GT-PSSM: Unified Probabilistic Framework for Stochastic Dynamics Modeling and Dependency Learning in Multivariate Time Series Anomaly Detection

Wonmo Koo; Jaeyeong Lee; Taeseong Yoon; Heeyoung Kim

stat.MLSep 28, 2026

GT-PSSM: Unified Probabilistic Framework for Stochastic Dynamics Modeling and Dependency Learning in Multivariate Time Series Anomaly Detection

Authors: Wonmo Koo, Jaeyeong Lee, Taeseong Yoon, Heeyoung Kim

Organizations: KAIST Daejeon, Republic of Korea

Abstract

Multivariate time series anomaly detection (MTAD) is crucial for ensuring the safe and reliable operation of complex systems. Many existing methods learn normal patterns by training reconstruction or forecasting models on predominantly normal data. However, a large portion of these approaches rely on deterministic models and their associated point-wise output errors for anomaly scoring. Since real-world multivariate time series are inherently stochastic due to measurement noise and intrinsic system randomness, purely error-based scores can be unreliable, as large errors may arise from benign fluctuations rather than true anomalies. Probabilistic approaches address this limitation by quantifying uncertainty in model outputs. In particular, probabilistic state-space models (PSSMs) provide a principled framework by modeling stochastic system dynamics through latent state transitions and measurement noise via emission models. Despite this advantage, existing PSSM-based MTAD methods often struggle to capture long-range temporal dependencies and inter-variable dependencies, as they typically rely on noise-sensitive recurrent architectures and lack explicit cross-variable structure modeling. To address these limitations, we propose Graph-Transformer-Enhanced Probabilistic State-Space Model (GT-PSSM), a novel PSSM-based MTAD method that tightly integrates PSSM-based probabilistic modeling of stochastic dynamics with Graph Transformer-based learning of temporal and inter-variable dependencies. By jointly modeling stochasticity, long-range temporal dependence, and variable interactions within a unified probabilistic framework, GT-PSSM enables more robust anomaly detection.

Figures & tables

Figure 1 . Overview of GT-PSSM and a graphical representation of its encoding and decoding processes at time step $t$ for reconstructing $x_{t}$ . The encoder and decoder correspond to the inference model in Eq.( 14 ) and the emission model in Eq.( 4 ), respectively. “Add” and “Norm” denote residual connections and layer normalization, respectively. Fully described in the text.

Method	WADI		PSM		SMD		SWaT
Method	F1	$\textbf{F1}_{\textbf{\text{PA}}}$	F1	$\textbf{F1}_{\textbf{\text{PA}}}$	F1	$\textbf{F1}_{\textbf{\text{PA}}}$	F1	$\textbf{F1}_{\textbf{\text{PA}}}$
MAD-GAN	0.3415 (0.0446)	0.4601 (0.0752)	0.4539 (0.0306)	0.7140 (0.0908)	0.2027 (0.0539)	0.3823 (0.1052)	0.6126 (0.1137)	0.7046 (0.1081)
USAD	0.5217 (0.0247)	0.6435 (0.0299)	0.4861 (0.0046)	0.7263 (0.0134)	0.6264 (0.0117)	0.9203 (0.0085)	0.7517 (0.0016)	0.8218 (0.0000)
MTAD-GAT	0.5302 (0.0084)	0.8059 (0.0511)	0.5806 (0.0073)	0.8158 (0.0222)	0.5259 (0.0302)	0.9978 (0.0001)	0.7722 (0.0029)	0.8766 (0.0076)
GDN	0.5306 (0.0272)	0.7564 (0.0323)	0.5488 (0.0368)	0.7806 (0.0159)	0.6807 (0.0935)	0.9953 (0.0100)	0.8186 (0.0143)	0.8797 (0.0021)
GTA	0.5165 (0.0079)	0.8333 (0.0047)	0.5398 (0.0110)	0.9670 (0.0039)	0.3793 (0.0104)	0.9996 (0.0002)	0.7768 (0.0085)	0.8954 (0.0062)

Table 1 . Anomaly detection performances of GT-PSSM and the baselines in terms of F1 and

\textbf{F1}_{\text{PA}}

. The baselines are ordered chronologically, with earlier methods listed at the top and more recent ones at the bottom. Bold and underlined values indicate the best and second-best performance, respectively.

