Pith. sign in

REVIEW 3 major objections 5 minor 55 references

Improving the Predictability of the Madden-Julian Oscillation at Subseasonal Scales with Gaussian Process Models

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Gaussian processes trained on RMM history alone give MJO forecasts with 12 days of correlation skill and calibrated uncertainty for several weeks.

desk verdict A GP-based MJO forecaster with a useful idea and a code deposit, but the reported skill and coverage numbers are unverifiable because the paper contradicts itself on the train/validation/test split. read the letter →

arxiv 2505.15934 v1 pith:ZSJC4BHG submitted 2025-05-21 math.NA cs.NAphysics.ao-phstat.ML

Haoyuan Chen , Emil Constantinescu , Vishwas Rao , Cristiana Stan This is my paper · ORCID
classification math.NAcs.NAphysics.ao-phstat.ML MSC 62M2060G1586A10
keywords Madden-JulianOscillationGaussianprocesssubseasonal-to-seasonalforecastRMMindexuncertaintyquantificationprobabilisticforecastingautoregressivetimeseriesempiricalcovariance
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

This paper claims that a Gaussian process trained only on the past values of the RMM index — the two-component summary of the Madden-Julian Oscillation's phase and amplitude — can forecast the MJO with skill that beats a neural network for the first five lead days while also supplying calibrated uncertainty intervals. This would matter because most machine-learning MJO forecasters return point predictions and require ensembles or special architectures to say anything about confidence, whereas a GP gives a predictive distribution directly. The authors estimate the GP's covariance from historical correlations rather than tuning hyperparameters, then add a validation-based covariance correction so the uncertainty widens as forecasts are iterated farther ahead. On their test period the corrected model reports correlation skill above the standard 0.5 threshold for 12 lead days and 68% confidence regions that cover observed MJO trajectories for about 60 days on average.

What carries the argument

The central object is a bivariate Gaussian process over (RMM1, RMM2) with an empirically estimated mean vector and covariance matrix. The covariance is built from cubic-spline interpolations of the historical cross- and autocorrelation functions of the two RMM components, which removes the need for kernel hyperparameter optimization. Forecasts are generated by iterated one-step-ahead GP regression, and the key correction is the posteriori covariance update (equations B7-B8): at each lead time the diagonal variances are inflated by the mean squared bias between validation predictions and observations, and the off-diagonal terms are rescaled consistently. The corrected covariance at lead time τ defines a 68% confidence ellipse via the chi-square quantile for two degrees of freedom, converting a fixed state-independent GP variance into a lead-time-dependent uncertainty estimate.

What would settle it

Retrain the GP on RMM data ending December 31, 2006, refit the covariance correction on the 2007-2011 period, and score 2012-2023; if the first-five-day advantage over the neural network or the three-week coverage extension disappears, the published skill numbers depend on the overlap between training and evaluation periods. A second check is to compute the 68% coverage separately for days with RMM amplitude above and below one; if coverage differs materially, the state-independent covariance correction is misspecified.

Watch

Extended reading notes

Core claim

The discovery is that the bivariate RMM time series can be treated as a Gaussian process whose mean and covariance are estimated empirically from training batches, and that iterating one-step-ahead GP forecasts with a lead-time-dependent covariance correction yields useful deterministic and probabilistic MJO predictions. The empirical covariance is obtained by cubic-spline interpolation of the observed cross- and autocorrelations of RMM1 and RMM2, so no kernel hyperparameters need to be optimized. Multistep forecasts are produced autoregressively: each one-day prediction becomes part of the input window for the next, and the posteriori covariance correction adds the validation mean-squared error to the predictive variance at each lead time to compensate for feeding predictions back as if they were observations. The paper reports that this model has higher correlation than a neural network baseline for the first five forecast days, lower RMSE overall, correlation skill above 0.5 through 12 lead days, and 68% confidence regions that cover the observations for roughly 60 days on average.

Load-bearing premise

The reported skill assumes that the validation years (2007-2011) and the early test years (2012-2016) are genuinely out-of-sample, even though the training period used to build the empirical mean and covariance runs through December 31, 2016.

Editorial extensions

If this is right

  • A GP using only the past 40 or 60 days of RMM values reaches the standard 0.5 correlation threshold at 12 lead days, so the deterministic mean is a usable MJO forecast for roughly two weeks.
  • The RMSE stays below the climatological 1.4 threshold through all 60 tested lead days, meaning the point forecasts do not drift to useless amplitudes at subseasonal range.
  • The posteriori covariance correction makes forecast variance grow with lead time, and the resulting 68% confidence ellipses cover observed RMM trajectories for about 60 days on average.
  • Because the GP yields a full predictive distribution directly, probabilistic MJO forecasting no longer requires ensembles or Monte Carlo dropout.
  • Compared with a neural network baseline, the GP's correlation skill is higher for the first five lead days and its RMSE is lower overall.

