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A single climate record, treated as the pullback attractor of a linear stochastic model driven by external forcing, can be decomposed exactly into forced response and internal variability, with the forced response matching or exceeding esta

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2026-08-02 06:43 UTC pith:PCMTDQJF

load-bearing objection A genuinely useful, clearly explained single-realization forced-response estimator with a mode-level decomposition; the headline skill claim is inflated by tuning the forcing variant on the test set, but the method and diagnostics are worth a serious referee. the 4 major comments →

arxiv 2607.18298 v1 pith:PCMTDQJF submitted 2026-07-13 stat.ML cs.LGphysics.ao-ph

Disentangling Forced and Internal Climate Variability in Single Realizations using Dynamic Mode Decomposition with Control

Nathan Mankovich , Andrei Gavrilov , Gustau Camps-Valls This is my paper
classification stat.ML cs.LGphysics.ao-ph MSC 37B5562M1086A10
keywords climate variabilityforced responseinternal variabilitydynamic mode decomposition with controlpullback attractorlinear stochastic modelsingle realizationEarth system model evaluation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims there is a way to separate the forced response of the climate — the part driven by greenhouse gases, volcanoes, and solar changes — from internal variability such as ENSO and the Atlantic Multidecadal Oscillation, using only a single observed record rather than a large ensemble of model runs. The method, PullbackDMDc, fits a linear stochastic model whose evolution depends on external forcing, x(t) = Ax(t−1) + By(t) + ξ(t), and interprets the record as that model's pullback attractor. Because the model is linear, the attractor is a single trajectory that splits exactly: the forced response is the whole forcing history filtered through the system's decaying memory, and internal variability is the same filter applied to the noise. The authors show this estimate matches or beats established baselines on four Earth system model large ensembles, and that the mode-by-mode decomposition gives a new lens for evaluating models — observations keep slow multidecadal modes in the lead, while models bury that variability in faster ENSO-like modes. If right, this gives climate science a practical single-realization tool for attribution, projection, and model evaluation without the cost of large ensembles.

Core claim

One climate record can be split exactly into forced response and internal variability by treating it as the pullback attractor of a linear stochastic model with forcing, x(t) = Ax(t−1) + By(t) + ξ(t). For stable A, pulling the initial time to −∞ leaves two sums: the forced response Σ A^j B y(t−j), the forcing history through the system's decaying memory, and internal variability Σ A^j ξ(t−j), the same filter applied to noise. Projecting onto the eigenvectors of A (dynamic mode decomposition with control) yields spatial patterns, each with its own forced and internal time series. Tested against large-ensemble means from four Earth system models, the forced response matches or beats existing b

What carries the argument

The load-bearing object is the pullback attractor of a forced linear stochastic system. Where an ordinary attractor is a time-independent limit set, the pullback attractor is a time-dependent family of limit sets: at each time t it is what remains after the initial condition is forgotten, under the same non-autonomous changes (the forcing history) leading up to t. Because the model x(t) = Ax(t−1) + By(t) + ξ(t) is linear and stable, its pullback attractor collapses to a single trajectory, and the forced–internal split becomes exact rather than statistical: the forced response is the infinite convolution Σ_{j≥0} A^j B y(t−j) and internal variability is the same kernel acting on the noise. An

Load-bearing premise

The decomposition is valid only if the observed record is the pullback attractor of a stable linear model driven by exactly the forcing history supplied — if the real climate is nonlinear, has drivers not included in y(t), or carries memory longer than the 100-year spin-up, then the residual labelled 'internal variability' is contaminated by model error and the split is no longer clean.

What would settle it

Run the algorithm on a long, weakly forced stretch of an ESM large ensemble where internal variability dominates (e.g., a multidecadal window without major volcanoes). If members' PullbackDMDc forced-response estimates remain tightly clustered near the ensemble mean, the single-realization separation is real; if their spread matches the raw member-to-member spread, the method is just reshuffling internal variability into the residual. The same test on a nonlinear toy model with known components would show how much of the separation survives outside the linear assumption.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Forced-response estimation no longer requires a large ensemble: one realization plus a forcing history yields estimates that match or beat methods that either ignore forcing or ignore dynamics.
  • Separating volcanic forcing from slow greenhouse-gas forcing measurably improves the estimated response around eruptions, identifying which forcing predictors actually matter for each variable.
  • The mode decomposition becomes an ESM evaluation tool: reanalysis places slow multidecadal, AMO-like modes first and projects most of the forced signal onto them, while the models put ENSO-scale oscillatory modes first and spread the forced response across faster modes.
  • The internal-component autocorrelations expose a non-stationary seasonal cycle in reanalysis that the models largely fail to reproduce.
  • Forced-response skill tracks model climate sensitivity: higher-sensitivity models with a stronger forced signal are reconstructed more accurately, so method comparisons across models must account for signal-to-noise differences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Implicit consequence: the method doubles as a check on forcing completeness — if the 'internal' series still carries low-frequency structure, some driver is missing from the supplied forcing history.
  • The forced time series of the leading modes could plausibly serve as fingerprints in detection-and-attribution regression, replacing ensemble-mean targets with single-realization estimates.
  • Because the authors note that noise and finite samples bias decay times toward faster decorrelation, part of the model-vs-observation reversal could be estimator bias rather than real model error; longer forcing histories or regularized operator fits would separate the two.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces PullbackDMDc, a linear stochastic forcing-plus-dynamics model (Eq. 2.3) that decomposes a single climate realization into forced and internal components via the pullback attractor of the fitted system (Eq. 2.5). The method is applied to monthly SAT, OSAT, and SLP anomalies from 20CRv3 and four MMLEA-v2 large ensembles, with the ensemble mean treated as ground truth. The authors report that PullbackDMDc matches or exceeds linear regression and other baselines in forced-response skill, and that the resulting mode decomposition reveals systematic ESM–reanalysis differences in multidecadal variability, ENSO/PDO structure, and seasonal-cycle non-stationarity.

