pith. sign in

arxiv: 2605.28531 · v1 · pith:PYJVHJ3Snew · submitted 2026-05-27 · 💻 cs.LG

Stabilizing distribution-free probabilistic forecasts

Jente Van Belle , Honglin Wen , Wouter Verbeke , Pierre Pinson This is my paper

Pith reviewed 2026-06-29 13:55 UTC · model grok-4.3

classification 💻 cs.LG
keywords probabilistic forecastingforecast stabilityquantile regressionneural networkstime seriesdistribution-freeregression splinesmulti-step forecasts
0
0 comments X

The pith

A neural network parameterizing regression splines for conditional quantiles can jointly optimize probabilistic forecast quality and stability.

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

The paper aims to reduce instability in multi-step probabilistic forecasts, where successive updates for the same target period cause unwanted variability that disrupts planning. It does so by training distribution-free models to penalize dissimilarities between updated quantile functions, using regression splines whose coefficients come from a neural network. This setup makes the stability penalty differentiable and allows different weights on different parts of the distribution. A reader would care because the resulting forecasts change less when new data arrives, without large losses in accuracy or calibration. Experiments on two datasets confirm that instability drops while quality holds and that stabilization can be focused on tails or center as needed.

Core claim

By representing forecasted conditional quantile functions through regression splines whose parameters are outputs of a neural network, the training objective can include an explicit penalty on the dissimilarity between quantile functions produced at successive forecast origins; minimizing this combined loss yields forecasts whose updates exhibit lower variability while the original quality metrics and coverage properties remain largely unchanged, and the penalty weights can be varied across quantile levels to emphasize stabilization in chosen regions of the distribution.

What carries the argument

Regression splines parameterized by a neural network for the forecasted conditional quantile functions, with an added dissimilarity penalty between successive forecast updates.

If this is right

  • Forecasts for any fixed target period show smaller changes when the forecast origin advances and new observations arrive.
  • The relative importance of stability versus quality can be tuned directly in the loss function during training.
  • Stabilization effort can be concentrated on central quantiles, tails, or any chosen subset by adjusting the penalty weights.
  • Probabilistic calibration and coverage properties of the base model are preserved to first order after the stability term is added.

Where Pith is reading between the lines

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

  • The same spline-penalty idea could be ported to other distribution-free architectures that already output quantiles, without requiring a full model redesign.
  • Inventory or scheduling systems that rely on upper-tail quantiles would see the largest practical benefit when the penalty is concentrated on those regions.
  • The approach might be combined with post-processing recalibration steps to recover any small calibration loss introduced by the stability term.
  • Testing on datasets with stronger seasonality or regime shifts would reveal whether the stability gains remain consistent when the underlying series are less stationary.

Load-bearing premise

Penalizing differences between spline-parameterized conditional quantile functions from updated forecasts will produce more stable outputs without substantially degrading calibration or accuracy.

What would settle it

Apply the stabilized training procedure to a new dataset and check whether the measured reduction in variance across forecast origins for fixed targets is accompanied by a large rise in pinball loss or by coverage probabilities that fall outside nominal intervals.

Figures

Figures reproduced from arXiv: 2605.28531 by Honglin Wen, Jente Van Belle, Pierre Pinson, Wouter Verbeke.

Figure 1
Figure 1. Figure 1: Empirical probability density functions of the ground truth distribution and the forecasts produced by the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the SQF forecaster model architecture and the optimization procedure to stabilize the forecasts [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pareto frontiers of forecast quality (sCRPS) versus stability (s [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Percentage differences in (quantile-weighted) forecast quality (sCRPS, sCRPS [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
read the original abstract

