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KoopAGRU: A Koopman-based Anomaly Detection in Time-Series using Gated Recurrent Units

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read KoopAGRU claims 90.88% average F1, beating 20 baselines on five benchmarks.

desk verdict Legitimate new architecture; the 90.88% SOTA claim is an artifact of per-dataset threshold tuning and needs a fair re-evaluation. read the letter →

arxiv 2501.17976 v2 pith:MNGWKFCD submitted 2025-01-29 cs.LG

Issam Ait Yahia , Ismail Berrada This is my paper · ORCID
classification cs.LG
keywords time-seriesanomalydetectionKoopmanoperatortheoryDeepDynamicModeDecompositionGatedRecurrentUnitsFastFourierTransformunsupervisedlearningprediction-errorthresholding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that Koopman-operator ideas, normally used for forecasting nonlinear dynamics, can be turned into a high-performing unsupervised time-series anomaly detector. It claims that KoopAGRU—an FFT split of each window into stable and changing parts, GRU encoders that learn Koopman observables, and fixed-size Koopman operators—can predict the next step and flag anomalies when prediction error exceeds a dataset-specific percentile threshold. The reported result is a new average F1-score of 90.88% on SMD, MSL, SMAP, SWaT, and PSM, above all 20 baselines (the introduction states 90.37%). The design would matter if true because removing the decoder and inverse mapping, while keeping the Koopman matrix small, gives faster inference and modest resource use. The paper itself notes in its limitations section that training time is longer than alternatives and that only five datasets have been tested.

What carries the argument

The central object is a finite-dimensional approximation of the Koopman operator: a fixed-size linear matrix $K$ that advances lifted observables one step in time. KoopAGRU builds the observable vector as $\Psi(X_t) = [X_t; \psi(X_t)]$, with $\psi$ learned by GRU encoders, so the raw measurements appear inside the lifted state and can be recovered without an inverse mapping. The FFT filter splits the input into dominant time-invariant frequencies $G_\alpha$ and the remaining time-variant frequencies, making the operator block-structured and letting $\beta$ weight the invariant branch. This machinery converts the nonlinear forecasting problem into a linear one, and the single forecasting loss doubles as the anomaly score: points whose prediction errors exceed the $r$-th percentile of validation errors are flagged.

What would settle it

Apply the paper's own validation rule to every baseline: for each dataset, choose $r$ (and any equivalent threshold or anomaly-ratio parameters) to maximize F1 on the validation split, then compute test F1. If KoopAGRU's average F1 lead over ModernTCN, TimesNet, and FEDformer drops to about one point or less, the claimed state-of-the-art result is explained by tuning protocol, not by the Koopman/GRU architecture.

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Extended reading notes

Core claim

KoopAGRU claims that a nonlinear, multi-sensor time series can be treated as approximately linear by lifting each window into an observable space and applying a fixed-size Koopman operator. The FFT split separates each look-back window into a time-invariant part (dominant frequencies shared across all windows) and a time-variant part (the remainder). Two GRU encoders learn Koopman observables for the two parts; because the observable vector includes the raw measurements themselves, the next-step prediction is read off directly and no decoder or inverse map is needed. Training minimizes the Frobenius distance between the lifted future window and the operator-applied lifted present window, regularized by the norms of the Koopman matrices. On SMD, MSL, SMAP, SWaT, and PSM, the paper reports an average F1-score of 90.88%, above all 20 baselines.

Load-bearing premise

The 90.88% average depends on choosing $r$, $\alpha$, and $\beta$ separately per dataset using validation data; if the baselines were not given the same per-dataset calibration, the reported lead is a tuning artifact rather than a model advantage.

Editorial extensions

If this is right

  • If the reported numbers hold, KoopAGRU would be the leading unsupervised detector on the SMD, MSL, SMAP, SWaT, and PSM benchmark, ahead of transformer and TCN baselines.
  • The measurement-inclusive observable design means future values are read directly from the lifted state, so inference needs no decoder and can run in one forward pass.
  • The fixed-size Koopman operator keeps parameter counts between roughly 0.7 million and 1.6 million across datasets, with lower GPU and RAM footprints than the transformer baselines in the reported runs.
  • The dataset-specific $\beta$ values suggest that weighting the time-invariant component is the main tuning lever; SWaT's F1 moves from 83.60 to 94.83 as $\beta$ goes from 0 to 0.8.
  • The model is strongest on datasets whose anomalies break periodic patterns and weakest on PSM's irregular, spike-like anomalies, consistent with the frequency-split design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: a direct stress test not reported in the paper is to freeze a single ($\alpha$, $\beta$, $r$) configuration across all five datasets and measure the average F1 loss; this separates architecture from per-dataset calibration.
  • Editorial extension: the global FFT split (one $\alpha$ per dataset) could be replaced by per-window or per-channel frequency masks; datasets like PSM, whose anomalies are spikes rather than periodicity breaks, might then show larger gains than the paper reports.
  • Editorial extension: since $K_{\mathrm{var}}$ is a learned linear operator, its eigenvalues can be inspected after training; a testable prediction is that anomaly onsets on SWaT and SMAP align with growing magnitude of specific Koopman modes, which would turn the anomaly score into a spectrally interpretable signal.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This manuscript proposes KoopAGRU, a forecasting-based unsupervised time-series anomaly detector that combines FFT-based decomposition into time-variant and time-invariant components, GRU encoders that learn Koopman observables, and fixed-size Koopman operators, trained with a single loss. The evaluation on SMD, PSM, MSL, SMAP, and SWaT reports an average F1 of 90.88% against 20 baselines, together with resource-efficiency comparisons. The central claim is that KoopAGRU achieves a new state of the art in unsupervised time-series anomaly detection with fast inference and modest parameter counts.

