CUSUM Chart

Cumulative-sum control chart — accumulates deviations from the target and alarms when a threshold $h$ is crossed.

Also known as: CUSUM, Cumulative Sum

The CUSUM chart (Cumulative Sum, Page 1954) accumulates each observation's deviation from the target and alarms as soon as the cumulative sum crosses a threshold $h$ . Like EWMA it is very sensitive to small persistent shifts.

C_{i}^{+} = max (0, x_{i} - μ_{0} - k + C_{i - 1}^{+}), C_{i}^{-} = max (0, μ_{0} - x_{i} - k + C_{i - 1}^{-})

k (reference / slack) — the shift you want to react to; typically $k = 0.5 σ$ (optimally tuned for 1σ: $k = Δ/2$ )
h (decision interval) — alarm when $∣ C_{i} ∣ > h$ ; typical $h = 4 σ$ or $5 σ$
V-mask — graphical equivalent, historically popular, today mostly abandoned in favour of the tabular form

Used in

Control Chart (Time-Weighted)

In the Algorithm Lab

control-charts/time-weighted-chart

Sources

Page, E. S. — Continuous Inspection Schemes, Biometrika, 41(1/2), 1954
Montgomery, D. C. — Introduction to Statistical Quality Control, 8th Edition, Chapter 9
Hawkins, D. M. & Olwell, D. H. — Cumulative Sum Charts and Charting for Quality Improvement, Springer