Towards Universal Representation-Based Process Control

Jinmyeong Choi; Taesup Kim; Artur Dubrawski

cs.LGSep 25, 2026

Towards Universal Representation-Based Process Control

Authors: Jinmyeong Choi, Taesup Kim, Artur Dubrawski

Organizations: Auton Lab, Robotics Institute, Carnegie Mellon University, Pittsburgh, USA · Graduate School of Data Science, Seoul National University, Seoul, Korea

Abstract

Many temporal process learning and monitoring pipelines operate in local windows, making window-level decisions unavoidable in practice. In such settings, classical statistical tests can be applied to individual windows, but they typically evaluate predefined parametric hypotheses-such as unit-root or moment-based conditions-thereby limiting flexibility when reference behavior is defined empirically from task- or domain-specific data. In this work, we view window-level monitoring as a process control problem and reformulate it as reference-based hypothesis testing, where the null hypothesis is specified by an empirical reference distribution rather than a fixed parametric model. We operationalize this perspective through a representation-based, nonparametric framework that combines pretrained time series encoders, kernel density estimation, and conformal calibration, yielding finite-sample valid inference in learned representation space. Classical notions such as stationarity and cyclostationarity arise as natural instantiations of empirical reference sets within this framework. Through experiments, we demonstrate sensitivity to window-level distributional deviations while maintaining well-calibrated inference under stable reference regimes, highlighting the applicability of the proposed approach to a broad class of time series process control and monitoring tasks.

Figures & tables

	AR(1) Reference Regime					Cyclostationary Reference Regime
Method	Compound	Mean	Trend	Unit-root	Variance	Compound	Frequency	Mean	Compound-Structural	Trend	Variance
Proposed	0.993 $\pm$ 0.002	0.992 $\pm$ 0.001	0.992 $\pm$ 0.002	0.988 $\pm$ 0.003	0.988 $\pm$ 0.002	0.980 $\pm$ 0.003	0.987 $\pm$ 0.002	0.974 $\pm$ 0.005	0.989 $\pm$ 0.002	0.967 $\pm$ 0.005	0.982 $\pm$ 0.004
ClaSP (CPD)	0.516 $\pm$ 0.018	0.500 $\pm$ 0.011	0.505 $\pm$ 0.017	0.568 $\pm$ 0.013	0.509 $\pm$ 0.018	0.582 $\pm$ 0.016	0.854 $\pm$ 0.015	0.738 $\pm$ 0.025	0.850 $\pm$ 0.014	0.553 $\pm$ 0.015	0.580 $\pm$ 0.018
CUSUM (CPD)	0.339 $\pm$ 0.009	0.392 $\pm$ 0.008	0.301 $\pm$ 0.008	0.612 $\pm$ 0.011	0.277 $\pm$ 0.002	0.700 $\pm$ 0.011	0.709 $\pm$ 0.009	0.682 $\pm$ 0.010	0.731 $\pm$ 0.005	0.686 $\pm$ 0.013	0.700 $\pm$ 0.014
ADF + DG	0.691 $\pm$ 0.023	0.937 $\pm$ 0.004	0.920 $\pm$ 0.008	0.831 $\pm$ 0.013	0.459 $\pm$ 0.007	0.531 $\pm$ 0.011	0.510 $\pm$ 0.012	0.532 $\pm$ 0.015	0.512 $\pm$ 0.016	0.534 $\pm$ 0.011	0.531 $\pm$ 0.012
KPSS + DG	0.746 $\pm$ 0.017	0.944 $\pm$ 0.003	0.941 $\pm$ 0.005	0.807 $\pm$ 0.009	0.478 $\pm$ 0.008	0.524 $\pm$ 0.013	0.498 $\pm$ 0.013	0.502 $\pm$ 0.009	0.498 $\pm$ 0.011	0.528 $\pm$ 0.007	0.520 $\pm$ 0.010

Table 1: Window-level regime consistency under stationary and cyclostationary reference regimes. AUC (mean

\pm

std over 10 seeds) aggregated across sequence lengths. Normal windows are defined by the reference regime; higher AUC indicates better reference-relative discrimination.

