Autokorrelations
Autocorrelation, also known as serial correlation, is a statistical measure of the relationship between observations in a time series and the same observations at a different time lag. In time series analysis, autocorrelation at lag k quantifies the linear relationship between X_t and X_{t-k}. The autocorrelation coefficient at lag k is rho_k = Cov(X_t, X_{t-k}) / Var(X_t), assuming finite variance. For stationary processes, rho_0 = 1, and |rho_k| ≤ 1.
For a stochastic process with mean mu and variance sigma^2, the autocovariance function gamma_k = Cov(X_t, X_{t+k}).
Estimation of autocorrelation in finite samples yields the sample autocorrelation function (SACF). At lag k, the
Applications of autocorrelation include model selection, forecasting, and diagnosing time series data. It is used to