COGNOS: Universal Enhancement for Time Series Anomaly Detection via Constrained Gaussian-Noise Optimization and Smoothing

Wenlong Shang; Shihao Tian; Xutong Wan; Peng Chang

cs.LGNov 10, 2025

COGNOS: Universal Enhancement for Time Series Anomaly Detection via Constrained Gaussian-Noise Optimization and Smoothing

Authors: Wenlong Shang, Shihao Tian, Xutong Wan, Peng Chang

Organizations: School of Information Science and Technology, Beijing University of Technology, China · Beijing Key Laboratory of Computational Intelligence and Intelligent System, Beijing University of Technology, China

Abstract

Reconstruction-based methods are a dominant paradigm in time series anomaly detection (TSAD), however, their near-universal reliance on Mean Squared Error (MSE) loss results in statistically flawed reconstruction residuals. This fundamental weakness leads to noisy, unstable anomaly scores, hindering reliable detection. To address this, we propose Constrained Gaussian-Noise Optimization and Smoothing (COGNOS), a universal, model-agnostic enhancement framework that tackles this issue at its source. COGNOS introduces a novel Gaussian-White Noise Regularization strategy during training, which directly constrains the model's output residuals to conform to a Gaussian white noise distribution. This engineered statistical property creates the ideal precondition for our second contribution: Adaptive Residual Kalman Smoother that operates as a statistically robust estimator to denoise the raw anomaly scores. Extensive experiments on multiple benchmarks demonstrate that COGNOS consistently enhances the performance of state-of-the-art backbones significantly, validating the efficacy of coupling statistical regularization with adaptive filtering.

Figures & tables

Figure 1 : We analyze the TimesNet model on the SWaT dataset: (a) The resulting anomaly scores are highly noisy, leading to a high noise floor during normal operation which makes actual anomalies difficult to detect. (b) The autocorrelation plot shows significant temporal correlations persist in the residuals, suggesting that the model failed to capture all predictable patterns. (c) The Q-Q plot reveals that reconstruction residuals are strongly non-Gaussian, indicating that the underlying noise was poorly modeled.

Figure 2 : COGNOS: Backbones are trained with GWNR Loss, and ARKS is used during inference to generate stable anomaly scores

Models	Datasets	MSL		PSM		SWAN
Models	Metircs	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours
Autoformer	Std-F1	0.7724	0.9321	0.9098	0.9759	0.7336	0.7950
	Aff-F1	0.4008	0.9388	0.5292	0.8743	0.1828	0.3604
	R-A-R	0.6942	0.7084	0.6667	0.6954	0.9398	0.9433
	R-A-P	0.2366	0.2446	0.4851	0.5235	0.9362	0.9382
	V-R	0.6684	0.6884	0.6349	0.6650	0.9532	0.9542

Table 1: Multi-metric evaluation of anomaly detection performance between the Vanilla method and COGNOS (Ours) across three datasets. Best results are highlighted in bold .

Datasets	GECCO		MSL		PSM		SMAP		SWAN		SWaT		UCR
Models	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours
Autoformer	0.3929	0.9426	0.4008	0.9388	0.5292	0.8743	0.6266	0.8521	0.1828	0.3604	0.3967	0.9571	0.4702	0.7427
CrossAD	0.3115	0.9320	0.4135	0.9386	0.5904	0.8628	0.4871	0.7857	0.1829	0.5374	0.2973	0.9511	0.3383	0.7412
DLinear	0.4549	0.9384	0.5654	0.9432	0.7728	0.8615	0.6728	0.8437	0.2268	0.4402	0.4283	0.9542	0.4782	0.7325
KANAD	0.3785	0.9465	0.5360	0.9157	0.6223	0.8628	0.6666	0.8491	0.2066	0.3936	0.2905	0.9543	0.4409	0.7279
LSTMAE	0.3265	0.9463	0.5501	0.9460	0.6798	0.8629	0.6183	0.7770	0.2193	0.4329	0.4677	0.9521	0.4074	0.7328

Table 2: Comparison of anomaly detection performance between the Vanilla method and COGNOS (Ours) across seven datasets. The table reports the Affiliated-F1 metric. Best results are highlighted in bold .

Datasets				GECCO		MSL		PSM
GWNR	ARKS	MA	LP	Std-F1	Aff-F1	Std-F1	Aff-F1	Std-F1	Aff-F1
✗				0.4738	0.3785	0.8115	0.5360	0.8946	0.6223
✗	✓			0.4740	0.4845	0.8083	0.5285	0.9433	0.6689
✓				0.3440	0.4098	0.8704	0.5917	0.9160	0.4910
✓		✓		0.3531	0.4123	0.1626	0.2564	0.7979	0.1845
✓			✓	0.3539	0.4236	0.1662	0.2584	0.8006	0.2302

Table 3: Ablation Study of COGNOS Components and Filtering Methods. The best results are highlighted in bold , and the second-best are underlined . The full experiment results and more details can be found in the Appendix D.3 .

