Advanced control charts

The Shewhart charts on the Control charts page react to one point at a time. A single X-bar or individual reading is compared against limits set three sigma out, and the run-rules engine on the Run rules page looks for patterns across a window of points. That design is good at catching a large, sudden shift. It is slow to catch a small shift that the process holds for many readings, because each individual point still looks plausible on its own.

This page covers the charts that fill that gap and a few that answer a different question entirely:

• EWMA and CUSUM accumulate information across observations, so a small sustained shift in the mean builds up into a signal.
• The short-run chart standardizes the measurement so several part numbers, each with its own target, share one chart.
• Pre-control is a shop-floor running check driven by the tolerance, not by control limits.
• The time-series screen characterizes drift and autocorrelation so you know whether the independence assumption behind a control chart holds at all.

The constants table, the variables/attributes limit formulas, and the run-rules catalog are not repeated here. See Control charts and Run rules.

A note that applies to EWMA and CUSUM both: each assumes the in-control mean mu0 and standard deviation sigma are known, estimated from a stable phase-I baseline. mfgQC does not silently assume that baseline is valid. Every EWMA and CUSUM result carries an in_control_parameters assumption check that states where mu0 and sigma came from and recommends validating phase-I stability (for example with an I-MR chart) before trusting the limits.

EWMA

What it is and when to use it

The exponentially weighted moving average smooths the series. Each plotted point is a weighted average of the current reading and all readings before it, with the weights decaying geometrically into the past. Because the statistic carries memory, a shift that is too small to push any single point past a Shewhart limit still accumulates in the EWMA until it crosses. Use an EWMA chart for individual measurements when you care about detecting small, sustained shifts in the mean (roughly 0.5 to 2 sigma) faster than a Shewhart chart would.

The formula

The EWMA statistic is the recursion

\[ z_t = \lambda\, x_t + (1 - \lambda)\, z_{t-1}, \qquad z_0 = \mu_0 . \]

The smoothing constant \(\lambda\) sets how much weight the current reading gets. A small \(\lambda\) remembers more history and reacts to smaller shifts; a \(\lambda\) of 1 reduces the chart to plotting the raw readings. mfgQC defaults to \(\lambda = 0.1\).

The control limits widen as history builds up. At observation \(t\) the half-width is

\[ \text{half-width}_t = L\,\sigma \sqrt{\frac{\lambda}{2 - \lambda}\,\bigl(1 - (1-\lambda)^{2t}\bigr)} , \]

with center line \(\mu_0\), so \(\mathrm{UCL}_t = \mu_0 + \text{half-width}_t\) and \(\mathrm{LCL}_t = \mu_0 - \text{half-width}_t\). The factor \(\bigl(1 - (1-\lambda)^{2t}\bigr)\) is the variance build-up term. The variance of \(z_t\) starts small and grows toward its steady-state value, so the early limits are tight and flare out over the first several points before settling. \(L\) is the limit width in sigma units; mfgQC defaults to \(L = 2.7\). As \(t\) grows the half-width approaches its asymptote \(L\,\sigma\sqrt{\lambda/(2-\lambda)}\); for \(\lambda = 0.1\), \(L = 2.7\), \(\sigma = 0.15\) that asymptote is about \(0.0929\), which the example below approaches from below.

mfgQC resolves the parameters this way (in mfgqc/timeseries_charts.py):

• mu0 defaults to the spec target if one is set, otherwise the sample mean.
• sigma defaults to the moving-range estimate \(\bar{R}_m / d_2\) with \(d_2 = 1.128\) for moving ranges of size 2, the same short-term estimate an I-MR chart uses. This isolates within-process variation and is not inflated by the very shift the chart is meant to catch.
• A point signals when \(z_t > \mathrm{UCL}_t\) or \(z_t < \mathrm{LCL}_t\).

A caution on the sample-mean default: if you let mu0 default to the sample mean of a record that already contains a sustained shift, the mean absorbs part of that shift and the chart can fail to signal. When you know the in-control mean, pass it as mu0= (and sigma= if known). The example below does.