Significance. If the state-of-the-art result were obtained under an evaluation protocol matched to the baselines, this would be a useful contribution: it is one of the few Koopman/DeepDMD formulations applied to unsupervised anomaly detection, it uses measurement-inclusive observables to avoid a decoder, and the training objective in Section III.G is clean and simple to implement. The paper also states its limitations explicitly in Section IV.G, which is good practice. However, the empirical claim is currently conditional on label-informed per-dataset threshold tuning, and the manuscript contains several numerical inconsistencies. No code release is mentioned, so the efficiency numbers are not independently checkable. I assess the significance as high if the protocol issue can be resolved, but the evidence in the present version does not yet support the headline claim.

major comments (4)
  1. [IV.B, IV.C.a, IV.F.4 (Eq. 22; Table IX)] The threshold ratio r is a per-dataset free parameter selected on validation labels: Eq. (22) sets delta as the r-th percentile of validation prediction errors, Section IV.C.a fixes r=0.5/1/1/4/4 per dataset, and Section IV.F.4 with Table IX confirms that r was chosen to maximize validation F1. This is label-informed threshold selection, and the paper's own Table IX shows it is decisive: fixing r=1 for all datasets gives F1 values of 97.16, 90.61, 85.31, 67.20, and 85.18 on PSM, SWaT, SMD, SMAP, and MSL, an average of 85.09, which is below the 86.62 average of ModernTCN in Table IV. The reported 90.88% average is therefore an argmax over a per-dataset r grid, and the manuscript does not state what threshold protocol was used for the 20 baselines. The central state-of-the-art claim is not established without a matched, pre-specified threshold protocol.
  2. [Abstract, Introduction, IV.C.b, IV.E.c, Table IV, Table V, Table VII] The numerical record is internally inconsistent. The abstract reports 90.88% average F1 while the introduction reports 90.37%; Section IV.E.c reports a PSM F1 of 97.03 while Table IV and Table IX report 97.16; Table VII pairs PSM with (alpha=0.1, beta=0.8) even though Tables V and VI identify the optimal PSM configuration as (alpha=0, beta=0.5); and Section IV.C.b gives alpha=0.1 for SMD while Table V's SMD column reaches its maximum at alpha=0.5. These inconsistencies must be corrected before the evaluation can be reproduced.
  3. [IV.C.c, Table IV] No error bars, multiple-seed results, or statistical tests are reported. Section IV.C.c describes a single training setup, and Table IV reports one F1 value per dataset. Because the headline advantage over ModernTCN is about 4.26 points and the threshold protocol is sensitive, single-run numbers are insufficient to support a state-of-the-art claim; please report mean and standard deviation over at least five seeds and, ideally, a significance test against the strongest baselines.
  4. [IV.G, Table X] Section IV.G honestly lists longer training time and limited dataset coverage as limitations, but Table X shows that KoopAGRU's training time on SWaT is 3039 seconds versus 1477 seconds for TimesNet and 862 seconds for Autoformer. If efficiency is part of the contribution, it should be framed as inference-time and parameter efficiency rather than overall efficiency, because the training-time picture is unfavorable on at least one dataset.
minor comments (6)
  1. [IV.D] Section IV.D states that 20 baselines are compared, but the enumeration contains 19 named models unless TimesNet (I) and TimesNet (R) are counted separately; please clarify the count.
  2. [IV.E.b, IV.C.b] Section IV.E.b contains the typo 'SWaR' for 'SWaT', and Section IV.C.b says 'SMAP and SMAP' where the second dataset name appears to be missing.
  3. [III.G, Eq. (18)] Equation (18) uses Phi X_t before it is defined in Eq. (20), and Phi X_{t+1} is defined later in Eq. (16); reorder the definitions or add forward references.
  4. [Tables V, VI, VII] The table captions use the informal headers 'Alpha' and 'Beta' and the abbreviation 'SW AT'; please use consistent dataset names and explain the parenthetical hyperparameter values in each caption.
  5. [References] References [65] and [26] are the same paper; citing it twice in different positions is confusing.
  6. [Reproducibility] No code release is mentioned; providing code or a configuration file would materially improve reproducibility, especially because many hyperparameters are dataset-specific.

Circularity Check

1 steps flagged · score 4.0 of 10

The reported SOTA F1 average is partly an evaluation-protocol artifact: the anomaly threshold is set per dataset by maximizing validation F1 over the anomaly ratio r, and the headline numbers are the best rows of that sweep; the underlying Koopman/GRU model is not itself circular.

  1. fitted input called prediction [Section IV.B (Eq. 22), Section IV.C.b (r values), Section IV.F.4 (Table IX)]
    "δ = Pr(e) ... The value of r is dataset-specific, chosen to calibrate the proportion of validation data labeled as anomalies. ... Anomaly Ratio r: The optimal α and β values for each dataset were selected, and the anomaly ratio was varied to examine its impact on F1-scores, as defined in Equation (22)."