Multi-step-ahead forecasts are often updated as new observations become available, since shorter forecast horizons typically improve forecast quality. However, such improvements come at the cost of forecast instability, i.e., variability in forecasts for the same target period. This instability can trigger costly changes to plans formulated based on the forecasts and may erode trust in the forecasting system. In this work, we integrate forecast stability alongside forecast quality into the training of distribution-free probabilistic time-series forecasting models, allowing us to control this trade-off. We propose a method for generating stabilized forecasted conditional quantile functions using regression splines parameterized by a neural network. This approach enables joint optimization of quality and stability, as it allows us to directly penalize dissimilarities arising from forecast updates. Furthermore, it allows assigning varying importance to stabilizing different parts of the forecast distributions (e.g., central parts vs. tails) to focus on the parts most relevant for the intended downstream use (e.g., the upper tail for inventory management). We empirically evaluate the proposed method on two datasets with different statistical properties and show that it can effectively reduce forecast instability without a substantial loss in forecast quality, and that it can target stabilization effort toward specific parts of the forecast distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that by parameterizing conditional quantile functions via regression splines in a neural network and adding a penalty on dissimilarities between successive forecast updates, one can jointly optimize for forecast quality (via pinball loss) and stability in distribution-free probabilistic time-series models. It further claims that importance weights can target stabilization to specific distribution regions, and that experiments on two datasets demonstrate effective instability reduction without substantial quality degradation.

Significance. If the central construction holds, the work would be useful for applications (e.g., inventory, planning) where forecast revisions trigger costly adjustments; the ability to differentially weight distribution regions is a practical feature. The empirical demonstration is limited to two datasets whose statistical properties are only qualitatively described, and no machine-checked proofs or parameter-free derivations are provided.

major comments (2)
  1. [Method (spline parameterization and combined loss)] The spline parameterization of the conditional quantile functions (described in the method) provides no explicit constraints or reparameterization to enforce monotonicity. Consequently, when the stability penalty dominates, the resulting functions can produce non-monotonic quantile curves, directly violating the ordering required for valid distributions and undermining any coverage guarantees inherited from the base pinball-loss estimator.
  2. [Method (combined loss) and Experiments] No derivation or bound is given showing that the combined objective (pinball loss + stability penalty) preserves the distribution-free calibration or coverage properties of the underlying quantile estimator. The central claim that stability can be added “without a substantial loss in forecast quality” therefore rests on an unproven assumption that the penalty term does not materially degrade probabilistic calibration.
minor comments (2)
  1. [Method] The abstract and method description refer to “regression splines parameterized by a neural network” without specifying the knot placement strategy, basis order, or how the neural network outputs are mapped to spline coefficients.
  2. [Experiments] The two evaluation datasets are described only by “different statistical properties”; quantitative characteristics (length, frequency, missingness, tail behavior) should be reported in a table.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive comments. Below we respond point-by-point to the major comments, indicating planned revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: [Method (spline parameterization and combined loss)] The spline parameterization of the conditional quantile functions (described in the method) provides no explicit constraints or reparameterization to enforce monotonicity. Consequently, when the stability penalty dominates, the resulting functions can produce non-monotonic quantile curves, directly violating the ordering required for valid distributions and undermining any coverage guarantees inherited from the base pinball-loss estimator.

    Authors: We agree that the manuscript does not describe explicit monotonicity constraints on the neural-network-parameterized regression splines. This omission leaves open the possibility of non-monotonic quantile functions under a dominant stability penalty. In the revised manuscript we will add a reparameterization (e.g., outputting non-negative increments and taking cumulative sums) to enforce monotonicity by construction while preserving the flexibility of the spline representation. The updated method section will include this change together with a brief verification that the reparameterization does not materially alter the optimization landscape. revision: yes

  2. Referee: [Method (combined loss) and Experiments] No derivation or bound is given showing that the combined objective (pinball loss + stability penalty) preserves the distribution-free calibration or coverage properties of the underlying quantile estimator. The central claim that stability can be added “without a substantial loss in forecast quality” therefore rests on an unproven assumption that the penalty term does not materially degrade probabilistic calibration.

