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:
$$Z_t = \lambda \cdot x_t + (1 - \lambda) \cdot Z_{t-1}$$
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**
$$UCL/LCL = \mu_0 \pm L \cdot \sigma \sqrt{\frac{\lambda}{2-\lambda} \left[1-(1-\lambda)^{2t}\right]}$$
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.