    The test decision rule is et > δ, with δ equal to the r-th percentile of validation prediction errors. Table IX shows r being swept over {0.5, 1, 4, 5} per dataset, and the F1 numbers used in the headline Table IV (SMD 90.12 at r=0.5, MSL 85.19 at r=1, PSM 97.16 at r=1, SWaT 94.83 at r=4, SMAP 87.09 at r=4) are exactly the best-r entries of that table. Thus the reported 90.88% average is the argmax of a label-informed threshold search rather than a fixed-protocol prediction. The paper's own Table IX shows that fixing r=1 lowers the average to roughly 85.09 (97.16, 90.61, 85.31, 67.20, 85.18), below the 86.62 of ModernTCN, so the claimed margin over baselines is substantially an artifact of per-dataset threshold selection.

full rationale

This is an empirical paper rather than a formal derivation, and the central Koopman/DeepDMD construction is not circular: the Koopman operator and GRU observables are learned on unlabeled training data through a least-squares-type loss (Eqs. 17-20), and the anomaly score is the prediction error of Eq. (21). Fitting a model to training data is normal, not circular. The one significant circularity is in the evaluation protocol: δ is defined as a percentile of validation errors (Eq. 22), r is declared dataset-specific in Section IV.C.b, and Section IV.F.4/Table IX explicitly varies r and selects the value that maximizes F1. Because the headline Table IV uses those best-r results, the 'state-of-the-art' number is partly constructed by the same validation labels that it is then used to beat baselines with. This is a fitted threshold called a detection result, and it undermines the fairness of the SOTA comparison unless the baselines were given an equivalent oracle threshold search. No load-bearing self-citation chain exists: reference [65] is used for the forecasting-plus-threshold recipe but is not an unverified premise; related-work citations [26]-[28] are background. There are also self-consistency errors (90.88 vs 90.37 average, PSM 97.03 vs 97.16, 'SMAP and SMAP' in the α list) that I treat as correctness risks, not circularity. Overall score 4: the model derivation is self-contained, but the headline empirical claim is partially circular in its threshold-selection protocol.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model introduces no new physical entities. Its central claim rests on standard Fourier and Koopman mathematics, plus domain assumptions that frequency-domain separation captures stable versus evolving dynamics and that anomalies inflate one-step forecast error. The per-dataset hyperparameters α, β, r and GRU depths are free parameters fitted on validation data, and they directly affect the reported F1.

free parameters (5)
  • α (dominant frequency ratio) = PSM 0; SMD 0.5; SWaT 0.1; SMAP 0.5; MSL 0.1
    Chosen per dataset to maximize F1 (Section IV.C.b, Table V). Controls the split into time-invariant and time-variant components.
  • β (time-invariant weight) = PSM 0.5; SWaT 0.8; SMD 0.1; MSL 0; SMAP 0.3
    Chosen per dataset to maximize F1 (Table VI). Scales the contribution of the time-invariant branch in the loss.
  • r (validation anomaly ratio) = SMD 0.5; MSL 1; PSM 1; SWaT 4; SMAP 4
    Sets the percentile threshold δ (Eq. 22). Selected per dataset, effectively calibrating the proportion of validation points flagged as anomalies and directly influencing F1.
  • GRU layer counts per encoder = Variant encoder: PSM 4, MSL 12, SMAP 8, SWaT 14, SMD 6; Invariant encoder: PSM 2, MSL 8, SMAP 2, SWaT 8, SMD 2
    Table II; selected per dataset, affecting model capacity and results.
  • λ regularization coefficient = 10^-3
    Fixed in main runs but ablation in Table VII shows strong sensitivity to this value; effectively selected as best.
assumptions (5)
  • standard math Fast Fourier Transform and its inverse provide an exact decomposition of the input signal into time-invariant and time-variant components as defined in Eq. (4).
    Fourier theory is well-established; the decomposition is exact. The paper relies on this to split the signal.
  • domain assumption The dominant-frequency subset Gα (top α% by average amplitude) corresponds to time-invariant dynamics, and its complement to time-variant dynamics.
    This is the central modeling assumption adopted from Koopa [56]; if wrong, the decomposition harms rather than helps.
  • domain assumption Anomalies produce larger one-step prediction errors in the learned Koopman model.
    The anomaly score is the L2 prediction error (Eq. 21); this is the standard forecasting-based detection assumption.
  • domain assumption The learned observable function ψ places the dynamics in a space where a fixed linear operator K yields accurate one-step predictions (Eq. 30).
    Core Koopman assumption; if the dynamics are not near-linear in the learned embedding, the model's Koopman component is just a nonlinear encoder with a linear readout.
  • domain assumption The validation set error distribution is representative of the test set, so the r-th percentile threshold δ generalizes.
    Evaluation protocol in Eq. (22); with r tuned per dataset, this is a fragile assumption.

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Cite this review

Pith. "Pith review of KoopAGRU: A Koopman-based Anomaly Detection in Time-Series using Gated Recurrent Units." pith.science (2026). https://pith.science/paper/MNGWKFCD

@misc{pith2026250117976,
  author       = {Pith},
  title        = {Pith review of: KoopAGRU: A Koopman-based Anomaly Detection in Time-Series using Gated Recurrent Units},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MNGWKFCD}},
  note         = {Machine review of arXiv:2501.17976}
}
read the original abstract

Anomaly detection in real-world time-series data is a challenging task due to the complex and nonlinear temporal dynamics involved. This paper introduces KoopAGRU, a new deep learning model designed to tackle this problem by combining Fast Fourier Transform (FFT), Deep Dynamic Mode Decomposition (DeepDMD), and Koopman theory. FFT allows KoopAGRU to decompose temporal data into time-variant and time-invariant components providing precise modeling of complex patterns. To better control these two components, KoopAGRU utilizes Gate Recurrent Unit (GRU) encoders to learn Koopman observables, enhancing the detection capability across multiple temporal scales. KoopAGRU is trained in a single process and offers fast inference times. Extensive tests on various benchmark datasets show that KoopAGRU outperforms other leading methods, achieving a new average F1-score of 90.88\% on the well-known anomalies detection task of times series datasets, and proves to be efficient and reliable in detecting anomalies in real-world scenarios.