Reading between the lines

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

  • The paper leaves implicit that the 12-day correlation ceiling is an estimate of how much MJO predictability is contained in the RMM history alone; adding exogenous predictors such as tropical winds or convection would test whether the ceiling is a property of the index or of the inputs.
  • A reader could extend the covariance correction into a full recalibration by fitting the validation bias as a smooth function of lead time, which would likely improve CRPS and log score without changing the deterministic mean.
  • Because the corrected covariance depends only on lead time, the model predicts the same uncertainty for quiescent and active MJO episodes; testing coverage separately for strong and weak MJO events would show whether state-dependent variance is needed.
  • The positive phase errors reported for the GP suggest that feeding back predicted values biases the phase toward faster propagation; a simple amendment would be to diagnose the phase-speed bias on validation and rotate the forecast vector by the average error.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a Gaussian process (GP) model for forecasting the two-component RMM index of the Madden–Julian Oscillation. The GP is built from an empirical mean and an empirically estimated covariance (cubic-spline-interpolated autocorrelations) rather than optimized kernels, and multi-day forecasts are generated by iterating one-step-ahead predictions. To account for the additional uncertainty created by iterating, the authors add a validation-based variance bias at each lead time (Eqs. B7–B8). They report deterministic skill (COR, RMSE, phase/amplitude errors, HSS), probabilistic scores (CRPS, log score), and two case studies of confidence regions, and they claim the model outperforms ANN models for the first five lead days and that the covariance correction extends probabilistic coverage by more than three weeks.

Significance. If the claims were established, the paper would offer a simple, hyperparameter-light probabilistic baseline for MJO forecasting with an explicit uncertainty estimate, complementing neural-network approaches. The authors provide code (Zenodo DOI) and evaluate against S2S models, which aids reproducibility. However, the current manuscript contains a critical inconsistency in the train/validation/test split, an indirect ANN comparison, and no quantitative coverage metric for the probabilistic claims; these issues must be resolved before the paper's central claims can be accepted.

major comments (3)
  1. [Section 2.1 vs. Table B1/B5] The manuscript gives two incompatible definitions of the training period. Section 2.1 states that the training set is January 1, 1979, to December 31, 2016, while the validation set is 2007–2011 and the test set begins in 2012; this places the validation period and the first half of the test period inside the training window. If that description is what was implemented, then the empirical mean and covariance in Eqs. (B3)–(B4) and the validation bias correction in Eq. (B7) are computed from the very days being forecast, so the reported COR, RMSE, CRPS, and coverage numbers are in-sample and the central claims are void. Table B1 and Eq. (B5), by contrast, construct the training data from the first L+n days (n=10,000, L=40 or 60), which would end around 2006, before the validation period. These two descriptions cannot both be true. The authors must state unambiguously which split was used, correct the other description, and confirm that no evaluation day contributes to the empirical mean/covariance or to the fitted variance bias. Until this is resolved, none of the quantitative claims can be assessed.
  2. [Section 4.2 and Abstract] The claim that the model has "better prediction skills than the ANN models for the first five lead days" is not supported by a direct experiment. Section 4.2 compares the GP results with Kim et al. (2021), which is a bias-correction study for a dynamical model rather than an ANN trained and tested on the same RMM data and period. No ANN baseline is implemented in this paper, and the comparison is therefore indirect and uncontrolled. The authors should either add a direct ANN baseline under the same data split and metrics or limit the claim to what the experiments actually show.
  3. [Section 4.3 and Abstract] The statement that the posteriori covariance correction "extends probabilistic coverage by more than three weeks" is not supported by a quantitative coverage statistic. Section 4.3 presents only two case studies (Figures 5–7) and a qualitative statement that observations lie within the confidence region. The paper should report, over the full test set, the empirical coverage of the nominal confidence regions as a function of lead time, with and without the correction in Eqs. (B7)–(B8), together with a reliability diagram or PIT histogram. Without this, it is impossible to determine whether the correction genuinely calibrates the forecast or merely inflates the variance on the validation set (and, given the first major comment, possibly on training days).
minor comments (5)
  1. [Section 2.2 and Figure 3 caption] There are typos: "covraiance" should be "covariance" in Section 2.2, and "degress" should be "degrees" in the Figure 3 caption.
  2. [Section 4.2] The text says "the black cross marker in Figure 1" but the HSS heatmap with the cross markers is Figure 4; please correct the cross-reference.
  3. [Section 3] The evaluation section states np=528 but does not specify how these 528 predictions are initialized or spaced in time; please state the initialization dates and whether the forecasts are overlapping or independent.
  4. [Appendix B2, Eq. (B7)] The notation "Bias(\hat z, z)^2" is not defined; please state explicitly that this term is the mean squared error over the validation set, and use "a posteriori" consistently throughout the paper.
  5. [Table B1] The workflow in Table B1 would be easier to follow if the exact date range used to build the X and y matrices were stated explicitly rather than implied by the indexing in Eq. (B5).

