autokovarians

Autokovarians, or autocovariance, is a statistical measure of how a time series values at one time point relate to values at a later time, after removing the mean. For a weakly stationary process X_t with mean mu and finite variance, the autocovariance at lag h is defined as gamma(h) = Cov(X_t, X_{t+h}) = E[(X_t - mu)(X_{t+h} - mu)]. When h = 0, gamma(0) equals the variance of the process.

If the process is stationary, gamma(h) depends only on the lag h, not on the specific time t. The sequence {gamma(h)} for h = 0, 1, 2, … is called the autocovariance function (ACVF). The autocovariance measures the linear association between values separated by lag h. The corresponding autocorrelation at lag h is rho(h) = gamma(h) / gamma(0).

Examples help illustrate behavior. For white noise, gamma(h) = 0 for all nonzero h. For an AR(1) process X_t = phi X_{t-1} + e_t with |phi| < 1 and e_t white noise, gamma(h) = phi^h gamma(0) for h >= 0, where gamma(0) = Var(X_t) = sigma_e^2 / (1 - phi^2).

Estimation in samples uses the sample autocovariance gamma_hat(h) = (1/(n-h)) sum_{t=1}^{n-h} (X_t - X_bar)(X_{t+h} - X_bar) if the mean is unknown, or with mu if the mean is known. Autocovariance is central to time series modeling and analysis, including identification of dependency structure, seasonality, and residual diagnostics in ARIMA-type models.