    Authors: The manuscript frames the combined objective as an empirical regularizer whose effect on forecast quality is assessed experimentally rather than through theoretical bounds. The central claim is therefore limited to the observed behavior on the two evaluated datasets, where pinball loss and empirical coverage remain close to the unregularized baseline. We will revise the introduction, method, and discussion sections to state explicitly that no theoretical guarantee is provided and to list the absence of such a bound as a limitation. In addition, the experimental section will be expanded with per-quantile coverage plots and a sensitivity analysis over the stability weight to strengthen the empirical support for the claim. revision: partial

Circularity Check

0 steps flagged

No significant circularity; new penalty term and empirical validation are independent of inputs

full rationale

The paper introduces a spline-parameterized neural network for conditional quantile functions and augments the training objective with an explicit dissimilarity penalty between successive forecast updates. This construction is not self-definitional: the stability term is an added regularizer whose effect is measured post-hoc on held-out data rather than being algebraically identical to the quality term. No load-bearing step reduces to a self-citation, fitted parameter renamed as prediction, or imported uniqueness theorem; the central empirical claim (reduced instability with limited quality loss) rests on external dataset evaluation and is therefore falsifiable outside the fitted values themselves.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The paper introduces a penalty term and spline parameterization, but specific values for free parameters are not detailed in the abstract. Standard assumptions about quantile functions and neural network optimization are used.

free parameters (2)
  • stability penalty coefficient
    The weight given to the stability penalty in the joint optimization is likely a hyperparameter tuned on data.
  • importance weights for distribution parts
    Weights to focus stabilization on central vs tail parts are introduced to customize the method.
axioms (1)
  • domain assumption The conditional quantile functions can be accurately represented by regression splines.
    The method relies on this to parameterize the forecasts.

pith-pipeline@v0.9.1-grok · 5742 in / 1163 out tokens · 51018 ms · 2026-06-29T13:55:14.738595+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

51 extracted references · 49 canonical work pages · 3 internal anchors

  1. [1]

    International Journal of Forecasting 39, 1502–1511

    On the evaluation of hierarchical forecasts. International Journal of Forecasting 39, 1502–1511. doi:10.1016/j.ijforecast.2022.08.003. Benidis, K., Rangapuram, S.S., Flunkert, V., Wang, Y., Maddix, D., Turkmen, C., Gasthaus, J., Bohlke-Schneider, M., Salinas, D., Stella, L., et al.,

    work page doi:10.1016/j.ijforecast.2022.08.003 2022

  2. [2]

    ACM Computing Surveys 55, 1–36

    Deep learning for time series forecasting: Tutorial and literature survey. ACM Computing Surveys 55, 1–36. doi:10.1145/3533382. Buizza, R.,

    work page doi:10.1145/3533382

  3. [3]

    Monthly Weather Review 136, 3343–3362

    Comparison of a 51-member low-resolution (t l 399l62) ensemble with a 6-member high-resolution (t l 799l91) lagged-forecast ensemble. Monthly Weather Review 136, 3343–3362. doi:10.1175/2008MWR2430.1. Caljon, D., Vercauteren, J., De Vos, S., Verbeke, W., Van Belle, J.,

    work page doi:10.1175/2008mwr2430.1

  4. [5]

    6989–6997

    N-HiTS: Neu- ral hierarchical interpolation for time series forecasting, in: Proceedings of the AAAI Conference on Artificial Intelligence, pp. 6989–6997. doi:10.1609/aaai.v37i6.25854. DeYong, G.D., Cattani, K.D.,

    work page doi:10.1609/aaai.v37i6.25854

  5. [6]

    European Journal of Operational Research 220, 93–105

    Well adjusted: Using expediting and cancelation to manage store replenishment inventory for a seasonal good. European Journal of Operational Research 220, 93–105. doi:10.1016/j.ejor.2012. 01.029. DeYong, G.D., Cattani, K.D.,

    work page doi:10.1016/j.ejor.2012 2012

  6. [7]

    International Journal of Production Economics 201, 173–192

    The unlimited newsvendor: A general solution to a class of two-period newsven- dor problems. International Journal of Production Economics 201, 173–192. doi:10.1016/j.ijpe.2018.04.018. Ehret, U.,

    work page doi:10.1016/j.ijpe.2018.04.018 2018

  7. [8]

