Interval Width Fuzzy Aggregation for Streaming Process Monitoring on the Tennessee Eastman

2.1 Fuzzy FMEA and Type-2 Fuzzy Risk Assessment

Type-2 fuzzy systems distinguish uncertainty in memberships from ordinary type-1 membership grades [2], [3], [10], [11]. Karnik–Mendel [12] centroid type reduction is a separate operation on type-2 sets. Recent FMEA and industrial risk-assessment studies use fuzzy representations to organize expert judgments [4], [5]. The present benchmark instead uses numerical process deviations and a fixed interval on Gaussian widths. It evaluates this particular anomaly-score construction rather than transferring claims about expert-panel ranking to streaming detection. Tadić and Komatina [13] use expert-linguistic interval type-2 risk-factor assessments with DEMATEL weights and MABAC failure ranking; this is a different evidential task from the fixed anomaly-membership pipeline studied here.

2.2 Dynamic Risk Assessment for Process Systems

Dynamic process-safety work considers how monitoring information informs evolving decisions [6]. That broader task may include accident scenarios, consequences, and probabilistic event models. The score studied here uses deviations from fault-free observations; it contains no severity model or calibrated accident likelihood. Dynamic risk aggregation is retained in the experimental label IT2-DRA, but the empirical task is fault detection.

2.3 Data-Driven Monitoring and the Tennessee Eastman Benchmark

The TE problem was introduced as a plant-wide industrial control benchmark [7]. Subsequent process-monitoring work includes canonical variate analysis and dynamic principal component analysis [9], while the monograph of Chiang et al. [8] provides broader fault-detection context. The extended TE dataset supports evaluation beyond the classic operating condition [14]. These sources motivate a multivariate baseline comparison without establishing a universal ranking of monitoring families.

2.4 Positioning of the Present Study

The present study isolates the effect of a width interval within a fixed anomaly pipeline. Its design choices are disclosed, and its type-1 and uniform-weight ablations examine two local modifications. The reported campaign supports a comparison within the classic TE release. It does not establish priority over every fuzzy detection method, validate FMEA risk semantics, or demonstrate a preregistered study; no registration record is supplied.

4.1 Shared Preprocessing

All methods (proposed and baselines) share the same preprocessing. The fault-free training record d00 (500 samples × 52 variables) yields per-variable means μj and standard deviations σj; every incoming sample is standardized and capped at ±10 to limit the influence of gross outliers on all subsequent statistics:

z_jt = clip((x_jt − μ_j) / σ_j, −10, +10), j = 1…52. (1)

Clipping is a disclosed preprocessing choice applied to training and incoming samples. It bounds each standardized coordinate and can alter covariance, reconstruction loss, and tail sensitivity. A clipping sensitivity study is not reported.

4.2 Per-Variable Fuzzy Risk Membership

For each variable j and sample t, the degree to which the variable is “normal” is expressed by a Gaussian primary membership centered at zero deviation with width σ, evaluated on the absolute standardized value:

μ_jt(σ) = exp( − z_jt² / (2σ²) ). (2)

The risk membership is the complement of normality:

r_jt(σ) = 1 − μ_jt(σ) = 1 − exp( − z_jt² / (2σ²) ). (3)

With σ = 1 this is a smooth, bounded, monotone transform of the standardized magnitude: r ≈ 0.39 at |z| = 1, r ≈ 0.86 at |z| = 2, and r → 1 for gross deviations. Each variable therefore contributes a graded, interpretable “risk degree” in [0, 1] rather than a raw quadratic magnitude.

4.3 Interval Width Layer and Midpoint Selection

The width band [0.8, 1.2] is a fixed modelling choice around nominal width 1. Because r_jt(σ) decreases with σ, its interval enclosure is

[rL_jt , rU_jt] = [ r_jt(σ = 1.2) , r_jt(σ = 0.8) ], rU_jt ≥ rL_jt. (4)

With fixed nonnegative weights that sum to one, linear interval arithmetic gives aggregate endpoints R_L,t = Σ_j w_j rL_jt and R_U,t = Σ_j w_j rU_jt. The chosen scalar output is their midpoint:

R_t = ½ ( Σ_j w_j rL_jt + Σ_j w_j rU_jt ). (5)

Eq. (5) defines midpoint selection after endpoint aggregation. It follows from this chosen interval-arithmetic construction, not from a general identity for Karnik–Mendel centroid type reduction [12]. No rule consequents, variable firing-strength ratios, or iterative switching calculation are present. The interval layer therefore changes the score through averaging two width-dependent memberships. We also record the aggregate interval width

W_t = Σ_j w_j ( rU_jt − rL_jt ) (6)

as a sensitivity indicator for the chosen width band. It has no empirically calibrated confidence or epistemic-probability interpretation. Section 6.6 reports its behavior in the benchmark.

