Exponentially Weighted Moving Average (Ewma)
Exponentially Weighted Moving Average (EWMA) is a statistical process control method that assigns exponentially decreasing weights to older data points, making it highly sensitive to small, gradual drifts in process parameters — drifts that traditional Shewhart charts might miss.
How EWMA Works
The EWMA statistic at time $t$ is:
Where:
- $x_t$ = Current observation.
- $Z_{t-1}$ = Previous EWMA value.
- $\lambda$ = Weighting factor (0 < λ ≤ 1), typically 0.05–0.25.
- $Z_0$ = Process target (initial value set to the process mean).
Each new EWMA value is a weighted combination of the current measurement and the accumulated history. Smaller λ gives more weight to history (better for detecting small drifts); larger λ gives more weight to the current point (more responsive, similar to Shewhart).
EWMA Control Limits
Where $L$ is typically 2.5–3.0 and $\sigma$ is the process standard deviation. The limits start narrow and widen, converging to steady-state values.
Why EWMA Excels at Drift Detection
- Shewhart charts evaluate each point independently — they need a large shift (typically >2σ) to trigger an alarm on a single point.
- EWMA accumulates information across multiple points. A sustained small drift (0.5–1.0σ) gradually pushes the EWMA statistic toward the control limits, triggering an alarm that Shewhart would miss.
- Think of EWMA as having "memory" — it remembers the trend, not just the latest point.
Applications in Semiconductor Manufacturing
- Etch Rate Drift: Detecting gradual etch rate changes due to chamber aging or consumable wear.
- Film Thickness Trends: Identifying slow drift in CVD deposition rate.
- CD Trending: Monitoring lithographic CD drift due to resist aging, environmental changes, or equipment degradation.
- Overlay Drift: Tracking gradual alignment degradation in lithography scanners.
EWMA vs. Other Methods
| Method | Best For | Sensitivity to Small Shifts |
|---|---|---|
| Shewhart | Large, sudden shifts | Low |
| EWMA | Small, sustained drifts | High |
| CUSUM | Small, sustained shifts | High |
Choosing λ
- λ = 0.05–0.10: High sensitivity to small drifts, but slow response to large shifts.
- λ = 0.20–0.30: Good balance between drift sensitivity and responsiveness.
- λ = 1.0: Reduces to a standard Shewhart chart (no memory).
EWMA is the preferred SPC method for semiconductor process control where gradual drift is the primary concern — it catches the slow changes that erode yield long before Shewhart charts raise an alarm.