Complexity- and Statistics-Guided Anomaly Detection in Time Series Foundation Models
Authors: Jongwon Kim
Samuel Yoon
Young Myoung Ko
Yerin Kim
Sung Il Kim
JAEUNG TAE
Venue: ICLR 2026 Poster (2026)
Keywords: Timeseries anomaly detection
Timeseries foundation model
Reconstruction based anomaly detection
Primary area: learning on time series and dynamical systems
TL;DR: We propose solutions based on a complexity measure α that captures high-frequency complexity and restores statistical features removed by RevIN, leading to theoretical and empirical improvements in anomaly detection.

Abstract: This paper introduces a methodology for anomaly detection in time series using Time Series Foundation Models (TFMs). While TFMs have achieved strong success in forecasting, their role in anomaly detection remains underexplored. We identify two key challenges when applying TFMs to reconstruction-based anomaly detection and propose solutions. 

The first challenge is overgeneralization, where TFMs reconstruct both normal and abnormal data with similar accuracy, masking true anomalies. We find that this effect often occurs in data with strong low-frequency components. To address it, we propose a complexity metric, $\alpha$, that reflects how difficult the data is for TFMs and design a Complexity-Aware Ensemble (CAE) that adaptively balances TFMs with a statistical model. 

The second challenge is overstationarization, caused by instance normalization layers that improve forecasting accuracy but remove essential statistical features such as mean and variance, which are critical for anomaly detection. We resolve this by reintroducing these features into the reconstruction process without retraining the TFMs. 

Experiments on 23 univariate and 17 multivariate benchmark datasets demonstrate that our method significantly outperforms both deep learning and statistical baselines. Furthermore, we show that our complexity-based metric, $\alpha$, provides a theoretical foundation for improved anomaly detection, and we briefly explore prediction-based anomaly detection using TFMs.

Published as a conference paper at ICLR 2026
COMPLEXITY- AND STATISTICS-GUDDE ANOMALY DETECTION IN TIME SERIES FOUNDATION MODELS
Jongwon Kim
Samuel Yoon' Young Myoung Ko
Yerin Kim?
Sung Il Kim
Jaeung Tae²
I POSTECH, (pioneer0517, muel2, youngko @postech. ac. kr
SS hynix Inc., (yerinl.. .imm, sungi12. kim, jaeung. tae (ssk. com
ABSTRACT
This paper introduces a methodology for anomaly detection in time series using Time Series Foundation Models (TFMs). While TFMs have achieved strong success in forecasting, their role in anomaly detection remains underexplored. We identify two key challenges when applying TFMs to reconstruction-bast anomaly detection and propose solutions. The first challenge is overgeneralization, where TFMs reconstruct both normal and abnormal data with similar accuracy, masking true anomalies. We find that this effect often occurs data with strong low-frequency components. To address it, we propose complexity metric, o, that reflects how difficult the data for TFMs and design a Complexity -ware Ensemble (CAE) that adaptively balances TFMs with statistical model. The second challenge is overstationarizztion, caused by instance normalization layers that improve forecasting accuracy but remove essential statistical features such as mean and variance, which are critical for anomaly detection. We resolve this by reintroducing these features into the reconstruction process without retraining the TFMs. Experiments on 23 univariate and 17 multivariate benchmark datasets demonstrate that our method significantly outperforms both deep learning and statistical baselines. Furthermore, we show that our complexity-basee metric, provides theoretical foundation for improved anomaly detection, and we briefly explore prediction-based anomaly detection using TFMs.
KEYWORDS
Time series anomaly detection, Time series foundation model
INTRODUCTION
Anomaly detection identifies events points that deviate from expected patterns and is context dependent (Chandola et al., 2009). time series, it plays a vital role in industrial monitoring (Zheng et al. 2017). healthcare (Kiranyaz et al. 2015). safety systems (Giraldo al., 2018), and finance (Ahmed et al., (0166). The emergence of foundation models, inspired by the success of large language models (LLMs), has opened new opportunities for time-series analysis. Recent advances in time-series foundation models (TFMs), whether LLM based or independently designed, have
demonstrated strong predictive capabilities (Wu et al. 2023 Gao al. 2024 Zhou et al. 2023 Woo
et al. 2024; Goswami et al., [2224] [Liang et al. 2024 Ansari et al.) 2024; Shi et al. [025] Liu et al.
2025). However, despite their superior performance on prediction tasks, applications of TFMs to anomaly detection remain limited.
This study addresses two primary limitations TFMS for time-series anomaly detection in reconstruc- tion settings. In reconstruction-bat based anomaly detection, anomalies are identified via reconstruction error, with larger errors indicating anomalous points. The first limitation, overgeneralization. occurs when a model reconstructs both normal and anomalous data with similarly high accuracy, yielding low anomaly scores even for truly anomalous cases. Prior work has largely attributed this behavior
"orresponning author.

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VUS-PR Scores for Ensemble Methods between TFM and statistical model
0.50
Nethod
standa one
Rancom Selection
(hoposee)
CAE por
VIIS.PR
TFM baselines
0.05
HOMENT
0.20
Sub-FFo
SA
Sub-PCA
POLY
Statistical
Figure 1: VUS-PR scores for five statistical models (Sub-IForest (Liu et al. 2008). SR Ren et al. 2019), Sub-HBOS (Goldstein & Dengel 2012), Sub-PCA Aggarwal & Aggarwal 2017) and POLY Li et al. 2007)) combined with MOMENT, time-series foundation model. Within each model group, bars indicate the standalone statistical method and four ensemble strategies: Random Selection, CAS, CAE (Proposed), and CAE per data (our model variants). The green and blue dashed lines represent the MOMENT-Stat and fine-tuned MOMENT (FT) baselines, respectively.
to high model capacity while paying less attention to the role of data complexity. We show that overgeneralization becomes more pronounced on simpler datasets, for example those dominated by low-frequency structure, where the capacity of TFMs enables easy reconstruction of anomalies (Gong et al. 2019 Park et al. 2020; Wang et al. 2022). The second limitation, overstationarization, arises when instance- normalizaion layers remove critical distributional characteristics such as mean and variance. Although instance normalization improves predictive performance (Kim et al., 2021 Liu ct al. 2022). can hinder anomaly detection.
To overcome these limitations, we propose two complementary approaches. First, we introduce complexity-awar ensemble (CAF) that uses a complexity metric a, derived from the TFM output, to adaptively balance TFMs with simpler statistical models based on dataset complexity. This mechanism prefers statistical methods for low-complexity data and TFMs for high- high-complexity data. We theoretically show that is sensitive high-frequency content and, under moderate assumptions, that larger a provides a provable guarantee of greater separation between normal and anomalous scores. Second, we improve reconstruction by explicitly reintroducing instance-level mean and variance into the encoder output, thereby mitigating overstationarization without retraining the TFM. Our experiments across 23 univariate benchmarks demonstrate the effectiveness of both approaches.
Figure [ shows that CAE outperforms the MOMENT-SOE Stat baseline, standalone models, and alter- TFM (MOMENT-Stal
native ensemble strategies. Moreover, the overstationariizaion-ccooe achieves higher anomaly-detection performance than fine-tuned MOMENT. By excluding the poorly performing Sub-HBOS variant, the CAE over the remaining four statistical models further surpasses the previous state-of offte-of-the the-hhe-rt Sub-PCA score of 0.4233 as well as the VUS-PR of 0.3857 achieved by the vanilla TFM, while our proposed method attains a score 0.4679.
We also evaluate forecasting-orienter TFMs for anomaly detection by interpreting higher forecasting errors as anomalies. While forecasting models have traditionally underperformed due to limited predictive accuracy, recent TFMs motivate a reexamination of this paradigm. In addition, many modern TFM architectures, often inspired by PSt do not explicitly model inter-channel dependencies despite strong multivariate forecasting performance. We evaluate these models on multivariate amomaly-detection tasks and show that capturing relationships among channels is critical for effective detection.
The remainder of the paper is organized as follows: Section 2 reviews related work, Section details the proposed methodology, Section 4 presents experimental results, and Section concludes.
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2
RELATED WORK
Prior work has investigated a variety of methods to mitigate overgeneralizatioo and overstationariza- models reconstruct both normal
tion in time-series anomaly detection. Overgeneralization- and anomalous data with similarly high aceuracy-has been addressed by several strategies. Notable
approaches include limiting the expressive capacity of decoders (Wang et al., 2022 Song et al. 2023
Goswami et al., 2024), using memory-base -bsed models (Song et al. |023] Liu et al. (2024). applying
contrastive learning with synthetic anomalies (Audibert al. |020:] Sun Sun et al.] 2024 Wang et al.)