Panel of Figure 2 — Figure 2 . Visualization of anomaly scores on WADI produced by GT-PSSM and its variant (“w/o Unc.”). The gray shaded regions indicate ground-truth anomalous segments, while the red horizontal lines denote the anomaly detection threshold. Purple and orange arrows indicate anomalous and normal data points, respectively, for which GT-PSSM and the variant produce contrasting detection outcomes.

Method	WADI	PSM	SMD	SWaT
GT-PSSM	0.7343 (0.0095)	0.6631 (0.0076)	0.7375 (0.0229)	0.8466 (0.0049)
w/o Unc.	0.7055 (0.0209)	0.6450 (0.0066)	0.7115 (0.0315)	0.8435 (0.0036)
w/o Trans.	0.7193 (0.0109)	0.6577 (0.0029)	0.7197 (0.0137)	0.8448 (0.0025)
w/o MHSA	0.6891 (0.0301)	0.6464 (0.0096)	0.7156 (0.0160)	0.8306 (0.0076)
w/o GConv	0.7127 (0.0083)	0.6377 (0.0136)	0.7154 (0.0269)	0.8405 (0.0078)
w/o LRMF	0.7026 (0.0088)	0.6054 (0.0089)	0.6954 (0.0121)	0.8314 (0.0171)

Table 2 . Results of the ablation study evaluated using F1 . Bold and underlined values indicate the best and second-best performance, respectively.

Figure 3 . (a)–(b): Observations (blue line) and mean reconstructions (orange line) for Variables 1 and 3 over a selected portion of the PSM test period. The normal period is highlighted by the green box; (c): Anomaly scores over the same period, together with the threshold that maximizes $\mathbf{F1}$ (red horizontal line); (d): Heat map of the adjacency matrix of the GConv layer (rows: target nodes, columns: neighboring nodes), where lighter colors indicate stronger dependencies. Fully described in the text.

Figure 4 . Average ELBO on training windows, together with $\mathbf{F1}$ and $\mathbf{F1}_{\text{PA}}$ on the test dataset, evaluated every five epochs during training for each dataset. Fully described in the text.

Figure 5 . The results of the sensitivity analysis on WADI. Fully described in the text.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	#Variables	#Train	#Test	%Anomalies
WADI	112	118,800	17,280	6
PSM	25	132,481	87,841	28
SMD	38	28,479	28,479	9
SWaT	49	47,520	44,991	12

Appendix

Table 3 . Statistics of the datasets used in the experiments: number of variables, number of training time steps, number of test time steps, and the ratio of anomalies in the test data.

Dataset	$w$	$d_{h}$	$d_{z}$	$d_{e}$	$d_{s}$	$M$	$\beta$	$L$
WADI	10	256	32	32	32	8	1.0	200
PSM	20	256	16	6	32	8	1.0	200
SMD	30	128	8	6	16	8	1.0	200
SWaT	10	256	32	8	32	8	1.0	200

Appendix

Table 4 . Detailed hyperparameter settings for GT-PSSM.

SMD (Full)
GDN	NPSR	SensitiveHUE	GT-PSSM
0.5209 (0.0064)	0.5233 (0.0114)	0.4250 (0.0084)	0.5272 (0.0113)

Appendix

Table 5 . Anomaly detection performance of GT-PSSM and the selected baselines on the full SMD dataset comprising all 28 machines, evaluated using F1 . We report the mean and standard deviation of the machine-averaged F1 scores across 10 repeated experiments. The value in bold indicates the best performance, and the underlined value denotes the second-best performance.

Dataset	Method	TPE	TPF	Memory Usage
Dataset	Method	(seconds)	(seconds)	(MiB)
WADI	GDN	25.82	0.017	2169
	NPSR	180.27	0.034	1466
	SensitiveHUE	11.25	0.003	134
	GT-PSSM	52.58	0.076	1832
SMD	GDN	3.31	0.001	72

Appendix

Table 6 . Computational cost comparison on WADI and SMD. The values in bold indicate the highest computational costs, while the underlined values denote the second-highest costs.

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Abstract

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