Figure 3 : Qualitative Analysis on Signal Purity and Statistical Physics. Top Row: Anomaly score dynamics on GECCO and PSM datasets. Bottom Row: Statistical diagnostics of reconstruction residuals.

Datasets	Models	Autoformer		KANAD		TimesNet
	Models	Vanilla	COGNOS	Vanilla	COGNOS	Vanilla	COGNOS
GECCO	Train	62.53	+41.648	39.94	+48.4	101.34	+171.49
	Inference	0.1782	+0.1274	0.0847	+0.1467	0.1694	+0.1513
PSM	Train	53.30	+82.66	97.31	+206.68	103.89	+257.55
	Inference	0.2150	+0.1031	0.2459	+0.1387	0.2054	+0.1327
SWAN	Train	87.64	+164.11	216.54	+355.01	153.57	+359.27

Table 4: Computational Efficiency Analysis. We report the average execution time (ms) per iteration (Training) and per sample point (Inference), (+) denotes the additional overhead introduced by our COGNOS framework. We set batch size

=128

and sequence length

=128

Figure 4 : The impact of the Confidence Level (

1-\alpha

) on KAN-AD backbone across the SMAP and SWAN datasets.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Category	Dimension	Training	Validation	Test	AR (%)
GECCO	Water treatment	9	55,408	13,852	69,261	1.25
MSL	Spacecraft	55	46,653	11,664	73,729	10.5
SMAP	Spacecraft	1	108,146	27,037	427,617	12.8
PSM	Server Machine	25	105,984	26,497	87,841	27.8
SWAN	Space Weather	38	48,000	12,000	60,000	23.8
SWaT	Water treatment	51	396,000	99,000	449,919	12.1

Appendix

Table 5 : Dataset Details

Miscellaneous Configurations
Datasets	Hyper-parameter						Traning Process
	Window Length	dModel	dMLP	Layers	Num Heads	Confidence	Learning Rate 6 6 6 Initial Learning Rate	Batch Size	Epochs
GECCO	128	128	128	3	8	0.9	1e-4	128	10
UCR									3
SWaT									10
MSL	96								3

Appendix

Table 6 : Hyperparameters Configurations

Panel of Figure 5 — Figure 5 : Residual analysis on the SWaT dataset using TimesNet. Left: vanilla method, Right: COGNOS.

Panel of Figure 6 — Figure 6 : Residual analysis on the PSM dataset using TimeMixer++. Left: vanilla method, Right: COGNOS.

Panel of Figure 7 — Figure 7 : Residual analysis on the SMAP dataset using CrossAD. Left: vanilla method, Right: COGNOS.

Panel of Figure 8 — Figure 8 : Qualitative comparison of anomaly scores on PSM using TimeMixer++. Left: vanilla method, Right: COGNOS.

Panel of Figure 9 — Figure 9 : Qualitative comparison of anomaly scores on GECCO using ModernTCN. Left: vanilla method, Right: COGNOS.

Panel of Figure 10 — Figure 10 : Qualitative comparison of anomaly scores on PSM using KAN-AD. Left: vanilla method, Right: COGNOS.

Models	Datasets	GECCO		MSL		PSM		SMAP		SWAN
Models	Metircs	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours
Autoformer	Std-F1	0.4826	0.7507	0.7724	0.9321	0.9098	0.9759	0.6847	0.7829	0.7336	0.7950
	Aff-F1	0.3929	0.9426	0.4008	0.9388	0.5292	0.8743	0.6266	0.8521	0.1828	0.3604
	R-A-R	0.9758	0.9825	0.6942	0.7084	0.6667	0.6954	0.5662	0.6109	0.9398	0.9433
	R-A-P	0.5918	0.5471	0.2366	0.2446	0.4851	0.5235	0.1796	0.1652	0.9362	0.9382
	V-R	0.9930	0.9920	0.6684	0.6884	0.6349	0.6650	0.5552	0.5882	0.9532	0.9542

Appendix

Table 7: Multi-metric evaluation of anomaly detection performance between the Vanilla method and COGNOS (Ours) across six datasets. Best results are highlighted in bold .