2023 Liu et al.
2024), and adding auxiliary scoring mechanisms beyond plain reconstruction error
(Su et al. 2019
Zhao et al. 2020 Tuli al., 2022 Xu et al. 2022 Song al. 2023).
Each strategy has limitations. Reducing decoder capacity does not eliminate overgeneralization even when the decoder is single fully connected layer (Goswami et al.) 2024). Memory-based methods typically require retraining and can be computationally expensive for large TFMs. Contrastive schemes struggle to produce meaningful synthetic anomalies because definitions of anomalies are dataset dependent; generic perturbations or noise may not reflect true anomalies. Finally, introducing auxiliary scores increases sensitivity to the choice and weighting of components. In contrast, we integrate statistical- model score and adaptively set the weights from data characteristics, which reduces manual tuning.
Overstationarization arises when instance-normalizznee normalization or RMSNorm layers remove essential statistics, such as mean and variance, from each input segment. These layers are common TFMs (Zhou et al., 2023: Goswami et al., 2024; Woo et al.
2024 Liang et al. 2024 Ansari et al. 2024
Gao et al. [2024] Shi et al. 2025; Liu et al. 2025) because they improve predictive accuracy, yet they can hinder anomaly detection. Prior work (Kim et al. 2021 Liu al. 2022) shows that re-injecting
the lost statisties can improve forecasting. However, this idea has been less explored for anomaly detection, and naive implementations may require costly retraining. A practical alternative is to append per-instance statistics directly to token embeddings, which supplies distributional cues without additional training.
In this study, we use MOMENT (Goswami et al. 2024 as our reconstruction-bant based anomaly detection model, and Chronos (Ansari al. 2024), Moirai (Woo al. 2024), and TimeMoE (Shi al. 2025 as forecasting- -based models. MOMENT provides released parameters and explicit support for anomaly detection. Other TFMs that support anomaly detection, including One-fits-all (Zhou et al.) 2023), TimesNet (Wu et al., 2023), and TimeMixer++ (Wang et al. 2025), do not release trained weights. MOMENT employs Transformer architecture trained with masked imputation and has shown strong performance among TFMs (Liu & Paparrizos 2024).
Forecasting- anomaly detection has traditionally underperformed reconstruction-bast methods because prediction errors on normal data can be substantial (Zamanzadeh Darban et al. [024]] Schmidl et al. 2022). Recent TFMs have improved forecasting quality; for example, PatchTST attains state-of-the-a results by treating each channel segment an individual token while ignoring cross-channel relations (Nie et al.
2023) Whether higher forecasting accuracy translates into better anomaly detection, remains open question. We address this gap through empirical evaluation.
recent review (Zhang et al. 2024) surveys efforts to apply LLMs to time series, while subsequent studies report limited gains in forecasting (Tan et al. 2024) and anomaly detection (Zhou & Yu 2025).
We therefore focus on time-series foundation models rather than
LLMs.
3 METHOD
3.1 ANOMALY DETECTION USING A TIME-SERIES FOUNDATION MODEL
We consider time series E RT with instance normalization such that ||>||| = T. We employ pre-trained Time Series Foundation Model (TFM) consisting of an encoder E R - linear decoder D: d T Given an input and binary mask M {0, 1) (where 0 indicates masked), the model output is ((M) D(E O M)).
e and a
In the standard reconstruction-bant based anomaly detection setting, utilize types of errors:
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Reconstruction Error (Crec): The computed when the model reconstructs the input with minimal or no masking (test-time configuration). Let Mies be the test mask (often all ones or boundary masked).
Tre (x) = | - - besa
Imputation Error (Limp): The expected error when the model reconstructs the input under a random masking scheme used during pre-training (e.g.. 30% masking). Let M be the distribution of random masks.
Cimp (a) EM~M [|| 2 (
The anomaly score for a test instance is typically defined based on (T).
3.2 COMPLEXITY-WAAR ENSEMBLE AND THEORETICAL GUARANTHES
We propose Complexity-Aware Ensemble (CAE) that dynamically balances the TFM-based score with a statistical model score. The core motivation is that TFMs tend to overgeneralize on low- complexity data (where Limp Lrec), masking anomalies.
Complexity Metric a. We define the raw complexity score w(x) as the normalized difference between the imputation error (hard task) and the reconstruction error (casy task):
Limp(x) (x) Lree
w (x)
||0:1
To handle varying scales across datasets, we compute the final complexity metric a [0, by applying Quantile Transform to w(x) over the dataset:
a DuuntiileTrasffommuufus
A larger implies that the model struggles significantly more with partial masking, indicating complex, unpredictable patterns. The final ensemble score given by:
CAE a TFM (1 a) stac
We now provide theoretical justifications connecting a to the spectral properties of the data and the anomaly detection gap. Detailed proofs are provided in Appendix A.
Theorem (a reflects High-Frequenc Energy). Let Φ(k) be the energy of the signal x at Haar wavelet scale h (where higher corresponds to higher frequencies). The raw complexity gap = Limp Lree decomposes as:
A A o(k)bb bk.
where by 2 0 is scale-dependent coefficient. Under standard assumptions on TFM masking, by is non-decreasing with respect to k.
Interpretation. Since by increases with k the gap A (and thus a) is a weighted sum of spectral energies where high-frequency components are weighted more heavily. Therefore, a higher indicates that the time series contains significant high-frequency detail, which is harder the TFM to interpolate from context.
Theorem (Gap Growth with High-Frequency Complexity). Consider gradient-based update of the decoder parameters W to minimize the reconstruction loss on normal data Lx Let La(W) be the loss on anomalous data, and let denote the training step index. Under the assumption that gradients for normal and anomalous data are not aligned, the change in the anomaly gap Agap LA Ln after small step p satisfies:
Agap (W++1) Agap (W.) 2 p (WW)
Furthermore, the gradient norm V Lx (W..)l is stricily increasing with the high-frequency information share of the normal data.
Interpretation. Theorem 2 suggests that for high-complexity data (large a),, the model's optimization landscape is steeper (Il (| LN I1 is large). This stems from the property that TFMs tend to preserve low-frequ www-freuunccy components while struggling with high-frequency details (Assumption Consequently, this leads to a larger separation (Agap) between normal and anomalous scores during the learning process.
2 in Appendix).
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3.3 PROPOSED ENHANCEMENT! INSTANCE STATISTICS AUGMENTATION
The second challenge, overstationarization, arises from instance normalization layers (e.g., RevIN, RMSNorm) inherent in TFMs. While these layers stabilize training, they discard first- and second- order statistics (mean and standard deviation r). We formalize the limitation of this discarding process follows:
Lemma 3 (Affine Invariance under Instance Normalization). Let (xx) (x p2)/ be the instance normalization, and let f(z) D(E(N(z))) be the reconstruction model composed of an encoder E and decoder D. For any input time series and its affine transformation x' oz (with a > 0), the normalized imputs are identical: N (x') Nxx). Consequently,
- (C)=D((BN(())) - = f(x).
This implies that the model f is theoretically invariant to shifts in mean and variance, making incapable of distinguishing statistical anomalies from normal data.
To resolve this, we propose MOMENT-Stat, which explicitly re-injects these statistics into the reconstruction process without retraining the massive encoder. Formally, let F; (x)) be the feature representation of the normalized input. We augment the decoding process by directly concatenating the instance statistics the feature vector before the linear decoder D:
i; ReVIN (D(E;;N:))));
This simple modification ensures that the reconstruction error free is sensitive to abnormal shifts in and a, significantly boosting performance on datasets where anomalies are statistical outliers rather than pattern violations.
4 EXPERIMENT
We evaluate our method on the univariate TSB-AD-U benchmark, which contains 23 datasets from diverse domains such as web services, stock trading, healthcare, and industrial systems. Several univariate series were extracted from multivariate sources by selecting informative channels based on correlation and anomaly-detection performance. Results are reported as averages across datasets, following benchmark procedures in (Liu & Paparrizos 2024 Detailed dataset descriptions are provided in the appendix.
MOMENT trained with learning rate 10 4 for epochs using window size of 256 and the Adam optimizer. Training data for each dataset consisted of the first 25% of the time span or data until the first anomaly. The reconstruction head was single fully connected layer with SiLU activation and dropout rate 0.1.
