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.