ExCal: Optimization-Free Multivariate Time Series Anomaly Detection via Excess Calibration

Kadir-Kaan Özer; René Ebeling; Markus Enzweiler

cs.LGOct 26, 2026

ExCal: Optimization-Free Multivariate Time Series Anomaly Detection via Excess Calibration

Authors: Kadir-Kaan Özer, René Ebeling, Markus Enzweiler

Organizations: Mercedes-Benz AG, Germany · Institute for Intelligent Systems, Esslingen University of Applied Sciences, Germany

Abstract

We introduce ExCal, a normal-only multivariate time series anomaly detector that combines raw state retrieval, a signal/derivative subspace, and robust statistics across temporal scales. Each expert map receives a high normality floor from the training prefix. During scoring, positive excess is retained, attenuated for broadly firing maps, and combined with a small rank term. The method fits reference memories and projections without gradient optimization or anomaly labels. Prevalence and ranks depend on the complete scored sequence, making inference batch and transductive rather than online. On 180 series from 17 TSB-AD-M families, ExCal obtains AUC-PR $0.421\pm0.37$ and VUS-PR $0.452\pm0.37$ over three seeds, ranking second among 24 methods. Holm-adjusted paired tests detect no significant difference from AxonAD or StreamVAE, which are the two leading non-pretrained methods on TSB-AD-M, on either primary metric, while finding significant positive differences against the other 21 baselines. A separate single-run probe over three series measures median model-wrapper time of $0.0662$ s. Ablations associate the largest primary-metric changes with raw retrieval and excess-magnitude preservation. The evaluated score is decomposable by expert, but its diagnostic utility and online suitability remain untested.

Figures & tables

Panel of Figure 1 — Fig. 1: Schematic expert-map responses to three distinct event motifs. (A) Sustained raw-retrieval excess, (B) localized subspace excess, and (C) intermittent statistics excess. Dashed lines denote map-specific normality floors, and the bottom trace illustrates their possible combination. These are not measured responses and do not establish expert specificity or diagnostic utility.

Panel of Figure 2 — Fig. 2: The ExCal pipeline. The normal prefix supplies normalization, a temporal scale from the data, exemplar memories, a cross-channel subspace, robust statistic references, and per-map normality floors. The dashed branch is implemented and evaluated as an ablation but is disabled in the current base model. The diagram describes the implemented data flow. It does not establish independent branch effects or guarantee a particular detection outcome.

Quantity	Value
Exemplar and memory cap	$10{,}000$ windows
PCA sample cap	$4{,}096$ windows
Distance chunk	$1{,}024$ queries
Nearest neighbors	$k=3$
Top coordinates/channels	$3$
Normality floor	$q_{0.995}$

TABLE II: Fixed implementation constants in the supplied ExCal configuration.

Model	AUC-PR	AUC-ROC	VUS-PR	VUS-ROC	Std-F1	PA-F1	Event-F1	R-F1	Aff-F
AxonAD [ 19 ]	$\mathbf{0.451\pm 0.32}$	$\mathbf{0.828\pm 0.17}$	$\mathbf{0.492\pm 0.32}$	$\mathbf{0.854\pm 0.16}$	$\mathbf{0.534\pm 0.29}$	$0.709\pm 0.31$	$\underline{0.615}\pm 0.33$	$\mathbf{0.483\pm 0.29}$	$0.862\pm 0.13$
ExCal	$\underline{0.421}\pm 0.37$	$0.772\pm 0.24$	$\underline{0.452}\pm 0.37$	$0.794\pm 0.24$	$\underline{0.466}\pm 0.34$	$0.635\pm 0.38$	$0.558\pm 0.38$	$\underline{0.441}\pm 0.30$	$0.856\pm 0.12$
PaAno_PAI [ 23 ]	$0.398\pm 0.34$	$\underline{0.784}\pm 0.21$	$0.404\pm 0.35$	$0.806\pm 0.21$	$0.433\pm 0.31$	$0.773\pm 0.30$	$0.612\pm 0.37$	$0.376\pm 0.27$	$\mathbf{0.870\pm 0.11}$
StreamVAE [ 20 ]	$0.383\pm 0.35$	$0.755\pm 0.24$	$0.424\pm 0.36$	$0.771\pm 0.24$	$0.451\pm 0.34$	$0.533\pm 0.38$	$0.469\pm 0.37$	$0.417\pm 0.32$	$0.830\pm 0.13$
USAD [ 5 ]	$0.363\pm 0.34$	$0.737\pm 0.26$	$0.403\pm 0.35$	$0.765\pm 0.25$	$0.433\pm 0.32$	$0.623\pm 0.35$	$0.519\pm 0.36$	$0.423\pm 0.29$	$0.837\pm 0.13$
CNN [ 24 ]	$0.351\pm 0.35$	$0.774\pm 0.17$	$0.353\pm 0.36$	$\underline{0.807}\pm 0.16$	$0.413\pm 0.33$	$\underline{0.826}\pm 0.27$	$\mathbf{0.645\pm 0.36}$	$0.302\pm 0.24$	$\underline{0.867}\pm 0.12$