Assumptions

EWMA assumes the in-control mu0 and sigma are known from a stable phase-I baseline. It assumes individual observations that are independent over time; autocorrelation distorts the limits. mfgQC reports the parameter source through the in_control_parameters check but does not test independence inside the EWMA path. To screen for autocorrelation first, use the time-series screen below.

Source

Montgomery, Introduction to Statistical Quality Control (see the Bibliography), the EWMA chapter.

CUSUM

What it is and when to use it

The tabular cumulative sum keeps two running totals: \(C^+\) accumulates how far the process has drifted above target, and \(C^-\) how far it has drifted below. As long as the process sits at mu0 the sums stay near zero, because a slack value is subtracted at each step. Once the mean shifts, the relevant sum climbs steadily until it crosses a decision interval. Like EWMA, CUSUM is built to detect small sustained shifts faster than a Shewhart chart. The two are alternatives; CUSUM is the choice when you want the explicit two-sided up/down accounting.

The formula

mfgQC computes the two-sided tabular CUSUM. With reference value \(K = k\sigma\) and decision interval \(H = h\sigma\):

\[ C^+_t = \max\!\bigl(0,\; x_t - (\mu_0 + K) + C^+_{t-1}\bigr), \qquad C^+_0 = 0, \]

\[ C^-_t = \max\!\bigl(0,\; (\mu_0 - K) - x_t + C^-_{t-1}\bigr), \qquad C^-_0 = 0 . \]

The reference value \(k\) is the slack, in sigma units. It is the amount of drift you are willing to ignore before the sum starts to grow; it is conventionally set to half the shift you want to detect, so \(k = 0.5\) targets a one-sigma shift. The decision interval \(h\) is the threshold the sum must cross to signal, also in sigma units. mfgQC defaults to \(k = 0.5\) and \(h = 5\), which is a standard pairing for a one-sigma shift.

Reading a signal: a point is out of control when \(C^+_t > H\) (an upward shift) or \(C^-_t > H\) (a downward shift). The two arms are checked separately, and the result labels the violating point with cusum_upper or cusum_lower so you know which direction the process moved.

mu0 and sigma default exactly as for EWMA (spec target or sample mean; then \(\bar{R}_m / d_2\)). The chart plots \(C^+\) on a positive track and \(C^-\) negated on its own track for readability, with the decision interval drawn at \(\pm H\).

Assumptions

Same as EWMA: known in-control mu0 and sigma from a stable phase-I baseline, and independent observations over time. The in_control_parameters check reports the parameter source. Independence is not tested inside the CUSUM path.

Source

Montgomery, Introduction to Statistical Quality Control (see the Bibliography), the CUSUM chapter.

Worked examples: EWMA and CUSUM

The data below holds 15 readings around an in-control mean of 10.0, then a sustained shift of about one unit to 11.0 starting at reading 16. The in-control mean and standard deviation are known from phase I, so they are passed explicitly.

import pandas as pd
import mfgqc

x = [10.1, 9.8, 10.0, 9.9, 10.2, 9.7, 10.1, 10.0, 9.9, 10.0,
     10.2, 9.8, 10.1, 9.9, 10.0,                       # in control at 10.0
     11.0, 10.9, 11.1, 11.0, 10.8, 11.2, 11.0, 10.9, 11.1, 11.0]  # shifted to 11.0
qc = mfgqc.load(pd.DataFrame({"width": x}), measure="width")

ewma = qc.ewma_chart(mu0=10.0, sigma=0.15)   # defaults: lam=0.1, L=2.7
print(ewma.report())

EWMA Chart: lambda=0.1, L=2.7
=============================
EWMA: CL=10  mu0=10  sigma=0.15
UCL: 10.04 -> 10.093 (time-varying)
LCL: 9.9595 -> 9.9073 (time-varying)

Out-of-control signals: 9
  point 17: ewma - EWMA statistic beyond control limit
  point 18: ewma - EWMA statistic beyond control limit
  point 19: ewma - EWMA statistic beyond control limit
  point 20: ewma - EWMA statistic beyond control limit
  point 21: ewma - EWMA statistic beyond control limit
  point 22: ewma - EWMA statistic beyond control limit
  point 23: ewma - EWMA statistic beyond control limit
  point 24: ewma - EWMA statistic beyond control limit
  point 25: ewma - EWMA statistic beyond control limit

Assumption checks:
  [PASS] in_control_parameters (phase-I baseline (assumed)): phase-I baseline (assumed); n=25

Recommendations:
  - EWMA/CUSUM assume the in-control mu0 and sigma are known from a stable
    phase-I baseline; here mu0=10 (user-specified) and sigma=0.15
    (user-specified). Validate phase-I stability (e.g. an I-MR chart) before
    trusting these limits.

