Winner-takes-all for Multivariate Probabilistic Time Series Forecasting

Adrien Cort\'es; R\'emi Rehm; Victor Letzelter

REVIEW 4 major objections 4 minor 19 references

Winner-takes-all for Multivariate Probabilistic Time Series Forecasting

T0 review · 4 major / 4 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 TimeMCL trains a multi-head forecaster so that its K outputs converge to the optimal K-point codebook for the conditional distribution of future trajectories, with each head equal to the mean of its Voronoi cell and each score equal to…

desk verdict A practical WTA multi-head forecaster that is let down by a mismatch between its quantization theorem (proved only for teacher-forcing-free training) and the teacher-forced experiments that produce its headline results; still worth refereeing. read the letter →

arxiv 2506.05515 v2 pith:ZE2ZV2RM submitted 2025-06-05 cs.LG cs.AIstat.ML

Adrien Cortés , Rémi Rehm , Victor Letzelter This is my paper

classification cs.LGcs.AIstat.ML

keywords multiplechoicelearningwinner-takes-alllossprobabilistictimeseriesforecastingfunctionalquantizationcentroidalVoronoitessellationconditionaldensityestimationdiversetrajectorymultivariate

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 introduces TimeMCL, a way to train a multi-head neural forecaster so that its K output trajectories act as a small, representative codebook for the distribution of possible futures. The central claim is that under the Winner-Takes-All (WTA) loss, which updates only the head that fits a training trajectory best, the K heads converge to a conditional stationary quantizer: each head equals the conditional mean of the future trajectories that fall in its Voronoi cell. That turns forecasting into an implicit K-means over paths, giving users a single-pass set of diverse scenarios together with a learned probability for each scenario. The authors argue this captures multimodality more directly than likelihood-based or generative baselines and at a fraction of their computational cost, and they support the claim with a formal proposition, toy experiments on Gaussian processes and an AR(5) process, and real-world benchmarks. The formal guarantee assumes an expressive network trained without teacher forcing, while the real experiments use teacher-forced training, so the theory is best understood as a design principle that the empirical results approximate.

What carries the argument

The load-bearing mechanism is the Winner-Takes-All risk over unrolled forecasters $F^1_\theta,\dots,F^K_\theta$: at each training step, a trajectory is assigned to its closest head (the generator step, which induces a Voronoi partition of future trajectories), and only that winning head is updated toward the trajectory (the centroid step). Because the $\min$ is non-differentiable, the two steps alternate, and the proof borrows the decoupling argument of Rupprecht et al. (2017): at a stationary point of the decoupled functional the generators and centroids coincide, so the heads must sit at cell-conditional means. Score heads $\gamma^k_\theta$, trained with binary cross-entropy against the indicator of being the winner, then provide unbiased estimates of $P(x_{t_0:T} \in \mathcal{X}_k \mid x_{1:t_0-1})$, which is how the method turns a codebook into a predictive distribution. This is the functional analogue of Lloyd's algorithm for K-means, with trajectories replacing points.

What would settle it

On the synthetic AR(5) setup of Appendix B, train TimeMCL to convergence, assign held-out futures to their nearest head, and compute for each head the squared distance between the head's output and the empirical mean of its cell: if this distance does not approach the sampling-noise level, Proposition 5.1 fails. A second check is scaling: with an expressive model, distortion should fall roughly as $K^{-2/d}$ as K grows, so a clear plateau or worse rate would refute the quantization interpretation.

Watch

Extended reading notes

Core claim

Proposition 5.1 is the paper's central discovery: with the WTA risk $\mathcal{L}_{\mathrm{WTA}}(\theta)=\mathbb{E}[\min_k \mathcal{L}^k_\theta(x_{1:t_0-1}, x_{t_0:T})]$ and squared error, each converged head of TimeMCL satisfies $F^k_\theta(x_{1:t_0-1}) = \mathbb{E}[x_{t_0:T} \mid x_{t_0:T} \in \mathcal{X}_k(x_{1:t_0-1})]$, where $\mathcal{X}_k$ is the Voronoi cell of trajectories closer to head $k$ than to any other head. Consequently the K heads form a centroidal Voronoi tessellation of the space of future paths, i.e., a conditional, gradient-based version of K-means over trajectories, and the score heads trained with binary cross-entropy estimate the probability mass of each cell (Proposition 5.2). The proof decouples the Voronoi generators, which partition the trajectory space, from the centroids, which are the points whose squared distance is minimized, and shows by alternating optimization that any stationary point has generators equal to centroids. On synthetic data the learned quantizers align with the theoretical optimal quantizers for Brownian motion, a Brownian bridge, and an AR(5) process, and on six real-world benchmarks TimeMCL reaches distortion competitive with diffusion- and copula-based baselines at orders-of-magnitude lower inference cost, while producing markedly smoother trajectories.