Circularity Check

2 steps flagged · score 6.0 of 10

Section 2.1 places the validation (2007-2011) and the first half of the test (2012-2016) inside the 1979-2016 training window; equations (B3)-(B5) then make those 'forecasts' GP interpolants of the observed targets, so the short-lead skill and the covariance-corrected coverage reduce to in-sample fit unless the contradictory Table B1 split was used.

  1. self definitional [Section 2.1 (Data) and Appendix B1, equations (B3)-(B5)]
    "The dataset is divided into three subsets: i) the training set used to determine the parameters of the prediction and corresponding variance, January, 1, 1979 to December 31, 2016; ii) the validation set used to obtain the corrected variance with increasing lags, January 1, 2007 to December 31, 2011; and iii) the test set used to verify the results, January 1, 2012 to December 31, 2023. ... In equations (B3) and (B4) we use the empirical mean and covariance of n batches of training data with lag L to approximate the expectation of the target and the covariance of the target and predictors."

    With this split, every validation date (2007-2011) and the first half of the test period (2012-2016) are inside the training window. Equation (B5) defines training inputs X as sliding windows z_{i:L+i-1} and outputs y as z_{L+i}. A forecast initialized in 2012-2016 has an input window and a target day that are both among these n training pairs, so the posterior mean (B3) is the zero-noise GP interpolant at a training point and returns the observed RMM value. The reported skill for the first five lead days, and the MSE used in the variance correction (B7), are therefore in-sample reconstructions rather than predictions. Table B1, which implies training ends around 2006, contradicts Section 2.1, so the paper never establishes which split was actually used.

  2. fitted input called prediction [Appendix B2 (Covariance update), equations (B7)-(B8), and Table B1 step 7]
    "For each lead time we use a validation set of size nv(L) with lag L to compute the averaged variance bias between the posterior mean and true observations. Hence, the corrected variance ... ≈ Kt∗[j, j] + 1/nv Σ (ˆz(j)t(τ) − z(j)t(τ))^2."

    Equation (B7) defines the corrected variance as the raw GP variance plus the validation mean-squared error at each lead time, and Table B1 step 7 then transfers these corrected covariances to the test set at the corresponding lead. Because the validation period (2007-2011) lies inside the training period of Section 2.1, the MSE in (B7) is computed from interpolated in-sample means. The corrected covariance is therefore fitted to data already seen by the empirical mean/covariance, and the claimed extension of probabilistic coverage beyond three weeks is a fitted property of that correction rather than an independent out-of-sample probabilistic prediction.

full rationale

The paper's central claims — better deterministic skill than ANN for the first five lead days and probabilistic coverage extended beyond three weeks by the posteriori covariance correction — are exactly the quantities affected by the training/validation/test split. Section 2.1 states that the training set runs to December 31, 2016, while the validation set (2007-2011) and the first half of the test set (2012-2016) fall entirely inside that window. Under equations (B3)-(B5), a forecast whose input window and target date are contained in the training set is a zero-noise GP interpolant, so the 'prediction' equals the observed RMM value by construction. The covariance correction (B7) is likewise calibrated on validation predictions that would be in-sample under that split. However, the algorithmic description in Table B1 and equations (B5)-(B6) define the training dataset as the first L+n days, which for n=10,000 and L=40 or 60 ends around 2006, before the validation period; if that is what was implemented, the evaluation may be out-of-sample and the circularity would be low. The manuscript is internally inconsistent at the exact point on which the empirical claims depend. No load-bearing self-citation chain or imported uniqueness theorem is present; the only self-citation is the code repository. The score of 6 reflects that, on the paper as written, one or more 'predictions' reduce to in-sample fits or interpolations for a substantial portion of the reported evaluation period, while an alternative reading of Table B1 could rescue the out-of-sample status. The findings cannot be verified from the manuscript as written.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The GP model and its confidence intervals depend on a fitted empirical mean/covariance, a per-lead-time variance calibration, and the assumption that the bivariate RMM process is stationary and Gaussian. No new physical entities are introduced. The most severe issue is that the fitted quantities are evaluated on overlapping time periods.