    Falcon, W., The PyTorch Lightning team,

    doi:10.1127/0941-2948/2010/0480. Falcon, W., The PyTorch Lightning team,

    work page doi:10.1127/0941-2948/2010/0480 2010

  8. [9]

    doi:10.5281/zenodo.3828935 , license =

    PyTorch Lightning. URL:https://github.com/Lightning-AI/ lightning, doi:10.5281/zenodo.3828935. Franses, P.H., Legerstee, R.,

    work page doi:10.5281/zenodo.3828935

  9. [10]

    doi:10.1002/for.1129

    Do experts’ adjustments on model-based sku-level forecasts improve forecast quality? Journal of Forecasting 29, 331–340. doi:10.1002/for.1129. Gasthaus, J., Benidis, K., Wang, Y., Rangapuram, S.S., Salinas, D., Flunkert, V., Januschowski, T.,

    work page doi:10.1002/for.1129

  10. [11]

    1901–1910

    Proba- bilistic forecasting with spline quantile function RNNs, in: International Conference on Artificial Intelligence and Statistics, pp. 1901–1910. URL:https://proceedings.mlr.press/v89/gasthaus19a.html. Gneiting, T., Balabdaoui, F., Raftery, A.E.,

    1901

  11. [12]

    Journal of the Royal Statistical Society Series B: Statistical Methodology 69, 243–268

    Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society Series B: Statistical Methodology 69, 243–268. doi:10.1111/j.1467-9868.2007.00587.x. Gneiting, T., Raftery, A.E.,

    work page doi:10.1111/j.1467-9868.2007.00587.x 2007

  12. [13]

    Journal of the American Statistical Association 102, 359–378

    Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association 102, 359–378. doi:10.1198/016214506000001437. Gneiting, T., Ranjan, R.,

    work page doi:10.1198/016214506000001437

  13. [14]

    Journal of Business & Economic Statistics 29, 411–422

    Comparing density forecasts using threshold- and quantile-weighted scoring rules. Journal of Business & Economic Statistics 29, 411–422. doi:10.1198/jbes.2010.08110. Godahewa, R., Bergmeir, C., Baz, Z.E., Zhu, C., Song, Z., García, S., Benavides, D.,

    work page doi:10.1198/jbes.2010.08110 2010

  14. [15]

    International Journal of Forecasting 41, 1539–1558

    On forecast stability. International Journal of Forecasting 41, 1539–1558. doi:10.1016/j.ijforecast.2025.01.006. Gouttes, A., Rasul, K., Koren, M., Stephan, J., Naghibi, T.,

    work page doi:10.1016/j.ijforecast.2025.01.006 2025

  15. [16]

    doi:10.48550/ arXiv.2107.03743

    Probabilistic time series forecasting with implicit quantile networks, in: 38th International Conference on Machine Learning, Time Series Workshop. doi:10.48550/ arXiv.2107.03743. 28 Hyndman, R., Athanasopoulos, G., Bergmeir, C., Caceres, G., Chhay, L., O’Hara-Wild, M., Petropoulos, F., Razbash, S., Wang, E., Yasmeen, F.,

    work page arXiv

  16. [17]

    URL: https://pkg.robjhyndman.com/forecast/, doi:10.32614/CRAN.package.forecast

    forecast: Forecasting functions for time series and linear models. URL: https://pkg.robjhyndman.com/forecast/, doi:10.32614/CRAN.package.forecast. R package version 8.24.0. Hyndman, R.J., Khandakar, Y.,

    work page doi:10.32614/cran.package.forecast

  17. [18]

    Journal of Statistical Software 27, 1–22

    Automatic time series forecasting: the forecast package for R. Journal of Statistical Software 27, 1–22. doi:10.18637/jss.v027.i03. In, Y., Jung, J.Y.,

    work page doi:10.18637/jss.v027.i03

  18. [19]

