A function that maps a time lag to the correlation between values of a sequence or stochastic process separated by that lag; it quantifies linear temporal dependence within a single series.

Principle

Autocorrelation at lag τ is the covariance between the series and its τ-shift divided by the variance (when variance is finite); for processes whose second-order statistics are time-invariant it depends only on the lag and is symmetric in sign reversal of lag.

Demonstration

For an autoregressive process X_t = φ X_{t-1} + ε_t with white-noise ε_t, the theoretical autocorrelation at lag k equals φ^k, showing exponential decay when |φ|<1.

Misapplication

Using sample autocorrelations from a short or strongly trending series to infer long-range dependence, or relying on autocorrelation to detect nonlinear coupling that it cannot reveal.

Consequence

Reveals characteristic timescales and periodicities, informs model identification (lags to include) for linear predictors, and guides filter and control design based on temporal memory.

Reversal

Cross-correlation compares two different series to detect lead–lag relationships, whereas autocorrelation compares a series with itself across lags to reveal internal memory.

Boundary

Requires finite second moments and a consistent ordering of observations; undefined for unordered data or for processes lacking well-defined second-order structure.

Semantic Tension

Partial autocorrelation isolates direct lagged effects removing intermediate lags; confusion arises when one uses autocorrelation to infer direct causation instead of chain-mediated dependence.

Synthesis

The autocorrelation function is the normalized covariance of a sequence with its lagged copies, mapping lag to linear dependence strength and exposing temporal structure and decay scales.