Figures

Figures reproduced from arXiv: 2501.17976 by the authors.

Figure 1
Figure 1. KoopAGRU: Overall Model Architecture. During training and inference, the FFT component of KoopAGRU separates the input Xt as follows: Xinv = F −1 (Ff (Gα, F(Xt))) Xvar = F −1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Time-Variant Component Architecture. The time-variant component of KoopAGRU captures the dynamic aspects of the data that fluctuate over time. Using the Fourier filter, these components are isolated, enhancing the model’s ability to encode the temporal dynamics of the series. The time-variant component is denoted by Xvar: Xvar = F −1 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Time-Invariant Component Architecture. The time-invariant component of KoopAGRU represents the stable aspects of the time series that remain constant over time. Isolating these components with the Fourier filter ensures the stability of the encoded representation. The time-invariant component is denoted by Xinv: Xinv = F −1 (Ff (Gα, F(Xt))) (11) The time-invariant component spans three main blocks: Nor￾malisation, E… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Positive-Allocation Companion Predictors for Nonlinear Dynamics and Their Finite-Difference Diagnostics

    math.DS 2026-07 conditional novelty 5.0 of 10

    A nonnegative, sum-to-one weighted average of past snapshots defines a companion predictor whose spectrum lies in the unit disk and includes 1 as an eigenvalue.

Reference graph

Works this paper leans on

88 extracted references · 52 canonical work pages · cited by 1 Pith paper

  1. [1]

    Koopman operator, geometry, and learning,

    I. Mezic, “Koopman operator, geometry, and learning,” 2020, arXiv:2010.05377. [Online]. Available: https://arxiv.org/abs/2010.05377

    arXiv 2020

  2. [2]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems,

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems,” Proceedings of the National Academy of Sciences , vol. 113, no. 15, pp. 3932–3937, 2016

    2016

  3. [3]

    Graph neural network and koopman models for learning networked dynamics: A comparative study on power grid transients prediction,

    S. P. Nandanoori, S. Guan, S. Kundu, S. Pal, K. Agarwal, Y . Wu, and S. Choudhury, “Graph neural network and koopman models for learning networked dynamics: A comparative study on power grid transients prediction,” 02 2022

    2022

  4. [4]

    Learning koopman invariant subspaces for dynamic mode decomposition,

    N. Takeishi, Y . Kawahara, and T. Yairi, “Learning koopman invariant subspaces for dynamic mode decomposition,” in Advances in Neural Information Processing Systems , vol. 30, 2017, pp. 1130–1140

    2017

  5. [5]

    Empirical evaluation of gated recurrent neural networks on sequence modeling,

    J. Chung, C. Gulcehre, K. Cho, and Y . Bengio, “Empirical evaluation of gated recurrent neural networks on sequence modeling,” arXiv preprint arXiv:1412.3555, 2014

    arXiv 2014

  6. [6]

    Exploiting autoencoder- based anomaly detection to enhance cybersecurity in power grids,

    F. Harrou, B. Bouyeddou, A. Dairi, and Y . Sun, “Exploiting autoencoder- based anomaly detection to enhance cybersecurity in power grids,” Future Internet, vol. 16, no. 6, p. 184, 2024

    2024

  7. [7]

    The fast fourier transform,

    E. O. Brigham and R. E. Morrow, “The fast fourier transform,” IEEE Spectrum, vol. 4, no. 12, pp. 63–70, 1967

    1967

  8. [8]

    Non-stationary transformers: Exploring the stationarity in time series forecasting,

    Y . Liu, H. Wu, J. Wang, and M. Long, “Non-stationary transformers: Exploring the stationarity in time series forecasting,” 2023. [Online]. Available: https://arxiv.org/abs/2205.14415

    arXiv 2023

Show all 88 references
  1. [9]

    Robust anomaly detection for multivariate time series through stochastic recurrent neural network,

    Y . Su, Y . Zhao, C. Niu, R. Liu, W. Sun, and D. Pei, “Robust anomaly detection for multivariate time series through stochastic recurrent neural network,” in Proceedings of the 25th ACM SIGKDD SUBMITTED TO IEEE TPAMI FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED. 11 I...

    2019

  2. [10]

    Detecting spacecraft anomalies using lstms and nonparametric dynamic thresholding,

    K. Hundman, V . Constantinou, C. Laporte, I. Colwell, and T. Soderstrom, “Detecting spacecraft anomalies using lstms and nonparametric dynamic thresholding,” in Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery amp; Data Mining , ser. KDD 18. A...

    2018

  3. [11]

    Swat: a water treatment testbed for research and training on ics security,

    A. P. Mathur and N. O. Tippenhauer, “Swat: a water treatment testbed for research and training on ics security,” in 2016 International Workshop on Cyber-physical Systems for Smart Water Networks (CySWater), 2016, pp. 31–36

    2016

  4. [12]

    Practical approach to asynchronous multivariate time series anomaly detection and localization,

    A. Abdulaal, Z. Liu, and T. Lancewicki, “Practical approach to asynchronous multivariate time series anomaly detection and localization,” in Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining , ser. KDD ’21. New York, NY , USA: Association for C...

