Definition:Autocorrelation
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Definition
Let $S$ be a stochastic process giving rise to a time series $T$.
The autocorrelation of $S$ at lag $k$ is defined as:
- $\rho_k := \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$
where:
- $z_t$ is the observation at time $t$
- $\mu$ is the mean of $S$
- $\expect \cdot$ is the expectation.
Autocorrelation Coefficient
$\rho_k$ is known as the autocorrelation coefficient of $S$ at $k$.
Also known as
Autocorrelation is also known as serial correlation.
Also see
- Results about autocorrelation can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): autocorrelation or serial correlation (Statistics)
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next): Part $\text {I}$: Stochastic Models and their Forecasting
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- Part $\text {I}$: Stochastic Models and their Forecasting:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes
- Part $\text {I}$: Stochastic Models and their Forecasting:
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- Part $\text {I}$: Stochastic Models and their Forecasting:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- Part $\text {I}$: Stochastic Models and their Forecasting:
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): autocorrelation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): autocorrelation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): autocorrelation (serial correlation)