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.