Cusum Chart
A CUSUM (Cumulative Sum) chart is an SPC tool that detects small, sustained shifts in a process mean by tracking the cumulative sum of deviations from a target value. Unlike Shewhart charts that evaluate each point independently, CUSUM accumulates evidence over time, making it highly sensitive to persistent drifts.
How CUSUM Works
- Define a target value $\mu_0$ (the desired process mean).
- For each observation $x_i$, calculate the deviation: $x_i - \mu_0$.
- Accumulate these deviations:
- Upper CUSUM: $C_i^+ = \max(0, C_{i-1}^+ + (x_i - \mu_0 - K))$ — detects upward shifts.
- Lower CUSUM: $C_i^- = \max(0, C_{i-1}^- - (x_i - \mu_0 + K))$ — detects downward shifts.
- $K$ is the reference value (allowance), typically set at half the shift size you want to detect: $K = \delta\sigma / 2$.
- Signal when $C^+$ or $C^-$ exceeds the decision interval $H$ (typically 4–5 times $\sigma$).
Why CUSUM Is Powerful
- Cumulative Memory: Small deviations that individually look normal accumulate over time. A consistent 0.5σ drift will eventually push the CUSUM past the threshold.
- Optimal for Small Shifts: CUSUM is theoretically the most efficient fixed-sample-size test for detecting a sustained shift of known magnitude.
- V-Mask Alternative: An equivalent graphical approach uses a V-shaped mask placed on the cumulative sum plot — the process is out of control if the plotted path crosses the mask boundaries.
CUSUM vs. EWMA vs. Shewhart
| Feature | Shewhart | EWMA | CUSUM |
|---|---|---|---|
| Small shift (0.5–1σ) | Poor | Good | Excellent |
| Large shift (>2σ) | Excellent | Good | Good |
| Simplicity | Simplest | Moderate | Moderate |
| Diagnostic | Easy | Moderate | Hard |
| Memory | None | Exponential decay | Full accumulation |
Semiconductor Applications
- Etch Rate Drift: Detecting gradual etch rate changes of 0.5–1% that accumulate over many lots.
- Film Thickness Trends: Identifying CVD deposition rate drift before it impacts yield.
- Overlay Monitoring: Detecting systematic overlay drift between lithography maintenance cycles.
- Tool Degradation: Monitoring gradual performance degradation that signals upcoming maintenance needs.
Practical Considerations
- Resetting: After an alarm and corrective action, the CUSUM is reset to zero.
- Two-Sided: Separate upper and lower CUSUMs detect shifts in both directions.
- ARL (Average Run Length): The key performance metric — how quickly (in number of samples) the CUSUM detects a shift. Smaller ARL = faster detection.
CUSUM is the mathematically optimal method for detecting small persistent process shifts — it is the gold standard when sensitivity to drift matters more than simplicity.