free parameters (4)
  • Lag length L = 40 and 60 days
    Choice of how much RMM history feeds the GP; results are reported for both and are similar.
  • Training sample count n = 10,000
    Number of overlapping windows used to compute the empirical mean and covariance; chosen without a sensitivity analysis.
  • Validation period start and end = Jan 1 2007 to Dec 31 2011
    Used to compute the covariance correction; overlaps the training period, making the correction in-sample.
  • Per-lead-time variance bias b_j(tau) = Mean squared forecast error on the validation set for each lead time and RMM component
    Added to the GP variance in Eq. (B7); this fitted quantity produces the reported coverage extension.
assumptions (5)
  • domain assumption RMM1 and RMM2 form a stationary bivariate Gaussian process over the full 1979-2023 period.
    GP predictive formulas (B3)-(B4) assume a joint normal distribution and a fixed mean/covariance estimated from training data; no stationarity or Gaussianity test is reported.
  • domain assumption Empirical mean and covariance from the training windows generalize to all forecast dates.
    The model uses historical correlations as the true covariance; if the MJO statistics drift, the GP posterior is misspecified.
  • ad hoc to paper The predictive covariance K_t* is independent of lead time and predictor values and can be repaired by adding validation mean squared error.
    Equation (B7) assumes the only misspecification is a lead-time-dependent variance inflation, with unchanged correlation structure; no theoretical justification is given.
  • domain assumption The forecast distribution remains Gaussian after iterating predictions as inputs.
    The authors acknowledge the normality assumption is a limitation and say it is supported by numerical results without showing formal tests.
  • domain assumption Cubic spline interpolation is an accurate smooth representation of the true cross/auto-correlation functions.
    The covariance blocks are not used directly but replaced by splines through sample correlations; spline fitting error is not quantified.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Improving the Predictability of the Madden-Julian Oscillation at Subseasonal Scales with Gaussian Process Models." pith.science (2026). https://pith.science/paper/ZSJC4BHG

@misc{pith2026250515934,
  author       = {Pith},
  title        = {Pith review of: Improving the Predictability of the Madden-Julian Oscillation at Subseasonal Scales with Gaussian Process Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSJC4BHG}},
  note         = {Machine review of arXiv:2505.15934}
}
read the original abstract

The Madden--Julian Oscillation (MJO) is an influential climate phenomenon that plays a vital role in modulating global weather patterns. In spite of the improvement in MJO predictions made by machine learning algorithms, such as neural networks, most of them cannot provide the uncertainty levels in the MJO forecasts directly. To address this problem, we develop a nonparametric strategy based on Gaussian process (GP) models. We calibrate GPs using empirical correlations and we propose a posteriori covariance correction. Numerical experiments demonstrate that our model has better prediction skills than the ANN models for the first five lead days. Additionally, our posteriori covariance correction extends the probabilistic coverage by more than three weeks.

Figures

Figures reproduced from arXiv: 2505.15934 by the authors.

Figure 1
Figure 1. Flowchart of the entire algorithm. Top: Diagram of the GP model for the MJO forecast. The blue arrows indicate the order of operations in the algorithm. t ∗ represents the predicted timestamp, Bias2 is the square of the bias between the predicted values and the true observations. Bottom: Iterated method for the multistep time series forecasting for two outputs with lag = L, lead time = τ (τ > L). z (1) t , z (2) t a… view at source ↗
Figure 2
Figure 2. Cross-correlations and auto-correlations of RMMs with maximum lag = 60 days. covariance of the RMMs is modeled using a cubic spline interpolation of the cross- and auto-correlations of the indices, shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Prediction skill quantifiers and errors of the GP model with lag L = 40, 60, respec￾tively, compared to three models in the sub-seasonal to seasonal prediction project (S2S). Top: COR (higher is better), RMSE (lower is better), and phase error (degress) (lower is better) over 528 predictions. Bottom: Amplitude error (lower absolute is better), CRPS (lower is better), and log score (negative log-likelihood) (lower is… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: HSS heatmap for the GP model over 528 predictions with lag L = 40 (higher HSS is better). The cells with black cross marker “X” represent the significant samples from Fisher’s exact test with the critical value α = 0.05. confidence intervals (CI) grow as the forecast l…
Figure 5
Figure 5. Figure 5: Left: 60-days MJO phase diagram for Nov–03–2012 to Jan–01–2013 with lag L = 40. Black lines are observations (truth). Olive lines are predictions in November, and olive shadings are 68% confidence regions (CR) in November. Dark blue lines are predictions in December, a…
Figure 6
Figure 6. Figure 6: 60-days time series of MJO for Nov–03–2012 to Jan–01–2013 for lag L = 40, 60. We denote observations (truth) from the BOM by black dots; predictions of the GP model for lag L = 40 and L = 60 by blue cross and orange cross, respectively; ±σ CI of the GP model for lag L …
Figure 7
Figure 7. Figure 7: 60-days time series of MJO for Jan–14–2013 to Mar–14–2013 for lag L = 40, 60. We denote observations (truth) from the BOM by black dots; predictions of the GP model for lag L = 40 and L = 60 by blue cross and orange cross, respectively; ±σ CI of the GP model for lag L …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 50 canonical work pages

  1. [1]