Significance. If validated, PullbackDMDc would be a practical, interpretable single-realization estimator of forced response and internal variability, with potential value for ESM evaluation and detection–attribution studies. The derivation is transparent and largely self-contained, the code is publicly available, and the paper is unusually candid about the linearity assumption, finite-sample bias, and spin-up limitations. The main empirical claim, however, is currently supported by a test-set selection procedure that inflates reported skill, and the absence of significance testing makes it difficult to assess robustness. These issues are fixable within the manuscript's scope, so the contribution is promising but not yet fully demonstrated.

major comments (4)
  1. [Sec. 3a, Figs. 1 and S1–S3] The headline claim that PullbackDMDc 'matches or exceeds' linear regression is based on selecting, for each variable, the best of three forcing-variant models (1D, 2D, 3D) after inspecting their skill against the same ensemble-mean ground truth used for scoring (OSAT/SAT → 3D, SLP → 2D). This is selection on the test set: the maximum of several correlated skill scores is biased upward relative to any pre-specified method. The comparison to LR is therefore against the best of three fitted models, not against a fixed method. Please add a holdout protocol—for example, choose the forcing dimension on a subset of ensemble members or a training period, then evaluate on held-out members—or pre-register the variant per variable. Without this, the abstract's 'matching or exceeding' overstates what is demonstrated.
  2. [Sec. 3a, Fig. 1] There is no significance testing or uncertainty quantification for the pooled skill comparison. The large points pool across ensemble members, and the text interprets small differences as meaningful (e.g., 'at least one PullbackDMDc variant matches or exceeds LR skill across all variables'). Report per-member skill distributions, paired tests or bootstrap confidence intervals for differences in MSE/correlation, and state decision thresholds. As written, the 'matching or exceeding' claim is not statistically grounded.
  3. [Sec. 2b2 and Sec. 4] The method's central separation of forced and internal components assumes that the 100-year spin-up makes initial-condition errors negligible (Sec. 2b2), but Sec. 4 later states that 'a≈100-year forcing history may be insufficient for full convergence of the pullback dynamics when intrinsic timescales are longer than those recovered here.' These statements need reconciliation. If the fitted A has eigenvalues with decay times well below 100 years, please state this explicitly and show supporting evidence; otherwise, provide a sensitivity analysis of the forced-response estimate to spin-up length (e.g., 200 and 500 years). This matters because a residual contaminated by initial-condition memory is not pure internal variability, which directly affects the decomposition-based ESM evaluation in Sec. 3b.
  4. [Sec. 2c and Supplementary Sec. g] Several key hyperparameters—PC truncation (20 vs 200), propagator lag τ=3y, the 100-year spin-up, and the forcing predictor set—are selected after inspecting results on the same data. Some of these are described in the text as chosen because they 'generally produce the most accurate' estimates. Even if the qualitative conclusions are insensitive to these choices, the reported quantitative skill should be accompanied by a clear statement of which choices were made a priori and which were tuned on the evaluation data. A systematic sensitivity table (e.g., varying τ and spin-up length) would strengthen the paper.
minor comments (5)
  1. [Fig. 1 caption] Typo: 'PullbackDMDc(2D) for PSL' should read 'SLP'.
  2. [Fig. S18 caption] Typo: 'ESNO' should be 'ENSO'.
  3. [Eq. (2.9)] The model equation uses \hat y(t−1), whereas Eq. (2.3) uses y(t). The indexing convention is explained in the supplementary, but a brief note in the main text near Eq. (2.9) would avoid confusion.
  4. [Sec. 2c] The statement that the 'optimal forcing dimension varies little across ESMs' is based on visual inspection of overlapping uncertainty regions. Please provide numerical support or a more formal comparison.
  5. [References] Some references are dated 2026 and may be in press or preprint; please ensure they are publicly accessible and cite the archival versions where available.

