Autocorrelation

Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at different points in time. The analysis of autocorrelation is a mathematical tool for identifying repeating patterns or hidden periodicities within a signal obscured by noise. Autocorrelation is widely used in signal processing, time domain and time series analysis to understand the behavior of data over time.

Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance.

Various time series models incorporate autocorrelation, such as unit root processes, trend-stationary processes, autoregressive processes, and moving average processes.

In statistics, the autocorrelation of a real or complex random process is the Pearson correlation between values of the process at different times, as a function of the two times or of the time lag. Let ${\displaystyle \left\{X_{t}\right\}}$ be a random process, and ${\displaystyle t}$ be any point in time (${\displaystyle t}$ may be an integer for a discrete-time process or a real number for a continuous-time process). Then ${\displaystyle X_{t}}$ is the value (or realization) produced by a given run of the process at time ${\displaystyle t}$. Suppose that the process has mean ${\displaystyle \mu _{t}}$ and variance ${\displaystyle \sigma _{t}^{2}}$ at time ${\displaystyle t}$, for each ${\displaystyle t}$. Then the definition of the autocorrelation function between times ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ is

${\displaystyle \operatorname {R} _{XX}(t_{1},t_{2})=\operatorname {E} \left[X_{t_{1}}{\overline {X}}_{t_{2}}\right]}$

where ${\displaystyle \operatorname {E} }$ is the expected value operator and the bar represents complex conjugation. Note that the expectation may not be well defined.

Subtracting the mean before multiplication yields the auto-covariance function between times ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$:

${\displaystyle {\begin{aligned}\operatorname {K} _{XX}(t_{1},t_{2})&=\operatorname {E} \left[(X_{t_{1}}-\mu _{t_{1}}){\overline {(X_{t_{2}}-\mu _{t_{2}})}}\right]\\&=\operatorname {E} \left[X_{t_{1}}{\overline {X}}_{t_{2}}\right]-\mu _{t_{1}}{\overline {\mu }}_{t_{2}}\\&=\operatorname {R} _{XX}(t_{1},t_{2})-\mu _{t_{1}}{\overline {\mu }}_{t_{2}}\end{aligned}}}$