    2021

  5. [13]

    Lof: Identifying density- based local outliers

    M. Breunig, P. Krger, R. Ng, and J. Sander, “Lof: Identifying density- based local outliers.” vol. 29, 06 2000, pp. 93–104

    2000

  6. [14]

    Enhancing effec- tiveness of outlier detections for low density patterns,

    J. Tang, Z. Chen, A. W.-c. Fu, and D. W. Cheung, “Enhancing effec- tiveness of outlier detections for low density patterns,” in Advances in Knowledge Discovery and Data Mining , M.-S. Chen, P. S. Yu, and B. Liu, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002, pp. 535–548

    2002

  7. [15]

    Deep autoencoding gaussian mixture model for unsupervised anomaly detection,

    B. Zong, Q. Song, M. R. Min, W. Cheng, C. Lumezanu, D. Cho, and H. Chen, “Deep autoencoding gaussian mixture model for unsupervised anomaly detection,” in International Conference on Learning Representations , 2018. [Online]. Available: https: //openreview.net/forum?id=BJJLHbb0-

    2018

  8. [16]

    A data-driven health monitoring method for satellite housekeeping data based on probabilistic clustering and dimensionality reduction,

    T. Yairi, N. Takeishi, T. Oda, Y . Nakajima, N. Nishimura, and N. Takata, “A data-driven health monitoring method for satellite housekeeping data based on probabilistic clustering and dimensionality reduction,” IEEE Transactions on Aerospace and Electronic Systems , vol. 53, n...

    2017

  9. [17]

    Support vector data description,

    D. Tax and R. Duin, “Support vector data description,” Machine Learn- ing, vol. 54, pp. 45–66, 01 2004

    2004

  10. [18]

    Deep one-class classification,

    L. Ruff, R. Vandermeulen, N. Grnitz, L. Deecke, S. Siddiqui, A. Binder, E. Mller, and M. Kloft, “Deep one-class classification,” 07 2018

    2018

  11. [19]

    Timeseries anomaly detection using temporal hierarchical one-class network,

    L. Shen, Z. Li, and J. Kwok, “Timeseries anomaly detection using temporal hierarchical one-class network,” in Advances in Neural Information Processing Systems , H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, Eds., vol. 33. Curran Associates, Inc., 2020, pp. 13 ...

    2020

  12. [20]

    Integrative tensor-based anomaly detection system for satellites,

    Y . Shin, S. Lee, S. Tariq, M. S. Lee, OkchulJung, D. Chung, and S. Woo, “Integrative tensor-based anomaly detection system for satellites,” 2020. [Online]. Available: https://openreview.net/forum?id=HJeg46EKPr

    2020

  13. [21]

    A multimodal anomaly detector for robot-assisted feeding using an lstm-based variational autoencoder,

    D. Park, Y . Hoshi, and C. C. Kemp, “A multimodal anomaly detector for robot-assisted feeding using an lstm-based variational autoencoder,”

  14. [22]

    Robust anomaly detection for multivariate time series through stochastic recurrent neural network,

    Y . Su, Y . Zhao, C. Niu, R. Liu, W. Sun, and D. Pei, “Robust anomaly detection for multivariate time series through stochastic recurrent neural network,” Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining , 2019. [Online]. Availab...

    2019

  15. [23]

    Multivariate time series anomaly detection and interpretation using hierarchical inter-metric and temporal embedding,

    Z. Li, Y . Zhao, J. Han, Y . Su, R. Jiao, X. Wen, and D. Pei, “Multivariate time series anomaly detection and interpretation using hierarchical inter-metric and temporal embedding,” in Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining , ser. KD...

    2021

  16. [24]

    f-anogan: Fast unsupervised anomaly detection with generative adver- sarial networks,

    T. Schlegl, P. Seebck, S. Waldstein, G. Langs, and U. Schmidt-Erfurth, “f-anogan: Fast unsupervised anomaly detection with generative adver- sarial networks,” Medical Image Analysis , vol. 54, 01 2019

    2019

  17. [25]

    Generative adversarial networks,

    I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y . Bengio, “Generative adversarial networks,” 2014. [Online]. Available: https://arxiv.org/abs/1406.2661

    2014 arXiv

  18. [26]

    Fed-anids: Federated learning for anomaly-based network intrusion detection systems,

    M. J. Idrissi, H. Alami, A. El Mahdaouy, A. El Mekki, S. Oualil, Z. Yartaoui, and I. Berrada, “Fed-anids: Federated learning for anomaly-based network intrusion detection systems,” Expert Systems with Applications , vol. 234, p. 121000, 2023. [Online]. Available: https://www.s...

    2023

  19. [27]

    Investigating domain adaptation for network intrusion detection,

    H. Alami, M. J. Idrissi, A. El Mahdaouy, A. Bouayad, Z. Yartaoui, and I. Berrada, “Investigating domain adaptation for network intrusion detection,” in 2023 10th International Conference on Wireless Networks and Mobile Communications (WINCOM) , 2023, pp. 1–7

    2023

  20. [28]

    Nf-nids: Nor- malizing flows for network intrusion detection systems,

    M. J. Idrissi, H. Alami, A. Bouayad, and I. Berrada, “Nf-nids: Nor- malizing flows for network intrusion detection systems,” in 2023 10th International Conference on Wireless Networks and Mobile Communi- cations (WINCOM), 2023, pp. 1–7

    2023

  21. [29]

    G. E. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung, Time Series Analysis: Forecasting and Control. John Wiley & Sons, 2015

    2015

  22. [30]

    J. D. Hamilton, Time Series Analysis. Princeton University Press, 1994

    1994

  23. [31]

