ACFs

ACFs, or auto-correlation functions, are statistical tools used to measure the relationship between observations in a time series at different time lags. They quantify how a signal or process is related to its own past values, providing a compact summary of serial dependence that is central to time series analysis and signal processing.

Mathematically, for a weakly stationary stochastic process Xt with mean mu, the autocovariance at lag k is gamma(k) = E[(Xt - mu)(Xt+k - mu)]. The autocorrelation function is rho(k) = gamma(k) / gamma(0). In practice, for a finite sample of size n, the sample autocorrelation is r(k) = sum_{t=1}^{n-k} (Xt - xbar)(Xt+k - xbar) / sum_{t=1}^n (Xt - xbar)^2, where xbar is the sample mean. The ACF is typically computed for a range of lags and often used alongside the partial autocorrelation function (PACF) for model identification.

Key properties include that rho(0) = 1, rho(-k) = rho(k), and |rho(k)| ≤ 1. The ACF can reveal whether a series is white noise (ACF near zero for nonzero lags) or exhibits persistence and periodicity (peaks at regular lags). The rate and pattern of decay inform assumptions about memory and are instrumental in selecting ARMA-type models, conducting residual diagnostics, and identifying seasonality.

Practical use involves estimating the ACF from data, interpreting its significance with approximate confidence bounds, and being mindful of nonstationarity, trends, or heteroskedasticity which can distort the ACF. Techniques such as prewhitening or differencing are often applied to address these issues before relying on ACF-based inferences.