Datasets	Models	Autoformer		CrossAD		DLinear		KANAD		LSTMAE
Datasets	Metircs	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours	Vanilla	Ours
GECCO	Std-F1	0.4826	0.7507	0.4401	0.7228	0.4326	0.7400	0.4738	0.7466	0.4442	0.7619
GECCO	Aff-F1	0.3929	0.9426	0.3115	0.9320	0.4549	0.9384	0.3785	0.9465	0.3265	0.9463
MSL	Std-F1	0.7724	0.9321	0.7320	0.9315	0.8196	0.9322	0.8115	0.9165	0.8108	0.9320
MSL	Aff-F1	0.4008	0.9388	0.4135	0.9386	0.5654	0.9432	0.5360	0.9157	0.5501	0.9460
PSM	Std-F1	0.9098	0.9759	0.9256	0.9740	0.9582	0.9747	0.8946	0.9744	0.9138	0.9759

Appendix

Table 8: Comparison of anomaly detection performance between the Vanilla method and Ours (Ours) across seven datasets. The table reports the Standard-F1 and Affiliated-F1 metric. Best results are highlighted in bold .

Datasets				GECCO		MSL		PSM		SMAP		SWAN
GWNR	ARKS	MA	LP	Std-F1	Aff-F1	Std-F1	Aff-F1	Std-F1	Aff-F1	Std-F1	Aff-F1	Std-F1	Aff-F1
✗				0.4738	0.3785	0.8115	0.5360	0.8946	0.6223	0.6237	0.6666	0.7384	0.2066
✗	✓			0.4740	0.4845	0.8083	0.5285	0.9433	0.6689	0.6035	0.6024	0.7403	0.2152
✗		✓		0.3886	0.4259	0.1601	0.2530	0.8546	0.3179	0.3529	0.3040	0.7321	0.1792
✗			✓	0.3985	0.4292	0.1951	0.2650	0.8566	0.4108	0.4372	0.4340	0.7321	0.1778
✓				0.3440	0.4098	0.8704	0.5917	0.9160	0.4910	0.6027	0.2980	0.7348	0.1872

Appendix

Table 9: Ablation Study of COGNOS Components and Filtering Methods. The best results are highlighted in bold , and the second-best are underlined .

Datasets	Models	Autoformer		KANAD		TimesNet
Datasets	Models	Vanilla	COGNOS	Vanilla	COGNOS	Vanilla	COGNOS
GECCO	Train	62.53	+41.648	39.94	+48.4	101.34	+171.49
GECCO	Inference	0.1782	+0.1274	0.0847	+0.1467	0.1694	+0.1513
MSL	Train	65.32	+210.196	146.68	+354.92	97.57	+308.209
MSL	Inference	0.1829	+0.0831	0.2660	+0.144	0.1463	+0.1425
PSM	Train	53.30	+82.662	97.31	+206.687	103.89	+257.55

Appendix

Table 10: Computational Efficiency Analysis. We report the average execution time (ms) per iteration (Training) and per sample point (Inference). The value following “+” denotes the additional overhead introduced by our COGNOS framework. Note that the training overhead varies due to the data-dependent dynamic masking mechanism. We set batch size

=128

and sequence length

=128

Dataset	Precision	Recall	Std-F1	Aff-P	Aff-R	Aff-F1
PSM	0.9739	0.9550	0.9644	0.5429	0.7366	0.6251
MSL	0.9189	0.9515	0.9349	0.5187	0.9648	0.6746
SWaT	0.9024	0.9982	0.9479	0.5576	0.9760	0.7097
SMAP	0.9247	0.9745	0.9489	0.5125	0.9814	0.6733
GECCO	0.3737	0.4781	0.4195	0.5630	0.8612	0.6809

Appendix

Table 11: Anomaly Transformer performance under the unified quantile threshold protocol across five representative datasets.

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Time-Series Anomaly Detection Time-Series Classification Outliers Autoencoder Architectures

Open paper AskPDF arXiv

Time series anomaly detection (TSAD) has long been a hot research topic in data mining due to its various applications. Recent studies challenge the effectiveness of popular deep learning methods for TSAD, suggesting their failure in detecting subtle and prolonged anomalies. Outlier Exposure (OE) and Masked Autoencoder (MAE) emerge as two promising paradigms (classification and reconstruction) for solving the above problems. However, OE-based methods are constrained by poor generalization, while MAE-based methods are limited by masking misalignment issues. To address these limitations, this paper proposes a novel framework, CoAD, which unifies the two paradigms to leverage their complementary strengths while mitigating their respective weaknesses. In this framework, the classification module generates probability-informed soft masks for the reconstruction module, which in turn alleviates the generalization problem of the classification module. This cooperative design enables CoAD to effectively detect subtle and complex anomalies that are often overlooked by existing methods. Additionally, the classification module is carefully designed to resolve issues related to improper classification granularity and the neglect of frequency information. Extensive experiments on high-quality benchmark datasets, conducted under rigorous evaluation protocols, demonstrate that CoAD significantly outperforms both state-of-the-art deep learning and traditional data mining methods, highlighting the potential of deep learning in TSAD. Moreover, CoAD is lightweight and substantially faster than existing SOTA methods, demonstrating its practical value for large-scale, real-time applications.

Abstract

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