Circularity Check

1 steps flagged

Headline skill claim is inflated by selecting the forcing variant on the same ensemble-mean ground truth used for evaluation; the core estimator itself is not circular.

specific steps
  1. fitted input called prediction [Section 3a ('Choice of forcing in PullbackDMDc'); abstract]
    "Because the ESM ensemble mean provides a per-model ground truth, this comparison doubles as a principled way to select forcing predictors for PullbackDMDc."

    The forcing dimension (1D/2D/3D) is a hyperparameter selected by comparing each variant's skill against the same ensemble-mean ground truth used to compute the reported skill. The abstract's claim that 'PullbackDMDc estimates the forced response with skill matching or exceeding established baselines' is therefore the best of three fitted variants, not the skill of a pre-specified method. The selection criterion is the evaluation metric itself, so the headline performance is optimistically biased by construction; a holdout protocol would be needed.

full rationale

The paper's central estimator is not circular: Eq. (2.3) defines a linear stochastic model, Eq. (2.5) defines the forced response as the deterministic filter of the chosen forcing and internal variability as the residual, and Algorithm 1 computes this filter from a single realization. The ensemble-mean ground truth from large ESM ensembles is external to the fit, so the estimator could in principle fail; the decomposition is a modeling assumption rather than a tautological prediction. No self-citation is load-bearing (the cited Mankovich et al. 2025 is descriptive, not a uniqueness or ansatz argument). The one genuine circularity is in the evaluation: Section 3a states that the ESM ensemble mean 'doubles as a principled way to select forcing predictors,' and the paper then reports the skill of the best-performing variant per variable (PullbackDMDc(3D) for OSAT/SAT, PullbackDMDc(2D) for SLP) as 'PullbackDMDc matches or exceeds baselines.' Because the same ground truth is used both to choose the forcing dimension and to score the method, the headline skill is the best of three fitted variants, not the skill of a pre-specified model. This is selection-on-the-test-set bias and inflates the central empirical claim; a holdout protocol would remove it. Overall circularity is partial (4/10), not fundamental.

Axiom & Free-Parameter Ledger

7 free parameters · 7 axioms · 0 invented entities

The paper contributes a decomposition that is exact for the fitted linear model, but the physical content rests on user choices (forcings, lag, PC truncation, spin-up) and on strong domain assumptions about linearity, forcing completeness, and ensemble-mean ground truth. No new physical entities are postulated.

free parameters (7)
  • PC truncation for SAT/OSAT = 20 PCs
    Retained for temperature fields, motivated by ForceSMIP but a user choice that changes A, the modes, and the forced response estimate.
  • PC truncation for SLP = 200 PCs
    Chosen to capture the weak SLP forced signal; an arbitrary threshold that affects all SLP results.
  • Propagator time lag tau = 3 months (main text) vs 3 years (supplement)
    Reported inconsistently; supplement says lags of 1, 3, 6, 12 years were tested and 3 years selected. Strongly affects decay times and forced-response skill.
  • Spin-up forcing history length = 100 years
    Chosen as a conservative spin-up; the paper itself notes this may be insufficient for long-timescale convergence.
  • Forcing predictor set = 1D total ERF; 2D CO2+volcanic; 3D GHG+volcanic+aerosol
    The variant is selected per variable using the ensemble-mean ground truth that is also used for scoring, so the reported 'best' variant is chosen on the evaluation target.
  • Number of modes retained for interpretation = 4
    Selected after an elbow in decay-time and ACF plots; affects the ESM comparison in Sec. 3b.
  • SLP ground-truth smoothing = 3-year running mean
    Applied to the ensemble-mean SLP before scoring; improves apparent skill for all methods on SLP.
axioms (7)
  • domain assumption The fitted linear operator A is stable (all eigenvalues have magnitude below 1).
    Required for the pullback limit in Eq. (2.5) to converge; if violated, the forced response sum diverges.
  • domain assumption The climate response to external forcing is adequately linear over 1850–2014.
    Eq. (2.3) assumes linear dynamics; the paper notes nonlinearity would make the separation set-valued and non-exact.
  • domain assumption The chosen forcing time series (total ERF, or CO2/GHG+volcanic+aerosol) captures all relevant forced drivers.
    Omitted forcings (e.g., land use, solar, ozone) would be absorbed into the internal residual and bias the decomposition.
  • ad hoc to paper Initial conditions are forgotten by the analysis period / the 100-year spin-up is sufficient.
    Algorithm 1 sets x̂_0 = 0 and iterates the deterministic filter; Sec. 2b2 asserts observations lie on the attractor. The paper later concedes 100 years may be insufficient for long timescales.
  • domain assumption The large-ensemble mean is a valid ground-truth forced response.
    Used for all skill scores; the authors acknowledge it is inaccurate for small ensembles or large internal variability.
  • domain assumption PCA truncation preserves the forced response and dominant internal modes.
    20 or 200 PCs are used; any forced signal in discarded PCs is lost by construction.
  • standard math The noise term is well-behaved enough for least-squares estimates of A and B to be meaningful.
    Least squares does not require white noise for consistency, but serial correlation or correlation with forcing would bias A/B; this is not tested.