4.4 Criticality Weights

The 41 measured variables (indices 1–41) receive weight 1.0 and the 11 manipulated variables (indices 42–52) receive weight 0.6, normalized to sum to one:

a_j = 1.0 for j = 1, …, 41; a_j = 0.6 for j = 42, …, 52; w_j = a_j / (41 + 11 × 0.6), Σ_j w_j =

The rationale is a fixed, disclosed domain-knowledge proxy: Manipulated variables are operator- or controller-controlled moves whose deviations often reflect compensatory action rather than primary process anomalies, so they are down-weighted as evidence of abnormality. This choice is deliberately simple and is itself tested by the uniform-weight ablation IT2-DRA-uw (all wj = 1/52) in Section 7. No weight tuning of any kind was performed on the faulty test data.

4.5 Temporal Layer

Risk evidence is accumulated over time by exponential smoothing with β = 0.2, i.e. 80% weight on the freshest aggregate risk:

S_t = β S_{t−1} + (1 − β) R_t, S_0 = 0. (8)

The recursion assigns 80% weight to the current scalar score; the sum of its geometrically weighted lag indices is β/(1−β) = 0.25 samples. T1-FRA, T², and reconstruction scores use this scalar smoother. MEWMA instead smooths the input vector with current weight λ = 0.2, as specified in Section 5.2. The methods therefore differ in temporal filtering as well as score geometry.

4.6 Alarm Rule and Per-Method Calibration

For each configuration, the threshold is the 99th percentile of its scores on the separate fault-free stream d00_te, after excluding 10 warm-up samples. That record serves as calibration data rather than an untouched evaluation holdout. Sustained detection requires three consecutive above-threshold samples after onset. The reported delay uses the completion sample t* minus 160; its minimum is 3 under this convention. If no sustained detection occurs, the recorded value is 800. The reported false-alarm rate is the fraction of samples 1–160 exceeding the threshold, not the rate of sustained false alarms. Quantile calibration uses fault-free observations only and makes score scales comparable without proving equivalence of evaluation alarm rates.

4.7 Computational and Interpretive Properties

For d variables, IT2-DRA costs O(d) per sample and does not require covariance inversion. Its aggregate is bounded in [0, 1] and decomposes into weighted midpoint memberships. For a single coordinate j, changing only that coordinate changes the instantaneous aggregate by at most wj. This bound describes localized evidence dilution; it does not bound a total stream-mean change when other coordinates also change. Determinism follows from the stated fixed transformations, while operational stability under regime drift remains untested.

5.1 Tennessee Eastman Challenge Data

The TE process is a realistic simulation of an industrial-scale chemical plant (reactor, condenser, separator, stripper, recycle) published by Downs and Vogel [7]. We use the classic curated release of Chiang et al. [8]: fault-free training record d00 (500 samples × 52 variables, stored transposed and loaded accordingly), a fault-free test stream d00_te (960 samples), and 21 faulty test streams d01_te–d21_te (960 samples each) in which the fault is present from sample 161 onward, following the evaluation protocol established by Russell et al. [9] and used consistently since. The 52 variables comprise 41 measured and 11 manipulated variables; the auxiliary faulty-training files of the release were not used, and the source reports that no faulty data was used for fitting or calibration. The data were obtained from the public mirror github.com/qiuruiyu/tennessee-eastman-PCA with reported checks of file shapes (44 files; expected dimensions for all used streams). The faults span step, slow drift, and random-variation disturbances in feeds, temperatures, and valve positions; faults 3, 9, and 15 are known to be notoriously difficult because their effect on the observed variables is minimal [8], [9].