    Multivariate time series forecast- ing with temporal attention mechanism,

    S.-H. Shih, F.-K. Sun, and H.-Y . Lee, “Multivariate time series forecast- ing with temporal attention mechanism,” in Proceedings of the 33rd International Conference on Neural Information Processing Systems , 2019, pp. 672–683

    2019

  24. [32]

    Arnet: Modeling temporal dependencies in time series forecasting using autoregressive networks,

    B. Lim and S. Zohren, “Arnet: Modeling temporal dependencies in time series forecasting using autoregressive networks,” arXiv preprint arXiv:2105.02423, 2021

    2021 arXiv

  25. [33]

    Hamiltonian systems and transformation in hilbert space,

    B. O. Koopman, “Hamiltonian systems and transformation in hilbert space,” Proceedings of the National Academy of Sciences , vol. 17, no. 5, pp. 315–318, 1931. [Online]. Available: https: //www.pnas.org/doi/abs/10.1073/pnas.17.5.315

    1931 doi

  26. [34]

    S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering . CRC press, 2018

    2018

  27. [35]

    Extract- ing spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition,

    B. W. Brunton, L. A. Johnson, J. G. Ojemann, and J. N. Kutz, “Extract- ing spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition,” Journal of neuroscience methods, vol. 258, pp. 1–15, 2016

    2016

  28. [36]

    Spectral analysis of nonlinear flows,

    C. W. Rowley, I. Mezi ´c, S. Bagheri, P. Schlatter, and D. S. Henningson, “Spectral analysis of nonlinear flows,” Journal of fluid mechanics , vol. 641, pp. 115–127, 2009

    2009

  29. [37]

    Dynamic mode decomposition: Theory and applications,

    J. H. Tu, “Dynamic mode decomposition: Theory and applications,” Ph.D. dissertation, Princeton University, 2013

    2013

  30. [38]

    A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,” Journal of Nonlinear Science , vol. 25, pp. 1307–1346, 2015

    2015

  31. [39]

    Learning koopman invariant subspaces for dynamic mode decomposition,

    N. Takeishi, Y . Kawahara, and T. Yairi, “Learning koopman invariant subspaces for dynamic mode decomposition,” Advances in neural infor- mation processing systems , vol. 30, 2017

    2017

  32. [40]

    Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the koopman operator,

    Q. Li, F. Dietrich, E. M. Bollt, and I. G. Kevrekidis, “Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the koopman operator,” Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 27, no. 10, 2017

    2017

  33. [41]

    Deep learning for universal linear embeddings of nonlinear dynamics,

    B. Lusch, J. N. Kutz, and S. L. Brunton, “Deep learning for universal linear embeddings of nonlinear dynamics,” Nature communications , vol. 9, no. 1, p. 4950, 2018

    2018

  34. [42]

    Learning deep neural network representations for koopman operators of nonlinear dynamical systems,

    E. Yeung, S. Kundu, and N. Hodas, “Learning deep neural network representations for koopman operators of nonlinear dynamical systems,” in 2019 American Control Conference (ACC) . IEEE, 2019, pp. 4832– 4839

    2019

  35. [43]

    Graph neural network and koopman models for learning networked dynamics: A comparative study on power grid transients prediction,

    S. P. Nandanoori, S. Guan, S. Kundu, S. Pal, K. Agarwal, Y . Wu, and S. Choudhury, “Graph neural network and koopman models for learning networked dynamics: A comparative study on power grid transients prediction,” IEEE Access, vol. 10, pp. 32 337–32 349, 2022

    2022

  36. [44]

    Decomposing build- ing system data for model validation and analysis using the koopman operator,

    B. Eisenhower, T. Maile, M. Fischer, and I. Mezic, “Decomposing build- ing system data for model validation and analysis using the koopman operator,” in Proceedings of the National IBPSAUSA Conference, New York, USA, 2010

    2010

  37. [45]

    Data driven online learning of power system dynamics,

    S. Sinha, S. P. Nandanoori, and E. Yeung, “Data driven online learning of power system dynamics,” in 2020 IEEE Power & Energy Society General Meeting (PESGM) . IEEE, 2020, pp. 1–5

    2020

  38. [46]

    Applied koopman operator theory for power systems technology,

    Y . Susuki, I. Mezic, F. Raak, and T. Hikihara, “Applied koopman operator theory for power systems technology,” Nonlinear Theory and Its Applications, IEICE , vol. 7, no. 4, pp. 430–459, 2016

    2016

  39. [47]

    Model-agnostic algorithm for real-time attack identification in power grid using koopman modes,

    S. P. Nandanoori, S. Kundu, S. Pal, K. Agarwal, and S. Choudhury, “Model-agnostic algorithm for real-time attack identification in power grid using koopman modes,” in 2020 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (Sma...

    2020

  40. [48]

    Koopman operator framework for time series modeling and analysis,

    A. Surana, “Koopman operator framework for time series modeling and analysis,” Journal of Nonlinear Science , vol. 30, no. 5, pp. 1973–2006, 2020

    1973

  41. [49]

    On analytical construc- tion of observable functions in extended dynamic mode decomposition for nonlinear estimation and prediction,

    M. Netto, Y . Susuki, V . Krishnan, and Y . Zhang, “On analytical construc- tion of observable functions in extended dynamic mode decomposition for nonlinear estimation and prediction,” in 2021 American Control Conference (ACC). IEEE, 2021, pp. 4190–4195. SUBMITTED TO IEEE TPA...

