \dotsb$
Also presented as
The summation operator, when applied to a time series, can be written in terms of the backward shift operator as:
- $\map S f = \map {\nabla^{-1} } f = \paren {1 - B}^{-1}$
and so:
- $\map S {z_t} = \ds \sum_{j \mathop = 0}^\infty z_{t - j} = z_t + z_{t - 1} + z_{t - 2} + \dotsb$
Also see
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: