cross-correlation analysis

**Cross-Correlation Analysis** is a **technique that measures the similarity between two different time series as a function of time lag** — identifying delayed cause-effect relationships between process variables, where changes in one variable predict changes in another after a time delay. **How Does Cross-Correlation Work?** - **Lag**: Compute the correlation between $x_t$ and $y_{t-k}$ for different lag values $k$. - **Peak Lag**: The lag with maximum cross-correlation indicates the time delay between cause and effect. - **Direction**: If peak occurs at positive lag, $x$ leads $y$. If negative, $y$ leads $x$. - **Magnitude**: The correlation value indicates the strength of the delayed relationship. **Why It Matters** - **Causal Relationships**: If precursor gas flow change (step $N$) correlates with film thickness (step $N+1$) at lag 3, the time delay is quantified. - **Fault Propagation**: Traces how upstream process disturbances propagate through the manufacturing flow. - **Optimal Timing**: Determines the optimal timing for feed-forward control corrections. **Cross-Correlation** is **finding the echo between signals** — measuring time-delayed relationships between process variables to identify cause and effect.