    2021

  42. [50]

    Physics-informed autoencoders for lyapunov-stable fluid flow prediction,

    N. B. Erichson, O. Azencot, A. Queiruga, A. Khanna, and M. W. Mahoney, “Physics-informed autoencoders for lyapunov-stable fluid flow prediction,” arXiv preprint arXiv:1905.10866 , 2019

    1905 arXiv

  43. [51]

    Deep learning for universal linear embeddings of nonlinear dynamics,

    B. Lusch, J. N. Kutz, and S. L. Brunton, “Deep learning for universal linear embeddings of nonlinear dynamics,” Nature Communications, vol. 9, no. 1, Nov. 2018. [Online]. Available: http://dx.doi.org/10.1038/s41467-018-07210-0

    2018 doi

  44. [52]

    Learning koopman invariant subspaces for dynamic mode decomposition,

    N. Takeishi, Y . Kawahara, and T. Yairi, “Learning koopman invariant subspaces for dynamic mode decomposition,” 2018. [Online]. Available: https://arxiv.org/abs/1710.04340

    2018 arXiv

  45. [53]

    Sparse identification of nonlinear dynamics for model predictive control in the low-data limit,

    E. Kaiser, J. N. Kutz, and S. L. Brunton, “Sparse identification of nonlinear dynamics for model predictive control in the low-data limit,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , vol. 474, no. 2219, p. 20180335, Nov. 2018. [Online...

    2018

  46. [54]

    From fourier to koopman: Spectral methods for long-term time series prediction,

    H. Lange, S. L. Brunton, and N. Kutz, “From fourier to koopman: Spectral methods for long-term time series prediction,” 2020, arXiv:2004.00574. [Online]. Available: https://arxiv.org/abs/2004.00574

    2020 arXiv

  47. [55]

    Koopman neural forecaster for time series with temporal distribution shifts,

    R. Wang, Y . Dong, S. . Arik, and R. Yu, “Koopman neural forecaster for time series with temporal distribution shifts,” 2023, arXiv:2210.03675. [Online]. Available: https://arxiv.org/abs/2210.03675

    2023 arXiv

  48. [56]

    Koopa: Learning non-stationary time series dynamics with koopman predictors,

    Y . Liu, C. Li, J. Wang, and M. Long, “Koopa: Learning non-stationary time series dynamics with koopman predictors,” in Advances in Neural Information Processing Systems , A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, Eds., vol. 36. Curran Associates, In...

    2023

  49. [57]

    Data-driven analysis and forecasting of highway traffic dynamics,

    A. M. Avila and I. Mezi ´c, “Data-driven analysis and forecasting of highway traffic dynamics,” Nature Communications, vol. 11, no. 1, pp. 1–16, 2020

    2020

  50. [58]

    Chaos as an intermittently forced linear system,

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Chaos as an intermittently forced linear system,” Nature Communications, vol. 8, no. 1, pp. 1–9, 2017

    2017

  51. [59]

    Koopman operator theory in epidemiological modeling,

    I. Mezi ´c, M. Fonoberova, and J. Hogg, “Koopman operator theory in epidemiological modeling,” Journal of Theoretical Biology , vol. 540, p. 111063, 2024

    2024

  52. [60]

    Exponentially decaying modes and long-term prediction of sea ice concentration using koopman mode decomposition,

    J. Hogg, M. Fonoberova, and I. Mezi ´c, “Exponentially decaying modes and long-term prediction of sea ice concentration using koopman mode decomposition,” Scientific Reports, vol. 10, no. 1, pp. 1–12, 2020

    2020

  53. [61]

    Self-organized operational neural networks with generative neurons: towards the next generation of deep neural networks,

    S. Kiranyaz, T. Ince, and M. Gabbouj, “Self-organized operational neural networks with generative neurons: towards the next generation of deep neural networks,” Neural Networks, vol. 139, pp. 17–30, 2021

    2021

  54. [62]

    Fast training of convolutional networks through ffts,

    M. Mathieu, M. Henaff, and Y . LeCun, “Fast training of convolutional networks through ffts,” in 3rd International Conference on Learning Representations (ICLR), 2014

    2014

  55. [63]

    Denoise feature representation learning with fourier transform for robust face recognition,

    X. Zhang, Y . Guo, X. Han, J. Zhang, K. Ma, and J. Zhang, “Denoise feature representation learning with fourier transform for robust face recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2018

    2018

  56. [64]

    A class of logistic functions for approxi- mating state-inclusive koopman operators,

    C. A. Johnson and E. Yeung, “A class of logistic functions for approxi- mating state-inclusive koopman operators,” in Proceedings of the Annual American Control Conference (ACC) . IEEE, 2018, pp. 4803–4810

    2018

  57. [65]

    Fed-anids: Federated learning for anomaly- based network intrusion detection systems,

    M. J. Idrissi, H. Alami, A. El Mahdaouy, A. El Mekki, S. Oualil, Z. Yartaoui, and I. Berrada, “Fed-anids: Federated learning for anomaly- based network intrusion detection systems,” Expert Systems with Appli- cations, vol. 234, p. 121000, 2023

    2023

  58. [66]

    Anomaly transformer: Time series anomaly detection with association discrepancy,

    C. Xu, Y . Li, Z. Wang, M. Long, and J. Wang, “Anomaly transformer: Time series anomaly detection with association discrepancy,” Ad- vances in Neural Information Processing Systems , 2021, arXiv preprint arXiv:2106.00189

    2021 arXiv

  59. [67]

    Adam: A method for stochastic optimization,

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in International Conference on Learning Representations (ICLR) , 2015

    2015

  60. [68]

    Pytorch: An imperative style, high- performance deep learning library,

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, “Pytorch: An imperative style, high- p...

