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On Gaussian AR(1) data, moving-range control charts have an in-control run length below 93 observations, not 370, once lag-1 autocorrelation reaches 0.4

We simulated four control charts and three remedies on autoregressive data, then tested registered alarm-count predictions on Tennessee Eastman data.

CheAI Research Inc. | Research article | February 2026

Abstract

Control charts that assume independence are often applied to autocorrelated process data. We simulated Shewhart, exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts on Gaussian first-order autoregressive (AR(1)) data with lag-1 autocorrelation φ from 0 to 0.95. The design in-control average run length (ARL0) was 370.4. Criteria were frozen before any run, and an independent implementation re-derived every verdict. With moving-range limits, the upper 95 % bound of ARL0 was below 93 for every chart at every φ ≥ 0.4. With the true process standard deviation, the Shewhart chart became conservative and EWMA and CUSUM stayed far too short. Residual charts, batch means and inflated limits each restored ARL0 in parts of the grid. For each remedy, at some φ ≥ 0.5 no chart both restored ARL0 and kept its detection speed for a one-sigma step. Batch means were not at a matched time-based false-alarm rate, and the inflated-limit CUSUM used our unsourced scaling. On Tennessee Eastman data, the registered alarm-count prediction held for 1 of 72 variable-chart pairs.

1. Introduction

Process dynamics, mixing and feedback control make consecutive measurements on a chemical process correlated [1]. A Shewhart individuals chart with three-sigma limits has an ARL0 of 370.4 observations on independent normal data, and EWMA and CUSUM charts are usually designed to the same value [2].

Positive autocorrelation shortens run lengths when the limits come from the moving range, and it changes the run-length distribution of EWMA and CUSUM charts [1], [3], [4]. The standard remedies chart model residuals [5], batch means [6] or limits inflated to the variance of the statistic [7]. A practitioner with a lag-1 estimate needs to know how short each run length becomes, and whether a remedy restores it without losing detection speed. On one grid with frozen criteria, moving-range charts fell below a quarter of the design value at φ ≥ 0.4, and every remedy failed for all four charts somewhere above φ = 0.5.

2. Methods

The in-control process was a stationary Gaussian AR(1) series,

Xt = μ + Ut,   Ut = φ Ut-1 + εt,

at 20 values of φ from 0 to 0.95, with process standard deviation σX. Step shifts of 0.5-3 σX measured detection speed. The four charts were Shewhart, EWMA with λ = 0.1 and 0.2, and CUSUM with k = 0.5, with constants solved for ARL0 = 370.40 on independent data.

The textbook moving-range design (D-MR) estimates sigma from the average moving range, which converges to

σ̂MR = σX √(1 - φ).

The standard-deviation design (D-SD) used the true σX. The remedies were residual charts with the true model (R1) or a model fitted on 500 observations (R1-est, secondary). They also included batch means (R2), with the smallest batch whose lag-1 autocorrelation is at most 0.10, and inflated limits from the variance of each statistic (R3). For CUSUM, R3 used a long-run scaling with no published source. A raw-data chart calibrated by simulation to ARL0 = 370 (B0) was the comparator for detection speed.

Each cell used 10,000 runs. All criteria and six predictions were frozen in a pre-registration [8]. A setting restored ARL0 when its 95 % interval lay inside 296-463. It kept power when the upper bound of its detection delay relative to B0 was at most 1.25. A remedy worked when both held at a one-sigma step. A second implementation, written without the primary code, re-simulated 72 cells and re-derived every verdict.

For the real-data test we used fault-free Tennessee Eastman process (TEP) runs [9], [10], as qualified for a companion study [11]. Each variable's lag-1 autocorrelation r1 and the grid ARL0 gave a predicted alarm count, committed before any test run was loaded.

3. Results

3.1 Textbook limits give short run lengths

The simulation reproduced all 20 reference values at φ = 0 within three standard errors. With moving-range limits, ARL0 fell steeply with φ (Figure 1a), and its upper bound was below 93 in all 48 cells at φ ≥ 0.4. At φ = 0.5, ARL0 was 33.4 for Shewhart, 22.3 for EWMA (λ = 0.1) and 16.9 for CUSUM.

Run length against autocorrelation, two designs.
Figure 1. In-control run length with (a) moving-range limits and (b) the true process standard deviation. The grey band is 296-463 and the line is 370.4. Ringed points were re-simulated independently and agreed.

With the true σX, the Shewhart chart became conservative, with ARL0 of 831.8 at φ = 0.9 (Figure 1b). EWMA (λ = 0.1) and CUSUM had 53.6 and 47.0 at φ = 0.5, and their upper bound stayed below 296 at every φ ≥ 0.2.

3.2 Every remedy fails somewhere above φ = 0.5

The residual chart with the true model restored ARL0 in all 80 cells, but its Shewhart version detected a one-sigma step more slowly than B0 from φ = 0.15 upwards. Batch-means Shewhart restored ARL0 in plotted points, while batch-means EWMA (λ = 0.1) stayed anti-conservative at φ ≥ 0.5. Inflated-limit EWMA charts were conservative at φ ≥ 0.8.