pith-pipeline@v1.3.0-alltime-deepseek · 42197 in / 15132 out tokens · 147966 ms · 2026-08-02T06:43:40.330615+00:00 · methodology

0 comments
read the original abstract

We show that a single climate realization can be decomposed into forced and internal components by treating external forcing as a dynamical driver within a linear stochastic system, an idea grounded in pullback attractor theory. In doing so, we address a central methodological challenge in climate science with direct implications for climate projection and the detection and attribution of the forced response, disentangling the forced climate response from internal variability in a single observed record. Statistical methods range from approaches trained on large ensembles to techniques operating on single realizations. The latter often rely on linear frameworks such as linear inverse models (LIMs) and linear regression. LIMs ignore forcing predictors, whereas linear regression omits climate system dynamics. Here we introduce PullbackDMDc, a method grounded in non-autonomous dynamical systems theory and dynamic mode decomposition with control (DMDc), incorporating pullback attractor estimation to decompose a single climate realization into spatial modes and their associated forced and internal components, yielding a physically interpretable picture of the underlying dynamics. We illustrate the utility of PullbackDMDc for Earth System Model (ESM) evaluation by applying it to near-surface air temperature and sea-level pressure from reanalysis and four ESM large ensembles. PullbackDMDc estimates the forced response with skill matching or exceeding established baselines and identifies optimal forcing predictors against model-based ground truth. Its internal variability components reveal that ESMs qualitatively capture interannual and decadal modes while exhibiting systematic differences relative to each other and to observations. Skillful forced response estimation and a novel decomposition position PullbackDMDc as a practical tool for single-realization climate analysis and ESM evaluation.

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Reference graph

Works this paper leans on

294 extracted references · 142 canonical work pages

  1. [1]

    doi:10.1137/1.9781611974508 , isbn =

    Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems , author =. doi:10.1137/1.9781611974508 , isbn =

    doi:10.1137/1.9781611974508

  2. [2]

    Journal of Computational Dynamics , volume = 1, number = 2, pages =

    On Dynamic Mode Decomposition: Theory and Applications , author =. Journal of Computational Dynamics , volume = 1, number = 2, pages =

  3. [3]

    and Kevrekidis, Ioannis G

    Williams, Matthew O. and Kevrekidis, Ioannis G. and Rowley, Clarence W. , year = 2015, journal =. A Data-Driven Approximation of the

    2015

  4. [4]

    Applied and Computational Harmonic Analysis , volume = 62, number = 2, pages =

    Data-Driven Spectral Decomposition and Forecasting of Ergodic Dynamical Systems , author =. Applied and Computational Harmonic Analysis , volume = 62, number = 2, pages =

  5. [5]

    Journal of Climate , volume = 29, number = 11, pages =

    Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case , author =. Journal of Climate , volume = 29, number = 11, pages =

  6. [6]

    Stochastic Climate Models,

    Hasselmann, Klaus , year = 1976, journal =. Stochastic Climate Models,

    1976

  7. [7]

    Climate Dynamics , volume = 15, number = 6, pages =

    Checking for Model Consistency in Optimal Fingerprinting , author =. Climate Dynamics , volume = 15, number = 6, pages =

  8. [8]

    Climate Change 2013: The Physical Science Basis

    Detection and Attribution of Climate Change: From Global to Regional , author =. Climate Change 2013: The Physical Science Basis. Contribution of Working Group. doi:10.1017/CBO9781107415324.022 , editor =

    doi:10.1017/cbo9781107415324.022 2013

  9. [9]

    npj Climate and Atmospheric Science , volume = 1, pages = 28, doi =

    A Signal-to-Noise Paradox in Climate Science , author =. npj Climate and Atmospheric Science , volume = 1, pages = 28, doi =

  10. [10]

    Global attractors of non-autonomous dissipative dynamical systems , author =

  11. [11]

    Proceedings of the National Academy of Sciences of the United States of America , volume = 119, pages =

    Internal variability and forcing influence model-satellite differences in the rate of tropical tropospheric warming , author =. Proceedings of the National Academy of Sciences of the United States of America , volume = 119, pages =

  12. [12]