4.1 EVALUATION METRIC
Our primary metric is the Volume Under the Precision-Recul Surface (VUS-PR), as used in recent benchmarks (Paparrizos et al., 2022, Liu & Paparrizos 2024 Goswami et al. 2024, Yang et al.) 2023). VUS-PR ranges from to 1, where larger values indicate stronger detection performance. Unlike threshold based scores such as adjusted F,, which may fluctuate due to arbitrary cutoffs misalignments, VUS-PR remains stable and threshold-fret We also report complementary threshold iiddddddddeedemem metrics: AUC-PR, AUC-ROC, and VUS-ROC. Full metric details appear in the appendix.
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4.2 STATISTICAL MODELING AND TIME COMPLEXITY
Among 25 baseline detectors (Liu & Paparrizos 2024h. we focus on statistical models with time complexity not exceeding (NN N. These include SR (Ren al.
Table 1: Time complexity of selected models.
2019), Sub-PCA (Aggarwal & Aggarwal 2017), Sub- HBOS (Goldstein & Dengel (0012) Fub-sForest (Liu al. 2008), and POLY (Li al.) 2007). Their asymptotic complexities are summarized Table I where N the number of points, d the number of channels, / the number of trees, and the polynomial order. The runtime of CAE equals the of the individual statistical models plus the cost of computing a. MOMENT-SEN adds negligible runtime compared to fine-tuned MOMENT. Empirical timings are given in the appendix.
Model
Time Complexity
SR
O(N log N) O(N + +dd) O(Nd) O(tN log O(Np" +p")
Sub-PCA
Sub-HIBOS
Sub-IForest
POLY
4.3 OVERGENERALIZATTION IN TEM RECONSTRUCTION
Figure 2 illustrates MOMENT's tendency to overgeneralize by reconstructing normal and anomalous data similarly, especially on lower-complexity datasets. We analyze three synthetic craasets-linees trends. quadratic functions, and Brownian motions-whie 0.1034, and 0.4053, respectively. While PCA shows clearer gaps on low-complexity series but struggles on complex signals, MOMENT sometimes fails to separate anomalies in simple datasets. CAE mitigates both issues by combining MOMENT-Sta with statistical model according dataset complexity.
average complexities ā are 0.0660,
Original
MOMENT (TRM)
PCA modeli 100 0%
CAE (ouns) (Acc: 100
14
(a) Trend timesriee (Q.0.0060)
Origial Timesories
(TRM) 76.0%)
PCA (stat. model 0%)
(ours) (Ace: 85.05)
(D) Quadralic timeserie (a-0.) (G.4.1004)
Original tmesorios
MOMENT (TFM) (Aoe: 100.0%)
PCA (Sta). model (Aoo: 0%00
CAE joun) (Ace: 100 100.0%
3
(4) Complex Smeseries (a-0 4053)
Figure 2: Three datasets are shown: (a) linear trends, (b) quadratic curves, and (c) complex Brownian- style signals with complexity measure In each row, the leftmost panel displays the original training and anomalous test series. The second panels compare histograms of TFM reconstruction-basee anomaly scores (dashed line = decision threshold). The next panels show PCA-based scores. Finally, our CAE ensemble clearly separates normal and anomalous samples.

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4.4 PERFORMANCE OF INSTANCE STATISTICS AUGMENTATION
To address overstationarization, MOMENT-Stat incorporates mean and variance into the reconstruc- tion phase. Table 2 compares MOMENT-Stal with fine-tuned MOMENT model (MOMENT (FT)) and zero-shot variant (MOMENT (ZS)) using a range of threshold free metrics. Ranks are computed across 25 baselines in the benchmark suite. We include the global rank for reference even subset of methods is displayed in the table.
MOMENT-Stat attains VUS-PR of 0.3913, outperforming MOMENT (FT) (0.3857) and MOMENT (ZS) (0.3790). This result suggests that reintroducing key statistical attributes (mean and variance) improves anomaly detection.
Table 2: Performance metrics comparison.
Method
AUC-PR
AUC-ROC
VUS-PR VUS-ROC
VUS-PR Rank
Sub-PCA
0.3700
0.7100
0.4233
0.7600
MOMENT-Stal
0,3040
0.7146
0.3913
0.7771
MOMENT (FT)
0.3000
0.6900
0.3857
0.7600
MOMENT (ZS)
0.3000
0.6800
0.3790
0.7500
4.5 PERFORMANCE EVALUATION OF COMPLEXITY- ENSEMBLE
CAE fuses MOMENT- Stat with statistical model to address overgeneralization. Table shows VUS-PR scores for several models under different values of a. a means only the statistical model is used, and a means only the MOMENT-based model is used. Intermediate a values form various weighted combinations.
CAE improves VUS-PR for most statistical backbones. An exception occurs with Sub-HBOS, where the MOMENT-only setting (a = 1) remains stronger than CAE This is because the standalone performance of Sub-HBOS is substantially lower than that of MOMENT-Stat (0.2283 vs. 0.3913), so the ensemble gains little from the statistical component; in such cases, relying on a single strong model is preferable to blending with weak Nevertheless, CAE still improves markedly the standalone Sub-HBOS (0.3734 vs. 0.2283, rank 28-33.. For example, Sub-PCA alone achieves VUS-PR of 0.4233 the highest among existing baselines aaseinns-whiih which increases 0.4679 when integrated within the CAE framework.
Table 3: VUS- SR- scores for various models under different settings, along with their rank changes relative to the 25 baselines in Table Rank change indicates the position at and the new position after applying CAE.
Model
0
a = 0.1
0.3
& = 0.5
0.7
CAE
Rank change
Sub-IForest
0.2230
0.3685
0.3903
0.4155
0,4194
0.3913
0.4318
29. +
SR
0.3237
0,4006
0.4342
0.4376
0,4129
0.3913
0.4596
14 +
Sub-HBOS
0,2283
0.2522
0.3126
0.3688
0.3812
0.3913
0.3734
28 + 3
Sub-PCA
0.4233
0.4405
0.4560
0.4557
0.4448
0.3913
0.4679
1
POLY 0.3897
0.4004
0.4139
0.4170
0.4127
0.3913
0.4274
3 3
Table 3 shows that CAE consistently raises baseline models to the top of the benchmark. For example, SR improves from 14th to 1st, and Sub-IForest from 29th to 2nd. Sub-PCA remains 1st, while POLY and Sub-HBOS both move into the top three. Overall, CAE markedly boosts the ranking of statistical backbones.
4.6 COMPARISON WITH OTHER COMPLEXITY MEASURES
We also tested the other Lime-seress complexity measures: Spectral Entropy (Inouye et al. 1991), Approximate Entropy (Pincus 1991). and Sample Entropy (Richman & Moorman, 2000) for CAE. Table 4 reports the VUS-PR values for CAE under different complexity measures. For instance, the Sub-PCA model achieves VUS-PR value of 0.4679 when using the proposed complexity
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measure, which is higher than the VUS-PR values obtained when using the other complexity metrics. This indicates that our proposed measure leads to better anomaly detection performance across the evaluated models.
Table 4: VUS-PR Values for CAE with MOMENT-Stat under different complexity measures.
Model Name
App Entropy
Sample Entropy
Spectral Entropy
Proposed Complexity
Sub-IFores,
0.3896
0.3928
0.3912
0.4318
SR
0.3928
0.3921
0,4515
0.4596
Sub-HBOS
0,3554
0,3548
0.3835
0.3734
Sub-PCA
0.4373
0.4409
0.4288
0.4679
POLY
0.3993
0.3987
0.4153
0.4274
4.7 EVALUATION OF COMPLEXITY-AWARE SELECTION
Model selection based on dataset complexity was also examined as an extreme case of ensemble weighting, where single model receives full weight. Specifically, MOMENT-Stat was chosen when the dataset-level complexity ã; exceeded the median complexity of that dataset; otherwise, a statistical model was used. As baseline, models were also selected at random with equal probability. In addition, we report a variant denoted as CAE per data, which replaces the dataset-level complexity a; with the instance-level complexity oss 5 shows that CAS improves VUS-PR scores compared to both the standalone and random-selection approaches. For example, Sub-PCA under CAS achieves a VUS-PR of 0.4400, outperforming vanilla Sub-PCA (0.4233) and random selection (0.4073). However, although CAS yields notable gains over simpler methods, CAE consistently achieves higher VUS-PR scores than CAS.
Table 5: Combined VUS-PR scores for merged models.