    , Hendon, H H

    bergman2001intraseasonal APACrefauthors Bergman, J W. , Hendon, H H. \ Weickmann, K M. APACrefauthors \ 2001 . Intraseasonal air--sea interactions at the onset of El Ni \ n o Intraseasonal air--sea interactions at the onset of El Ni \ n o . Journal of Climate 14 8 1702--1719

    work page 2001

  2. [2]

    , Wheeler, M C

    camargo2009diagnosis APACrefauthors Camargo, S J. , Wheeler, M C. \ Sobel, A H. APACrefauthors \ 2009 . Diagnosis of the MJO modulation of tropical cyclogenesis using an empirical index Diagnosis of the MJO modulation of tropical cyclogenesis using an empirical index . Journal of the Atmospheric Sciences 66 10 3061--3074

    work page 2009

  3. [3]

    , Ling, J

    chen2022mjo APACrefauthors Chen, G. , Ling, J. , Zhang, R. , Xiao, Z. \ Li, C. APACrefauthors \ 2022 . The MJO from CMIP5 to CMIP6: Perspectives from tracking MJO precipitation The MJO from CMIP5 to CMIP6: Perspectives from tracking MJO precipitation . Geophysical Research Letters 49 1 e2021GL095241

    work page 2022

  4. [4]

    , Constantinescu, E

    hchen19_2024_13654353 APACrefauthors Chen, H. , Constantinescu, E. , Rao, V. \ Stan, C. APACrefauthors \ 2024 09 . Reproduce the results in the paper ``Improving the Predictability of the Madden-Julian Oscillation at Subseasonal Scales with Gaussian Process Models'' . Reproduce the results in the paper ``Improving the Predictability of the Madden-Julian O...

    work page doi:10.5281/zenodo.13654353 2024

  5. [5]

    , Tan, P N

    cheng2006multistep APACrefauthors Cheng, H. , Tan, P N. , Gao, J. \ Scripps, J. APACrefauthors \ 2006 . Multistep-ahead time series prediction Multistep-ahead time series prediction . Advances in Knowledge Discovery and Data Mining: 10th Pacific-Asia Conference, PAKDD 2006, Singapore, April 9-12, 2006. Proceedings 10 Advances in knowledge discovery and da...

    work page 2006

  6. [6]

    APACrefauthors \ 2009

    clivarmjo2009 APACrefauthors CLIVAR, M J O W G. APACrefauthors \ 2009 . MJO Simulation Diagnostics MJO Simulation Diagnostics . Journal of Climate 22 11 3006--3030

    work page 2009

  7. [7]

    \ Christensen, H M

    delaunay2022interpretable APACrefauthors Delaunay, A. \ Christensen, H M. APACrefauthors \ 2022 . Interpretable deep learning for probabilistic MJO prediction Interpretable deep learning for probabilistic MJO prediction . Geophysical Research Letters 49 16 e2022GL098566

    work page 2022

  8. [8]

    APACrefauthors \ 1922

    fisher1922interpretation APACrefauthors Fisher, R A. APACrefauthors \ 1922 . On the interpretation of ^2 from contingency tables, and the calculation of P On the interpretation of ^2 from contingency tables, and the calculation of P . Journal of the Royal Statistical Society 85 1 87--94

    work page 1922

Show all 55 references
  1. [9]

    \ Isasi, P

    galvan2001multi APACrefauthors Galv \'a n, I M. \ Isasi, P. APACrefauthors \ 2001 . Multi-step learning rule for recurrent neural models: an application to time series forecasting Multi-step learning rule for recurrent neural models: an application to time series forecasting ....

    2001

  2. [10]

    , Leung, L R

    hagos2022observationally APACrefauthors Hagos, S. , Leung, L R. , Zhang, C. \ Balaguru, K. APACrefauthors \ 2022 . An observationally trained Markov model for MJO propagation An observationally trained Markov model for MJO propagation . Geophysical Research Letters 49 2 e2021GL095663

    2022

  3. [11]

    he2021sub APACrefauthors He, S. , Li, X. , DelSole, T. , Ravikumar, P. \ Banerjee, A. APACrefauthors \ 2021 . Sub-seasonal climate forecasting via machine learning: Challenges, analysis, and advances Sub-seasonal climate forecasting via machine learning: Challenges, analysis, ...

    2021

  4. [12]

    u te der Windst \

    heidke1926berechnung APACrefauthors Heidke, P. APACrefauthors \ 1926 . Berechnung des Erfolges und der G \"u te der Windst \"a rkevorhersagen im Sturmwarnungsdienst Berechnung des Erfolges und der G \"u te der Windst \"a rkevorhersagen im Sturmwarnungsdienst . Geografiska Anna...