    Forced Component Estimation Statistical Method Intercomparison Project (Force

    Wills, Robert CJ and Deser, Clara and McKinnon, Karen A and Phillips, Adam and Po-Chedley, Stephen and Sippel, Sebastian and Merrifield, Anna L and B. Forced Component Estimation Statistical Method Intercomparison Project (Force. Journal of Climate , publisher =

  13. [13]

    Science Advances , volume = 7, number = 43, pages =

    Robust detection of forced warming in the presence of potentially large climate variability , author =. Science Advances , volume = 7, number = 43, pages =. doi:10.1126/sciadv.abh4429 , url =

    work page doi:10.1126/sciadv.abh4429

  14. [14]

    ArXiv , volume =

    Fair Regression: Quantitative Definitions and Reduction-based Algorithms , author =. ArXiv , volume =

  15. [15]

    SIAM Journal on Applied Dynamical Systems , publisher =

    Dynamic mode decomposition with control , author =. SIAM Journal on Applied Dynamical Systems , publisher =

  16. [16]

    Journal of climate , volume = 8, number = 8, pages =

    The optimal growth of tropical sea surface temperature anomalies , author =. Journal of climate , volume = 8, number = 8, pages =

  17. [17]

    Journal of fluid mechanics , publisher =

    Dynamic mode decomposition of numerical and experimental data , author =. Journal of fluid mechanics , publisher =

  18. [18]

    Nature Communications , publisher =

    An increase in marine heatwaves without significant changes in surface ocean temperature variability , author =. Nature Communications , publisher =

  19. [19]

    N. Demo, M. Tezzele, and G. Rozza , year = 2018, journal =. Py

    2018

  20. [20]

    Climate Change 2021: The Physical Science Basis

    The Earth's Energy Budget, Climate Feedbacks, and Climate Sensitivity , author =. Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change , publisher =. doi:10.1017/9781009157896.009 , url =

    doi:10.1017/9781009157896.009 2021

  21. [21]

    and Marotzke, J

    Lee, J.-Y. and Marotzke, J. and Bala, G. and Cao, L. and Corti, S. and Dunne, J. P. and Engelbrecht, F. and Fischer, E. and Fyfe, J. C. and Jones, C. and Maycock, A. and Mutemi, J. and Ndiaye, O. and Panickal, S. and Zhou, T. , title =. Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the I...

    2021

  22. [22]

    Demo, Nicola and Tezzele, Marco and Rozza, Gianluigi , year = 2018, journal =. Py

    2018

  23. [23]

    Ichinaga, Sara M and Andreuzzi, Francesco and Demo, Nicola and Tezzele, Marco and Lapo, Karl and Rozza, Gianluigi and Brunton, Steven L and Kutz, J Nathan , year = 2024, journal =. Py

    2024

  24. [24]

    Pattern recognition methods to separate forced responses from internal variability in climate model ensembles and observations , author =. J. Climate , volume = 33, number = 20, pages =

  25. [25]

    Nature Climate Change , publisher =

    Reconciling disparate twentieth-century Indo-Pacific ocean temperature trends in the instrumental record , author =. Nature Climate Change , publisher =

  26. [26]

    Physica D: Nonlinear Phenomena , volume = 240, pages =

    Stochastic climate dynamics: Random attractors and time-dependent invariant measures , author =. Physica D: Nonlinear Phenomena , volume = 240, pages =. doi:10.1016/j.physd.2011.06.005 , issn =

    doi:10.1016/j.physd.2011.06.005 2011

  27. [27]

    Climate Dynamics , publisher =

    Linear dynamical modes as new variables for data-driven ENSO forecast , author =. Climate Dynamics , publisher =. doi:10.1007/s00382-018-4255-7 , issue =

    work page doi:10.1007/s00382-018-4255-7

  28. [28]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , publisher =

    Data-driven stochastic model for cross-interacting processes with different time scales , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , publisher =. doi:10.1063/5.0077302 , issn =

    work page doi:10.1063/5.0077302

  29. [29]

    Physical Review E , volume = 85, pages =

    Random dynamical models from time series , author =. Physical Review E , volume = 85, pages =. doi:10.1103/PhysRevE.85.036216 , issn =

    work page doi:10.1103/physreve.85.036216

  30. [30]

    Annual Review of Control, Robotics, and Autonomous Systems , publisher =

    Koopman operators for estimation and control of dynamical systems , author =. Annual Review of Control, Robotics, and Autonomous Systems , publisher =

  31. [31]

    Linear predictors for nonlinear dynamical systems:

    Korda, Milan and Mezi. Linear predictors for nonlinear dynamical systems:. Automatica , publisher =

  32. [32]

    The updated

    Maher, Nicola and Phillips, Adam S and Deser, Clara and Wills, Robert C Jnglin and Lehner, Flavio and Fasullo, John and Caron, Julie M and Brunner, Lukas and Beyerle, Urs and Jeffree, Jemma , year = 2025, journal =. The updated