Model
Vanilla
Random Selection
CAS
CAE
CAE per data
Sub-IForest
0.2230
0.3071
0,3092
0.4318
0.4156
0.3237
0.3575
0.3764
0.4596
0.4414
Sub-HBOS
0,2283
0,3097
0.3182
0.3734
0.3331
Sub-PCA
0.4233
0.4073
0.4400
0.4679
0.4582
POLY
0.3897
0.3905
0.3973
0.4274
0.4146
4.8 FORECASTING-ORIENT TFM FOR ANOMALY DETECTION
Many prior studies have observed that reconstruction-basse anomaly detectors generally outperform forecasting-base models, primarily because limited prediction accuracy results high errors even for normal sequences. However, the emergence of large-scale TFMs has significantly enhanced forecasting capabilities.
Table compares forecasting-oriented TFMs-Chronee (Ansari al. 2024). TimeMoE (hi al.
2025), and Moirai (Woo et al.) 2024 under zero-shot anomaly detection setting. Each model's anomaly detection performance metrics (AUC-PR, AUC-ROC, VUS-PR, VUS-ROC) and forecasting error (sMAPH) were evaluated individually. Here, the sMAPE is defined as sMAPE(r, à)
# - 2 2 by TomeMoF. Moirai, despite higher forecasting errors, shows notably lower anomaly detection performance.
E - Chronos variants consistently achieve the best performance, followed closely
Additionally, Figure] visually illustrates the relationship between forecasting accuracy (mean sMAPE) and anomaly detection performance (mean VUS-PR) for each model variant. A clear inverse relationship between forecasting error and anomaly detection performance is observed, suggesting that models with higher forecasting accuracy (lower sMAPE) generally yield better anomaly detection performance. This supports the hypothesis that improvements in predictive accuracy directly contribute enhanced forecasting-baase anomaly detection.
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Model-wise Mean VUS-PR vs Mean SMAPE
Model
0.47
Chronou bolt base boit
0.40
0.88
Nanaewmral 85x 3.52
VUS
0.32
0.30
12755
Mean sMAPE
Figure 3: Relationship between forecasting accuracy (sMAPE) and anomaly detection performance
(VUS-PR) for forecasting-ories TFMs.
Table 6: Forecasting- anomaly detection performance under rero-shot setting.
Model
AUC-PR
AUC-ROC
VUS-PR
VUS-ROC
sMAPE
Mhronos-bolt-bant
0.3901
0.7416
0.4160
0,8131
0.1063
Chronos-bold bootssllllttt
0.4057
0.7478
0.4172
0.8169
0.1073
TimeMoE-larr
0.3725
0.7224
0.3954
0.7980
0.1562
TimcMooE-aas
0.3766
0.7236
0,3986
0.7971
0.1560
Moirai-base
0.2361
0.6914
0.2775
0.7865
0.2826
Moirai-small 0.2471
0.7056
0.2869
0.7916
0.2584
4.9 PERFORMANCE ON MULTIVARIATE TIME SERIES
We evaluated our framework on six multivariate datasets (MSL, SMD, SMAP, SWaT, Genesis, TAO). A limitation of MOMENT is its
independencee assumption, which treats multivariate series as independent univariate channels, ignoring inter-channel dependencies. Consequently, simply averaging TFM and statistical scores (Naive Ensemble, compared to using Statistical Models alone (a = 0), as shown in Table (e.g., PCA VUS-PR drops from 0.3878 to 0.3132).
0.5) often degrades performance
In contrast, our Complexity-Awee Aware Ensemble (CAE) consistently achieves higher YUS-PR than the naive approach, validating the effectiveness of adaptive weighting. By adaptively regulating the TFM contribution based on data complexity, CAE effectively mitigates the drawbacks channel- independence. Notably, CAE significantly improves VUS-PR for HBOS (0.175) - 0.2535) and LOF (0.1091 + 0.2122), demonstrating that an intelligent weighting strategy is crucial for robust multivariate detection.
In contrast to the reconstruction experiments focused on a representative subset, Table reports zero-shot anomaly detection results across the entire TSB-AD-M benchmark (17 datasets) for forecasting- based TFMs (Moirai, TimeMoE, Chronos) alongside reconstruction-bee based MOMENT. Among these models, only Moirai explicitly incorporates multi-channel information. Although Moirai uses flattened multivariate sequence format similar to PatchTST, it uniquely constructs attention matrices considering channel-specific information. Notably, Moirai shows relatively low performance on univariate datasets but demonstrates significantly improved performance in multivariate scenarios. These results highlight the necessity of explicitly modeling ihter-channel relationships in multivariate time-sert anomaly detection, requirement less critical in forecasting tasks. Rankings computed based on VUS-PR, where lower values indicate superior performance relative to other baseline models.

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Table Performance comparison on multivariate datasets.
VUS-PR
VUS-ROC
Detector
Stat
Naive (a 0.5)
CAE
Stat
Naive (ar = 0.5)
CAE
HBOS
0.1869
0.1751
0.2535
0.7148
0.7099
0.7324
IForest
0.3320
0,2553
0.2996
0.8139
0,8070
0.7981
LOF
0.1005
0.1091
0.2122
0.6300
0.6389
0.6796
PCA
0.3878
0,3132
0.3433
0.8026
0.7834
0.7730
MOMENT
0.1887
0.6352
Table 8: Multivariate anomaly-detection results under zero-shot setting.
Model
AUC-PR
AUC-ROC
VUS-PR
VUS-ROC
Multi-channel
Rank
Chronos-bolt-asse
0.1605
0.5675
0.1672
0.6149
False
22
TimeMoE-bbse
0.2164
0.6074
0.2276
0.6611
False
13
Moirai-bas base
0.2023
0.6210
0.2142
0.6731
True
MOMENT
0.1427
0.5553
0.1989
0.6369
False
17
CONCLUSION
This study addresses two significant challenges in reconstruction-based TFM anomaly detection: overgeneralization and overstationarization. To mitigate these limitations, we proposed two main enhancements. First, we introduced a Complexity-Awee integrates MOMENT-Stat with a statistical model based on dataset complexity. Second, Instance Statistics Augmentation preserves critical distributional information (mean and variance) during reconstruction, forming the MOMENT-Star model. Importantly, this augmentation is easily adaptable to other TFMs without the need for retraining. Experimental evaluations on 23 univariate and 17 multivariate benchmark datasets confirmed that both MOMENT-SE and CAE achieve state-of-the-art performance in anomaly detection, significantly outperforming existing baseline models.
Ensemble (CAE) framework that adaptively
Furthermore, evaluated forecasting- based TFMs (Chronos, Moirai, TimeMoE) and found that forecasting accuracy correlates with anomaly-detectioo performance, with recent high high-accuracy models achieving competitive, state-of-the-art results. In multivariate settings, only models that explicitly capture inter-channel dependencies (such as Moirai) perform effectively, highlighting the importance of cross-channed modeling.
One limitation of our work is that, when selecting the simple statistical model for the CAE frame- work, arbitrarily restricted our choices methods with O(N log N) time complexity. Another limitation is our exclusive focus on reconstruction based anomaly detection for univariate time series. Nonetheless, our concise yet meaningful experiments on forecasting basedd based detection and multivariate series suggest that TFMs can be extended to more general anomaly detection settings.
ACKNOWLEDGMENTS
This research was supported in part by SK Hynix AICC (P24.01); in part by the Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea government (MOTIE) (No. 00224000,,9092, 2024 HRD Program for Industrial Innovation); in part by the Korea Planning Evaluation Institute of Industrial Technology (KEIT) grant funded by the Ministry of Trade, Development of Core Technologies for Manufacturing Foundation Models); in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2025-1607298 and in part by the Institute of Information Communications Technology Planning & Evaluation (IIP)) grant funded by the Korea government (MSIT) (No. ITTP-2025-R 20--2025--S2222--0444 244, Global Data-X Leader HRD Program).
Industry and Energy (No.
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APPENDIX A. PROOFS AND TECHNICAL DETAILS
This appendix provides the formal definitions, rigorous setups, and proofs for the theorems presented in Section We establish the connection between the complexity metric and the high uighfrequency spectral properties of the time series (Theorem ! and provide the guarantee for the anomaly gap growth (Theorem
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APPENDIX A.I. COMMON SETUP AND NOTATION
Signal, mask, and projections. Let x = (ZI) E IR" be zero-mean time series with instance normalization E ||=:1 T. Fix masking ratio E (0, and an interval mask 7 C
(1....,T) with [ = [pT]. Define the coordinate projections
(Prz), [t di:
Sic.
P1= Id Pr-
Haar wavelet expansion and energies. On [0. 11. let (sh..s) be the compactly supported Haar ONB (Daubechies)
1992). Identify t E (I...., ... with t/T E [o, 1] and expand
Σ
Ck.j Wk.s
Σ EasePIa
Ckj = (z, vk.)), ase (2, s...))
Rej
For each scale ki define the scale energy and normalized weight
(kk
1 D Var(aj1)) =
(kk) =
Σ Elc.s
Wk =
Σ
wk + T
T
Define the detail and scaling energies
D:= Σ (k))
S S Σ Var(au,))
T= D
Let N& 2 be the number of detail coefficients scale k. For Haar, the support fraction is 8k == 2-k
Encoder, decoders, and the complexity Fix encoder E R - Rd and set Y E(x). Y'()== = E(Prox) Given F E EY the population-optimal linear decoder is
D3 E arg min E || DFI1?
D linear
Let D be the trained linear decoder and set mF == D F. By L -roojection, each error decomposes into linear Bayes risk and an approximation term. In our framework, the Reconstruction Error (froe) and Imputation Error (Limp) correspond to:
Lree = UE == E my 1 E | DE + -
- myl1?
Limp EM E | ml - ay r(nl + -
(1))-m((s))
The raw complexity metric is defined as the normalized gap:
Limp Lree
EM UE
Olyaw
T
T
APPENDIX A.2. EXACT ONB RISK DECOMPOSITION
To analyze Cryaw, we use the exact ONB risk decomposition. Let c == arg min, E[(ck, a F and define the coefficient-wiss R²
(2; = E((c 2F)
E 10.
Bic²]
Then the Bayes risk decomposes as:
E| D1FI = Σ E [CKJ oF
Σ E B
Assuming the approximation errors cancel out or are negligible (Mask-independent approximation assumption), the gap A == EM - UE satisfies:
A -
Σ o(k) by + Sb,
(2)
where the scale-wise gap coefficient by defined as:
by
o(k) Σ . (R² R2(ck.) Y')))..
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APPENDIX A.3. KEY PROPERTIES OF SCALE COEFFICIENTS
To prove Theorem D we establish that by is non-decreasing in k. This relies on the geometry of the Haar wavelet and spectral properties.
1. Refinement Monotonicity of Coverage AR. Let Jin (in (k; =- {ú :upp((re)) C mm.mmm))).)...))))))))))))))
# (k; I), and define A₃ := Er (m.))
Lemma 4 (Refinement monotonicity). For any fixed interval I and all & 2 0.
m)11(1
(
No+1
N2
Consequently,
Ak+1 A& 2 and S Ak+1 As 22
Proof. The refinement property of the Haar basis ensures that every interval fully containing the support of scale-k atom also contains at least one child atom at scale k+1. Thus m8+1 (T) 2 m)) Taking expectations yields the claim.
Monotone Envelope for Masked Explainability. Let La be defined as
LA == sup sup Rig (ck.j) Y I jeJia(k;))
Proposition 5. Assume is zero mean, weakly stationary with absoluiely summable autocovariance Y and bounded spectrum < o S f(w) S Cu. < 0o. Then there exists non-increasing envelope g(k) such that La g(k) for all k. Specifically, for Haar atoms:
7 2
72 T 6
g(k)
l1, 112
where To
Eb(r)
which decreases k increases.
Proof. We apply suboptimality under reduced masking to compare ('))) with raw predictor Pa where J C I. The standard bound for linear R² gives R (ck.j; PJ=x) < the spectral lower bound Var(ck.) 2 o and the -n-mo of covariance ||Cov]l2 S lig, we derive the bound g(k) x ||hs, |1? 2
[Co((k (Peaa) Using
Varies, Amiin
The following complementary result establishes that fully masked atoms suffer strictly positive information loss at every scale, not only at high frequencies.
Lemma (Spectral Predictability Bound). Under the bounded spectrum assumption S f(w) S Cu. for any atom with supp(()k.) C:: R²(Cn,gi 27o) S
Ce Cu
Proof. Decompose Σ + Eg where Ee > 0. The white-noise component / contributes zero to Er,rs, so the Schur complement satisfies Egpro 2 ce J. For any unit-norm &b supported on R = - v Enp S/= En eb < 1 ce/c..
Remark 1 (Combined picture). Proposition and Lemma 6 are complementary: the former shows L2 S g(k) 0 + 0 high frequencies, while the latter guarantees L S ct/c, at all frequencies. Together, the information loss (1 - L&) is bounded below by C2/Cu. > 0 everywhere and approaches as k + oo, strongly supporting monotone decrease of T.k.
3. Monotonicity of by. The coefficient by be modeled as by = A Lt), where M4 is the unmasked explainability. Proposition provides a non-increasing upper bound g(k) on Lx. but an upper bound decaying does not by itself force La be pointwise monotone. We therefore introduce the following regularity condition, which is well-supported by the combined picture above.
Assumption (Frequency-Ordee Preservation). For all scales k' k. the masked explainability satisfies Tup S Tak-
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This is justified by three observations: (i) the data processing inequality prevents the encoder from creating information beyond [ (ii)
atoms have narrower support, reducing correlation with unmasked data; (ii) the envelope g(k) = O k) from Proposition shrinks exponentially, confining any mn-moootone fluctuation within rapidly vanishing band, while Lemma ensures information loss is strictly positive at every scale.
Proposition 7. If Pk is non-decreasing (usually constant R: for reconstruction) and Lg is non- increasing (Assumption]) 7 then by is non-decreasing in k.
Proof. Since Aş increasing (Lemma 4) and positive, and (PR Ls) is increasing (due non- increasing LR:,, their product by. is non-decreasing.
The condition μı = 1 for l is a natural consequence of overparameterized TFM encoders (d > T), which possess sufficient capacity to preserve information across all frequency scales in the unmasked (reconstruction) setting.
APPENDIX A.4. PROOF OF THEOREM!
REFLECTS HIGH-FREQUENC ENERGY)
2 have A - Σn ((k) by + Sb. The scaling term S b, corresponds to low-frequency components which are robust to local masking (b, is small). The detail term Σn <bk) by dominates. By Proposition
Proof. From the decomposition in equation
7 by a non-decreasing sequence.
Since instance normalization fixes ||0| T, the total energy is constrained: Σ. d(k) + S T.
Define the normalized weights w/ == (b /T, so that Σ, w/ S/T 1. Now consider two signals with the same total energy but different spectral distributions {wx} } and (w satisfying 2x wh
En wt-- If [w/,} J places more weight on higher frequencies in the sense that LA2K w/ 2 2 WK for all K (first-order stochastic dominance), then since ly is non-decreasing:
Σ W by N Σ wkbk.
This follows from the standard monotone rearrangement property: when non-decreasing sequence {b& } paired with weights whose cumulative distribution shifts rightward (toward higher k),, the inner product increases. Thus, higher concentration of energy in high frequencies (larger k) results larger gap A, and consequently a larger complexity metric a x A.
APPENDIX A.5. PROOF OF THEOREM 2 (GAP GROWTH)
We analyze the gradient update of the decoder parameters W. Let LN(W) and (A(W) be the reconstruction losses for normal and anomalous data. The update rule is W = W. - Lx
Definitions and Assumptions. Let f be the energy-weighted average explainability at scale k for the current model We
Assumption (Decoder Preservation Decay). We assume the current explainability P4 decreases with frequency Equivalently, the non-preservation coefficients al p are non-decreasing in k.
((()
This implies the reconstruction error lower bound is higher for
data:
LN (w,))
Σ ((:)(1 - (l)))
Lemma 8 (Residual to Gradient Transmission). Assume the encoder output covariance satisfies ΣFF 2 La with of >> 0. For a linear decoder, the gradient norm satisfies:
(x) 2 or (Lx(WW) - Lx(W*)),
where W* is the optimal parameter.
Proof. The gradient is VL (W -W Let C W Wt. Then | |[CEF]l N = ([W)))(()))
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Using Lemma 8 and Assumption 2 see that V/V L increases with the high-frequency share of the data (since LN does).
Assumption (Gradient Direction). We assume the gradients for normal and anomalous data are aligned: LV (Wa), Vw I O.
(Gap Growth). We analyze the change in the anomaly gap Agap = LA IN after single gradient update step W W, vV Ln (W,).
Proof of Theorem 2
First, a first-order Taylor expansion of the loss functions yields:
A gap Saap(Weun ( A gap (Wi) P (I Lw (WWL,(W)) Vw V [x(W))) + Opp2).
Applying Assumption 3 bound:
(Gradient Direction Misalignment, (VLA, VLN) S 0), we obtain the lower
gaa Agap (W;) 2 [w O(p²).
(3)
Next, we link this gradient norm to the spectral properties of the data. By Lemma Transmission), the gradient norm is lower-bounded by the residual loss:
g
(Gradient
Lw 2 o (Ln(W)) Lx(W))).
Finally, incorporating Assumption 2 (Decoder Preservation Decay), the current loss is dominated by high-frequency components:
Lx(W,) 2 Σ ((k) (1- &
2
Since the con-preservation coefficient (1 1 is non-decreasing with respect to frequency scale k a higher concentration of energy ((k) in high frequencies (i.c., high complexity a) strictly increases the lower bound of the loss (N(W), which in turn increases the gradient norm in equation 3
Conclusion:
Consequently, for datasets with higher complexity a, the lower bound the gap growth is strictly larger. This guarantees that the separation between normal and anomalous scores increases faster during training for ligh-frequecyy data.
Remark (Tightness of the O(p²) remainder). Since the decoder is linear, Lx(W)) and LA (W) quadratic in W with constant Hessians, the Taylor expansion exact second order with no higher-order terms. In our experiments, p = 10-4, so p² 10-8, making the second-order correction negligible.
APPENDIX B DATASET DESCRIPTION
APPENDIX B. UNIVARIATE TIMESERIES DATASET
Table reports key statistics for the TSB-AD-U univariate benchmark. For each dataset, we list the total number of timeseries (TS), the average of timeseries length, the overall anomaly ratio (percentage of points labeled as anomalies), and the type of anomalies present (point "p" versus contiguous sequence "Seq").
Datasets marked with "(U)" are originally multivariate but have been converted to univariate by select- ing the most informative channel, based on correlation analysis and anomaly detection performance Liu & Paparrizos (202)).
A APPENDIX B.2 MULTIVARIATE TIMESERIES DATASET
For the multivariate experiments in Section 4.9 we use the TSB-AD-M benchmark. Table [0]lortss the statistics as Table adding the number of dimensions for each timeseries.
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Table 9: Summary statistics of univariate datasets from TSB-AD-U benchmark.
Dataset
Number of TS
Average of TS length
Anomaly ratio (%)
Anomaly type
UCR
228
67818
0.6
P&Seq
NAB
28
5099
10,6
Seq
YAHOO
259
1560
0.6
P&Seq
IOPS
17
72792
1.3
Seq
MGAB
97777
0.2
Seq
WSD
111
17444
0.6
Seq
SED
3
23332
4.1
Seq
TODS
35
5000
6.3
Seq
NEK
1073
8.0
S&
Stock
20
15000
9.4
Se
Power
1
35040
8.5
Seq
Daphnet (U)
38774
5.9
Seq
CATsV2 (U)
300000
4.9
Seq
SWaT (U)
419919
12.1
Seq
LTDB (U)
9
990700
18.6
Seq
TAO (U)
3
10000
9.4
S&
Exathlon (U)
32
44075
11.0
Seq
MITDB (U)
631250
4.2
Seq
MSL (U)
9
3492
5.8
Seq
SMAP (U)
19
7700
2.8
Seq
SMD (U)
38
24207
2.0
Seq
SVDB (U)
20
171380
3.6
Seq
OPP (U)
16544
6.4
Seq
Table 10: Summary statistics of multivariate datasets from TSB-AD-M benchmark.
Name
Number of
Average of length
Dim.
Anomaly Ratio (%)
Anomaly type
GHL
25
199001
19
1.1
Seq
Daphnet
38774
5.9
Seq
Exathlon
27
60878
21
9.8
Seq
Genesis
16220
18
0.3
Seq
OPP
17426
248
4.1
Seq
SMD
25466
3.8
Seq
SWaT
2
207457
59
12.7
Seq
PSM
217624
25
11.2
P&Seq
SMAP
27
7855
25
2.9
Seq
MSL
3119
55
5.1
Seq
CreditCard
284807
29
0.2
P&Seq
GECCO
138521
9
1.2
Seq
MITDB
13
336153
2
2.7
Seq
SVDB
31
207 122
4.8
Seq
LTDB
100000
15.5
Seq
CATSV2
240000
17
3.7
Seq
TAO
13
10000
3
8.7
P&Seq
18

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Published as a conference paper at ICLR 2026
APPENDIX C: FORMAL DEFINITION OF EVALUATION METRIC
Detailed derivations and discussion of evaluation metrics (AUC-PR, AUC-ROC, VUS-PR, and VUS-ROC) are provided in this appendix.
Precision and Recall. Given decision threshold т, let
TF()), FP(r), FN(T), TN(7)
denote the counts of true positives, false positives, false negatives, and true negatives, respectively, Then
(P(T) Procision (T) TP(T) + FP(7)' Recall(r) TP( + EN(T)
TP(T)
Area Under the Precision-Recall Curve (AUC-PR). Let (R, denote the parametric curve (Recall(7), Precision(r)) as varies over all possible score thresholds. Then the AUC-PR is the integral
AUC-PR = S. P( dr,
where R (r) is the threshold 7 achieving recall r.
Area Under the ROC Curve (AUC-ROC), Define the false positive rate and true positive rate as FP(T) FP(T) + TN(T)' TPR(T) TP(T) + FN(T)
TP(
FPR(7)
Then the AUC-ROC is
AUC-ROC = L TPR((PR- af.
with FPR (f) the threshold attaining false positive rate f-
Volume Under the Precision-Recal Surface (VUS-PR). Following Paparrizos et al. Paparrizos et al. (2022), let {/i]=0 be an increasing sequence of score thresholds (0 = ho < hı < ... < hr
1), and let { be a set of tolcrance-winde window sizes around each true anomaly (0 lo < 8 <
Eg). Denote the precision and recall threshold h, and buffer Ey by
Pa.i = Precision(h;,,)) Eg; R = Recall(h,, fg).
Then VUS-PR is defined the volume under this 2-D surface:
1
VUS-PR
RRy R1-15) (Prs + P.-1.).
Volume Under the ROC Surface (VUS-ROC), Analogously, define
TPR,, TPR(,, Ez), FPR., FPR(h,, ej).
Then
VUS-ROC
FPR., FPR;-15) (TPRi,) + TPR:11))
APPENDIX D: EXPERIMENTAL SETTING
In this experiment, we used the following default settings:
Statistical algorithms: window size determined automatically based on the auto-correlation
function of the benchmark.
Sub-IForest: number of estimators 100
19

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Published as a conference paper at ICLR 2026
SR: no hyperparameters
Sub-HBOS: number of bins 10
Sub-PCA: number of components = None (automatically determined)
POLY: power = (linear model)
For timeseries foundation models, we set: |t,l = 2W = 256, epochs = 2, learning rate =
0.0001, optimizer Adam, training loss MSE. The evaluated models were:
Chronos-holt-base, Chronos-bolt-small
TimeMoE-large, TimeMoE-base
Moirai-base, Moirai-small
Our proposed MOMENT-Stat has no hyperparameters. For CAE, the hyperparameters are K, P. and the window size; we set K' = 3 and P = 8.
All experiments were conducted on an HP DL380 GenlO Plus server equipped with Intel Silver 4310 CPU and single NVIDIA RTX 4090 GPU. The system ran Ubuntu 24.04 with hardware RAID configuration.
APPENDIX E: TABLE OF BASELINE MODEL PERFORMANCE
Table I1 presents an overview of various anomaly detection models, detailing VUS-PR scores, time complexity, and model type. The VUS-PR values in this table were directly taken from the results reported in previous research Liu & Paparrizos (2024). Deep learning and foundation-basel models show time complexity as "_", indicating that explicit complexity derivations are absent. N is the number of points, the number channels, p the polynomial order, m is the number clusters, and t the number of trees.
VUS-ROC and Fi-score values are directly taken from the original TSB-AD-U benchmark paper Lu
& Paparrizos 2024). In particular, the VUS-ROC and F1-score therefore appear with two decimal places only, reflecting the precision reported the original work.
Our model, which combines PCA and MOMENT-Stat, achieves VUS-PR of 0.4679, VUS-ROC of 0.8030, and an F1-score of 0.4330, ranking first, second, and first, respectively, Table [1
APPENDIX F: RUNTIME ANALYSIS
MOMENT-SHE incurs virtually no additional runtime compared to MOMENT (FT). In the CAE methodology, the total processing time equals the of the base statistical model's time plus the one-time cost to compute a. Once a has been computed for given dataset, no further overhead is incurred.
Table 12 reports the computational cost for MOMENT (Stat), Isolation Forest, and MOMENT (FT). For convenience, we denote the train/test series length by T and present the mean and standard deviation of both training and inference times. All runs used a sliding window of 256, two epochs, and the same optimizer settings as our experiments. As shown, MOMENT (Stat) and MOMENT (FT) exhibit statistically equivalent performance.
Table 13 presents the timing breakdown for CAE (Complexity- Selection) using MOMENT and Isolation Forest. The training/inference times equal the combined MOMENT (Stat) isolation forest computational times. T, denote the calculation time for complexity a with K = and P
20

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Published as a conference paper at ICLR 2026
Table 11: VUS-PR, VUS-ROC, and FI-scores comparison with time complexity.
Model
VUS-PR VUS-ROC F1-score Time Complexity
Model Type
Sub-PCA
0,4233
0.76
0.42
O(N d² d²)
Statistical
KShapcAD
0.4007
0.76
0.39
O(N3)
Statistical
POLY
0.3897
0.76
0.37
O(N + pz)
Statistical
Serpes2Graph
0.3881
0.80
0.38
O(N
Statistical
MOMENT (FT)
0.3857
0.76
0.35
O(N3)
Foundation-basee baaedd
MOMENT (ZS)
0.3790
0.75
0.35
O(N2)
Foundation-baase
K.MeanAD
0.3664
0.76
0.37
ON md)
Statistical
USAD
0,3611
0.71
0.37
Deep Learning
KNb-KNN
0.3501
0.79
0.34
O(N2)
Statistical
MatrixProfile
0.3496
0.76
0.33
O(N2)
Statistical
SAND
0,3437
0.76
0.35
O(N2)
Statistical
CNN
0.3428
0.79
0.38
Deep Learning
LSTMAD
0,3250
0.76
0.37
Deep Learning
0.3237
0.81
O(N Nog N)
Statistical
TimesFM
0,3003
0.74
0.34
O(N")
Foundation-basse based
IForest
0,2979
0.78
0.35
OCt log N)
Statistical
OmniAnomaly
0.2913
0.72
0.31
Deep learning
Lag-Llama
0.2733
0.72
0.30
O(N2)
Foundation-
Chronos
0,2722
0.73
0.32
O(N²)
Foundation based
TimesNet
0.2623
0.72
0.24
Deep Learning
AutoEncoder
0.2609
0.69
0.25
Deep Learning
TranAD
0.2572
0.68
0.25
Deep Learning
FITS
0.2553
0.73
0.23
Deep I.crning
Sub-LOF
0,2531
0.73
0.24
O(
Statistical
OFA
0,2369
0.71
0.22
O(N²)
Foundation-
Sub-MCD
0.2365
0.72
0.23
O(N3)
Statistical
Sub-HBOS
0.2283
0.67
0.23
O(N
Statistical
Sub-OCSVM
0,2251
0.73
O(N2)
Statistical
Sub-IForest
0,2230
0.72
0,22
O(L log
Statistical
Donut
0,1980
0.68
0.20
Deep Learning
LOF
0.1687
0.68
0.21
O(N*)
Statistical
AnomalyTransfome Transformer
0,1195
0.56
0.12
Deep Learning
APPENDIX G: COMPARISON WITH ENSEMBLES OF PURELY STATISTICAL MODELS
To verify that the performance improvement of our proposed Complexity-Awar Ensemble (CAB) stems from the unique contribution the Time Series Foundation Model (TFM) rather than the ensemble technique itself, evaluated ensembles composed solely of statistical models. We tested pairwise combinations of five statistical baselines: Sub-HBOS, Sub-IForest, POLY, Sub-PCA, and
SR. The ensemble score calculated Sensemble Smodell (1 a) 8model2, where varied a from 0.0 to 1.0.
Table 14 presents the VUS-PR scores for representative pairs. The results show that while combin- ing two statistical models can yield marginal improvements over single models, they consistently underperform compared the TFM-based ensemble (CAE). Specifically, the best statistical pair, Sub-PCA + SR, achieved maximum VUS-PR of 0.4398, which is lower than the 0,4679 achieved by our Sub-PCA + MOMENT (CAB). This demonstrates that TFMs provide complementary infor- mation- likely related to high-level semantic patterns- that statistical models alone cannot capture, thereby justifying the necessity of TFMs the anomaly detection pipeline.
21

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Published as a conference paper at ICLR 2026
Table 12: Computational cost for MOMENT-Stat, isolation forest, and MOMENT (FT).
MOMENT-Stal Training std
T
Training mean
Inference
Inference std
1,000
0,4339
0.0159
0.2415
0,0122
5,000
2.7602
0.0366
1.4430
0,0461
10,000
5.5242
0,1127
2.9577
0.1226
30,000
16.7726
0.5825
9.0411 1
0.2646
100,000
56.3860
1.2118
29.7477
0.7494
Isolation forest Training std
T
Training mean
Inference
Inference std
1,000
0.1807
0.0359
0.0151
0.0032
5,000
0.2479
0.0339
0.0403
0.0052
10,000
0.3480
0.0439
0.0786
0.0131
30,000
0.7912
0.1440
0.2155
0.0391
100,000
1.8050
0.2179
0.6502
0.1051
MOMENT (FT)
T
Training mean
Training std
Inference mean
Inference std
1,000
0,4273
0.0112
0.2398
0.0104
5,000
2,7302
0.0727
1.4959
0,0438
10,000
5,5857
0.0817
2.9848
0,0841
30,000
16.8385
0,3423
9.1684
0.1786
100,000
56.6425
0,8425
30.7839
0.4554
Table 13: Complexity selection (CAE) timing analysis.
CAE Inference mean
T
Training mean
Training std
Inference std
Ta mean
T, std
1,000
0.6145
0.0294
0.2567
0.0110
0.8696
0.0989
5,000
3.0081
0.0441
1.4833
0.0472
5.1230
0.1783
10,000
5.8723
0.1229
3.0363
0.1165
10.4340
0.2848
30,000
17.5637
0.4989
9.2566
0.2438
31.6091
0.5157
100,000
58.1910
1.0276
30.3979
0.6790
105.7010
1.7233
Table 14: VUS-PR scores for ensembles of two purely statistical models across varying weights (a)). The column 'eey" denotes the pair (Model Model 2), where represents the weight of Model 1. The best Stat+Stat performance (0.4398) is lower than the proposed CAE (0.4679).
