autocorrelace

Autocrarelace, or autocorrelation, is a statistical measure that describes the linear relationship between observations of a time series at different times, separated by a given lag. In a stationary series, correlations depend only on the lag, not on the time.

For a process X_t with mean μ and variance σ^2, the autocovariance at lag k is γ_k = Cov(X_t, X_{t-k}) = E[(X_t - μ)(X_{t-k} - μ)]. The autocorrelation (or autocorrelation function, ACF) is ρ_k = γ_k / γ_0, where γ_0 = σ^2. Note that ρ_0 = 1. The ACF is typically plotted against k and helps reveal persistence and periodicities.

Estimation from data: for a sample of size n with mean x̄, the sample autocorrelation at lag k is r_k = ∑_{t=k+1}^n (x_t - x̄)(x_{t-k} - x̄) / ∑_{t=1}^n (x_t - x̄)^2. For non-stationary series, trends can create spurious correlations; stationarity or differencing is often required.

Uses and interpretation: autocorrelation is used to identify dependence structure, seasonality, and to select models in time-series analysis, notably in ARIMA modeling, where the ACF and PACF aid model order selection. Statistical tests such as the Ljung-Box test assess whether a series behaves like white noise. Caution is needed when nonlinear relationships or changing variance are present.

See also: autocovariance, partial autocorrelation, ACF, PACF.