    International Journal of Forecasting 38, 1386–1399

    Simple averaging of direct and recursive forecasts via partial pooling using machine learning. International Journal of Forecasting 38, 1386–1399. doi:10.1016/j.ijforecast.2021.11.007. Januschowski, T., Gasthaus, J., Wang, Y., Salinas, D., Flunkert, V., Bohlke-Schneider, M., Callot, L.,

    work page doi:10.1016/j.ijforecast.2021.11.007 2021

  19. [20]

    International Journal of Forecasting 36, 167–177

    Criteria for classifying forecasting methods. International Journal of Forecasting 36, 167–177. doi:10.1016/j.ijforecast. 2019.05.008. Kingma, D.P.,

    work page doi:10.1016/j.ijforecast 2019

  20. [21]

    Adam: A Method for Stochastic Optimization

    Adam: A method for stochastic optimization, in: 3rd International Conference on Learning Representations. doi:10.48550/arXiv.1412.6980. Krishnan, J., Kleindorfer, P.R., Heching, A.,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1412.6980

  21. [22]

    INSEAD Business School Research Paper 2007/44/TOM/ACGRD

    Demand distortions and capacity allocation policies. INSEAD Business School Research Paper 2007/44/TOM/ACGRD. doi:10.2139/ssrn.1021950. Laio, F., Tamea, S.,

    work page doi:10.2139/ssrn.1021950 2007

  22. [23]

    Hydrology and Earth System Sciences 11, 1267–1277

    Verification tools for probabilistic forecasts of continuous hydrological variables. Hydrology and Earth System Sciences 11, 1267–1277. doi:10.5194/hess-11-1267-2007. Lashley, S.L., Fisher, L., Simpson, B., Taylor, J., Weisser, S., Logsdon, J., Lammers, A.,

    work page doi:10.5194/hess-11-1267-2007 2007

  23. [24]

    International Journal of Forecasting 37, 1748–1764

    Temporal fusion transformers for interpretable multi-horizon time series forecasting. International Journal of Forecasting 37, 1748–1764. doi:10.1016/j.ijforecast.2021.03.012. Makridakis, S., Spiliotis, E., Assimakopoulos, V.,

    work page doi:10.1016/j.ijforecast.2021.03.012 2021

  24. [25]

    International Journal of Forecasting 36, 54–74

    The M4-competition: 100,000 time series and 61 forecasting methods. International Journal of Forecasting 36, 54–74. doi:10.1016/j.ijforecast.2019.04.014. Makridakis, S., Spiliotis, E., Assimakopoulos, V.,

    work page doi:10.1016/j.ijforecast.2019.04.014 2019

  25. [26]

    International Journal of Forecasting 38, 1325–1336

    The M5 competition: Background, organization, and imple- mentation. International Journal of Forecasting 38, 1325–1336. doi:10.1016/j.ijforecast.2021.07.007. Morales-Brotons, D., Vogels, T., Hendrikx, H.,

    work page doi:10.1016/j.ijforecast.2021.07.007 2021

  26. [27]

    Mukherjee, S., Shankar, D., Ghosh, A., Tathawadekar, N., Kompalli, P., Sarawagi, S., Chaudhury, K.,

    doi:10.48550/arXiv.2411.18704. Mukherjee, S., Shankar, D., Ghosh, A., Tathawadekar, N., Kompalli, P., Sarawagi, S., Chaudhury, K.,

    work page doi:10.48550/arxiv.2411.18704

  27. [28]

    ARMDN: Associative and Recurrent Mixture Density Networks for eRetail Demand Forecasting

    AR- MDN: Associative and recurrent mixture density networks for eRetail demand forecasting. arXiv preprint doi:10. 48550/arXiv.1803.03800. Nikolopoulos, K.,

    work page internal anchor Pith review Pith/arXiv arXiv

  28. [29]

    European Journal of Operational Research 291, 549–559

    We need to talk about intermittent demand forecasting. European Journal of Operational Research 291, 549–559. doi:10.1016/j.ejor.2019.12.046. Nordhaus, W.D.,

    work page doi:10.1016/j.ejor.2019.12.046 2019

  29. [30]

