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.

exponentially weighted moving average (ewma)

Go deeper with CFSGPT