5.2 Baseline Methods

Four baselines were specified. 1) T1-FRA is the type-1 ablation of the proposed method: identical pipeline with σ fixed at 1.0, no interval layer, same weights and smoothing. 2) T2-HOT computes the Hotelling T² statistic of the capped standardized sample against the fault-free training covariance C (ridge 1e−3 added to the diagonal)

T²_t = z_tᵀ (C + 10⁻³ I)⁻¹ z_t. (9)

The scalar T² score uses the same β = 0.2 smoother. MEWMA instead uses y_t = λz_t + (1−λ)y_{t−1}, y_0 = 0, with λ = 0.2. Its reference covariance below is the standard formula for temporally uncorrelated inputs; serially dependent TE observations need not satisfy that covariance identity.

Σ_y(t) = (λ/(2−λ)) (1 − (1−λ)^{2t}) C, Q_t = y_tᵀ Σ_y(t)⁻¹ y_t. (10)

AE-DEEP is a shallow 52→8→52 autoencoder with a ReLU encoder and sigmoid decoder. Capped inputs are mapped to [0, 1] by (z/10 + 1)/2. The source reports training on the fault-free record with Adam, 200 epochs, learning rate 0.01, mini-batch size 64, and seeds 1–5. Scores are scalar-smoothed mean squared reconstruction errors. Each seed has its own fault-free calibration threshold; per-fault metrics are seed averages. Reported thresholds range from 0.006329 to 0.006389 and reported total training time is 0.7 s. This comparator represents one modest reconstruction architecture rather than a tuned deep-learning benchmark.

5.3 Metrics

For each fault f ∈ {1, …, 21} and method m, three metrics are computed on the 960-sample stream. (i) AUC: the rank-based (Mann–Whitney) area under the ROC curve of the risk score for classifying the 800 faulty samples (161–960) against the 160 normal samples (1–160)

AUC = ( R₁ − n₁(n₁+1)/2 ) / ( n₀ n₁ ). (11)

R₁ is the sum of average ranks of the 800 faulty scores among all 960 samples, with n₁ = 800 and n₀ = 160. Detection delay is t*−160, where t* completes the first three-sample above-threshold run wholly within samples 161–960. Undetected streams receive 800. The false-alarm metric counts individual pre-onset threshold exceedances. AUC uses temporally dependent observations; its across-fault dispersion is not a confidence interval.

5.4 Statistical Analysis

The reported analysis uses Friedman tests across 21 faults for the five primary methods and Nemenyi mean-rank comparisons [15]. For k = 5 and n = 21, CD = q₀.₀₅√(k(k+1)/(6n)) = 1.3311 with q₀.₀₅ = 2.728; the six-configuration analysis reports CD = 1.6454. Two-sided Wilcoxon signed-rank comparisons against IT2- DRA use Holm correction over four comparisons per metric. Reported r values use |Z|/√21. Ablation AUC tests are also reported. These are exploratory comparisons over the stated fault set; independent fault replications and a registration record are not supplied.

5.5 Determinism, Seeds, and Software

The source reports deterministic runs for all methods except the autoencoder, whose results average seeds 1–5. Reported software versions are Python 3.12.14, NumPy 2.1.3, SciPy 1.14.1, and Matplotlib 3.9.2. The reported campaign time is about one second. Hardware, timing boundaries, raw per-seed outcomes, and the executable environment are needed to assess reproducibility and runtime comparisons.

6.1 Aggregate Performance

Table 1 reports the aggregate metrics over the 21 faults for all six configurations, and Fig. 1 previews the raw score dynamics for one representative hard fault (fault 4). The proposed IT2-DRA is not competitive with the classical multivariate baselines on this benchmark. Its mean AUC is 0.8099 ± 0.1667, against 0.9183 ± 0.1586 for Hotelling T², 0.9117 ± 0.1553 for MEWMA, and 0.8758 ± 0.1678 for the shallow autoencoder; its mean detection delay is 251.4 samples (median 101) against 98.0 (median 13) for T² and 98.3 (median 15) for MEWMA. The type-1 ablation T1-FRA (0.8105 ± 0.1665, delay 282.3) and the uniform-weight ablation IT2- DRA-uw (0.8122 ± 0.1657) sit within two and a half thousandths of an AUC unit of the proposed method — the changes are small on the AUC scale. Mean false-alarm rates are low for every method (0.0036–0.0054), i.e. roughly 0.6–0.9 false-alarm samples per 160-sample pre-onset window, under the 99th-percentile per-method calibration.