    The Review of Economics and Statistics 69, 667–674

    Forecasting efficiency: Concepts and applications. The Review of Economics and Statistics 69, 667–674. doi:10.2307/1935962. Olivares, K.G., Challu, C., Marcjasz, G., Weron, R., Dubrawski, A.,

    work page doi:10.2307/1935962

  30. [31]

    International Journal of Forecasting 39, 884–900

    Neural basis expansion analysis with exogenous variables: Forecasting electricity prices with nbeatsx. International Journal of Forecasting 39, 884–900. doi:10.1016/j.ijforecast.2022.03.001. Oreshkin, B.N., Carpov, D., Chapados, N., Bengio, Y.,

    work page doi:10.1016/j.ijforecast.2022.03.001 2022

  31. [32]

    doi:10.48550/ arXiv.1905.10437

    N-BEATS: Neural basis expansion analysis for in- terpretable time series forecasting, in: 8th International Conference on Learning Representations. doi:10.48550/ arXiv.1905.10437. Oreshkin, B.N., Carpov, D., Chapados, N., Bengio, Y.,

    work page arXiv 1905

  32. [33]

    9242–9250

    Meta-learning framework with applications to zero- shot time-series forecasting, in: Proceedings of the AAAI Conference on Artificial Intelligence, pp. 9242–9250. doi:10.1609/aaai.v35i10.17115. Pappenberger, F., Cloke, H.L., Persson, A., Demeritt, D.,

    work page doi:10.1609/aaai.v35i10.17115

  33. [34]

    On forecast (in)consistency in a hydro-meteorological chain: curse or blessing?

    HESS Opinions “On forecast (in)consistency in a hydro-meteorological chain: curse or blessing?". Hydrology and Earth System Sciences 15, 2391–2400. doi:10. 29 5194/hess-15-2391-2011. Park, Y., Maddix, D., Aubet, F.X., Kan, K., Gasthaus, J., Wang, Y.,

    2011

  34. [35]

    Computational Optimal Transport.Found

    Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning 11, 355–607. doi:10.1561/2200000073. Pritularga, K., Kourentzes, N.,

    work page doi:10.1561/2200000073

  35. [36]

    Ruth, D.P., Glahn, B., Dagostaro, V., Gilbert, K.,

    doi:10.2139/ssrn.4711817. Ruth, D.P., Glahn, B., Dagostaro, V., Gilbert, K.,

    work page doi:10.2139/ssrn.4711817

  36. [37]

    Weather and Forecasting 24, 504–519

    The performance of MOS in the digital age. Weather and Forecasting 24, 504–519. doi:10.1175/2008WAF2222158.1. Salinas, D., Flunkert, V., Gasthaus, J., Januschowski, T.,

    work page doi:10.1175/2008waf2222158.1

  37. [38]

    International Journal of Forecasting 36, 1181–1191

    DeepAR:Probabilisticforecastingwithautoregressive recurrent networks. International Journal of Forecasting 36, 1181–1191. doi:10.1016/j.ijforecast.2019.07.001. Spiliotis, E., Petropoulos, F.,

    work page doi:10.1016/j.ijforecast.2019.07.001 2019

  38. [39]

    European Journal of Operational Research 314, 111–121

    On the update frequency of univariate forecasting models. European Journal of Operational Research 314, 111–121. doi:10.1016/j.ejor.2023.08.056. Sweeney, C., Bessa, R.J., Browell, J., Pinson, P.,

    work page doi:10.1016/j.ejor.2023.08.056 2023

  39. [40]

    WIREs Energy and Environment 9, e365

    The future of forecasting for renewable energy. WIREs Energy and Environment 9, e365. doi:10.1002/wene.365. Syntetos, A.A., Babai, Z., Boylan, J.E., Kolassa, S., Nikolopoulos, K.,

    work page doi:10.1002/wene.365

  40. [41]

    European Journal of Operational Research 252, 1–26

    Supply chain forecasting: Theory, practice, their gap and the future. European Journal of Operational Research 252, 1–26. doi:10.1016/j.ejor. 2015.11.010. Syntetos, A.A., Boylan, J.E., Croston, J.,

    work page doi:10.1016/j.ejor 2015

  41. [42]