    1926

  5. [13]

    APACrefauthors \ 2000

    hersbach2000decomposition APACrefauthors Hersbach, H. APACrefauthors \ 2000 . Decomposition of the continuous ranked probability score for ensemble prediction systems Decomposition of the continuous ranked probability score for ensemble prediction systems . Weather and Forecas...

    2000

  6. [14]

    , Leordeanu, C

    horelu2015forecasting APACrefauthors Horelu, A. , Leordeanu, C. , Apostol, E. , Huru, D. , Mocanu, M. \ Cristea, V. APACrefauthors \ 2015 . Forecasting techniques for time series from sensor data Forecasting techniques for time series from sensor data . 2015 17th international...

    2015

  7. [15]

    APACrefauthors \ 2014

    hyvarinen2014probabilistic APACrefauthors Hyv \"a rinen, O. APACrefauthors \ 2014 . A probabilistic derivation of Heidke skill score A probabilistic derivation of Heidke skill score . Weather and Forecasting 29 1 177--181

    2014

  8. [16]

    , Wichern, D W

    johnson2002applied APACrefauthors Johnson, R A. , Wichern, D W. \ . APACrefauthors \ 2002 . Applied multivariate statistical analysis Applied multivariate statistical analysis . Prentice hall Upper Saddle River, NJ

    2002

  9. [17]

    , Ham, Y

    kim2021deep APACrefauthors Kim, H. , Ham, Y. , Joo, Y. \ Son, S. APACrefauthors \ 2021 . Deep learning for bias correction of MJO prediction Deep learning for bias correction of MJO prediction . Nature Communications 12 1 3087

    2021

  10. [18]

    , Janiga, M A

    kim2019mjo APACrefauthors Kim, H. , Janiga, M A. \ Pegion, K. APACrefauthors \ 2019 . MJO propagation processes and mean biases in the SubX and S2S reforecasts 1 MJO propagation processes and mean biases in the SubX and S2S reforecasts 1 . Journal of Geophysical Research: Atmo...

    2019

  11. [19]

    , Vitart, F

    kim2018prediction APACrefauthors Kim, H. , Vitart, F. \ Waliser, D E. APACrefauthors \ 2018 . Prediction of the Madden--Julian oscillation: A review Prediction of the Madden--Julian oscillation: A review . Journal of Climate 31 23 9425--9443

    2018

  12. [20]

    , Bengio, Y

    lecun1995convolutional APACrefauthors LeCun, Y. , Bengio, Y. \ . APACrefauthors \ 1995 . Convolutional networks for images, speech, and time series Convolutional networks for images, speech, and time series . The handbook of brain theory and neural networks 3361 10 1995

    1995

  13. [21]

    , Son, S W

    lim2018mjo APACrefauthors Lim, Y. , Son, S W. \ Kim, D. APACrefauthors \ 2018 . MJO prediction skill of the subseasonal-to-seasonal prediction models MJO prediction skill of the subseasonal-to-seasonal prediction models . Journal of Climate 31 10 4075--4094

    2018

  14. [22]

    \ Matthews, A J

    love2009real APACrefauthors Love, B S. \ Matthews, A J. APACrefauthors \ 2009 . Real-time localised forecasting of the Madden-Julian Oscillation using neural network models Real-time localised forecasting of the Madden-Julian Oscillation using neural network models . Quarterly...

    2009

  15. [23]

    \ Stan, C

    lybarger2019revisiting APACrefauthors Lybarger, N D. \ Stan, C. APACrefauthors \ 2019 . Revisiting MJO, Kelvin waves, and El Ni \ n o relationships using a simple ocean model Revisiting MJO, Kelvin waves, and El Ni \ n o relationships using a simple ocean model . Climate Dynam...

    2019

  16. [24]

    \ Julian, P R

    madden1971detection APACrefauthors Madden, R A. \ Julian, P R. APACrefauthors \ 1971 . Detection of a 40--50 day oscillation in the zonal wind in the tropical Pacific Detection of a 40--50 day oscillation in the zonal wind in the tropical Pacific . Journal of Atmospheric Scien...

    1971

  17. [25]

    \ Julian, P R

    madden1972description APACrefauthors Madden, R A. \ Julian, P R. APACrefauthors \ 1972 . Description of global-scale circulation cells in the tropics with a 40--50 day period Description of global-scale circulation cells in the tropics with a 40--50 day period . Journal of Atm...

    1972

  18. [26]

    \ Hartmann, D L

    maloney2000modulation APACrefauthors Maloney, E D. \ Hartmann, D L. APACrefauthors \ 2000 . Modulation of eastern North Pacific hurricanes by the Madden--Julian oscillation Modulation of eastern North Pacific hurricanes by the Madden--Julian oscillation . Journal of Climate 13...

