Auto-Correlation Analysis is a statistical technique that measures how a time series is correlated with lagged versions of itself — revealing periodicity, persistence, and memory effects in process data that indicate systematic patterns rather than random variation.

How Does Auto-Correlation Work?

• Lag: Compute the correlation between $x_t$ and $x_{t-k}$ for different lag values $k$.
• ACF (Auto-Correlation Function): Plot correlation vs. lag to visualize temporal structure.
• PACF: Partial ACF removes indirect correlations to show only direct lag dependencies.
• Significance Bands: $pm 1.96/sqrt{N}$ confidence bands identify statistically significant lags.

Why It Matters

• Process Memory: Significant autocorrelation at lag 1 means consecutive runs are not independent — SPC assumptions violated.
• Periodicity: Peaks in ACF at lag $L$ reveal periodic patterns with period $L$.
• Model Selection: ACF/PACF guide the choice of ARIMA model orders for time series modeling.

Auto-Correlation is asking how today predicts tomorrow — measuring the memory in process data to identify systematic patterns and temporal dependencies.