    Journal of the Operational Research Society 56, 495–503

    On the categorization of demand patterns. Journal of the Operational Research Society 56, 495–503. doi:10.1057/palgrave.jors.2601841. Taieb, S.B., Atiya, A.F.,

    work page doi:10.1057/palgrave.jors.2601841

  42. [43]

    IEEE Transactions on Neural Networks and Learning Systems 27, 62–76

    A bias and variance analysis for multistep-ahead time series forecasting. IEEE Transactions on Neural Networks and Learning Systems 27, 62–76. doi:10.1109/TNNLS.2015.2411629. Tashman, L.J.,

    work page doi:10.1109/tnnls.2015.2411629 2015

  43. [44]

    International Journal of Forecasting 16, 437–450

    Out-of-sample tests of forecasting accuracy: An analysis and review. International Journal of Forecasting 16, 437–450. doi:10.1016/S0169-2070(00)00065-0. Terwiesch, C., Ren, Z.J., Ho, T.H., Cohen, M.A.,

    work page doi:10.1016/s0169-2070(00)00065-0 2070

  44. [45]

    Management Science 51, 208–220

    An empirical analysis of forecast sharing in the semiconductor equipment supply chain. Management Science 51, 208–220. doi:10.1287/mnsc.1040.0317. Tunc, H., Kilic, O.A., Tarim, S.A., Eksioglu, B.,

    work page doi:10.1287/mnsc.1040.0317

  45. [46]

    International Journal of Production Economics 141, 619–625

    A simple approach for assessing the cost of system nervousness. International Journal of Production Economics 141, 619–625. doi:10.1016/j.ijpe.2012.09.022. Van Belle, J., Crevits, R., Caljon, D., Verbeke, W.,

    work page doi:10.1016/j.ijpe.2012.09.022 2012

  46. [47]

    IEEE Transactions on Neural Networks and Learning Systems 35, 18872–18885

    Probabilistic forecasting with modified N-BEATS net- works. IEEE Transactions on Neural Networks and Learning Systems 35, 18872–18885. doi:10.1109/TNNLS.2024. 3450832. Van Belle, J., Crevits, R., Verbeke, W.,

    work page doi:10.1109/tnnls.2024 2024

  47. [48]

    International Journal of Forecasting 39, 1333–1350

    Improving forecast stability using deep learning. International Journal of Forecasting 39, 1333–1350. doi:10.1016/j.ijforecast.2022.06.007. Villani, C.,

    work page doi:10.1016/j.ijforecast.2022.06.007 2022

  48. [49]

    IEEE Transactions on Sustainable Energy 13, 2250–2263

    Continuous and distribution-free probabilistic wind power forecasting: A conditional normalizing flow approach. IEEE Transactions on Sustainable Energy 13, 2250–2263. doi:10.1109/TSTE.2022.3191330. Wen, R., Torkkola, K., Narayanaswamy, B., Madeka, D.,

    work page doi:10.1109/tste.2022.3191330 2022

  49. [50]

    A Multi-Horizon Quantile Recurrent Forecaster

    A multi-horizon quantile recurrent forecaster, in: 31st Conference on Neural Information Processing Systems, Time Series Workshop. doi:10.48550/arXiv.1711.11053. Yeo, I.K., Johnson, R.A.,

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1711.11053

  50. [51]

    Biometrika 87, 954–959

    A new family of power transformations to improve normality or symmetry. Biometrika 87, 954–959. doi:10.1093/biomet/87.4.954. Zsoter, E., Buizza, R., Richardson, D.,

    work page doi:10.1093/biomet/87.4.954

  51. [52]

    Jumpiness

    “Jumpiness" of the ECMWF and Met Office EPS control and ensemble- mean forecasts. Monthly Weather Review 137, 3823–3836. doi:10.1175/2009MWR2960.1. 30

    work page doi:10.1175/2009mwr2960.1