The EWMA statistic climbs through the shift and crosses the upper limit at point 17, two readings after the shift began. The build-up is visible in the statistic: \(z\) goes \(9.986,\ 10.087,\ 10.169,\ 10.262,\ 10.336,\ 10.382\) across points 15 through 20, while the upper limit has settled near \(10.092\). The notice on the report is the in_control_parameters check confirming the parameters were user-specified, not a warning that anything failed.

The same data through a CUSUM chart at the defaults:

cusum = qc.cusum_chart(mu0=10.0, sigma=0.15)   # defaults: k=0.5, h=5
print(cusum.report())

CUSUM Chart: k=0.5, h=5
=======================
CUSUM: mu0=10  sigma=0.15
K (reference)=0.075  H (decision interval)=0.75

Out-of-control signals: 10
  point 16: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 17: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 18: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 19: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 20: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 21: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 22: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 23: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 24: cusum_upper - C+ exceeds decision interval H (upward shift)
  point 25: cusum_upper - C+ exceeds decision interval H (upward shift)

Assumption checks:
  [PASS] in_control_parameters (phase-I baseline (assumed)): phase-I baseline (assumed); n=25

Here \(K = k\sigma = 0.5 \times 0.15 = 0.075\) and \(H = h\sigma = 5 \times 0.15 = 0.75\). The upper sum \(C^+\) stays at \(0\) through the in-control stretch, then climbs \(0.925,\ 1.75,\ 2.775,\ 3.7\) over points 16 through 19 and crosses \(H = 0.75\) at point 16, the first shifted reading. On this dataset CUSUM signals one reading earlier than EWMA, and the cusum_upper label tells you the shift was upward.

Short-run charts

What it is and when to use it

A standardized short-run chart lets several part numbers, each with a different target, share one control chart. Instead of charting the raw measurement, it charts how far each piece sits from its part's target, in that part's own standard deviation units. That puts a 12 mm shaft and a 40 mm bore on the same vertical scale. Use it for low-volume or mixed production where no single part runs long enough to build a stable chart of its own.

The formula

For each part \(p\) with target \(T_p\) and within-part standard deviation \(\sigma_p\), every reading is standardized:

\[ z = \frac{x - T_p}{\sigma_p} . \]

The pooled \(z\) values are plotted on one chart with center line \(0\) and limits at \(\pm 3\). The target is resolved per the target argument to short_run_chart: a scalar applies one target to every part, a {part: target} mapping gives each part its own, and None (the default) uses each part's own sample mean as its target. \(\sigma_p\) is the sample standard deviation of that part's readings (with one degree of freedom; a part with a single reading contributes \(z = 0\)). The run-rules engine runs on the standardized series; the default rule set is Nelson.

Assumptions

The standardized scale is only shared if the parts have comparable within-part variability. A part whose spread departs from the pool breaks the common scale. mfgQC checks this: when two or more parts have usable spread it runs a homogeneity- of-variance check across them and attaches the result, so a discordant part is flagged rather than quietly distorting the chart.

Source

Montgomery, Introduction to Statistical Quality Control (see the Bibliography), the short-run and standardized-chart material.

Pre-control

What it is and when to use it

Pre-control, also called stoplight control, is a running shop-floor check tied to the tolerance, not to control limits. It splits the tolerance into colored zones and uses a simple count of greens, yellows, and reds to decide whether to keep running, adjust, or stop. It needs no subgroups and no computed control limits, which is why it is used at the machine for a quick go/no-go on a process that is already known to be capable and centered.

That last condition matters. Pre-control assumes a capable, centered process. It is not a substitute for a control chart, and it is unreliable on an incapable or off-center process. mfgQC estimates Cpk from the data and flags when capability has not been established.