	$\alpha=0.01$		$\alpha=0.05$
Method	TPR $\uparrow$	FPR $\downarrow$	TPR $\uparrow$	FPR $\downarrow$
Proposed	0.984 $\pm$ 0.005	0.006 $\pm$ 0.003	0.998 $\pm$ 0.002	0.063 $\pm$ 0.012
ADF	0.744 $\pm$ 0.008	0.045 $\pm$ 0.006	0.699 $\pm$ 0.009	0.019 $\pm$ 0.005
KPSS	0.669 $\pm$ 0.009	0.018 $\pm$ 0.004	0.729 $\pm$ 0.010	0.091 $\pm$ 0.009
PP	0.526 $\pm$ 0.012	0.029 $\pm$ 0.005	0.456 $\pm$ 0.011	0.005 $\pm$ 0.002

Table 2: Operating-point performance under stationary AR(1) reference regimes. Window-level detection results at fixed false-alarm rates (

\alpha\in\{0.01,0.05\}

), aggregated across all sequence lengths. True positive rate (TPR) and false positive rate (FPR) are reported as mean

\pm

std over 20 random seeds.

Panel of Figure 1 — Figure 1: QQ plots of null p-values under mixed window lengths. Left: Globally pooled calibration appears approximately uniform marginally, but masks systematic length-dependent miscalibration. Middle and right: Length-conditional calibration yields near-uniform p-values for both short and long windows, explaining the stable type-I error control across lengths reported in Table 3 .

seq_len	Type-I error @ $\alpha=0.05$		Max. calibration deviation
	Global	Conditional	Global	Conditional
64	0.0800	0.0436	0.1818	0.0256
96	0.0509	0.0400	0.0800	0.0339
128	0.0400	0.0400	0.0366	0.0458
192	0.0436	0.0400	0.0327	0.0220
256	0.0545	0.0582	0.0691	0.0406

Table 3: Length-wise calibration diagnostics under mixed window lengths. We report the empirical type-I error at

\alpha=0.05

and the maximum deviation from nominal calibration,

\sup_{\alpha}\lvert\widehat{\mathbb{P}}(p\leq\alpha)-\alpha\rvert

, for global pooling and length-conditional calibration.

Rule of Thumbs	Mult.	ROC-AUC	PR-AUC	$\tau_{\alpha=0.05}$	F1 ( $\alpha=0.05$ )	Recall
Scott	0.50	0.9514	0.9670	2.5267	0.9241	0.8882
	1.00	0.9519	0.9672	2.4551	0.9243	0.8905
	1.50	0.9525	0.9676	2.3665	0.9244	0.8923
	2.00	0.9524	0.9679	2.2672	0.9243	0.8936
Silverman	0.50	0.9513	0.9670	2.5371	0.9240	0.8873
	1.00	0.9518	0.9671	2.4725	0.9243	0.8905

Table 4: Ablation study on KDE bandwidth selection. We report performance under different bandwidth rules and scaling factors. Results show stable performance across a wide range of bandwidth choices.

$n_{\text{ref}}$	ROC-AUC	PR-AUC	$\tau$ $(\alpha=0.05)$	F1-Score	Precision	Recall
5,000	0.9816	0.9788	3.3722	0.9353	0.9550	0.9164
10,000	0.9810	0.9745	3.1031	0.9315	0.9551	0.9091
15,000	0.9880	0.9820	3.1457	0.9475	0.9560	0.9391
20,000	0.9859	0.9827	3.1346	0.9399	0.9486	0.9314
25,000	0.9827	0.9799	3.1794	0.9253	0.9541	0.8982
50,000	0.9827	0.9791	3.1555	0.9286	0.9505	0.9077

Table 5: Ablation study on the size of the reference set (

n_{\text{ref}}

). All metrics are reported at a fixed false-alarm rate

\alpha=0.05

using length-conditional calibration. Performance remains stable across a wide range of reference sizes, with both detection accuracy and decision thresholds exhibiting minimal sensitivity to

n_{\text{ref}}

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Discrimination		Detection ( $\alpha=0.05$ )
	ROC-AUC	PR-AUC	F1-Score	Precision	Recall
Proposed	0.952 $\pm$ 0.000	0.967 $\pm$ 0.001	0.924 $\pm$ 0.001	0.961 $\pm$ 0.001	0.890 $\pm$ 0.001
KPSS	0.770 $\pm$ 0.002	0.765 $\pm$ 0.004	0.702 $\pm$ 0.005	0.795 $\pm$ 0.006	0.628 $\pm$ 0.006
ADF	0.742 $\pm$ 0.009	0.753 $\pm$ 0.013	0.686 $\pm$ 0.009	0.709 $\pm$ 0.009	0.664 $\pm$ 0.010
PP	0.741 $\pm$ 0.008	0.754 $\pm$ 0.013	0.684 $\pm$ 0.009	0.707 $\pm$ 0.010	0.663 $\pm$ 0.011

Appendix

Table 6: Fixed-length setting (AR(1) reference regime). Window lengths are fixed per run and results are aggregated across lengths (mean

\pm

std over 8 runs: 8 lengths

\times

1 seed in this ablation).

