Autocorrelation Function (ACF)
The ACF is the normalized version of the Autocovariance Function (ACVF), giving a dimensionless measure of linear dependence between a stationary process and its lagged values.
1. Definition
$$\rho_X(h) = \frac{\text{Cov}(X_t, X_{t+h})}{\sqrt{\text{Var}(X_t)\text{Var}(X_{t+h})}} = \frac{\gamma_X(h)}{\gamma_X(0)}$$2. Properties
- $\rho_X(0) = 1$
- $|\rho_X(h)| \leq 1$
- $\rho_X(h) = \rho_X(-h)$ (symmetric)
3. Diagnostic Value
The shape of the ACF helps identify the underlying process:
| Process | ACF Behavior |
|---|---|
| White Noise Process | $\rho(h) = 0$ for all $h \neq 0$ |
| AR(1) | Exponential decay: $\rho(h) = \phi^h$ |
| MA(q) | Cuts off after lag $q$: $\rho(h) = 0$ for $ |
| ARMA(p,q) | Tails off (mixture of exponentials/oscillations) |
The cutoff vs tail-off distinction is the primary tool for model identification.
Interactive reference: Theoretical ACF / PACF simulator (AR/MA) (standalone page; Chart.js via CDN).