The zones and the rules

The tolerance \([\text{LSL}, \text{USL}]\) is divided at the pre-control lines

\[ \text{PC}_{\text{lo}} = \text{LSL} + \tfrac{1}{4}(\text{USL} - \text{LSL}), \qquad \text{PC}_{\text{hi}} = \text{USL} - \tfrac{1}{4}(\text{USL} - \text{LSL}), \]

which is the central half of the tolerance. A piece is green if it falls in that central half, yellow if it is between a pre-control line and its spec limit (the outer quarters), and red if it is outside spec. The target defaults to the midpoint of the tolerance when no target is set.

Qualification and the running rule:

• Qualify the run when five consecutive pieces are green. Until then the run is not qualified.
• After qualification, a two-piece rule decides on each consecutive pair:
  • a red piece means STOP (a piece is out of spec);
  • two yellows on the same side mean ADJUST (the process has drifted off-center toward that limit);
  • two yellows on opposite sides mean STOP (variation is too wide for the tolerance).

precontrol() requires both spec limits (set with .spec(lower=, upper=)); it raises a MissingPrerequisiteError if either is absent.

Assumptions

The method presumes the process is capable and centered. mfgQC estimates \(\widehat{\text{Cpk}} = \min(\text{USL} - \mu,\ \mu - \text{LSL}) / (3\hat\sigma)\) from the pooled data and attaches a capability_prerequisite check; if the estimate is below 1.33 the result recommends running a capability study and a control chart first. Pre-control is a running check, not a capability study.

Source

Pre-control is shop-floor practice; the specific zone definitions and the five- green qualification plus two-piece running rule implemented here are not pinned to a source listed in the Bibliography. Treat this method's source as not yet cited there, rather than assuming one.

Time-series screen

What it is and when to use it

The time-series screen answers a prerequisite question for any control chart: are the observations independent and stationary over time, or do they drift and carry memory? Control charts (Shewhart, and to a degree EWMA and CUSUM) assume the readings are not autocorrelated. When they are, the limits are unreliable. The screen characterizes that structure. It is deliberately not a forecaster: there is no ARIMA or machine-learning model here, and nothing predicts future values.

timeseries() reports two things:

• Trend. It regresses the measure on time order (a linear slope with a t-test) and also runs the nonparametric Mann-Kendall trend test (the \(\tau\) statistic and its p-value). Reporting both means a monotone but nonlinear drift is still caught. The direction is read off Mann-Kendall: increasing, decreasing, or no trend at the chosen \(\alpha\) (default 0.05).
• Autocorrelation. It computes the sample autocorrelation function ACF(\(k\)) on the mean-centered series and flags every lag whose ACF falls outside the band \(\pm z_{1-\alpha/2}/\sqrt{n}\) (with \(z_{1-\alpha/2} = 1.96\) at the default \(\alpha\)). A flagged lag is evidence the observations are not independent.

mfgQC surfaces both as assumption checks. If a trend is present, the recommendation says to detrend before control charting or to model the trend. If autocorrelation is present, the recommendation says Shewhart limits are unreliable and points you toward EWMA, CUSUM, or a time-series model. The result's own summary text states plainly that it is an exploratory screen that complements, and does not replace, the CUSUM and EWMA charts for monitoring. The screen needs at least four observations.

There is also a richer set of characterization tools in mfgqc/timeseries.py (a standalone trend result, a full ACF/PACF view via the Durbin-Levinson recursion, and a classical additive trend-plus-seasonal-plus-residual decomposition). Those are reached through the module functions rather than the timeseries() dispatcher.

Source

The linear-trend regression follows mfgQC's own regression machinery (ordinary least squares). The Mann-Kendall test and the \(\pm 1.96/\sqrt{n}\) ACF band are standard time-series methods; they are not pinned to a text listed in the Bibliography; the NIST/SEMATECH e-Handbook listed there covers these methods at the reference level.

See also

• Control charts: the Shewhart variables/attributes charts, the constants table, and the limit formulas.
• Run rules: the Western Electric and Nelson out-of-control rules.
• Bibliography: the sources cited above.
• API reference: ewma_chart, cusum_chart, short_run_chart, precontrol, and timeseries.