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.