autokorelace

Autokorelace, also known as autocorrelation or serial correlation in statistics, is a measure of the relationship between observations in a time series and past values of the same series. It quantifies how the current value of a process relates to its past values at different lags, helping to reveal patterns such as persistence, seasonality, or structure in the data.

Mathematically, for a stochastic process Xt with mean μ, the autocovariance at lag k is gamma(k) = E[(Xt − μ)(Xt+k − μ)], and the autocorrelation (or autocorrelation function) is rho(k) = gamma(k) / gamma(0), provided gamma(0) > 0. The function rho(k) depends on the lag k and, for a stationary process, on k alone rather than on time t. Non-stationary data can exhibit spurious autocorrelation unless appropriately transformed (for example by differencing).

In practice, autocorrelation is estimated from data as the sample autocorrelation. For a series {xt} of length n, the sample autocorrelation at lag k is rk = sum_{t=1}^{n−k} (xt − x̄)(xt+k − x̄) / sum_{t=1}^n (xt − x̄)^2. The autocorrelation function (ACF) and its companion, the partial autocorrelation function (PACF), are key diagnostics in time series analysis, notably in identifying components for ARIMA models. The Durbin-Watson statistic is one test related to first-order autocorrelation.

Common cautions include interpreting autocorrelation as causation, distinguishing true dependence from trends or seasonality, and ensuring stationarity to avoid misleading results. Autokorelace is the term used in several languages for this concept, with autocorrelation as the standard English terminology.