    2000

  19. [27]

    , Hendon, H H

    marshall2016visualizing APACrefauthors Marshall, A G. , Hendon, H H. \ Hudson, D. APACrefauthors \ 2016 . Visualizing and verifying probabilistic forecasts of the Madden-Julian Oscillation Visualizing and verifying probabilistic forecasts of the Madden-Julian Oscillation . Geo...

    2016

  20. [28]

    , Barnes, E A

    martin2022using APACrefauthors Martin, Z K. , Barnes, E A. \ Maloney, E. APACrefauthors \ 2022 . Using simple, explainable neural networks to predict the madden--Julian oscillation Using simple, explainable neural networks to predict the madden--Julian oscillation . Journal of...

    2022

  21. [29]

    \ Winkler, R L

    matheson1976scoring APACrefauthors Matheson, J E. \ Winkler, R L. APACrefauthors \ 1976 . Scoring rules for continuous probability distributions Scoring rules for continuous probability distributions . Management science 22 10 1087--1096

    1976

  22. [30]

    \ Jain, L

    medsker2001recurrent APACrefauthors Medsker, L R. \ Jain, L. APACrefauthors \ 2001 . Recurrent neural networks Recurrent neural networks . Design and Applications 5 64-67 2

    2001

  23. [31]

    , O’Brien, T A

    molina2023review APACrefauthors Molina, M J. , O’Brien, T A. , Anderson, G. , Ashfaq, M. , Bennett, K E. , Collins, W D. Ullrich, P A. APACrefauthors \ 2023 . A review of recent and emerging machine learning applications for climate variability and Wweather phenomena A review ...

    2023

  24. [32]

    \ Lawrence, E

    myren2021comparison APACrefauthors Myren, S. \ Lawrence, E. APACrefauthors \ 2021 . A comparison of Gaussian processes and neural networks for computer model emulation and calibration A comparison of Gaussian processes and neural networks for computer model emulation and calib...

    2021

  25. [33]

    , Subramanian, S

    pathak2022fourcastnet APACrefauthors Pathak, J. , Subramanian, S. , Harrington, P. , Raja, S. , Chattopadhyay, A. , Mardani, M. others APACrefauthors \ 2022 . Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators Fourcastnet: A...

    2022 arXiv

  26. [34]

    , Sanchez-Gonzalez, A

    price2023gencast APACrefauthors Price, I. , Sanchez-Gonzalez, A. , Alet, F. , Ewalds, T. , El-Kadi, A. , Stott, J. Willson, M. APACrefauthors \ 2023 . GenCast: Diffusion-based ensemble forecasting for medium-range weather GenCast: Diffusion-based ensemble forecasting for mediu...

    2023 arXiv

  27. [35]

    \ Juang, B

    rabiner1986introduction APACrefauthors Rabiner, L. \ Juang, B. APACrefauthors \ 1986 . An introduction to hidden Markov models An introduction to hidden Markov models . IEEE ASSP Magazine 3 1 4--16

    1986

  28. [36]

    , Seeger, M W

    rangapuram2018deep APACrefauthors Rangapuram, S S. , Seeger, M W. , Gasthaus, J. , Stella, L. , Wang, Y. \ Januschowski, T. APACrefauthors \ 2018 . Deep state space models for time series forecasting Deep state space models for time series forecasting . Advances in Neural Info...

    2018

  29. [37]

    , Hendon, H H

    rashid2011prediction APACrefauthors Rashid, H A. , Hendon, H H. , Wheeler, M C. \ Alves, O. APACrefauthors \ 2011 . Prediction of the Madden--Julian oscillation with the POAMA dynamical prediction system Prediction of the Madden--Julian oscillation with the POAMA dynamical pre...

    2011

  30. [38]

    , Osborne, M

    roberts2013gp APACrefauthors Roberts, S. , Osborne, M. , Ebden, M. , Reece, S. , Gibbson, N. \ Aigrain, S. APACrefauthors \ 2013 . Gaussian processes for time--series modeling Gaussian processes for time--series modeling . Phylosophical Transactions of the Royal Society A 371

    2013

  31. [39]

    \ Gallo, G M

    rossi2006volatility APACrefauthors Rossi, A. \ Gallo, G M. APACrefauthors \ 2006 . Volatility estimation via hidden Markov models Volatility estimation via hidden Markov models . Journal of Empirical Finance 13 2 203--230

    2006

  32. [40]

    \ Smith, L A

    roulston2002evaluating APACrefauthors Roulston, M S. \ Smith, L A. APACrefauthors \ 2002 . Evaluating probabilistic forecasts using information theory Evaluating probabilistic forecasts using information theory . Monthly Weather Review 130 6 1653--1660

    2002

  33. [41]

    , Barreiro, M

    silini2021machine APACrefauthors Silini, R. , Barreiro, M. \ Masoller, C. APACrefauthors \ 2021 . Machine learning prediction of the Madden--Julian Oscillation Machine learning prediction of the Madden--Julian Oscillation . npj Climate and Atmospheric Science 4 1 57

    2021

  34. [42]

    , Lerch, S

    silini2022improving APACrefauthors Silini, R. , Lerch, S. , Mastrantonas, N. , Kantz, H. , Barreiro, M. \ Masoller, C. APACrefauthors \ 2022 . Improving the prediction of the Madden--Julian oscillation of the ECMWF model by post-processing Improving the prediction of the Madde...