TABLE III: TSB-AD-M results on the official M/eval split. Entries are mean

\pm

standard deviation over series results pooled across seeds. The best value in each column is in bold. The second-best value is underlined. All entries have 540 valid observations for the area metrics. Affiliation-F has 537 valid observations for LOF and 528 for Left_STAMPi. Method citations are shown beside the method names and use the supplied bibliography.

Panel of Figure 5 — Fig. 3: Accuracy-latency trade-off across 23 evaluated detectors on TSB-AD-M. The horizontal axis shows the median model-wrapper runtime per time series on a logarithmic scale, and the vertical axis shows VUS-PR. Solid circles denote semisupervised methods, and hollow squares denote unsupervised methods. The dashed gray line indicates the Pareto front. ExCal ( $0.0662\,\text{s}$ , $0.452$ VUS-PR) lies on the Pareto front and executes $99.6\times$ ( $\sim 100\times$ ) faster than AxonAD ( $6.5936\,\text{s}$ , $0.492$ VUS-PR).

Metric	Mean $\pm$ SD	Median	Q1–Q3	Min–Max	Rank / #1 / Top-3
AUC-PR	$0.421\pm 0.37$	$0.344$	$[0.061,0.753]$	$[0.003,1.000]$	$7.75$ / $30$ / $71$
VUS-PR	$0.452\pm 0.37$	$0.411$	$[0.067,0.845]$	$[0.003,1.000]$	$8.09$ / $29$ / $74$

TABLE IV: Per-dataset distribution of current ExCal scores after averaging the three seeds. Ranks are computed against all 24 methods for each dataset.

Baseline	AUC-PR				VUS-PR
	$\Delta$	95% CI	W–L	$p_{\mathrm{Holm}}$	$\Delta$	95% CI	W–L	$p_{\mathrm{Holm}}$
AxonAD [ 19 ]	$-0.031$	$[-0.075,0.014]$	94–86	0.177	$-0.041$	$[-0.084,0.001]$	92–88	0.276
StreamVAE [ 20 ]	$+0.038$	$[0.010,0.067]$	88–92	0.111	$+0.027$	$[0.001,0.055]$	84–96	0.276
PaAno_PAI [ 23 ]	$+0.022$	$[-0.023,0.066]$	106–74	0.042	$+0.047$	$[0.009,0.087]$	106–74	0.016
USAD [ 5 ]	$+0.058$	$[0.023,0.094]$	120–60	0.001	$+0.049$	$[0.014,0.085]$	122–58	0.002
CNN [ 24 ]	$+0.070$	$[0.020,0.120]$	114–66	0.0002	$+0.099$	$[0.055,0.143]$	113–67	$8.94{\times}10^{-5}$

TABLE V: Per-dataset effects for selected baselines.

\Delta

is ExCal minus the baseline after averaging seeds within each dataset. Bootstrap intervals use 10,000 resamples. W–L counts ExCal wins and baseline wins.

Variant	AUC-PR	AUC-ROC	VUS-PR	VUS-ROC	Std-F1	PA-F1	Event-F1	R-F1	Aff-F
Base, shape disabled	$\underline{0.421}\pm 0.37$	$0.772\pm 0.24$	$\mathbf{0.452\pm 0.37}$	$0.794\pm 0.24$	$0.466\pm 0.34$	$0.635\pm 0.38$	$0.558\pm 0.38$	$0.441\pm 0.30$	$0.856\pm 0.12$
With shape expert	$0.415\pm 0.35$	$\mathbf{0.786\pm 0.24}$	$0.446\pm 0.36$	$\mathbf{0.804\pm 0.24}$	$\mathbf{0.469\pm 0.32}$	$0.637\pm 0.38$	$0.559\pm 0.38$	$0.377\pm 0.28$	$0.842\pm 0.13$
Without raw retrieval	$0.376\pm 0.36$	$0.733\pm 0.25$	$0.405\pm 0.37$	$0.763\pm 0.24$	$0.424\pm 0.34$	$\underline{0.687}\pm 0.33$	$0.522\pm 0.38$	$0.397\pm 0.30$	$0.841\pm 0.12$
Without derivative subspace	$\mathbf{0.421\pm 0.37}$	$0.771\pm 0.24$	$0.450\pm 0.37$	$0.790\pm 0.24$	$\underline{0.468}\pm 0.34$	$0.624\pm 0.38$	$0.541\pm 0.38$	$\underline{0.444}\pm 0.31$	$0.853\pm 0.12$
Without multi-scale statistics	$0.403\pm 0.35$	$\underline{0.780}\pm 0.22$	$0.438\pm 0.37$	$\underline{0.801}\pm 0.22$	$0.457\pm 0.32$	$0.667\pm 0.38$	$\mathbf{0.602\pm 0.38}$	$0.413\pm 0.28$	$\mathbf{0.862\pm 0.12}$
Without excess calibration	$0.389\pm 0.34$	$0.759\pm 0.25$	$0.417\pm 0.36$	$0.780\pm 0.25$	$0.432\pm 0.32$	$\mathbf{0.752\pm 0.30}$	$\underline{0.588}\pm 0.37$	$0.403\pm 0.29$	$\underline{0.861}\pm 0.11$

