auto-correlation analysis

**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.