    2022

  35. [43]

    , Hao, J

    sorjamaa2007methodology APACrefauthors Sorjamaa, A. , Hao, J. , Reyhani, N. , Ji, Y. \ Lendasse, A. APACrefauthors \ 2007 . Methodology for long-term prediction of time series Methodology for long-term prediction of time series . Neurocomputing 70 16-18 2861--2869

    2007

  36. [44]

    , Straus, D M

    stan2017review APACrefauthors Stan, C. , Straus, D M. , Frederiksen, J S. , Lin, H. , Maloney, E D. \ Schumacher, C. APACrefauthors \ 2017 . Review of tropical-extratropical teleconnections on intraseasonal time scales Review of tropical-extratropical teleconnections on intras...

    2017

  37. [45]

    APACrefauthors \ 1999

    stein1999interpolation APACrefauthors Stein, M L. APACrefauthors \ 1999 . Interpolation of spatial data: some theory for kriging Interpolation of spatial data: some theory for kriging . Springer Science & Business Media

    1999

  38. [46]

    , Nakai, K

    suematsu2022machine APACrefauthors Suematsu, T. , Nakai, K. , Yoneda, T. , Takasuka, D. , Jinno, T. , Saiki, Y. \ Miura, H. APACrefauthors \ 2022 . Machine learning prediction of the MJO extends beyond one month Machine learning prediction of the MJO extends beyond one month ....

    2022 arXiv

  39. [47]

    , Kashinath, K

    toms2019testing APACrefauthors Toms, B A. , Kashinath, K. , Yang, D. \ . APACrefauthors \ 2019 . Testing the reliability of interpretable neural networks in geoscience using the Madden--Julian Oscillation Testing the reliability of interpretable neural networks in geoscience u...

    2019 arXiv

  40. [48]

    , Markou, S

    vaughan2024aardvark APACrefauthors Vaughan, A. , Markou, S. , Tebbutt, W. , Requeima, J. , Bruinsma, W P. , Andersson, T R. Turner, R E. APACrefauthors \ 2024 . Aardvark Weather: end-to-end data-driven weather forecasting Aardvark Weather: end-to-end data-driven weather foreca...

    2024 arXiv

  41. [49]

    \ Kyrki, V

    verdoja2020notes APACrefauthors Verdoja, F. \ Kyrki, V. APACrefauthors \ 2020 . Notes on the behavior of MC dropout Notes on the behavior of MC dropout . arXiv preprint arXiv:2008.02627

    2020 arXiv

  42. [50]

    , Smola, A

    wang2019deep APACrefauthors Wang, Y. , Smola, A. , Maddix, D. , Gasthaus, J. , Foster, D. \ Januschowski, T. APACrefauthors \ 2019 . Deep factors for forecasting Deep factors for forecasting . International conference on machine learning International conference on machine lea...

    2019

  43. [51]

    APACrefauthors \ 2018

    weigend2018time APACrefauthors Weigend, A S. APACrefauthors \ 2018 . Time series prediction: forecasting the future and understanding the past Time series prediction: forecasting the future and understanding the past . Routledge

    2018

  44. [52]

    \ Hendon, H H

    wheeler2004all APACrefauthors Wheeler, M C. \ Hendon, H H. APACrefauthors \ 2004 . An all-season real-time multivariate MJO index: Development of an index for monitoring and prediction An all-season real-time multivariate MJO index: Development of an index for monitoring and p...

    2004

  45. [53]

    \ Rasmussen, C E

    williams2006gaussian APACrefauthors Williams, C K. \ Rasmussen, C E. APACrefauthors \ 2006 . Gaussian processes for machine learning Gaussian processes for machine learning \ ( 2)\ ( 3). MIT Press Cambridge, MA

    2006

  46. [54]

    yunpeng2017multi APACrefauthors Yunpeng, L. , Di, H. , Junpeng, B. \ Yong, Q. APACrefauthors \ 2017 . Multi-step ahead time series forecasting for different data patterns based on LSTM recurrent neural network Multi-step ahead time series forecasting for different data pattern...

    2017

  47. [55]

    APACrefauthors \ 2013

    zhang2013madden APACrefauthors Zhang, C. APACrefauthors \ 2013 . Madden--Julian oscillation: Bridging weather and climate Madden--Julian oscillation: Bridging weather and climate . Bulletin of the American Meteorological Society 94 12 1849--1870

    2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.