Method	Discrimination		Detection ( $\alpha=0.05$ )
	ROC-AUC	PR-AUC	F1-Score	Precision	Recall
Proposed	0.952 $\pm$ 0.000	0.967 $\pm$ 0.001	0.924 $\pm$ 0.001	0.961 $\pm$ 0.001	0.890 $\pm$ 0.001
ClaSP	0.658 $\pm$ 0.010	0.650 $\pm$ 0.014	0.478 $\pm$ 0.015	0.695 $\pm$ 0.020	0.364 $\pm$ 0.015
CUSUM	0.609 $\pm$ 0.004	0.613 $\pm$ 0.006	0.497 $\pm$ 0.007	0.598 $\pm$ 0.009	0.425 $\pm$ 0.007
ADF+DG	0.537 $\pm$ 0.004	0.542 $\pm$ 0.005	0.482 $\pm$ 0.005	0.499 $\pm$ 0.006	0.466 $\pm$ 0.004
KPSS+DG	0.524 $\pm$ 0.005	0.530 $\pm$ 0.006	0.484 $\pm$ 0.006	0.499 $\pm$ 0.007	0.469 $\pm$ 0.006

Appendix

Table 7: Fixed-length setting (combined stationary–cyclostationary reference regime). Window lengths are fixed per run and results are aggregated across lengths (mean

\pm

std over 8 runs).

Method		Discrimination		Detection ( $\alpha=0.05$ )
Type	Dim ( $d$ )	ROC-AUC	PR-AUC	F1-Score	Precision	Recall
LDA (Proposed)	1	0.952	0.967	0.924	0.961	0.890
PCA	1	0.842	0.862	0.688	0.919	0.550
	2	0.868	0.890	0.727	0.927	0.598
	4	0.862	0.890	0.727	0.927	0.598
	8	0.871	0.872	0.709	0.918	0.578

Appendix

Table 8: Projection ablation study using the Chronos2 encoder. We compare supervised LDA (

d=1

) with unsupervised PCA and random projections under mixed window lengths. Performance is reported using standard detection metrics (ROC-AUC, PR-AUC, F1-score, precision, and recall), with decision thresholds determined via conformal calibration at

\alpha=0.05

Method	Dim	ROC-AUC	PR-AUC	F1	Precision	Recall
LDA	1	0.8598	0.9001	0.7721	0.9557	0.6477
PCA	1	0.3819	0.4081	0.0000	0.0000	0.0000
PCA	2	0.3598	0.3984	0.0000	0.0000	0.0000
PCA	4	0.5104	0.5504	0.0000	0.0000	0.0000
PCA	8	0.5697	0.6204	0.0063	0.8750	0.0032
Random	1	0.5097	0.5062	0.0345	0.6290	0.0177

Appendix

Table 9: Projection ablation study (MOMENT). We compare supervised LDA with unsupervised PCA and random projections under mixed sequence lengths. Detection metrics (ROC-AUC, PR-AUC, F1, precision, and recall) are computed from window-level decisions, with decision thresholds determined via conformal calibration at

\alpha=0.05

Encoder	ROC-AUC	PR-AUC	F1	Precision	Recall
Chronos2	0.9974	0.9961	0.9664	0.9421	0.9920
MOMENT	0.9968	0.9974	0.9660	0.9515	0.9810
TOTEM	0.0458	0.3176	0.4887	0.3929	0.6460

Appendix

Table 10: Encoder ablation results under View 1 (stationary / cyclostationary reference regimes). Chronos2 and MOMENT produce highly discriminative representations, while TOTEM fails to separate reference and out-of-regime windows.

Encoder	ROC-AUC	PR-AUC	F1	Precision	Recall
Chronos2	0.9859	0.9827	0.9404	0.9478	0.9332
MOMENT	0.8557	0.8836	0.7013	0.9342	0.5614
TOTEM	0.3564	0.3957	0.1755	0.2477	0.1359

Appendix

Table 11: Encoder ablation results under View 2 (cyclostationary deviation settings). Chronos2 remains robust, while MOMENT degrades gracefully and TOTEM performs poorly.

