Cusum Ewma Monitoring

CUSUM and EWMA Monitoring (manufacturing-quality/as9100/cusum-ewma-monitoring)

Use when the task is sequential monitoring of a production process for small sustained mean shifts that a single-point check on the raw values would miss. This leaf implements the tabular CUSUM (accumulating standardized deviations against a slack k and a decision interval h) and the EWMA recursion with time-varying sigma limits in pure Python, stdlib only. It pairs with the large-shift charting sibling manufacturing-quality/as9100/statistical-process-control for the point rule side of process monitoring and with manufacturing-quality/as9100/corrective-action for the downstream response once a signal fires. Observations are assumed independent and normal around the in-control mean mu0 with known sigma (or sigma estimated from an in-control study).

Domain quick reference

• Standardized observation: z_i = (x_i - mu0) / sigma.
• Tabular CUSUM (one-sided): S+i = max(0, z_i - k + S+{i-1}), S-i = max(0, -z_i - k + S-{i-1}), starting S+_0 = S-_0 = 0. A signal fires when S+_i > h or S-_i > h. Defaults k = 0.5, h = 5.0, the standard choice for detecting a sustained 1-sigma shift with good average run length behaviour.
• EWMA recursion: e_0 = mu0, e_i = lam * x_i + (1 - lam) * e_{i-1}.
• Time-varying EWMA sigma: sigma_e_i = sigma * sqrt(lam / (2 - lam) * (1 - (1 - lam)^(2 i))), so the limits widen from the first sample toward the steady state sigma * sqrt(lam / (2 - lam)) as i grows.
• EWMA limits: UCL_i = mu0 + L * sigma_e_i, LCL_i = mu0 - L * sigma_e_i. Defaults lam = 0.2, L = 3.0.
• First signal index is the 1-based sample number where the statistic first exceeds its limit, or None for an in-control sequence.
• AS9100 frames the monitoring and data-analysis context; the chart methodology above is standard engineering method, summary-only.

Workflow

1. Confirm the in-control reference: the target mean mu0 and the process sigma (known or from an in-control study) for the sequence xs.
2. Run cusum_statistics(xs, mu0, sigma) and read sp_plus, sp_minus and first_signal_index; S+ accumulates positive standardized deviations beyond the slack k, S- the negative ones.
3. Run ewma_statistics(xs, mu0, sigma) and read ewma_series, ucl, lcl and first_signal_index; e smooths the raw values with weight lam and is compared against the per-sample limits.
4. Compare the two first-signal samples: EWMA with lam 0.2 usually fires first on gentle drifts, CUSUM on slightly stronger sustained steps; both beat a single-point check for small shifts.
5. Combine with monitoring_verdict(cusum_index, ewma_index, n) to get cusum_signaled, ewma_signaled, any_signal and the earlier first_signal_index.
6. For a one-call run use small_shift_monitoring_report(xs, mu0, sigma) which returns the cusum dict, the ewma dict and the verdict dict.
7. Confirm the deterministic checks with the contract test scripts/test_cusum_ewma_monitoring.py.

Worked example

mu0 = 10, sigma = 1, xs = [10.2, 10.5, 11.0, 10.8, 11.5, 11.9, 12.2, 11.6, 12.0, 11.4, 11.8], k = 0.5, h = 5.0, lam = 0.2, L = 3.0.

• CUSUM S+ path: [0, 0, 0.5, 0.8, 1.8, 3.2, 4.9, 6.0, 7.5, 8.4, 9.7]; S- stays 0 everywhere. First signal at sample 8 (x = 11.6, S+ = 6.0

  5.0); sample 7 sits at 4.9, just below the decision interval.

