autokorrelation

Autocorrelation, also known as autokorrelation in some languages, is a measure of the correlation of a signal with a delayed copy of itself. It is used to detect repeating patterns and to characterize the dependence structure of stochastic processes.

For a real-valued process X(t) that is wide-sense stationary, the autocorrelation function (ACF) is R_xx(τ) = E[X(t) X(t+τ)]. For a discrete-time process X[n], the autocorrelation is R_xx[k] = E[X[n] X[n+k]]. The value at zero lag, R_xx(0), equals the variance of the process. The ACF is even: R_xx(-τ) = R_xx(τ). The normalized form, ρ_xx(τ) = R_xx(τ)/R_xx(0), lies in the interval [-1, 1].

In the frequency domain, the power spectral density S_xx(ω) is the Fourier transform of R_xx(τ). This relationship links time-domain dependence to frequency components of the signal.

Autocorrelation is distinct from autocovariance. The autocovariance γ_xx(τ) = Cov(X(t), X(t+τ)) = E[(X(t)−μ)(X(t+τ)−μ)] uses the mean μ. The normalized autocorrelation is ρ_xx(τ) = γ_xx(τ)/γ_xx(0). For zero-mean processes, autocorrelation and autocovariance coincide.

Estimation is typically done via the sample autocorrelation function (SACF): r[k] = ∑_{n=0}^{N−k−1} (x[n]− x̄)(x[n+k]− x̄) / ∑_{n=0}^{N−1} (x[n]− x̄)². The ACF aids in identifying dependence, seasonality, and appropriate models (for example, ARIMA) in time series analysis. Confidence bands for r[k] are often used to assess significance under the assumption of independence.

Applications span signal processing, econometrics, geophysics, and climate science, wherever understanding the memory and periodic structure of a signal or series is important.