$n_{\text{ref}}$	ROC-AUC	PR-AUC	$\tau$ ( $\alpha=0.05$ )	F1	Precision	Recall
5,000	0.8457	0.8877	2.7547	0.7748	0.9578	0.6505
10,000	0.8598	0.9001	2.8033	0.7721	0.9557	0.6477
15,000	0.8467	0.8898	2.7968	0.7758	0.9542	0.6536
20,000	0.8509	0.8956	2.8516	0.7727	0.9545	0.6491
25,000	0.8369	0.8876	2.8951	0.7652	0.9583	0.6368
50,000	0.8410	0.8875	2.8802	0.7579	0.9532	0.6291

Appendix

Table 12: Sensitivity to the number of reference windows (

n_{\text{ref}}

). Performance is reported under mixed sequence lengths using conformal calibration at

\alpha=0.05

. by MOMENT

	Chronos2 (Proposed)			MOMENT			TOTEM
Seq Len	ROC-AUC	F1-Score	Recall	ROC-AUC	F1-Score	Recall	ROC-AUC	F1-Score	Recall
64	0.952	0.905	0.857	0.908	0.826	0.755	0.452	0.437	0.415
96	0.965	0.923	0.886	0.940	0.865	0.795	0.478	0.530	0.535
128	0.969	0.930	0.897	0.955	0.899	0.860	0.490	0.538	0.546
192	0.972	0.929	0.895	0.960	0.910	0.855	0.465	0.584	0.612
256	0.974	0.937	0.904	0.963	0.907	0.862	0.474	0.589	0.619

Appendix

Table 13: Overall detection performance comparison across different encoders and sequence lengths. Chronos2 demonstrates the best performance, peaking at sequence length 336. TOTEM fails to extract meaningful anomaly features.

Component	Parameter	Range / Description
AR(1) Reference Regime
AR(1)	Autoregressive coefficient $\phi$	$\mathrm{Unif}(-0.9,\,0.9)$
	Mean $\mu$	$\mathrm{Unif}(-0.5,\,0.5)$
	Noise std. $\sigma$	$\mathrm{Unif}(0.02,\,0.30)$
AR(1) Deviations
Mean shift	Shift magnitude	$\mathrm{Unif}(0.2,\,0.6)$ (random sign)

Appendix

Table 14: Parameter ranges used for synthetic data generation. All parameters are independently sampled per window unless stated otherwise. Ranges are chosen to induce diverse but well-controlled stationary and cyclostationary regimes, as well as structured deviations within individual windows.

Component	Configuration
Window length ( seq_len )	Mixed lengths drawn from $\{64,96,128,192,256,336,512,720\}$ , covering short- to long-range temporal contexts.
Window length sampling	Discrete mixture over predefined lengths to reflect heterogeneous monitoring conditions.
Reference regime	Stationary or cyclostationary windows only; no anomalous samples included.
Representation model	Pretrained time-series foundation models used as frozen encoders (Chronos2 ( Ansari et al., 2025 ) as primary backbone).
Fine-tuning	None. Encoders are used without task-specific training.
Projection method	Linear Discriminant Analysis (LDA)

Appendix

Table 15: Implementation details and experimental configuration checklist.

p_{\text{conf}}=\frac{1+\sum_{x_{i}\in\mathcal{R}_{\text{cal}}^{(L_{\text{test}})}}\mathbb{I}[s_{i}\geq s_{\text{test}}]}{|\mathcal{R}_{\text{cal}}^{(L_{\text{test}})}|+1}

Appendix

Algorithm 1 Reference-Relative Consistency Testing (Length-Conditional Conformal)

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Distribution shift detection identifies when data or model behavior changes between reference and deployment settings.

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Time Series Foundation Models for Process Model Forecasting

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Research Center for Information Systems Engineering (LIRIS), KU Leuven, Belgium

Zero-Shot Time Series Forecasting Process Mining Time Series Forecasting Time-Series Foundation Models

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Process Model Forecasting (PMF) aims to predict how the control-flow structure of a process evolves over time by modeling the temporal dynamics of directly-follows (DF) relations, complementing predictive process monitoring that focuses on single-case prefixes. Prior benchmarks show that machine learning and deep learning models provide only modest gains over statistical baselines, mainly due to the sparsity and heterogeneity of the DF time series. We investigate Time Series Foundation Models (TSFMs), large pre-trained models for generic time series, as an alternative for PMF. Using DF time series derived from real-life event logs, we compare zero-shot use of TSFMs, without additional training, with fine-tuned variants adapted on PMF-specific data. TSFMs generally achieve lower forecasting errors (MAE and RMSE) than traditional and specialized models trained from scratch on the same logs, indicating effective transfer of temporal structure from non-process domains. While fine-tuning can further improve accuracy, the gains are often small and may disappear on smaller or more complex datasets, so zero-shot use remains a strong default. Our study highlights the generalization capability and data efficiency of TSFMs for process-related time series and, to the best of our knowledge, provides the first systematic evaluation of temporal foundation models for PMF.

Abstract

Distribution Shift Detection

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