    2025

  33. [33]

    Journal of Climate , volume = 31, number = 14, pages =

    On the choice of ensemble mean for estimating the forced signal in the presence of internal variability , author =. Journal of Climate , volume = 31, number = 14, pages =

  34. [34]

    Part I: Forced response , author =

    Global-scale multidecadal variability in climate models and observations. Part I: Forced response , author =. Journal of Climate , publisher =

  35. [35]

    Climate Dynamics , volume = 62, pages =

    Global-scale multidecadal variability in climate models and observations, part II: The stadium wave , author =. Climate Dynamics , volume = 62, pages =. doi:10.1007/s00382-024-07451-4 , issn =

    work page doi:10.1007/s00382-024-07451-4

  36. [36]

    Durack, Paul J and Taylor, Karl E and Eyring, Veronika and Ames, Sasha K and Doutriaux, Charles and Hoang, Tony and Nadeau, Denis and Stockhause, Martina and Gleckler, Peter J , year = 2017, institution =. input4

    2017

  37. [37]

    Swart, Neil C and Cole, Jason NS and Kharin, Viatcheslav V and Lazare, Mike and Scinocca, John F and Gillett, Nathan P and Anstey, James and Arora, Vivek and Christian, James R and Hanna, Sarah and others , year = 2019, journal =. The

    2019

  38. [38]

    Description and basic evaluation of simulated mean state, internal variability, and climate sensitivity in

    Tatebe, Hiroaki and Ogura, Tomoo and Nitta, Tomoko and Komuro, Yoshiki and Ogochi, Koji and Takemura, Toshihiko and Sudo, Kengo and Sekiguchi, Miho and Abe, Manabu and Saito, Fuyuki and others , year = 2019, journal =. Description and basic evaluation of simulated mean state, internal variability, and climate sensitivity in

    2019

  39. [39]

    Developments in the

    Mauritsen, Thorsten and Bader, J. Developments in the. Journal of Advances in Modeling Earth Systems , publisher =

  40. [40]

    A higher-resolution version of the

    M. A higher-resolution version of the. Journal of Advances in Modeling Earth Systems , publisher =

  41. [41]

    Journal of Climate , volume = 39, pages =

    Forced Component Estimation Statistical Method Intercomparison Project (ForceSMIP) , author =. Journal of Climate , volume = 39, pages =. doi:10.1175/JCLI-D-25-0326.1 , issn =

    work page doi:10.1175/jcli-d-25-0326.1

  42. [42]

    The Kernelized Taylor Diagram , author =

  43. [43]

    Environmental Data Science , volume = 4, pages =

    Analyzing climate scenarios using dynamic mode decomposition with control , author =. Environmental Data Science , volume = 4, pages =. doi:10.1017/eds.2025.8 , issn =

    doi:10.1017/eds.2025.8 2025

  44. [44]

    Geoscientific Model Development , volume = 17, pages =

    New model ensemble reveals how forcing uncertainty and model structure alter climate simulated across CMIP generations of the Community Earth System Model , author =. Geoscientific Model Development , volume = 17, pages =. doi:10.5194/gmd-17-1585-2024 , issn =

    work page doi:10.5194/gmd-17-1585-2024 2024

  45. [45]

    Applied Soft Computing , volume = 68, pages =

    Physics-aware Gaussian processes in remote sensing , author =. Applied Soft Computing , volume = 68, pages =

  46. [46]

    Reviews of Modern Physics , volume = 93, pages =

    Emergent constraints on climate sensitivities , author =. Reviews of Modern Physics , volume = 93, pages =. doi:10.1103/RevModPhys.93.025004 , issn =

    doi:10.1103/revmodphys.93.025004

  47. [47]

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume = 365, pages =

    The use of the multi-model ensemble in probabilistic climate projections , author =. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume = 365, pages =. doi:10.1098/rsta.2007.2076 , issn =

    arXiv 2007

  48. [48]

    Earth System Dynamics , volume = 10, pages =

    ESD Reviews: Model dependence in multi-model climate ensembles: weighting, sub-selection and out-of-sample testing , author =. Earth System Dynamics , volume = 10, pages =. doi:10.5194/esd-10-91-2019 , issn =

    work page doi:10.5194/esd-10-91-2019 2019

  49. [49]

    Russian Journal of Numerical Analysis and Mathematical Modelling , volume = 40, pages =

    Reproducibility of atmospheric circulation regimes over the winter Northern Hemisphere by the INMCM5 Earth system model , author =. Russian Journal of Numerical Analysis and Mathematical Modelling , volume = 40, pages =. doi:10.1515/rnam-2025-0011 , issn =

    work page doi:10.1515/rnam-2025-0011 2025

  50. [50]

    arXiv:2412.06933 , abstract =

    Metastability, atmospheric midlatitude circulation regimes and large-scale teleconnection: a data-driven approach , author =. arXiv:2412.06933 , abstract =

    arXiv

  51. [51]