    2019

  61. [69]

    Long short-term memory,

    S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural computation, vol. 9, no. 8, pp. 1735–1780, 1997

    1997

  62. [70]

    Attention is all you need,

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in Advances in neural information processing systems , 2017, pp. 5998–6008

    2017

  63. [71]

    En- hancing the locality and breaking the memory bottleneck of transformer on time series forecasting,

    S. Li, X. Jin, Y . Xuan, X. Zhou, W. Chen, Y . Wang, and X. Yan, “En- hancing the locality and breaking the memory bottleneck of transformer on time series forecasting,” Advances in Neural Information Processing Systems, vol. 32, pp. 1–11, 2019

    2019

  64. [72]

    An empirical evaluation of generic convolutional and recurrent networks for sequence modeling,

    S. Bai, J. Z. Kolter, and V . Koltun, “An empirical evaluation of generic convolutional and recurrent networks for sequence modeling,” arXiv preprint arXiv:1803.01271, 2018

    2018 arXiv

  65. [73]

    Reformer: The efficient trans- former,

    N. Kitaev, Ł. Kaiser, and A. Levskaya, “Reformer: The efficient trans- former,” in International Conference on Learning Representations, 2020

    2020

  66. [74]

    Informer: Beyond efficient transformer for long sequence time-series forecasting,

    H. Zhou, S. Zhang, J. Peng, S. Zhang, J. Li, H. Xiong, and W. Zhang, “Informer: Beyond efficient transformer for long sequence time-series forecasting,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 12, 2021, pp. 11 106–11 115

    2021

  67. [75]

    Anomaly transformer: Time series anomaly detection with association discrepancy,

    J. Xu, H. Wu, J. Wang, and M. Long, “Anomaly transformer: Time series anomaly detection with association discrepancy,” 2022

    2022

  68. [76]

    Pyraformer: Low-complexity pyra- midal attention for long-range time series modeling and forecasting,

    H. Liu, J. Liu, L. Sun, and J. Zhou, “Pyraformer: Low-complexity pyra- midal attention for long-range time series modeling and forecasting,” arXiv preprint arXiv:2106.01950 , 2021

    2021 arXiv

  69. [77]

    Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting,

    H. Wu, J. Xu, J. Wang, and M. Long, “Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting,” in Advances in Neural Information Processing Systems , 2021

    2021

  70. [78]

    Long sequence time-series learning with structured state space for human action recognition,

    X. Liu, H. Zhang, and J. Feng, “Long sequence time-series learning with structured state space for human action recognition,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 12 506–12 516

    2022

  71. [79]

    Deep stationary time-series modeling,

    T. Zhou, X. Qi, Y . Shen, and H. Yang, “Deep stationary time-series modeling,” in Advances in Neural Information Processing Systems , 2022

    2022

  72. [80]

    Dlinear: Modeling long-term dependence and variability for time series forecasting,

    A. Zhang, C. Liu, C. Zhang, and Y . Wang, “Dlinear: Modeling long-term dependence and variability for time series forecasting,” arXiv preprint arXiv:2301.02005, 2023

    2023 arXiv

  73. [81]

    Etsformer: Exponential smoothing transformers for time-series forecasting,

    S. Woo and J. Baek, “Etsformer: Exponential smoothing transformers for time-series forecasting,” arXiv preprint arXiv:2302.06637 , 2023

    2023 arXiv

  74. [82]

    Lightts: Lightweight transformers for time series forecasting,

    H. Zhou and S. Zhang, “Lightts: Lightweight transformers for time series forecasting,” arXiv preprint arXiv:2301.11511 , 2023

    2023 arXiv

  75. [83]

    Fedformer: Frequency enhanced decomposed transformer for time-series forecasting,

    T. Zhou, Z. Ma, Y . Shen, and H. Yang, “Fedformer: Frequency enhanced decomposed transformer for time-series forecasting,” in Advances in Neural Information Processing Systems , 2023

    2023

  76. [84]

    Timesnet: Tem- poral neural networks for time-series forecasting,

    J. Wu, J. Xu, J. Wang, Z. Zhang, and Y . Wang, “Timesnet: Tem- poral neural networks for time-series forecasting,” arXiv preprint arXiv:2301.07045, 2023

    2023 arXiv

  77. [85]

    Crossformer: Cross-attention transformer for time- series forecasting,

    J. Wang and X. He, “Crossformer: Cross-attention transformer for time- series forecasting,” arXiv preprint arXiv:2303.15020 , 2023

    2023 arXiv

  78. [86]

    Patchtst: Patch time-series transformer for long-term forecasting,

    Q. Nie and X. Liu, “Patchtst: Patch time-series transformer for long-term forecasting,” arXiv preprint arXiv:2303.05797 , 2023

    2023 arXiv

  79. [87]

    ModernTCN: A modern pure convolution structure for general time series analysis,

    L. donghao and wang xue, “ModernTCN: A modern pure convolution structure for general time series analysis,” in The Twelfth International Conference on Learning Representations , 2024. [Online]. Available: https://openreview.net/forum?id=vpJMJerXHU SUBMITTED TO IEEE TPAMI FOR P...

    2024

  80. [2017]

    Available: https://arxiv.org/abs/1711.00614

    [Online]. Available: https://arxiv.org/abs/1711.00614

    arXiv

Pith tools

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