Map of where each remedy works.
Figure 2. Where each remedy works (white). Shading gives the reason for failure. Verdicts at φ ≥ 0.5 were compared one by one with the independent implementation, and those below 0.5 were not.

For each remedy there were values of φ ≥ 0.5 at which no chart worked (Figure 2). These were 0.90-0.95 for R1, 0.50-0.95 for R2 and 0.70-0.95 for R3. Above 0.5, R1 worked for EWMA (λ = 0.1) from 0.50 to 0.85, the widest range of any remedy and chart. R3 worked only for Shewhart, from 0.50 to 0.65. The R2 verdicts are not at a matched time-based false-alarm rate, and the R3 failures include our unsourced CUSUM scaling.

3.3 The alarm-count prediction failed on TEP data

Of 32 screened TEP variables, 18 had r1 inside the grid. For the moving-range design, the predicted count lay inside the observed 95 % interval for 1 of 72 variable-chart pairs, against a target of 70 % (Figure 3). The independent implementation reproduced all 72 predictions and counts.

Observed against predicted alarm counts.
Figure 3. Predicted against observed false-alarm counts on the TEP test runs, moving-range design. Horizontal bars show the prediction uncertainty and vertical bars the run-bootstrap 95 % interval of the count. Each count covers 480,000 observations. The ringed point is the one pair that held.

4. Discussion

The two designs separate two mechanisms. The moving range underestimates σX by √(1 - φ), which tightens the limits. Autocorrelation also inflates the variance of statistics that average observations, which affects EWMA and CUSUM but not Shewhart. These directions agree with earlier studies [1], [3], [4]. These run lengths measure the wait to the first alarm. Under autocorrelation their reciprocal need not equal the repeated-alarm rate, because restarting a chart does not reset the process.

A residual reflects the full shift only in the first observation after a step. Each batch mean represents many observations. Inflated limits correct the variance of the statistic but not its serial correlation. At φ ≥ 0.9 no design in this study worked, and limits there should be calibrated by simulation or on long in-control records.

The analysis cannot separate three possible sources of the TEP prediction error. They are the registered approximation that converts run length into alarm count, the finite runs of 960 samples, and the lag-1 AR(1) description. For XMEAS(1), the moving-range to total standard deviation ratio was 0.41, against 0.56 for an AR(1) series with the same r1, which points to structure beyond AR(1). A lag-1 estimate warns of short run lengths, but alarm rates should be measured on in-control data.

5. Limitations

Results hold for Gaussian AR(1) data with a known mean, large-sample limits and the stated constants. Finite training samples were studied only for R1-est. Non-normal noise was not studied, and power assumes the shift is present from the start. Batch means are not at a matched time-based false-alarm rate, and the inflated-limit CUSUM uses our own unsourced scaling. The combined criterion gives a binary verdict, so excessive conservatism and slower detection, which are separate reasons for failure, both count as failure.

The literature full texts were not available. The formulas were derived for this study, the references support only the method families, and no published run-length table was replicated. The Siegmund approximation checked only the CUSUM ARL0, because it misses by 7.6 % at a three-sigma shift. Remedy ranges below φ = 0.5 and quoted values outside the 72 re-simulated cells were not verified independently. One cell (φ = 0.5, EWMA λ = 0.1, moving range) disagreed and then agreed on re-run. The independent estimates lie about 3.6 % below ours, possibly from the EWMA start state, which was not resolved, although no verdict depends on it. The residual Shewhart power result holds by 0.006 in one cell (1.256 against 1.25), so one cell flipping would flip that part. Three deviations from the pre-registration concerned computation only. They were recorded before the affected outcomes and changed no cell, run count or threshold.

The TEP test covered the whole registered procedure, including the count conversion and the 960-sample runs, which were not separated from model error. Four points of the protocol admit more than one reading. Using the exact d2 instead of 1.128 changes the limits by 0.03 %. Restricting the TEP test to 0 ≤ r1 ≤ 0.95 excludes six nearly white variables. The TEP hold fraction has no uncertainty bound, and the protocol does not state whether one EWMA prediction covers both λ. The TEP data are simulator output, and the 72 pairs are not independent, because four charts share each variable and some variables nearly duplicate others. Finally, a SKAB laboratory data set [12] was used as a low-precision illustration only, not verified independently. It is not stationary between its segments.

6. Conclusions

Charts with moving-range limits had in-control run lengths far below design on autocorrelated data. With the true standard deviation, Shewhart became conservative while EWMA and CUSUM stayed short. Residual charts, batch means and inflated limits, in the designs studied, each worked in part of the grid. Each failed for all charts at some autocorrelation of 0.5 or more. Batch means were not at a matched time-based false-alarm rate, and the inflated CUSUM used our unsourced scaling. On Tennessee Eastman data, the registered procedure that converts lag-1 autocorrelation into alarm counts did not predict them.

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