Control Charts

Time-Weighted Control Charts (EWMA, CUSUM)

EWMA and tabular CUSUM with transient/fixed control limits for small drifts.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Control Charts > Time-Weighted Charts > EWMA / CUSUM

Description

Engine for the two standard time-weighted control charts. EWMA smooths the series via zᵢ = λ·xᵢ + (1−λ)·zᵢ₋₁ with z₀ = μ₀; control limits are transient (narrow at first via σ_z(i) and approach the steady-state σ·√(λ/(2−λ)) asymptotically). Tabular CUSUM forms two one-sided sums C⁺ and C⁻ with slack k and decision interval h (both in σ units); a signal fires as soon as either sum exceeds h. The analyze(inputs) dispatcher accepts modes 'ewma' and 'cusum'.

Formulas

EWMA

z_{i} = λ x_{i} + (1 - λ) z_{i - 1}, z_{0} = μ_{0}, σ_{z_{i}} = σ \frac{λ}{2 - λ} (1 - (1 - λ)^{2 i}), UCL_{i} / LCL_{i} = μ_{0} \pm L σ_{z_{i}}

EWMA with transient limits (asymptotically σ·√(λ/(2−λ)))

Tabular CUSUM

C_{i}^{+} = max (0, x_{i} - (μ_{0} + K) + C_{i - 1}^{+}), C_{i}^{-} = max (0, (μ_{0} - K) - x_{i} + C_{i - 1}^{-}), K = kσ, H = hσ

Two one-sided sums — signal when C⁺ or C⁻ > H

Assumptions

Observations independent and (approximately) normally distributed
Target μ₀ and σ known from stable baseline — preferably σ from MR̄/d₂

Limitations

Both charts react slower to single large outliers than Shewhart charts
Autocorrelation produces false-positive signals — decorrelate first

References

Montgomery, Introduction to Statistical Quality Control, 8th Ed. — Chapter 9
Hunter, J. S. (1986), The Exponentially Weighted Moving Average, Journal of Quality Technology
Page, E. S. (1954), Continuous Inspection Schemes, Biometrika