Note (Fig. 1): Dashed lines show method-specific thresholds; the dotted line at index 160 marks the boundary preceding one-based sample 161. Display normalization is not used for fitting or calibration. The multivariate score crosses its threshold promptly, while fuzzy scores remain closer to their thresholds.

6.2 Global Rank Analysis Friedman and Nemenyi

The reported Friedman tests reject equality on AUC (χ²(4) = 29.288, p = 6.83e−06) and capped delay (χ²(4) = 66.327, p = 1.35e−13), and do not reject equality on per-sample false-alarm rate (p = 0.2483). Table 2 gives mean ranks and CD = 1.3311. On AUC, the rank gap from IT2-DRA exceeds CD for MEWMA, T², and AE-DEEP but not T1-FRA. On delay, T² is best ranked and IT2-DRA fourth. Non-rejection for false alarms is not an equivalence test and cannot certify exactly matched operating points.

Fig. 2 shows the per-fault AUC distributions; Fig. 3 shows the six-method Nemenyi critical-difference diagram on AUC (supplementary analysis with the uniform-weight ablation included, k = 6, CD = 1.6454): The three non-fuzzy baselines form the leading group, and the two fuzzy configurations plus the uniform-weight ablation form a trailing cluster. In the six-method analysis MEWMA is significantly ahead of all three fuzzy configurations, and T² and the autoencoder are significantly ahead of IT2-DRA and T1-FRA (the difference to the uniform-weight variant falls just below CD).

Note (Fig. 2): Boxes show medians and interquartile ranges; whiskers show non-outlier ranges, with more extreme observations plotted separately. The non-fuzzy baselines have higher central AUCs.

6.3 Pairwise Comparisons against IT2-DRA Wilcoxon with Holm Correction

Table 3 reports the two-sided Wilcoxon signed-rank tests of each baseline against IT2-DRA with Holm correction and effect sizes. On AUC, the reported paired tests favor every baseline: MEWMA (p_Holm = 7.90e−04, r = 0.75, mean ΔAUC = +0.1018), AE-DEEP (p_Holm = 0.0015, r = 0.71, +0.0659), T2-HOT (p_Holm = 0.0086, r = 0.60, +0.1084), and — notable for the ablation analysis — even the type-1 fuzzy ablation T1-FRA is significantly different (p_Holm = 0.0303, r = 0.47), albeit with a practically negligible mean advantage of +0.0006 AUC. On detection delay, T2-HOT (p_Holm = 7.63e−06, r = 0.87, mean advantage 153.4 samples), MEWMA (p_Holm = 2.63e−04, r = 0.85, 153.0 samples) and AE-DEEP (p_Holm = 2.63e−04, r = 0.85, 121.1 samples) are all significantly faster, with large effect sizes. The approximate delay comparison against T1-FRA gives p_Holm = 0.317, but is uninformative with one nonzero pair: the two fuzzy configurations have identical delays on 20 of 21 faults, the single exception being fault 4 (Section 7).

6.4 Detection Delay

Fig. 4 compares capped IT2-DRA and MEWMA delays. IT2-DRA is slower on 20 faults, with one tie at 800. Median delays are 13 samples for T², 15 for MEWMA, and 101 for IT2-DRA. IT2-DRA records 800 for faults 9, 14, and 19; T1-FRA also records 800 for fault 4. T² has a recorded delay below 800 for every fault, with maximum 733 on fault 9, while MEWMA records 800 only for fault 14. Fig. 1 illustrates the fault-4 threshold-crossing contrast.

Note (Fig. 4): Twenty points are above the diagonal and one is tied at the censoring value 800. The complete title is “below diagonal = IT2-DRA faster (0 below / 1 equal / 20 above)”; its right edge is truncated in the supplied raster.

