Definition:Linear Filter
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Definition
Let $S$ be a stationary stochastic process governed by a white noise process:
- $\map z t = \mu + a_t$
where:
- $\mu$ is a constant mean level
- $a_t$ is an independent shock at timestamp $t$.
A linear filter takes the terms of $S$, and uses a weight function $\psi$ to apply a weighted sum of the past values so that:
\(\ds \map z t\) | \(=\) | \(\ds \mu + a_t + \psi_1 a_{t - 1} + \psi_2 a_{t - 2} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mu + \map \psi B a_t\) |
where $B$ denotes the backward shift operator, hence:
- $\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$
Transfer Function
The operator:
- $\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$
is the transfer function of $L$.
Stable
Consider the sequence $\sequence {\psi_k}$ formed by the weight function $\psi$ of $L$.
Suppose that:
- $\ds \sum_k \size {\psi_k} < \infty$
Then $L$ is said to be stable, and the model for $S$ is stationary.
Hence $\mu$ is the mean about which $S$ varies.
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: Linear filter model: $(1.2.1)$
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: