Contextual and Seasonal LSTMs for Time Series Anomaly Detection
Authors: Lingpei Zhang
Qingming Li
Yong Yang
Jiahao Chen
Rui Zeng
Chenyang Lyu
Shouling Ji
Venue: ICLR 2026 Poster (2026)
Keywords: time series anomaly detection
Primary area: learning on time series and dynamical systems
TL;DR: We present CS-LSTMs that accomplishes anomaly detection for univariate time series with unified framework.

Abstract: Univariate time series (UTS), where each timestamp records a single variable, serve as crucial indicators in web systems and cloud servers. Anomaly detection in UTS plays an essential role in both data mining and system reliability management. However, existing reconstruction-based and prediction-based methods struggle to capture certain subtle anomalies, particularly small point anomalies and slowly rising anomalies. To address these challenges, we propose a novel prediction-based framework named Contextual and Seasonal LSTMs (CS-LSTMs). CS-LSTMs are built upon a noise decomposition strategy and jointly leverage contextual dependencies and seasonal patterns, thereby strengthening the detection of subtle anomalies. By integrating both time-domain and frequency-domain representations, CS-LSTMs achieve more accurate modeling of periodic trends and anomaly localization. Extensive evaluations on public benchmark datasets demonstrate that CS-LSTMs consistently outperform state-of-the-art methods, highlighting their effectiveness and practical value in robust time series anomaly detection.

Published as a conference paper at ICLR 2026
CONTEXTUAL AND SEASONAL LSTMs FOR TIME SE- RIES ANOMALY DETECTION
Lingpei Zhang', Qingming Li Yong Yang', Jiahao Chen', Rui Zeng', Chenyang Lyu". Shouling Ji 1:3
'Zhejiang University, HHuawei Technologies *Zhejiang Key Laboratory Decision Intelligence {lingpz, ligm, yangyong2022, xaddwell]eezj.edde quizeng24@oute com, Lyuctenyangiihuuwee.. com, sjiszju.edu. en
ABSTRACT
Univariate time series (UTS), where each timestamp records a single variable, serve as crucial indicators in web systems and cloud servers, Anomaly detec- tion in UTS plays an essential role both data mining and system reliability management. However, existing reconstruction-bee based and prediction-based meth- ods struggle to capture certain subtle anomalies, particularly small point anoma- lies and slowly rising anomalies. To address these challenges, we propose a novel prediction-based framework named Contextual and Seasonal LSTMs (CS- LSTMs). CS-LSTMs built upon noise decomposition strategy and jointly leverage contextual dependencies and seasonal patterns, thereby strengthening the detection of subtle anomalies. By integrating both time-domain and frequency- domain representations, CS-LSTMs achieve more accurate modeling of peri- odic trends and anomaly localization. Extensive evaluations on public bench- mark datasets demonstrate that CS-LSTMs consistently outperform state-of-the- art methods, highlighting their effectiveness and practical value in robust time series anomaly detection.
INTRODUCTION
In cloud services, IoT, and monitoring systems (Faloutsos et al. 2019 Wen et al. 2022 Dai et al. 2014 Huang et al. 2022 Ka-
202 Deldari al. 2021 Ganatra al., 2023 Günnemann
marthi et al. 2022: Tuli et al. [2022] Xu et al.) 2018), data is often represented as univariate time se- ries (UTS) (Faloutsos et al. al.]|211) 2019). where each timestamp corresponds to a single variable. Anoma- lies may arise when a website is subject to malicious attacks or experiences excessive server load. These anomalies are typically reflected in UTS derived from traffic and system load monitoring. Missed or delayed detection of anomalous events may lead to severe consequences (Ruff et al.) 2021 As illustrated Figure anomalies UTS are typically categorized into two types: point anomalies, which refer to abrupt deviations (either global or contextual), and segment anomalies, which reflect more subtle shifts in trends or periodic patterns (Lai et al. 2021).
Unsupervised approaches, such reconstruction based and prediction based methods, have become the dominant paradigm. Both aim to model normal behavior and detect anomalies by comparing pre- dicted reconstructed values against actual observations. Reconstruction-ben bbasdd techniques (Wang jet al. 2024| Kingma et al.) 2013) attempt to map abnormal segments to a normal space. However, lacking ground truth for anomalies leads poor reconstruction and degraded performance. These methods also treat time windows independently. ignoring temporal dependencies. Prediction-based method (Zhou al. 2021; Xu et al.) 2021; Kingma et al. (013) shows strong results in multivariate time series (MTS) using Transformers (Vaswani et al. (017) 7) or LSTMs (Hochreiter & Schmidhuber, 1997] to model complex dependencies. However, UTS, each timestamp contains only one value, which restricts the representation of temporal context and results in suboptimal performance of these methods on UTS tasks.
"oorresooniigg author.

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Published as a conference paper at ICLR 2026
1/ MM 11
ww
(a) Seasonal anomaly
(b) Trend anomaly
(c) Global anomaly
(d) Context anomaly
Figure 1: Anomaly types: (a), (b) segment anomalies; (c), (d) point anomalies. (a) Abrupt period
shortening; (b) Unexpected trend drop; (c) Global outlier; (d) Contextual outlier within range but locally deviant.
To examine the limitations of existing anomaly detection methods in UTS tasks, we conduct a re-evaluation of several state-of-the-art approaches. including FCVAE (Wang et al., 2024). KAN- AD (Zhou et al.] 2024), and TFAD (Zhang et al.] (022a). We find that two anomaly types remain largely undetected (Figure 2). (1) Small Point Anomalies: short-term spikes that appear normal over longer windows; (2) Slowly Rising Anomalies: gradual segment deviations from periodic patterns. Root cause analysis reveals that these hard-to-deted anomalies arise from existing meth- ods' limitations. For example, FCVAE (Wang et al.] 2024) focuses on frequency components rather than local dependencies, failing capture subtle point anomalies gradual segment anomalies;
KAN-AD Zhou et al. 2024) uses timing information for prediction but ignores frequency details.
To effectively detect the two types of anomalies, highlight the following three key challenges:
Challenge 1: Capturing Local Trends Rather Than Absolute Values. As illustrated in Figure the absolute magnitude of value changes is insufficient determine whether point is anomalous. For instance, in Figure 3al changes between 0.5 and 1 are considered abnormal, whereas in Fig- ure 3b similar changes are deemed normal-only variations exceeding are treated as anomalies. This observation underscores the importance of incorporating local information. i.e., data within a short time window, to predict future values. By modeling the value distribution and change patterns within the current local window, we can better estimate the expected range of future normal values.
Challenge Capturing the Evolution of Periodicity Rather Than Treating it as Static. Time series data exhibit evolving trends and periodicity (Wen et al.] 202I). Without future points, distin- guishing slow anomalies from normal periodic changes is difficuit. As shown in Figure periodic- ity varies over time and between datasets, including period length and frequency changes. Previous methods focusing on adjacent historical periodicity neglect this evolution. Modeling dynamic yet heterogencous periodicity is challenging. The Fourier transform (Cohen 2020) effectively captures periodicity and its changes by converting time-domain data to the frequency domain.
Challenge 3: Capturing Normal Patterns Amid Anomalies and Noise. In real-world scenarios, time series often contain long-term noise and sudden anomalies, both of which hinder the learning of normal patterns. Noise introduces significant fluctuations, making it difficult for models to fit the underlying normal behavior accurately. Meanwhile, the lack of ground-truth labels for anomaly positions prevents the model from computing reliable prediction losses during training, further com- plicating the learning process.
To address these challenges, we propose CS-LSTMs, novel prediction-based framework. CS- LSTMs are built upon a noise decomposition strategy to mitigate the impact of anomalies, and leverage both the local trends and the evolution of periodic patterns to predict the expected nor- mal values at future time steps, and they detect anomalies via discrepancies between prediction and observation. CS-LSTMs employ a dual-branch architecture: one models long-term evolution in periodic, while the other captures short-term local fluctuations. By jointly modeling these comple- mentary aspects, CS-LSTMs better capture time series data evolution and richer context, yielding more accurate and robust anomaly detection. Our contributions are summarized as follows:
We analyze hard-to-deted anomalies in UTS and identify their root cause as the inability of existing methods to integrate local trends with periodic variations effectively.
We introduce noise decomposition strategy better capture normal patterns by eliminat- ing the non-stationarity of time series effectively, thereby improving predictive accuracy and anomaly detection robustness.
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(a) Undetected anomaly
point
(b) Undetected segment anomaly
Small value
(h) Big value
Figure 3: Point anomalies at different scales. Normal points are black: anomalies are red. In (a), value of 0.5 is anomalous due to the smooth background, while in (b), only value around 5 is considered anomalous amid larger fluctuations. The key lies context sensitivity.
Figure 2: Two anomaly types existing meth- ods fail to detect. Normal points are shown in black, detected anomalies in blue, and detected anomalies red. (a) smaller point anomaly; (b) gradual rising segment anomaly.
We design CS-LSTM, with dual-branch architecture: one branch leverages frequency do- main analysis Hamilton 2020) to model long-term periodicity and its evolution, while the other captures short-term local dynamics. The combination of the two effectively enhances the ability of anomaly detection.
Extensive experiments demonstrate that CS-LSTMs achieve superior F1 scores and im- prove time efficiency by 40% over SOTA.
The replication package for this paper, including all data, source code, and documentation, publicly available online at https://sihlbbooo//SSSAA..CCooot Sen-ssssonann--S-nnaa-aa--a----a Seasonnl-LSSTMs.. TSAD.
RELATED WORK
Time Series Anomaly Detection (TSAD) has been extensively studied in both univariate and multi- variate settings, Existing approaches can be divided into traditional and deep learning methods, and further categorized into supervised and unsupervised depending on label availability.
Mahimkar et al. 2011 Rasheed et al. 2009 Rosner 1983 Bailey, 1993, Vallis et al.) 2014 Zhang et al. 2005 Guan et al. 2016 Lane al. 1997
Traditional methods (Lu & Ghorbani 2008;
Peterson (2099) Muthukrishnan et al. 2004) are computationally efficient and interpretable, making them suitable when data scarce or temporal structures are clear. They include similarity- approaches (e.g., kNN (Peterson] 2009)), window-based approaches using subsequence matching or HMMs (Gao et al. 2002), decomposition-
approaches (e.g.. STL (Cleveland et al., 1990), deviation-basse based methods such as SPOT (Vallis et al. 2014) based on Extreme Value Theory, and frequency-domain methods like FFT (Rasheed et al. 2009).
Deep learning approaches (Yang et al. 2023 Zhao et al. 2023b Shen et al. 2020 Zong et al.
2018 have attracted increasing attention due to their strong representation learning capabilities and ability to capture complex temporal dependencies. Supervised methods (Laptev al. 2013 et al. 20155 Ren et al. 2019: Zhao et al. (2023b) achieve high accuracy with sufficient labels, of- leveraging CNNs, random forests, or pseudo-labeling strategies. However, their generalization degrades under label scarcity or distribution shifts. Unsupervised methods (Chen et al.) 2019) Kieu al. 2022 Li al. 2018 2022b Zhao al. 2023a Wang al.) (024) are more common in practice, typically classified into reconstruction- based and prediction based. Reconstruction-bart approaches (e-g., FCVAE, Donut (Wang et al.) 2024 Xu et al., 2018)) detect anomalies through reconstruction losses, while prediction-based methods forecast future values and flag deviations. Representative prediction models include LSTMAD and recent Transformer-based methods such as Informer (Zhou et al.) 2021), Autoformer (Wu et al. [202I. and Fedformer (Zhou et al. (0022. More advanced architectures like Anomaly- Anomaly-Transformer (Xu et al.) 202I) employ minimax strategies to enhance robustness, while TimesNet (Wu et al. 2022) integrates frequency- domain modeling for multi-scale and periodic signals. These approaches demonstrate strong adaptability in roal-world scenarios with limited labeled data.
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indicating normal point and c 1 indicating an the objective of USS anomaly detection is to utilize the Published as a conference paper at ICLR 2026
19:03
700
189
1961
1100
1303
1900
39941
1390
(a) Yahoo
(b) WSD
Figure 4: Periodicity visualization in Yahoo and WSD. Time series shifted and overlaid using different colors. Stronger overlap indicates stronger periodicity; reduced overlap over time suggests periodic changes.
3 PRELIMINARY
3.1 PROBLEM STATEMENT
This paper primarily addresses the issue of anomaly detection in single time series curves, also known UTS. To elaborate on the problem more comprehensively, consider the following UTS observational data: 20.: = {ro, I1, .2..... Øt and anomaly labels C' {co, C1, C2..... Cs where I, R. C E {0,1).
{0. 1..,, t}. with anomaly. Given = [o, ........ :x),
data [ro, I1,..... - preceding each point xy to predict C₁.
3.2 LSTM AND COVARIATE
In our method, the LSTM is utilized to learn both the periodic trends and local variations of the time series data, while the value range information is incorporated covariate to assist in prediction. The combination these two aspects effectively enhances the accuracy of the forecasting.
LSTM is special type of Recurrent Neural Network (RNN) (Elman 1990 designed address the problem of vanishing gradient, which is difficult for traditional RNNs to capture long-tern de- pendencies. LSTM is effective in handling time series data, natural language processing tasks, and other problems that involve temporal dependencies.
Covariate is variable used input in statistical models, machine learning, and associated research studies. It may be associated with the target variable (dependent variable) and could have direct indirect influence on Covariates (Hyndman Khandakar] 2008) are typically used explain variations in the target variable to control confounding factors.
METHODOLOGY
Our method involves three main steps: (1) Data Preprocessing, where normalization and missing
value imputation (Wen al. 2020 Gao et al. 2020 applied without relying data augmen- tation (Le Guennec et al. 20 Li et al. 2021 x (2) Dual-Branch Prediction Network. where the S-LSTM branch learns periodic patterns from the frequency domain of the historical sequence, and the C-LSTM branch captures local trends and short-term variations; (3) Anomaly Scoring with Masked Probabilistic Loss, where we use masked negative log-likelihood loss to focus learn- ing the distribution of normal data and predict future values. Anomalies are detected when the deviation between predicted and actual values exceeds the normal range.
4.1 NETWORK ARCHITECTURE
Our proposed CS-LSTMs model is demonstrated Figure 5 The model consists of two branches: the first branch is the seasonal prediction branch, which predicts future points by learning the peri- odicity historical sequences; and the second branch is the local prediction branch, which predicts future points by learning the changing trends and distributions in the short term.
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Published as a conference paper at ICLR 2026
4.1.1 NOISE DECOMPOSITION
The proposed noise decomposition strategy (Eq. D is designed to address two key challenges: (1) the absence of ground-truth labels for anomalous points; (2) the difficulty in fitting normal patterns of time series. In the context of time series forecasting, DLinear (Zeng et al., 2023) introduces simple pooling-based decomposition method that splits the series into trend and residual components. In time series anomaly detection, TFAD (Zhang al. 2022a) uses Robust STL (Wen al. 2019) to decompose the series into trend and residual components for anomaly detection.
However, the objective of CS-LSTMs is to predict both periodic and trend components separately. Therefore, we propose a noise decomposition strategy that formulates time series
Time Series = (TTennd - Season) + Noise
(1)
where only the noise is filtered out without further decomposition. DLinear's strategy is insufficient address this formulation, while STL is overly complex and computationally expensive. To achieve
efficient decomposition, we adopt wavelet transform (Donoho & Johnstone 1994):
(CA, (L)
0 wavedec(z, v,
(2)
median ( (CD
0, (-1(0..5)
A = o, 2 n, Vi L....,b
(3)
(1) D = sign(c)) (1) max(1c)) -1, 0), Vi L,,....4
(4)
= waverec CA. ((1)
(1)
(5)
We employ a method based on the Median Absolute Deviation (MAD) (Rousseeuw Hubert 201
This approach is less sensitive outliers compared to methods using the mean and standard devia- tion, allowing for a more accurate estimation of the data's 's dispersion. We assume that the underlying data (without outliers) follows a normal distribution db
The original signal x R" is first decomposed into a set of wavelet coefficients using a selected wavelet basis va and predefined decomposition level L (Eq. 2 Specifically, this decomposition yields one set of approximation coefficients at the coarsest level and multiple sets of detail coef- ficients at each level from 1 to L. The noise level is then estimated based on the MAD of the finest-scale detail coefficients (Eq. 3 where the factor 3 ensures consistency with the standard deviation under the assumption of Gaussian distribution.
Next, a universal threshold À is computed (Eq.3), which is subsequently applied to all detail coef- ficients using soft-thresholding (Eqq. 4). This process suppresses noise while preserving significant signal components. Finally, the denoised signal is reconstructed via the inverse wavelet transform (Eq. 5). The derivation of the denoising method is presented in the appendix D
4.1.2 S-LSTM
The S-LSTM branch focuses on learning the periodic information and evolving trends in historical data to detect segment-level anomalies. As highlighted in Challenge 2, time series periodicity is dynamic rather than static. Unlike previous methods that only capture periodicity near the point of interest, S-LSTM models the temporal evolution of periodic patterns across the entire historical sequence, providing a more comprehensive representation of the underlying structure.
Specifically, we divide the historical sequence before the detection point into several equal-sized, non-overlapping windows. Each window transformed from the time domain to the frequency do- main via the Fourier transform, yiclding frequency components that intuitively reflect the periodicity of the data. Since periodic characteristics vary across different segments, we model the relationships among these frequency representations capture the evolution of periodicity over time.
To achieve this, a single-layer LSTM is employed to leam the variation trends of consecutive fre- quency vectors and predict future periodic patterns. Assuming a window size of w, the Fourier transform produces frequency vectors a dimension w.. Extracting n such windows results in the input 2, E Raxw, Furthermore, to enhance the model's representation of temporal distribution,
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Published as a conference paper at ICLR 2026
Concotenate
Frqquracy
Origical
S-LSTM
w
MMMWV
LSTM
Scorts
Nous De Decomppoonnte
Anomaly Score
LSTM
Score,
0g
Covanass
C-LSTM
PC-
Figure 5: Architecture of CS-LSTMs.
the raw time-domain values are treated as covariates and concatenated with the frequency features before being fed into the LSTM. This combination enables the model capture both distributional and periodic correlations effectively.
4.1.3 C-LSTM
The C-LSTM branch is designed to learn locally changing trends and distributions the adjacent historical data to identify sudden, mutational anomalies. Due to the limited information available at individual points in UTS, conventional point- prediction methods often struggle in anomaly detection tasks. Therefore, the C-LSTM branch captures richer local temporal dependencies by focusing on overlapping time segments.
Similarly, extract shorter historical sequence before the point of interest and divide it into eral equal-sized, overlapping windows. This transformation shifts the task from learning relation- ships between individual points to learning relationships between segments, effectively alleviating the scarcity of information at single time points. Each overlapping window is further converted to the frequency domain via the Fourier transform capture finer-grained details.
Next, single-layer LSTM is employed to model the temporal relationships among these over- lapping frequency segments, using this local information predict future values. When a sud- den change occurs, the predicted points will significantly deviate from the normal trend, enabling anomaly detection. Given the need to capture short-term local variations, we select smaller win- dow size whe, intercepting n overlapping windows to form the input 2e E Raxw, for the model, thus achieving precise modeling of local temporal dynamics.
4.2 LOss FUNCTION
We adopt a noise-decomposed Negative Log-Likelihood (NLL) loss function perform anomaly detection by modeling the probabilistic distribution of normal points. This approach emphasizes filtering anomalous regions and accurately capturing normal patterns. Since normal behavior is inherently probabilistic rather than deterministic, a key challenge lies in how to represent its variation range appropriately.
To address this, we design loss function with three core components: (1) noise decomposition, (2) anomaly masking, and (3) range-based prediction. During loss calculation, we leverage anomaly masks and noise decomposition results to obtain the reference value for each point under normal conditions. This enables the model to focus on learning normal patterns without being misled by unreliable ground truth in anomalous regions.
Compared with MSE [Thompson] 1990) or MAE (Hodson 2022), the NLL loss better reflects uncer- tainty by predicting both mean and variance, naturally adapting to the probabilistic characteristics of normal behavior.
The final loss function D (Eq. 6) incorporates the predicted mean μ4, variance o², ground truth I, noise-decomposed value a, and the anomaly mask. By optimizing only on reliable data, our model achieves more robust normal pattern learning. We provide the proof of the loss function's

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Published as a conference paper at ICLR 2026
Table I: The performance of anomaly detection on the test datasets. Fl means best Fl and FI* means delay FI. The best-performing results are shown in bold, while the second-bes best results are underlined.
Method
Yahoo
WSD
SINTT
0.17 03322
0032
0417
0.572
0.325
0.160
0.33
1
0.07
0.45
0061
0.315
0.27 0.8
01.37
0.99
0.113
0.33
0.00
6012
SRCNN
0.251 0.268
0.256
0.195
0.219
0.181
0.786
0.573
0,94
0.675
0667
0.253
0.176
0.093
0.003
0.053 0.025
0.361
0.625
0.825
0.822
0.575
0,460
0.766
BONUT
0.315 0:381 0.150
0.215
0.58
1800
0.454
0.778 0.56 0.964
0.407
0.334
0.26
195
0.15% 19999 0.131
0.915
1111 919
797
03811 0.774
VORAF
0.500
0.19
0.493
0.911
831
0.372
0.300
0.418
111 0.167
0111
0.113
0.51
0.231
B
0.341
0066
0.913
919 8888 0.89
0,800 1800
Anotransfer
0.507 0.912 0413
0.495 0.575 0.437
0:655
0.515
9 0.461 0.557
0.39
0.674
0.095
0.611
0.379 0.331 0.444
0.965
D.902 0 0.871 0337 0998
Informer
0.94 0.447 0663 0.671 0.731 0.619 0.918
0:927 9.910 0.22 1100
03555 0557
0.52 0,583 0.395 0.002 0.355
0.973
[1971 0.574 0.892 0307 0.907
TEAD
0.792 03355 0.723 0.7M1 03579 0.719 0.753 0.664
9.834 01600 01650
0.71
0.028
0.54 0.759
0.25 0.831 0.452
0.734 1.789 0.719 0.348 0.200 0.333
Amomal- Transforree
0.214 0.588 0.179 0,029 0.004 0.020 0.868 0.990 0.514 0.546 0.622
0.240
0.728
0.861 0.630
0.137 0.144 0.129
0.97] 0.944 1,000 0911 0.91 0.932
FCVAE
0.654 0.913 0.803 0.53 0.91] 0.778 0.924 0.929 0.919 0.51 0.894 0.82
0,905 0.757 0,859 0,995 0.562
0.13
0.972 B.94 1,000 0.899 0,888 0.910
KAN-AD
0.79 0.644 0.726 0.861 0,830
0.71 0.912 0.919 0.005 0.0
744
0.237
03756
0.881 0.963
0.60
0:540
0.572
0.993
0.981
0.900
0.800 6.911
CS-LSTMs
0.885 0,919 0.533
6.678
0.915
0.555
0.836
0.915
0.955
0.879
0.905
0.855
8.918
0.898
0.921
0.57
0.991
0.825
0.99%
0.992
1,000
0,918
0.936 0.901
effectiveness and convergence in the appendix F
(x mask f mask 11)
D(e, 2, = log o² +
(6)
Ø²
4.2.1 THE OVERALL FRAMEWORK
Both branches predict future values from historical inputs. Let U, S-LSTM(22) and Pe: de C-LSTM((z) denote the predicted values from each branch, respectively. Their respective losses, L. and Le, computed based the discrepancy between predicted values and actual future points, respectively. The total loss is designed as follows:
L (ps, Ts Js, Z, Le = De, L- Le
As shown in Figure 8 the S-LSTM branch segments the time series into windows of size N x Was transforms them into the frequency domain the Fourier Transform, and predicts the next cycle. The result is converted back to the time domain using the Inverse Fourier Transform to produce lss and a. As shown in Figure in the C-L.STM branch, the series is divided into overlapping windows of size N x we, with we slightly larger than the stride to retain local patterns. These are concatenated into be and processed by C-LSTM to yield pe and de.
EXPERIMENTS
5.1 EXPERIMENTS SETTINGS
5.1.1 DATASETS
To evaluate the effectiveness of our proposed framework, we conducted experiments on four benchmark datasets. The Ya- hoo dataset (n. d., () an open dataset specifically de- signed for anomaly detection, released by Yahoo Labs. The KPI dataset (Li et al. (2022a) consists of data collected from eBay, Baidu, Tencent, and Ali Ali-capturing key performance indicators (KPI). The WSD dataset (n. d.j a) contains real-world KPIs collected from three prominent Internet companies: Baidu, Sogou, and eBay, representing large-scale web service usage. Finally, the NAB dataset (Lavin & Ahmad 2013), created by Numenta, is an open benchmark designed to evaluate the performance of TSAD algorithms. For all publicly available datasets, we di- vide them into 35% training set, 15% validation set, and 50% testing set.
Table 2: Dataset statistics. "To- tal Length" indicates the total num- ber of time series data. "Anomaly Ratio" represents the proportion of anomaly points in the entire time series data.
five major internet companies-Soupe
Datasets
Total Length
Anomaly Ratio
Yahoo KPI
573K
0.68%
5923K
2.26%
WSD
7511K
1.60%
NAB
159K
9.89%
5.1.2 BASELINES
We compare CS-LSTMs against ten representative methods: SPOT Siffer et al., 2017). SR- CNN (Ren et al, 2019), TFAD (Zhang et al., (222a),, DONUT (Xu et al.) 2018). Informer (Zhou al. 2021), Anomaly- Transformer moomaly-Transformas (Xu et al. 2021). AnoTranseer (Zhang al. 2022b).

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Published as a conference paper at ICLR 2026
Table Comparison of transferability among five mainstream models. The left side reports results when trained on Yahoo and tested on KPI, WSD, and NAB; the right side shows results when trained on KPI and tested on Yahoo, WSD, and NAB. F1 denotes the best FI score, P denotes Precision, and R denotes Recall.
Yahoo WSD
KPI
Methud
KPI
NAB
Yaboo
WSD
NAB
F1
P
R
F1
P
R
F1
P
R
F1
P
R
F1
P
R
F1
P
R
Informer
0.901
0.895
0.907
0.571
0.552
0.591
0.969
0.970
0.971
0.536
0.552
0.521
0.564
0,545
0.584
0.971
0,964
0.978
TEAD
0,810 0.815
0.805
0.615
0.554
0.690
0.714
0.729
0.699
0.594
0.864
0.453
0.635
0,570
0.717
0.727
0.751
0.706
Mnomaly-Transsorrne
0.791 0.701
0.746
0.616
1.000
0.408 0.576 0.435
0.731
0.598
0.918
FOVAE
0.915 0.914 0.916
0.829
0.916 0.739
0.968
0.938 1.000
0.574 0.838 0.431
0.859
0.833
0.886
0.967
0.935 1.000
KAN-AD
0.905 0.906 0.903
0.743
0.883
0.641
0,991
0.982
1.000
0.507
0.439
0.601
0.754
8.893
0.652
0.990
0,981 1.000
CS-LSTMs
0.920
0.670
0.927
0.980
1,000
VQRAE (iiuu et al. 2022). KAN-AD (Zhou et al. 2024) and FCVAE (Wang et al. 2024 SPOT
statistical method based on extreme value theory. SRCNN and TEAD are supervised methods requiring high-quality labels. DONUT, VQRAE. and Ano are unsupervised reconstruction- based VAEs. Informer is a prediction- -based model using attention, while Anomaly-Transfore applies Transformers for unsupervised outlier detection. FCVAE leverages frequency-conditioned VAEs for sequence reconstruction, treating all points in a window as normal.
5.1.3 EVALUATION METRICS
In industrial scenarios, anomalies often appear as segments rather than isolated points, making point-wise precision less critical. Fol- lowing prior work (Xu et al. 2018 Ren et al., 2019), we use two metrics: Best F1 and Delay FI, both employing point adjustment strategies. Best F1 counts a segment as detected any point within it is identified, while Delay Fl requires detection within maxi- mum waiting time k with smaller k enforcing stricter evaluation.
0
Figure 6: Illustration of the ad- justment strategy.
We set dataset-specific h values based on complexity and noise: Yahoo (k = 3) and KPI (k = 7) have accurate annotations: NAB (k 150) contains many continuous segment anomalies; WSD
(k 40) has substantial label noise around anomalies, requiring a moderate increase in l to reduce impact.
5.2 EFFECTIVENESS AND EFFICIENCY
Table compares CS-LSTMs with nine baselines. Our model consistently outperforms all competitors. achieving Best Fl gains of 3.1% 9, 1.2%. 10.5%, and 0.6%, and Delay Fl gains of 3.9%. 2.8%, 16.1%, and 0.7% across four datasets.
Table 4: Inference efficiency of 512 itera- tions.
GPU 3090 24G
Method
GPU Memory
GPU Time
CPU Time
Shown in Table 4 CS-LSTMs are also efficient, with 600K parameters less than half of SOTA's L.4M-reduiing inference time by 40%, enabling faster anomaly alerts. Baseline limitations in- clude SPOT (Siffer et al.) 2017) misclassifying mild anomalies, SRCNN (Ren et al., 2019) struggling with implicit features, Informer (Zhou et al.] 2021) missing frequency changes, Anomaly- Transformer (Xu et al. (002)) showing lower Delay FI, TEAD (Zhang et al. 2022a) suffering la- tency, and FCVAE (Wang al. 2024) failing subtle spikes gradual periodic shifts.
Informer
147 MB
2280 ms
2280 ms
TEAD
MB
Ano-Transformer
130 MB
1539 ms
1539 ms
FCVAE
37 MB
7.73 ms
7.73 ms
CS-LSTMs
44 MB
4.62
3.82
By jointly modeling time and frequency domain information and forecasting future normal points, CS-LSTMs more effectively detect subtle abrupt changes and smooth periodic variations.
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5.3 TRANSFERABILIT EXPERIMENTS
In industrial scenarios, retraining TSAD models for each new domain is costly and often restricted by privacy concern,, making cross-domain transferability essential. Considering that WSD and NAB contain significant label noise, we select Yahoo and KPI as source datasets to train models on these datasets, and directly test them on the remaining three to evaluate their robustness under varying anomaly types and noise levels.
As shown in 3 we selected the best performing model Table I for the transferabilityy ex- periment. Baseline models including Informer (Zhou et al., (021). TFAD (Zhang et al. (022a)) Anomaly-Transfomne (Xu et al.] 202T), and FCVAE (Wang et al.] 2024). suffer significant perfor- mance drops unseen datasets, while CS-LSTMs consistently achieve the best Fl scores across all transfer tasks. This advantage stems from its dual-branch design: S-LSTM captures global peri- odic patterns in the frequency domain, while C-L.TTM models local trend variations via overlapping windows. Combined with unsupervised and noise-resilient training, CS-LSTMs demonstrate strong cross-domain adaptability and practical deployment potential.
5.4 ABLATION STUDY
To validate our model design, we conduct ablation studies on three key components and the window size. Results show that each component significantly contributes to improved performance in time series prediction and anomaly detection. Meanwhile, the window size also has a certain degree of influence the model's capability. The appropriate sizes of the seasonal window and the contextual window can improve the model's performance.
5.4.1 KEY COMPONENTS
We conducted ablation studies on different components of CS-LSTMs, including the co-branch architecture, covariate, and the noise decomposition module. Regarding the architecture, we eval- uated the S-LSTM branch for modeling periodic patterns and the C-LSTM branch for capturing local dynamics. Using either branch alone yielded inferior results compared the combined model, indicating their complementarity. Meanwhile, the results show that incorporating covariates leads to better performance than excluding them, since frequency-domain analysis can capture period- icity, while temporal information provides complementary local variation features. For the noise decomposition module, the results demonstrate that this design facilitates the modeling of normal patterns and enhances anomaly detection performance. Overall, the results in Table 5 confirm that the dual-branch architecture, the introduction of covariates, and the noise decomposition strategy are all indispensable for achieving superior performance.
Table Ablation experiments across Yahoo, KPI, and WSD datasets.
Yahoo
KPI
WSD
NAB
Method
Delay
Delay
Delay
Delay
wño C-LSTM
0.864
0.862
0.923
0.849
0.856
0.725
0.985
0.905
w/o S.LSTM
0.717
0.768
0.904
0.809
0.762
0.431
0.987
0.902
w/o Covariate
0.826
0.816
0.925
0.850
0.840
0.576
0.986
0.901
w/o Noise Decomposition and Mask
0.868
0.858
0.913
0.864
0.858
0.812
0.972
0.897
CS-I.TMss
0.885
0.878
0.936
0.879
0.910
0.857
0.996
0.918
5.4.2 DENOISE STRATEGY
To evaluate the impact of different denoising strategies, we conducted set of ablation studies. Specifically, we tested two alternative approaches, Pooling Decomposition and STL Decomposition, as substitutes for our proposed denoising module, and compared all three strategies across four benchmark datasets. As shown in Table 6 our method consistently achieves the best anomaly- detection performance on all datasets. Table 7urther further reports the per-epoch training time under each strategy, where approach again demonstrates superior efficiency.
Upon analysis, we find that Pooling Decomposition overly coarse-grained although fast, it strug- gles to capture fine-level temporal structure, leading suboptimal accuracy. In contrast, STL.

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Table Comparison of denoising strategies across Yahoo, NAB, WSD, and AIOPS datasets.
Yahoo
NAB
WSD
AIOPS
Method
Best
Delay
Best
Delay
Best
Delay
Best
Delay
Noise Decomposition
0.878
Pooling Decomposition
0.872
0.860
0.988
0.904
0.865
0.812
0.923
0.871
STL Decomposition
0.877
0.871
0.989
0.890
0.874
0.816
0.914
0.875
Decomposition offers more fine-grained modeling but incurs substantial computational overhead. Overall, our denoising module strikes a more favorable balance between accuracy and efficiency, achieving the best performance among all compared methods.
Table 7: Training time per epoch under different denoising strategies.
Method
Time
Noise Decomposition
52.383s
Pooling Docompsition
59.352s
STI Decompasition
458.389s
5.4.3 WINDOW SIZE
We also study how window sizes affect CS-LSTMs' performance across four datasets (Yahoo, KPI, WSD, NAB), as shown in Figure The seasonal branch (S-LSTM) benefits from windows spanning a full period, improving frequency-domain modeling, but degrades when longer windows introduce noise. The context branch (C-I.STM) improves with larger windows until an optimal point, after which fine-grained changes are diluted. Overall, CS-LSTMs remain robust across window sizes, achieving strong Fl scores. In practice, periodicity guides seasonal window selection, while context windows show good cross-dataset generalizability, reducing manual tuning.
3 2,m
6.00
5.100
030
0.90
20mm
2000
250
030
500
(a) Yahoo
(b) KPI
(c) WSD
(d) NAB
Figure 7: Impact of window sizes Fl score. Top: Varying context window (blue) with fixed
seasonal window. Bottom: Varying seasonal window (orange) with fixed context window.
CONCLUSION
This paper presents CS-LSTMs, novel unsupervised prediction-based method for anomaly detec- tion UTS. The model adopts a dual-branch architecture: S-LSTMM captures long-term periodic pat- terns via rrquenncydomail modeling with FFT, while C-LSTM focuses short-term local trends through context dependencies. The combination of the two branches improves the capabilities the model. We introduce noise decomposition during training reduce noise from abnormal points and propose a "normal distance" metric to quantify anomaly severity. CS-LSTMs address key challenges in UTS anomaly detection by effectively modeling both periodic structures and local variations. It achieves SOTA results on four public datasets (Yahoo, KPI, WSD, NAB), and improves inference efficiency by 40%.
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ACKNOWLEDGEMENTS
This work was partly supported by New Generation Artificial Intelligence- National Science and Technology Major Project (20252D0123503), NSFC under No. U2441239, U24A20336, 62172243, 62402425, 62402418, 62502432 and 62502433, the China Postdoctoral Science Foundation under No. 2024M762829 and 2025M781522, the Zhejiang Provincial Natural Science Foundation un- der No. LD24F020002, the "Pioneer and Leading Goose" R&D Program of Zhejiang under No. 2025C02033 and 2025C01082.
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APPENDIX
A HYPERPARAMETER SETTINGS
To ensure reproducibiliity Table 8 summarizes the hyperparameter settings used in all experiments. CS-LSTMs adopt dataset-specific hyperparameters mainly due to differences in intrinsic informa- tion density (c.g., Yahoo: 1-hour sampling, NAB: 5-minute sampling. WSD/AIOps: 1-minute sam-
pling). Similar practices are followed in FCVAE (Wang et al. 2024) and TFAD (Zhang et al., 2022ah.
In practice, only three byperparameters need to be set: fotal.windowwsie size (input sequence length), seasonal window size (S-LSTM window), and contextual window size (C-LSTM window). The first and last can be derived from seasonal window size using simple overlapping windows, without requiring extensive tuning. Practical rules are:
To capture periodicity evolution:
total 1 _wii ndow_s ze = m
m E (5,7)
To capture contextual variations:
sessonal n (5,7)
These settings consistently yield stable performance across datasets. As shown in Figure 7 perfor- mance varies by only about 2% within a reasonable range of window sizes, indicating robustness and ease of deployment.
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Table 8: Hyperparameter settings used in all experiments.
Hpperparameter batch max epochs
Value
30
gesonal_windon ssize
48
lotal sw indoow siize
240
wontext_winddww _size
4
d.model
256
num layers
1
FFT
Step We
X glow
X, Rnxw,
X, RToKi,
Z5 Rix(le+m))
Figure 8: Architecture of S-LSTM.
B ARCHITECTURE OF C-LSTM AND S-LSTM
C LSTM
LSTM is special type of Recurrent Neural Network (RNN) (Eman 1990) designed to address the problem vanishing gradient, which is difficult for traditional RNNs capture long-term depen- dencies. LSTM is effective in handling time series data, natural language processing tasks, and other problems that involve temporal dependencies. The key components of LSTM are the Memory Cell and Gating Mechanism, which allow the network to selectively retain or discard information. Forget Gate: The forget gate determines which information in the memory cell should be discarded. The mathematical expression is:
ft = (W [M4-1, ma] + by)
(8)
where fi the output of the forget gate, and represents the sigmoid activation function. Input Gate: The input gate decides what new information should be added to the memory cell:
i = (W, [/4-1, 27] + b:)
The input gate is combined with the candidate memory cell:
C. tanh(w, [/4-1; 22] | + b.)
(10)
Memory Update: The memory cell is updated by combining the forget gate and input gate:
C. = ft Ct-1 + i C.
(11)
Output Gate: The output gate controls the information output at the current time step:
0t a (W.. [A,-1, mx] + b.)
(12)
The final output is:
h tCnh
(13)
D NOISE DECOMPOSING WITH MAD-BASED THRESHOLDING
To mitigate noise in one-dimensional time series, we adopt wavelet shrinkage with thresholds esti- mated via the Median Absolute Deviation (MAD). MAD uses the median instead of the mean, min- imizing the impact of extreme values on dispersion. Let x = [rr) denote the observed signal, which is assumed be corrupted by additive white Gaussian noise. The signal first decomposed into approximation and detail coefficients using discrete wavelet transform (DWT):
xtt)
DWT
() 1.(o G
A
where c are approximation coefficients at level j. ( are detail coefficients, and L is the decom- position level. High-frequency detail coefficients (6) predominantly capture noise.
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FFT
Step & < We
X glxw
X E RunM:
Xe E Ranse
Ze E Ruxtletad
Figure 9: Architecture of C-LSTM.
Noise estimation via MAD. Generally, assume that the noise follows normal distribution, the noise standard deviation o; is estimated from the detail coefficients cD by
median
0 = + (0.75)
Vi vi=1,... 1,..., L.
(14)
The Median Absolute Deviation (MAD) is defined
MAD(z) = median (|x - median(x)))
and serves as robust alternative the standard deviation. To calibrate MAD under Gaussian noise, consider Z ~ NN(0, 1). Then
MAD(Z) median(1Z))
Let Y Z|. Its distribution satisfies
P(Y E) = P 2 < 21 -
where (-) denotes the standard normal cumulative distribution function. The median of Y solves
(m( 1 0.5
(mm) 0.75
yielding
m (0.75) ~ 0.6745
Thus, for Gaussian noise with variance d²
MAD z 0.6745g
To obtain a consistent estimator the noise standard deviation, MAD normalized as
MAD
σ Φ (0.75)
Hence, constant (0.75) arises as the 0.75-quantile the standard normal distribution, en- suring that MAD yields asymptotically unbiased estimate of under Gaussian assumptions.
Universal threshold. Following VisuShrink, the universal threshold (Donoho Johnstone 1994 defined as
X; = logn, =i= 1,..,
(15)
where is the signal length. This threshold is then applied to each set of detail coefficients.
We consider wavelet coefficients {2) )ー= generated purely by Gaussian noise, i.i.d. 2, ~ W0
and define the maximum absolute coefficient as Mn := maXişjss 1zyl- To control the probability that any noise coefficient exceeds a threshold & 0, we apply the union bound:
Pr(M, > t) S
Pr(1z,] > t) = n Pr[ > t/o), ~~0((11)
2 -222 and invoke the standard Gaussian tail bound Pr([Z) > u) S V
yielding
20
22
Pr(M, > 1) S m
exp
π
2
17

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Published as a conference paper at ICLR 2026
Setting & o 0 leads to
Pr (Mr > a / log $
log n-
which implies that, with high probability, all noise coefficients remain below o / log n. Equiva- lently, threshold of this order asymptotically removes coefficients generated solely by noise. From the perspective of extreme-value theory, the maximum of n i.i.d. Gaussian samples satisfies
Mn /v 20g + O log log n
Vlog
in probability, and, after centering and scaling, converges in distribution the Gumbel law. The leading term 2 log n thus captures the characteristic growth of the maximum and justifies the scaling of the universal threshold.
Practical implications follow directly: coefficients exceeding À = 2 n are unlikely to orig- inate from noise, while smaller coefficients are predominantly noise and can be suppressed. The universal threshold is therefore theoretically grounded, yet conservative; it guarantees noise removal may introduce over-smer weerssoothing by discarding small yet informative coefficients. Adaptive alter- natives, including SURE-base based or level-dependen thresholds, can mitigate this bias. Refinements such as / log n log log n further improve finite-sample accuracy, while level-dependent thresh- olds A; o.v/ threshold arises from controlling the maximal deviation of Gaussian noise coefficients, providing a principled and asymptotically optimal rule to discriminate between noise and signal in the wavelet domain.
2 log accommodate different variances across wavelet scales. Overall, the universal
Thresholding operator.
For coefficient c, the soft-thresholding operator is given by
(8) e sign(c)) ( max (1o81-Ar,
Nr, a). Vi L
(16)
which shrinks small coefficients towards zero while preserving larger structures. Hard-thresholding can also be applied, though it may introduce discontinuities.
Reconstruction.
The denoised signal is reconstructed via the inverse wavelet transform (IDWT):
*tt) IDWT (cf, 121
Implementation.
In practice, this procedure can be implemented using the Algorithm 1
Algorithm 1: Noise decomposing with MAD-based Thresholding
Input: Noisy signal & E R wavelet basis 1h decomposition level L Output: Denoised signal à (T)
{CA, D ..... ..........
(1)
wavedec(a, L)
for i f- to do
median(Ic
f + (0.75)
/* MAD noise estimate */
A f- 0i / logn:
* Universal threshold
(8)
sign( (
max
(1 (8) X,, 0):
/* soft-threshholing */
f <- waverec (c,, D ..... (1)
(L)
return à
This approach effectively suppresses high- figh-frequency noise while preserving salient temporal struc- tures, making it well-suited for anomaly detection and other downstream time series tasks.
18

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Published as a conference paper at ICLR 2026
E EVALUATING CS-LSTMS UNDER KEY CHALLENGES
To further demonstrate the effectiveness of CS-LSTMs, conduct case study focusing the two key challenges previously discussed. Specifically, we evaluate the model's performance in two representative scenarios: (I) detecting Small Point Anomalies, which relates to the challenge of Capturing Local Trends, and (2) detecting Slowly Rising Anomalies, which reflects the challenge of Capturing Periodicity and Its Evolution.
As shown in Figure 10 we compare the detection results of CS-LSTMs with those of a state-of-the- art baseline, FCVAE (Wang al. 2024). this figure, black lines represent normal time points, blue lines indicate correctly detected anomalies, and red lines mark missed anomalies. FCVAE (Wang al. (024) struggles simultaneously detect both point anomalies and segment anomalies within the same time series data. In contrast, CS-LSTMs achieve precise detection in both scenarios, successfully identifying anomalous points and segments.
This superior performance is attributed to the model's dual-branch design: the C-LSTM branch ef- fectively captures localized patterns and short-term variations, enabling accurate detection of muta- tional anomalies; meanwhile, the S-LSTM branch models the evolving periodic structures in the fre- quency domain, allowing the model to capture complex changes in periodicity and identify segment- level anomalies. These capabilities directly address the limitations highlighted in Challenge 1 and Challenge 2, thereby explaining the overall advantage of CS-LSTMs over prior methods.
-075-
-104
2.5-
-1.5
0.0
CS-LSTMs undetected points
(b) CS-LSTMs undetected segments
0.
-05
-100
-125
-10
-15
(c) FCVAE undetected points
(d) FCVAE undetected segments
Comparison of CS-LSTMs and FCVAE detecting specific anomaly types. Normal points are shown in black, detected anomalies in blue, and undetected anomalies in red. Subfig- ures (a) and (b) present the results of CS-LSTMs on Small Point Anomalies and Slowly Rising Anomalies, respectively, while (c) and (d) illustrate the corresponding results from FCVAE.
Figure 10:
F Loss FUNCTION
In this section, investigate the impact of loss function design model performance, with particular focus on the application NLL and anomaly masking. We conduct ablation studies com- paring commonly used loss functions, including MAE and MSE, with the NLL employed in this work. As shown in T the results validate the effectiveness of our proposed improvements, sig- nificantly enhancing the model's ability capture normal patterns and accurately detect anomalies.
Table 9: Branch ablation experiments across Yahoo, KPI, and WSD datasets.
Yahoo
KPI
WSD
NAB
Method
Best
Delay
Best
Delay
Best
Delay
Best
Delay
MSE
0.722
0.676
0.828
0.495
0.852
0.604
0.985
0.889
MAE
0.712
0.695
0.909
0.757
0.850
0.602
0.983
0.884
NLL
0.885
0.878
0.936
0.879
0.910
0.857
0.996
0.918
19

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Published as a conference paper at ICLR 2026
We consider the noise-decomposed Negative Log-Likelihood (NLL) loss for time series of length
72:
o(,, Σ log of (x) C mask, + t O mäsk, me)
(17)
of
where Ih" denote the original and denoised sequences, μ., R the predicted means and standard deviations, and mask E {0, 1 indicates normal points.
Defining the residual vector
mask + X O mask μ
The loss can be compactly rewritten as
2
0(,,,,x,8)
Σ log of +
Optimal mean. For fixed Ø, the loss is strictly convex in with gradient
OD
-
0p
o²
Setting this to zero yields the unique global minimizer
mask + & @ mask
(18)
Intuitively, the optimal mean corresponds the hybrid sequence in which original values are tained at normal points and replaced by denoised estimates at abnormal points.
Optimal standard deviation. For fixed μr, each oe can be optimized independently, Differentiat- ing with respect to 8t = of and solving ös, (log 8t + r9 /s:) = gives
ril
Hence, the optimal standard deviation at each time step equals the absolute residual between the hybrid sequence and the predicted mean. In practice, a small lower bound Fmin > is imposed to prevent degenerate solutions.
Extension to composite loss. For the combined seasonal and contextual loss
L L, + Le D(ps, x, D(ue, Ce, x, x)
the global optima (1i, o and (ne, o:) are obtained independently using the same elementwise formulae.
Convergence remark. The loss is smooth for a; Guin > 0, strictly convex in μ., and coercive in Standard first-order optimization methods with appropriate step sizes converge stationary points, which coincide with the elementwise global minimizers. Consequently, the masked Gaussian NLL admits well-defined and attainable optima under the hybrid masking scheme.
G TRAINING EFFICIENCY EVALUATION
To further evaluate training efficiency, we conducted additional experiments recording resource con- sumption for training one epoch under the same dataset and experimental settings as the baselines, including overhead introduced by windowing and FFT operations. The results shown in Table I0 demonstrate that CS-LSTMs maintain competitive training efficiency compared to strong baselines. Despite the additional computations for Noise-Decaa DiiiseeDccooosttionnn processing, the lightweight design of both C-LSTM and S-LSTM branches, together with efficient batching strategies, ensures that the overall training time per epoch remains advantageous.
20

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Published as a conference paper at ICLR 2026
Table 10: Training efficiency comparison of CS-LSTMs and baseline models.
Model
CPU Time
GPU Time
GPU Memory
CS-LSTMs FOVAR
39
MB
Informer
$
316
Anotramsformer
483 s
483
TFAD
1606
75 s
4048 MB
H ADDITIONAL EVALUATION ON MEDICAL AND ENVIRONMENTAL. DATASETS
To further address the reviewer's concern regarding the generalizability of the proposed method, conducted additional experiments on two widely used non-IT univariate time series datasets from the UCR Archive: ECG (medical) and CIMIS44AirrTemperatuue (environmental temperature).
Although CS-LSTMs are primarily motivated by AIOps scenarios where univariate anomaly detec- tion central challenge, our method is designed to model both periodicity evolution and local contextual variations, which naturally extend to diverse time series domains. Consistent with results on IT monitoring datasets, CS-LSTMs continue to outperform strong baselines on these two new domains, demonstrating strong cross-domain robustness. The detailed performance comparison is shown in Table IT
Table 11: Performance comparison ECG and CIMIS444Airfeemperatuue.. F1 denotes the best F1
score, P denotes Precision, and denotes Recall.
ECG
CIMIS44AirTeepeertuure
Method
Best
Delay
Best
Delay
F1
P
R
F1
R
F1
P
R
F1
R
CS-LSTMs
0.918
0.849
1,000
0.823
0.699
1.000
0.950
0.910
0.995
0.809
0.883
0.746
FCVAE
0.914
0.842
0.819
0.735
0.820
0.667
0.850
0.741
0.995
0.850
0.741
0.995
Anomaly-Tramm Transformer
0.594
0.741
0.496
0.153
0.218
0.118
0.810
0.805
0.815
0.313
0.638 0.207
TFAD
0.835
0.783
0.894
0.713
0.684
0.745
0.772
0.718 0.835
0.756
0.733 0.780
Informer
0.736 0.682
0.799
0.715
0.658
0.783
0.842
0.768 0.932
0.836
0.771 0.913
KAN-AD
0.916 0.846 1.000
0.797 0.991 0.667
0.842
0.855 0.870
0.842
0.815 0.870
AnoTraanser
0.695 0.803 0.613
0.545
0.525
0.567
0.562
0.752 0.449
0.540
0.608 0.486
VQRAE
0.316 0.524 0.226
0.145
0.185
0.119
0.504
0.688 0.398
0.494
0.672 0.391
DONUT
0.513 0.420 0.391
0.445
0.405
0.494
0.255
0.410 0.185
0.255
0.410 0.185
SRCNN
0.740 0.705 0.779
0.636
0.602
0.674
0.285
0,310
0.264
0.212 0.246 0.186
SPOT
0,412 0.932 0.264
0.136
0.885
0.074
0.432
0.560
0.352
0.448
0.542 0.382
I THE USE OF LLMs
The authors affirm that throughout this study, LLMs were solely for the translation and polishing of the manuscript. L.LMs were not involved in the literature search for the related work section in the formulation of the research idea. The authors take full responsibility for this declaration.
21


Reviews:
[{"review_id": "KXYl1dj13h", "reviewer": "Reviewer_3pdC", "summary": "This paper addresses anomaly detection in univariate time series (UTS) by identifying three key challenges that limit existing methods' ability to detect subtle anomalies. The authors propose CS-LSTMs, a dual-branch framework where S-LSTM captures long-term periodic patterns through frequency-domain analysis and C-LSTM models short-term local dynamics. A noise decomposition strategy is introduced to improve robustness against unlabeled anomalies during training. Experimental results on four benchmark datasets demonstrate that CS-LSTMs achieves superior F1 scores compared to ten baseline methods while reducing inference time.", "questions": "- How is the decomposition level L in Equation 2 determined Is it already set as hyperparameters? or is it just automatically derived based on discrete wavelet transform (DWT)?\n- In equation 7, there’s Kronecker product in the equation while Appendix E shows element-wise multiplication for the same operation. Which is correct? Couldd the author please clarify and ensure consistent notation.\n- In Section 5.4.2, the presentation is confusing. If the top graphs vary seasonal window with fixed context window (and bottom panels do the reverse), why do the fixed window values differ across datasets (e.g., context=4 for Yahoo vs. context=48 for NAB)? Could the author please clarify the experimental setup and using different baseline window sizes across datasets.\n- Table 4 reports overall inference time but doesn't break down the wavelet decomposition overhead. How much time does the noise decomposition step consume relative to the dual-branch prediction?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- **Systematic analysis of detection failures in existing methods.** The paper identifies why current approaches fail on small point anomalies and slowly rising segment anomalies, attributing this to inadequate integration of local trends with periodic variations.\n- **Principled noise decomposition strategy.** The wavelet-based approach effectively filters noise while preserving trend and seasonal components, enabling robust learning of normal patterns despite unlabeled anomalies in training data.\n- **Well-motivated dual-branch architecture.** S-LSTM captures long-term periodic evolution via frequency-domain analysis while C-LSTM models short-term local dynamics, providing complementary temporal perspectives that enhance detection capability.\n- **Strong empirical results with efficiency gains.** CS-LSTMs consistently outperform ten baselines across four datasets while reducing inference time and using fewer parameters, demonstrating practical deployment value.", "weaknesses": "- Equation 6 appears to rely on the function or definition presented later in Equation 7. For clarity and logical coherence, the authors should introduce Equation 7 before Equation 6.\n- The paper lacks critical implementation details including specific hyperparameters such as  batch size, number of training epochs,window sizes, decomposition Level L.\n- The paper provides limited explanation for why LSTMs are better over other recent architectures (Transformers, state-space models) that have also demonstrated superior performance on time series tasks.\n- While Section 5.4.1 shows noise decomposition helps, the paper doesn't compare the MAD against other alternatives (pooling-based decomposition ,Robust STL or recent decomposition approaches).\n- All datasets come from web/IT monitoring domains, limiting generalizability claims to other time series applications (medical, financial, environmental).", "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": "uIWLFSAod5", "reviewer": "Reviewer_S6ae", "summary": "The paper introduces CS-LSTMs, a dual-branch prediction framework for detecting subtle anomalies in univariate time series (UTS). It consists of two complementary LSTM branches, S-LSTM (Seasonal branch) captures evolving periodicity via wavelet-based time–frequency representations, and C-LSTM (Contextual branch) models short-term dependencies and local dynamics in the time domain.\n\nA wavelet transform first decomposes the signal into trend, seasonal, and noise components, allowing each branch to process denoised inputs. Their outputs are fused for final anomaly prediction, optimized by a masked probabilistic loss that focuses learning on normal patterns while suppressing anomalous effects.\n\nAcross multiple benchmarks, CS-LSTMs consistently outperform reconstruction- and prediction-based baselines. Ablation and transfer experiments verify that the dual-branch design and noise-decomposition mechanism jointly enhance robustness and generalization. The approach effectively overcomes prior models’ difficulty in detecting small point and gradually rising anomalies by integrating time- and frequency-domain representations and modeling the evolution of periodicity.", "questions": "- How exactly does the wavelet-based noise decomposition feed into the dual-branch architecture? Need clear data flow and mathematical formulation.\n- How are mask and ~mask generated and applied in the loss function to handle anomalies during training?\n- How is ‘evolution of periodicity’ defined and quantified within the S-LSTM branch?\n- Why does the paper choose LSTM over Transformer-based alternatives (Informer, Autoformer) for subtle anomaly modeling?\n- Can CS-LSTMs handle multivariate or cross-correlated series, and what modifications would be required?\n- Recent works such as Dual-TF (Nam et al., 2024), TFAD (Zhang et al., 2022), FAD (Li et al., 2023), and WaveletAE (Zhao et al., 2020) also integrate time–frequency representations for anomaly detection. How do CS-LSTMs differ from these dual-TF or frequency-aware models in terms of architectural design, ability to capture evolving periodicity, and computational efficiency?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The separation of seasonal and contextual learning provides complementary modeling of periodic and local behaviors, enabling precise detection of small or slowly evolving anomalies.\n- The use of wavelet-based decomposition allows simultaneous handling of non-stationarity and noise suppression, improving interpretability and robustness compared with pure time-domain models.\n- Demonstrates consistent superiority over SOTA baselines across several datasets, supported by transferability and ablation experiments showing each module’s distinct contribution.\n- Addresses a realistic gap in anomaly detection—capturing weak, evolving irregularities that traditional reconstruction or forecasting models often miss.", "weaknesses": "- Key mechanisms—especially the noise-decomposition pipeline, fusion process, and masked loss formulation—lack detailed mathematical description and ablation justification.\n- The paper does not analyze how the method scales to multivariate time series or handles cross-variable dependencies.\n- Efficiency metrics are presented in relative terms without absolute computational cost (e.g., GPU time, memory).", "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": "iNpMHjvMrC", "reviewer": "Reviewer_Bshn", "summary": "The paper identifies key shortcomings of prior approaches to univariate time-series (UTS) anomaly detection and proposes CS-LSTM. This dual-branch architecture explicitly separates long-term seasonal dynamics from short-term contextual fluctuations. By assigning distinct roles to the two LSTM branches, the method aims to detect both *small point anomalies* and *slowly rising interval anomalies.* Across several benchmarks, the approach is reported to outperform prior methods in both accuracy and runtime.", "questions": "- **Training efficiency.** Table 4 reports inference latency. How does training time compare to strong baselines under the same hardware and data conditions (including windowing/FFT overhead)?\n- **Mask estimation.** Please describe the procedure to construct the mask used in the masked Gaussian NLL in full detail (initialization, update rule, thresholds/criteria).", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- **Clear problem framing**. This paper is, to my knowledge, the first to explicitly explain the two challenges of *small point anomalies* and *slowly rising anomalies* in UTS, and to analyze why popular reconstruction- and prediction-based baselines struggle with them.\n- **Probabilistic predictions**. Instead of deterministic outputs, the model predicts $\\mu$ and $\\sigma$, allowing $\\sigma$ to act as a data-driven tolerance band and enabling a principled anomaly score.\n- **Architectural novelty with complementary roles**. The design assigns seasonal/long-term dynamics to the S-LSTM and contextual/short-term dynamics to the C-LSTM, yielding strong empirical performance. The dual-branch decomposition is simple, interpretable, and appears novel in this setting.", "weaknesses": "- **Generalization beyond the targeted anomaly types.** The evaluation focuses on cases where the two targeted anomaly types are present. It remains unclear how the model behaves when anomalies fall outside these categories (e.g., regime shifts without seasonality, changes in variance, or adversarial bursts).\n- **Scope limited to univariate data.** Many real-world series are multivariate. It is not apparent how this UTS-specific design would extend to multivariate settings.\n- **Hyperparameter sensitivity.** The method introduces several windowing hyperparameters (e.g., $w_{s}$, $w_{c}$, $k$). The relative importance of these choices, guidance for tuning, and robustness across datasets and tasks are not fully clarified; optimal values are likely to vary substantially across domains, which may complicate deployment.", "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)