Pair (Model Model 2)
0.0
0.1
0.3
0.5
0.7
1.0
Best Score
HBOS IForest
0.2230
0.3536
0.3513
0,3300
0.2751
0.2283
0,3536
HBOS_POW PPLLY
0,3897
0.3761
0.3744
0.3454
0.2910
0.2283
0,3897
HBOS SR
0.3237
0.3252
0.3274
0.3160
0.2879
0.2283
0.3274
Fooeett_
|ooret_PPOLY POLY
0.3897
0.3827
0.3869
0.3909
0.3849
0.2230
0.3909
Forest SR
0.3237
0.3318
0.3573
0.3832
0.3822
0.2230
0.3832
PCA_HBOS HBOS
0.2283
0.2265
0.2914
0.3736
0.4098
0.4233
0,4233
PCA IForest
0.2230
0.3615
0.3824
0.3960
0.4035
0.4233
0.4233
POLY
0,3897
0.3845
0.3974
0.4207
0.4355
0.4233
0.4355
PCA SR
0.3237
0.3440
0.3876
0.4287
0.4398
0.4233
0,4398
SR POLY
0.3897
0.4092
0.4171
0.4037
0.3798
0.3237
0.4171
Proposed CAE (Sub-PCA + MOMENT)
0.4679
22


Reviews:
[{"review_id": "MbYAg7lWDc", "reviewer": "Reviewer_BRgs", "summary": "The paper introduces methods which enhance the performance of time series foundation models (TFMs) in two different ways. The first is a direct adjustment on the networks so that global timeseries statistical information (mean, variance) which is lost in normalization steps is re-introduced in the end. The second method, called compexity-aware ensemble (CAE), ensembles the TFM with a statistical methods in an adaptive way to improve performance.\n\nAfter a short description of the task, the CAE method and the statistical augmenting method applied on the RevIN layer are introduced. The TFM used is MOMENT, which was along the high scoring NN-based methods in the TSB-AD-U benchmark. When statistially enhanced, it is called MOMENT-Stat.\n\nIn the experiments section, the effect of CAE is demonstrated on different types of synthetic series. Then MOMENT-stat is compared with the original versions of MOMENT and shown to achieve slightly better scores. Afterwards, MOMENT-stat is fixed as the TFM used and CAE is applied on different statistical methods and shown to consistently increase their performance. An ablation is also present comparing different fixed values of alpha with CAE which results to the best performance. Additionally, alternative methods to weight the ensemble based on entropy are compared with CAE, again yielding lower performance. Finally, on an additional section some forecasting TFMs are briefly studied. It is observed that their forecasting performance is correlated with their anomaly detection performance and finally they are compared with each other and with MOMENT on multi-variate time series.", "questions": "- (More a remark) Section 3.1 is quite messy. A lot of symbols and operations are used without being introduced and details are missing. It would be great to revise and improve it. The usage of intervals and their complements to filter series is nicely used though.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper is quite readable and the method and goals are clear.\n\n- Descent datasets and evaluation methods where used, something often missing in works on timeseries anomaly detection\n\n- The paper touches different topics on TFMs and their usage in anomaly detection.", "weaknesses": "- It is not very clear from the paper that the CAE method is much better than simply ensembling statistical models with MOMENT. On table 3, one can see that the scores when using the value alpha = 0.5 are relatively close to the scores of CAE. To be convinced that CAE is preferable to simple ensembling one would expect a thorough analysis on how the values of alpha vary across datasets, where they peak and how close to this peak CAE is, compared to constant values of alpha. \n\n- The relation of alpha to the ensembling weight w is not adequately explained. It is only mentioned that it is an increasing function. It is not clear if it is the same for all datasets or also adaptive or how it is defined. This also raises some more questions about the difference CAE with simple ensembling, as no information is present on e.g. how ensembles with fixed weights w (e.g. w = 0.5) would perform. It could be that those are close to the columns of table 3, but it is not clear. Finally, this makes it also difficult to judge which mixture of MOMENT with the statistical models is tendentially better.\n\n- Though MOMENT-Stat performs slightly better than MOMENT as a standalone, MOMENT with CAE is never compared with MOMENT-Stat with CAE. One could imagine that datasets dominated by low frequency anomaly signal would tendentially also rely on high level statistics, so the enhancement of the TFM might not provide any significant advantage when ensembled. \n\n- The paper is lacking some focus. The CAE seems to be the central topic, but at the same time the statistical enhancement is introduced and near the end there is a quick study of forecasting TFMs. We feel it would have been better to have focused on CAE and try to support it more thoroughly, e.g. provide cleaner and more thorough definitions, add more details, like the relation of alpha with w, also study simple ensembles and properly compare the to CAE, further study how CAE is behaving on different datasets and picking its weighting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vPbxCxWgzk", "reviewer": "Reviewer_zNh6", "summary": "This paper proposes a framework that leverages complexity- and statistics-guided ensembling (CAE) to improve the anomaly detection capability of TFMs. The approach aims to mitigate the overgeneralization and overstationarization issues commonly observed in pretrained TFMs by integrating a statistical anomaly detector with foundation-model representations.", "questions": "* The ensemble strategy indeed improves performance, but the claim that this improvement demonstrates the effectiveness of TFMs is not fully convincing. It is important to support the argument by comparing against ensembles of two purely statistical methods, thereby isolating the specific contribution of TFMs.\n* The paper mentions overgeneralization and overstationarization issues, which also affect forecasting-based TFMs. Why are the proposed methodologies not extended or tested in this setting?\n* The discussion on the multivariate case is incomplete. Could the CAE framework be generalized to such case?\n* The difference between CAE and CAE-per-data should be more clearly articulated. What causes CAE-per-data to underperform, and what insights does this provide about model generalization?\n* How is the ensemble factor w determined? Have the authors conducted a sensitivity analysis to assess its stability across datasets?\n* Including more visual examples or case studies would improve readability and help contextualize the motivation behind the method.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The paper is among the first to systematically investigate the use of time-series foundation models for anomaly detection tasks.\n* The proposed complexity- and statistics-guided ensemble framework is well-motivated.", "weaknesses": "* Some claims of effectiveness, particularly regarding the superiority of TFMs within the CAE framework, are insufficiently supported by experiments.\n* The evaluation scope is limited to univariate cases and reconstruction-based TFMs.\n\nPlease find the detailed comments in the following section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "t3C6Sj5y6s", "reviewer": "Reviewer_aW8c", "summary": "In this paper, the authors propose an anomaly detection method built on time-series foundation models (TFMs). The authors found that when using TFMs (pre-trained for forecasting) in a reconstruction-based anomaly detection setting, the model may overgeneralize. Also, the use of instance normalization layers removes statistical features that are actually useful for anomaly detection. To address these issues, the authors propose a complexity metric  that measures how difficult a segment is for the TFM. Also, a method to reintroduce lost statistical features into the reconstruction process without retraining the TFM is introduced. Comprehensive experiments are conducted on 23 univariate benchmark datasets", "questions": "- How are anomalies labelled? Are there multiple anomaly types per dataset? \n- Are anomalies artificially injected or real events?\n- Does the complexity metric α help provide insight and is that interpretably presented?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The use of foundation models (TFMs) in time‐series is growing, but the paper identifies important pitfalls when using them for anomaly detection\n- The introduction of the complexity metric α and the ensemble approach is novel\n- The authors run experiments on 23 univariate time-series datasets", "weaknesses": "- The experiments are only on univariate time series. Many real-world anomaly detection tasks are multivariate. The method’s applicability to multivariate TFMs is unclear. The paper does not sufficiently discuss how the method scales to multivariate or high-dimensional data.\n- While the idea of a “complexity” metric is intuitive, the exact definition and properties of α need to be clearly explained. What does “complexity” measure exactly?\n- How sensitive are results to the choice of α. \n- Stronger empirical evidence regarding the claim that  \"instance normalization (and other normalization) removes statistical features useful for anomaly detection\" is needed.\n- Reintroducing statistics is consistent across models and domains?\n- If the underlying TFM is trained for forecasting, reconstruction may not be the most natural anomaly detection mechanism. Predictio-based anomaly detection might be better in some cases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UK19UBfyhI", "reviewer": "Reviewer_q2N4", "summary": "This paper aims to address the issues of overgeneralization and overstationarity in Time Series Foundation Models (TSFMs). To mitigate overgeneralization, it introduces an adaptive weighting mechanism that combines anomaly scores from TSFMs and traditional statistical methods. To alleviate overstationarity, the model integrates mean and variance information into the encoder’s input representation.", "questions": "1. Could the authors provide both theoretical and experimental analyses to clarify how normalization operations affect anomaly detection performance?\n\n2. I suggest comparing the proposed CAE with TSFMs such as One-Fits-All and Moment, as well as anomaly detection models like Anomaly Transformer and DCDetector, in terms of VUS-PR to more comprehensively evaluate its effectiveness.\n\n3. I recommend adding a comparison of recall values under the condition of achieving the best F1 score for each method, to better demonstrate that the proposed approach effectively mitigates the overgeneralization problem.\n\n4. Could the authors elaborate on the underlying principle of how incorporating mean and variance into the encoder input can effectively address the overstationarization problem?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The paper tackles the overgeneralization issue, which is a very important yet insufficiently addressed problem in anomaly detection using Time Series Foundation Models (TSFMs).\n2. The introduction is well-written, clearly presenting the motivation, the research problem, and the methodological approach, making it easy for readers to follow the authors’ reasoning.\n3. The experimental results convincingly demonstrate that TSFMs indeed suffer from the overgeneralization problem, providing strong empirical support for the paper’s premise.", "weaknesses": "1. The paper’s notation system is somewhat confusing, and the methodological presentation could be improved. For example, in lines 153–156, several symbols with different subscripts (e.g., P with varying indices) appear without clear definitions, making the section difficult to follow. Moreover, key background concepts such as scale energy and detail energy should be introduced in a dedicated Preliminaries section to help readers better understand the core methodological design.\n\n2. While the paper points out that TSFMs suffer from overstationarization, it lacks further discussion or empirical evidence on how normalization operations affect anomaly detection accuracy.\n\n3. The experimental setup is not fully convincing:\n3.1. Since TSFMs exhibit more severe overgeneralization than time-series models specifically designed for anomaly detection, it is essential to compare the proposed CAE method not only with widely used TSFMs (e.g., One-Fits-All, Moment) but also with task-specific anomaly detection models. The current experiments do not sufficiently demonstrate that CAE outperforms existing anomaly detection methods.\n3.2. To substantiate the claim that CAE alleviates overgeneralization, the paper should also compare the recall values of different methods (TSFMs and dedicated anomaly detection models such as Anomaly Transformer and DCDetector) under the condition of achieving their respective best F1 scores.\n\n4. It is doubtful whether mainly combining the mean and variance with the input representation in the encoder and optimizing it by reconstruction error can solve the problem of overstationarity. The experiments on this question are also unconvincing as Moment-stat only makes a marginal improvement compared with Moment (FT).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)