• EWMA e series: [10.040, 10.132, 10.306, 10.404, 10.624, 10.879, 11.143, 11.234, 11.388, 11.390, 11.472]; UCL series: [10.600, 10.768, 10.859, 10.912, 10.945, 10.965, 10.978, 10.986, 10.991, 10.994, 10.996] widening toward the steady-state limit 11.000. First signal at sample 7 (x = 12.2, e = 11.143 > UCL 10.978).
• Contrast: every raw value stays within 10 +- 3, so a single-point 3-sigma check on this sequence finds nothing; the accumulation charts flag it, which is the gap this leaf fills.
• Verdict: any_signal True, first_signal_index 7 (EWMA fires before CUSUM on this drift).
• The module outputs above are the exact recursion results, rounded to three decimals; they agree with the spec anchor lists at display precision (the exact closed-form UCL values are asserted to 1e-9 in the contract test).

Verification

• Confirm cusum_statistics returns the S+ path above to 1e-9, all-zero S- for the worked sequence, and first_signal_index 8.
• Confirm a constant mu0 series leaves S+ and S- at 0 with no signal, and that a sustained negative shift fires on the S- side.
• Confirm ewma_statistics reproduces the e and UCL series above, e_1 = lam * x_1 + (1 - lam) * mu0 exactly, first_signal_index 7, and that the last sigma_e of a 200-sample in-control run sits within 0.1 percent of sigma * sqrt(lam / (2 - lam)).
• Confirm monitoring_verdict and small_shift_monitoring_report report any_signal True with first_signal_index 7 for the worked sequence and False/None for a flat sequence.
• Confirm ValueError rejection of empty xs, sigma <= 0, k <= 0, h <= 0, lam <= 0 or lam > 1, L <= 0 and n < 0.
• Run the contract test offline: python3 scripts/test_cusum_ewma_monitoring.py (34 tests, deterministic).

Related leaves

• manufacturing-quality/as9100/statistical-process-control: the large-shift charting sibling (point rules and index-based process scoring); the boundary is point rules for large shifts here vs accumulation statistics for small shifts in this leaf.
• manufacturing-quality/as9100/corrective-action: downstream response planning once a monitoring signal fires.

Pitfalls

• Running the charts without a validated in-control reference: the statistics assume independent, normal observations around mu0 with known sigma, so a mu0 or sigma pulled from a drifting process fires signals that are artefacts of the reference, not of the shift.
• Declaring the process fine from a single-point check: every raw value in the worked sequence sits within mu0 +- 3 sigma, yet EWMA signals at sample 7 and CUSUM at sample 8 - the accumulation statistics exist precisely because the point check misses small sustained shifts.
• Reading only the S+ side: a sustained negative shift accumulates on S- (which stays at 0 for the worked example) and fires only there; both sides must be read before declaring in control.
• Judging charts by which fires first: EWMA with lam 0.2 usually leads on gentle drifts and CUSUM on slightly stronger sustained steps (7 vs 8 in the worked example), so an earlier first signal is a behaviour of the statistic, not proof of a larger shift.
• Comparing EWMA values against steady-state limits: the per-sample sigma_e widens from the first sample toward sigma * sqrt(lam / (2 - lam)), so early samples face narrower limits and late-sample comparisons against the steady state misplace the signal.
• Passing non-physical parameters: empty xs, sigma <= 0, k <= 0, h <= 0, lam outside (0, 1], L <= 0 and n < 0 raise ValueError, and silent defaults (k 0.5, h 5.0, lam 0.2, L 3.0) only suit the 1-sigma-shift detection context they were designed for.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_cusum_ewma_monitoring.py

The test covers the worked-example anchors (CUSUM S+ path to 1e-9, first CUSUM signal at sample 8, EWMA e and UCL display series, first EWMA signal at sample 7), the in-control all-zero CUSUM behaviour, the EWMA steady-state sigma_e limit within 0.1 percent, the exact closed-form UCL identity with LCL mirroring, outlier jump and decay, lambda = 1 passthrough, dict key contracts, deterministic run-to-run floats, and ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: AS9100 frames the quality management and data-analysis context (standards-map.yaml); the CUSUM and EWMA relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.