# Definition:Autocorrelation

## 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

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes
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