    Russian Journal of Numerical Analysis and Mathematical Modelling , volume = 40, pages =

    Analysis of forced response and internal climate variability in the INMCM Earth system model , author =. Russian Journal of Numerical Analysis and Mathematical Modelling , volume = 40, pages =. doi:10.1515/rnam-2025-0008 , issn =

    work page doi:10.1515/rnam-2025-0008 2025

  52. [52]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume = 32, doi =

    Revealing recurrent regimes of mid-latitude atmospheric variability using novel machine learning method , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume = 32, doi =

  53. [53]

    The Cryosphere , volume = 18, pages =

    Sources of low-frequency variability in observed Antarctic sea ice , author =. The Cryosphere , volume = 18, pages =. doi:10.5194/tc-18-2141-2024 , issn =

    doi:10.5194/tc-18-2141-2024 2024

  54. [54]

    The Cryosphere , volume = 17, pages =

    Forced and internal components of observed Arctic sea-ice changes , author =. The Cryosphere , volume = 17, pages =. doi:10.5194/tc-17-4133-2023 , issn =

    work page doi:10.5194/tc-17-4133-2023 2023

  55. [55]

    Earth System Dynamics , volume = 14, pages =

    The future of the El Niño–Southern Oscillation: using large ensembles to illuminate time-varying responses and inter-model differences , author =. Earth System Dynamics , volume = 14, pages =. doi:10.5194/esd-14-413-2023 , issn =

    work page doi:10.5194/esd-14-413-2023 2023

  56. [56]

    EGU General Assembly Conference Abstracts , publisher =

    Forced Component Estimation Statistical Methods Intercomparison Project (ForceSMIP): First Results , author =. EGU General Assembly Conference Abstracts , publisher =

  57. [57]

    Journal of Advances in Modeling Earth Systems , volume = 11, pages =

    The Max Planck Institute Grand Ensemble: Enabling the Exploration of Climate System Variability , author =. Journal of Advances in Modeling Earth Systems , volume = 11, pages =. doi:10.1029/2019MS001639 , issn =

    work page doi:10.1029/2019ms001639

  58. [58]

    Russian Journal of Numerical Analysis and Mathematical Modelling , volume = 39, pages =

    ENSO phase locking, asymmetry and predictability in the INMCM Earth system model , author =. Russian Journal of Numerical Analysis and Mathematical Modelling , volume = 39, pages =. doi:10.1515/rnam-2024-0004 , issn =

    work page doi:10.1515/rnam-2024-0004 2024

  59. [59]

    Featurizing

    David Aristoff and Jeremy Copperman and Nathan Mankovich and Alexander Davies , year = 2024, month = 8, journal =. Featurizing. doi:10.1063/5.0220277 , issn =

    work page doi:10.1063/5.0220277 2024

  60. [60]

    Nonlinear Dynamics , volume = 41, pages =

    Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , author =. Nonlinear Dynamics , volume = 41, pages =. doi:10.1007/s11071-005-2824-x , issn =

    doi:10.1007/s11071-005-2824-x

  61. [61]

    Nature Climate Change , volume = 10, pages =

    Insights from Earth system model initial-condition large ensembles and future prospects , author =. Nature Climate Change , volume = 10, pages =. doi:10.1038/s41558-020-0731-2 , issn =

    doi:10.1038/s41558-020-0731-2

  62. [62]

    Astronomy Letters , volume = 50, pages =

    The Method of Periodic Principal Components for the Dynamic Spectrum of Radio Pulsars and Faraday Rotation of Nine Pulse Components of PSR B0329\# , author =. Astronomy Letters , volume = 50, pages =. doi:10.1134/S1063773724700051 , issn =

    work page doi:10.1134/s1063773724700051

  63. [63]

    Climate Dynamics , publisher =

    Forced response and internal variability in ensembles of climate simulations: identification and analysis using linear dynamical mode decomposition , author =. Climate Dynamics , publisher =. doi:10.1007/S00382-023-06995-1 , issn = 14320894, abstract =

    work page doi:10.1007/s00382-023-06995-1

  64. [64]

    Nature Methods 2022 19:2 , publisher =

    Identifying temporal and spatial patterns of variation from multimodal data using MEFISTO , author =. Nature Methods 2022 19:2 , publisher =. doi:10.1038/s41592-021-01343-9 , issn =

    doi:10.1038/s41592-021-01343-9 2022

  65. [65]

    29th International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics , publisher =

    Identification of dynamic variables capturing interannual behavior of ENSO based on ocean heat content data , author =. 29th International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics , publisher =. doi:10.1117/12.2690490 , isbn = 9781510668096, abstract =

    work page doi:10.1117/12.2690490

  66. [66]