TABLE VI: ExCal component ablations on TSB-AD-M. Entries are mean

\pm

standard deviation over 540 observations across series and seeds. The best value in each column is in bold. The second-best value is underlined.

Variant	AUC-PR		VUS-PR		R-based-F1
	$\Delta$	W–L	$\Delta$	W–L	$\Delta$	W–L
With shape expert	$+0.006$	102–76	$+0.005$	96–81	$+0.064$	114–51
Without raw retrieval	$+0.045$	138–37	$+0.047$	136–39	$+0.044$	117–29
Without derivative subspace	$-0.000$	73–102	$+0.001$	83–89	$-0.004$	62–90
Without multi-scale statistics	$+0.017$	76–104	$+0.014$	70–109	$+0.028$	100–78
Without excess calibration	$+0.032$	118–61	$+0.035$	114–63	$+0.038$	119–58

TABLE VII: Per-dataset ablation effects after averaging the three seeds.

\Delta

is base ExCal minus the ablation. W–L counts base wins and ablation wins, with ties omitted.

Seed	AUC-PR	VUS-PR	AUC-ROC	VUS-ROC
2024	0.421	0.452	0.772	0.794
2025	0.421	0.452	0.772	0.794
2026	0.420	0.451	0.772	0.793
Pooled	0.421	0.452	0.772	0.794

TABLE VIII: Current ExCal mean scores by execution seed.

Family	Series	AUC-PR	VUS-PR	AUC-ROC
CATSv2	5	$0.157\pm 0.13$	$0.107\pm 0.09$	$0.665\pm 0.17$
CreditCard	1	$0.054$	$0.115$	$0.764$
Daphnet	1	$0.407$	$0.425$	$0.944$
Exathlon	25	$0.944\pm 0.12$	$0.915\pm 0.19$	$0.965\pm 0.08$
GECCO	1	$0.274$	$0.046$	$0.750$
GHL	23	$0.013\pm 0.01$	$0.014\pm 0.01$	$0.450\pm 0.24$

TABLE IX: ExCal family-level macro results. The count is the number of unique series. Each series is first averaged over the three execution seeds, and the displayed standard deviation is then computed across series. A dash denotes a family with one series.

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CALAD: Channel-Aware contrastive Learning for multivariate time series Anomaly Detection

Jaehyeop Hong, Youngbum Hur

Department of Industrial Engineering, Inha University, Incheon, Republic of Korea

Contrastive Learning Self-Supervised Time Series Representation Learning Time-Series Anomaly Detection

Open paper AskPDF arXiv

Multivariate time series anomaly detection has become increasingly important in real-world applications, where labeled data are often scarce. Many existing approaches rely on unsupervised learning to model normal patterns, but they often treat all channels equally. This design can dilute anomaly-relevant signals, since not all channels contribute equally to anomaly detection. In this paper, we propose CALAD, a channel-aware contrastive learning framework for multivariate time series anomaly detection. CALAD governs the construction of contrastive samples using estimated channel relevance, allowing the learning process to reflect anomaly semantics rather than generic similarity. Channel relevance is estimated from reconstruction errors of a transformer-based autoencoder and is used to distinguish channels that are more influential to anomalous behaviors. Using this information, we design a channel-wise augmentation strategy in which positive and negative samples are constructed based on whether anomaly-relevant channels are preserved or perturbed. This encourages invariance to changes in irrelevant channels while being sensitive to changes in anomaly-relevant channels. Furthermore, CALAD combines contrastive learning and an auxiliary reconstruction head, allowing the model to learn discriminative representations while retaining normal structures. Experiments on multiple real-world datasets shows that CALAD consistently outperforms existing methods, particularly under distribution shift scenarios. We provide the code for reproducibility at https://github.com/hirundo1218/CALAD

Abstract

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