6.5 False-Alarm Behavior

Fig. 5 shows mean pre-onset exceedance rates with across-fault standard deviations. The configuration means lie between 0.0036 and 0.0054. The reported Friedman test does not reject equality (p = 0.2483), and no Nemenyi rank gap exceeds CD. These observations support broadly similar measured tail behavior in this campaign, with substantial uncertainty; they do not rule out meaningful false-alarm differences. AUC is threshold-independent, while the delay comparison depends on the chosen quantile and persistence rule.

6.6 Per-Fault Analysis and Failure Mechanism

Table 4 lists the per-fault AUC of all six configurations and Table 5 the per-fault detection delays. Three regimes are clearly separated.

On faults 1, 2, 6, 7, 8, 10, 12, 13, 18, and 20, IT2-DRA has AUC ≥ 0.87; five of these exceed 0.99. Broad deviations can therefore be detected well by the aggregate. This per-fault observation does not imply small gaps for every broad-signature fault.

Faults 3 and 9 produce AUCs near chance for all configurations. IT2-DRA has the highest displayed fault-9 AUC (0.4891), and the uniform-weight ablation has the highest fault-3 AUC (0.5677); neither establishes useful detection. Fault 15 is different: Displayed AUCs range from 0.6761 to 0.7543, so it should not be grouped with near-chance results without qualification.

Table 6 reports the largest persistent mean shift for fault 4 as 7.23 training standard deviations in variable 51, with only one variable exceeding 1σ. Its normalized weight is 0.6/47.6 ≈ 0.0126, so its own instantaneous contribution cannot change the aggregate by more than 1.26 percentage points. The reported total stream means nevertheless differ by 0.049 (0.304 before onset and 0.353 after onset); that difference also reflects changes in other coordinates and their distributions. The squared standardized magnitude 7.23² ≈ 52 is illustrative, not the exact contribution to zᵀ(C+10⁻³I)⁻¹z, which depends on covariance and cross terms. T² attains displayed AUC 1.0000 and delay 3 for this fault, against 0.7434 and 151 for IT2-DRA. Low mean shifts on faults 11, 14, 16, and 19 are consistent with changes beyond location, but mean-shift summaries alone cannot distinguish variance, temporal structure, and correlation changes. The fuzzy score has no explicit cross-variable interaction; its distribution can nonetheless respond to some dependence changes. Fault 21 remains intermediate in the reported comparison.

The aggregate width W_t has reported mean 0.1423 before onset and 0.1272 after onset, increasing on only 8 of 21 faults. For the fixed endpoint widths a = 0.8 and b = 1.2, rU−rL = exp(−z²/(2b²))−exp(−z²/(2a²)); it vanishes at z = 0 and as |z| grows. Differentiation gives its maximum at |z| = √[4a²b² log(b/a)/(b²−a²)] ≈ 1.37. Thus large excursions can shrink this width. Neither these summaries nor the chosen band validate W_t as an onset or confidence indicator.

6.7 Summary of Findings

The reported campaign shows lower mean AUC and longer capped delays for IT2-DRA than for the three non-fuzzy baselines. The interval layer changes mean AUC by less than 0.001 and its delay effect is concentrated in one fault. Uniform weighting changes mean AUC by about 0.0023. Bounded single- *Note: A low mean shift does not establish a covariance-only fault mechanism.

7.1 Interval Type-2 Layer IT2-DRA versus T1-FRA

T1-FRA has mean AUC 0.8105 versus 0.8099 for IT2-DRA. The source reports W = 47, p = 0.0303, and r = 0.47 for the comparison, indicating a small negative AUC change for the interval layer. The reported mean delay decreases from 282.3 to 251.4 entirely because fault 4 changes from the censoring value 800 to 151; other delays are identical. The table’s approximate p = 0.3173 for this one-nonzero-pair delay comparison is not informative: an exact two-sided signed-rank test on one nonzero difference has p = 1. These observations assess endpoint-midpoint aggregation, not Karnik–Mendel inference.

7.2 Criticality Weights IT2-DRA versus Uniform Weights

Uniform weighting raises the reported mean AUC from 0.8099 to 0.8122. The source reports W = 67, p = 0.0958, and r = 0.37. This comparison does not establish equivalence or validate the criticality rationale. Fault 4 has the same recorded delay (151) under both weightings, with AUC 0.7434 versus 0.7575. The available ablation is limited to one alternative weight vector.