    Journal of Theoretical Biology , publisher =

    Learning pharmacokinetic models for in vivo glucocorticoid activation , author =. Journal of Theoretical Biology , publisher =. doi:10.1016/J.JTBI.2018.07.025 , issn =

    work page doi:10.1016/j.jtbi.2018.07.025 2018

  67. [67]

    Proceedings of the International Joint Conference on Neural Networks , publisher =

    Classification of sparsely and irregularly sampled time series: A learning in model space approach , author =. Proceedings of the International Joint Conference on Neural Networks , publisher =

  68. [68]

    Journal of Climate , volume = 32, pages =

    Uncovering the Forced Climate Response from a Single Ensemble Member Using Statistical Learning , author =. Journal of Climate , volume = 32, pages =. doi:10.1175/JCLI-D-18-0882.1 , issn =

    doi:10.1175/jcli-d-18-0882.1

  69. [69]

    Climate Dynamics , volume = 42, pages =

    Role for Eurasian Arctic shelf sea ice in a secularly varying hemispheric climate signal during the 20th century , author =. Climate Dynamics , volume = 42, pages =. doi:10.1007/s00382-013-1950-2 , issn =

    work page doi:10.1007/s00382-013-1950-2 1950

  70. [70]

    Journal of Climate , volume = 14, pages =

    Discriminants of Twentieth-Century Changes in Earth Surface Temperatures , author =. Journal of Climate , volume = 14, pages =

  71. [71]

    Journal of Climate , volume = 28, pages =

    Dynamical Adjustment of the Northern Hemisphere Surface Air Temperature Field: Methodology and Application to Observations* , author =. Journal of Climate , volume = 28, pages =. doi:10.1175/JCLI-D-14-00111.1 , issn =

    doi:10.1175/jcli-d-14-00111.1

  72. [72]

    Proceedings of the National Academy of Sciences , volume = 109, pages =

    Simulated versus observed patterns of warming over the extratropical Northern Hemisphere continents during the cold season , author =. Proceedings of the National Academy of Sciences , volume = 109, pages =. doi:10.1073/pnas.1204875109 , issn =

    work page doi:10.1073/pnas.1204875109

  73. [73]

    Journal of Climate , volume = 29, pages =

    Forced and Internal Components of Winter Air Temperature Trends over North America during the past 50 Years: Mechanisms and Implications* , author =. Journal of Climate , volume = 29, pages =. doi:10.1175/JCLI-D-15-0304.1 , issn =

    doi:10.1175/jcli-d-15-0304.1

  74. [74]

    Journal of Climate , volume = 24, pages =

    A Significant Component of Unforced Multidecadal Variability in the Recent Acceleration of Global Warming , author =. Journal of Climate , volume = 24, pages =. doi:10.1175/2010JCLI3659.1 , issn =

    work page doi:10.1175/2010jcli3659.1

  75. [75]

    Journal of Climate , volume = 30, pages =

    Estimation of the SST Response to Anthropogenic and External Forcing and Its Impact on the Atlantic Multidecadal Oscillation and the Pacific Decadal Oscillation , author =. Journal of Climate , volume = 30, pages =. doi:10.1175/JCLI-D-17-0009.1 , issn =

    work page doi:10.1175/jcli-d-17-0009.1

  76. [76]

    Physics Letters A , volume = 234, pages =

    Optimal filtering in singular spectrum analysis , author =. Physics Letters A , volume = 234, pages =. doi:10.1016/S0375-9601(97)00559-8 , issn =

    work page doi:10.1016/s0375-9601(97)00559-8

  77. [77]

    Reviews of Geophysics , publisher =

    A Review of the Role of the Atlantic Meridional Overturning Circulation in Atlantic Multidecadal Variability and Associated Climate Impacts , author =. Reviews of Geophysics , publisher =. doi:10.1029/2019RG000644 , issn =

    work page doi:10.1029/2019rg000644

  78. [78]

    Journal of Climate , volume = 29, pages =

    The Pacific Decadal Oscillation, Revisited , author =. Journal of Climate , volume = 29, pages =. doi:10.1175/JCLI-D-15-0508.1 , issn =

    doi:10.1175/jcli-d-15-0508.1

  79. [79]

    Geophysical Research Letters , publisher =

    Disentangling Global Warming, Multidecadal Variability, and El Niño in Pacific Temperatures , author =. Geophysical Research Letters , publisher =. doi:10.1002/2017GL076327 , issn =

    work page doi:10.1002/2017gl076327

  80. [80]

    Journal of Climate , volume = 22, pages =

    Forced and Internal Twentieth-Century SST Trends in the North Atlantic* , author =. Journal of Climate , volume = 22, pages =. doi:10.1175/2008JCLI2561.1 , issn =

    work page doi:10.1175/2008jcli2561.1

Showing first 80 references.