Load-bearing premise

The theory assumes the network is expressive enough that the global WTA risk can be minimized context-by-context and that training uses the unrolled forecaster rather than teacher forcing; the real benchmark experiments are teacher-forced on random context windows, so the proven centroid property is not guaranteed for the model actually evaluated.

Editorial extensions

If this is right

If Proposition 5.1 holds, a single forward pass yields the optimal K-point codebook of the conditional distribution, so K futures are guaranteed to be the most representative scenarios available for that budget.
The score heads turn the codebook into a calibrated mixture: the weighted average of hypotheses estimates the conditional mean, and each score is the posterior probability of its scenario.
Distortion should improve predictably with K, following quantization-theoretic rates, so users can choose K based on accuracy or compute targets.
Smooth predictions are an expected consequence: cell-conditional means average out centered noise, so smoothness is a sign of convergence rather than a defect.
TimeMCL strikes a favorable accuracy-versus-cost trade-off relative to diffusion and flow baselines, making diverse probabilistic forecasts practical in real-time settings.

Reading between the lines

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

Because the quantization argument only needs the WTA objective, the same training scheme could be transplanted to transformer backbones and would plausibly inherit the codebook interpretation, a direction the paper lists as future work.
The learned Voronoi cells give a natural, model-based way to read multimodality: the number of cells with non-negligible score and their separation indicate the number and distinctness of plausible regimes.
Removing teacher forcing in real experiments (as the theory requires) may bring the measured distortion closer to the theoretical quantization bound; this is directly testable by comparing teacher-forced and unrolled training on the same benchmarks.
The WTA codebook can be viewed as a hard-assignment latent-variable model of futures, connecting TimeMCL to Gaussian-mixture and discrete-VAE formulations of forecasting.

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.

Desk Editor's Note

read the letter

Two things to know. First, the paper's core contribution is a clean, practical adaptation of Multiple Choice Learning to multivariate probabilistic time series forecasting: a shared RNN backbone with K prediction heads and score heads, trained with a WTA loss, giving K diverse trajectories in a single forward pass. That is a useful addition to the toolbox, especially given the low inference cost compared with diffusion baselines. Second, the quantization theorem is real but narrower than the main text presents. Proposition 5.1 holds only under Assumption A.2, which explicitly requires teacher forcing to be disabled, i.e., the unrolled network is directly optimized. The real experiments in Section 6 use standard teacher-forced training and random context windows, so the guarantee does not apply to the model that produced the headline numbers.

What is genuinely new is the time-series instantiation, the synthetic validation against Karhunen-Loève and Lloyd quantizers, and the released code. The appendix is honest that the toy experiments adopt the simplified setup, and the score-head analysis is a nice extension.

The soft spots are real but not fatal. The main text states Proposition 5.1 without carrying the teacher-forcing caveat, and Section 5.2 leans on the theorem to interpret smoothness on real data, where it does not directly apply. The Distortion metric is essentially the WTA objective, so strong distortion numbers are partly expected; the CRPS and RMSE tables in the appendix are more mixed. The Total Variation comparison also samples TimeMCL by its predicted scores while baselines are sampled uniformly, which is not a like-for-like comparison. And the relaxed/annealed variants used in the headline tables fall outside the hard-min, beta=0 statement of the proposition. These are scope limitations rather than internal contradictions, and they can be addressed by either running the unrolled variant on real data or explicitly restricting the theoretical claims.

This paper is for researchers who want a lightweight multi-scenario forecaster and who care about the quantization interpretation. It deserves a serious referee: the gap between theory and experiments is fixable, the code is available, and the synthetic experiments are informative. I would engage with it, but the authors should be asked to reconcile the theoretical assumptions with the actual experimental protocol.

Referee Report

4 major / 4 minor

Summary. The paper introduces TimeMCL, a multi-head Winner-Takes-All (WTA) method for multivariate probabilistic time-series forecasting. The method trains K prediction heads with a WTA loss so that each head specializes in a different mode of the conditional future distribution. The central theoretical claim is Proposition 5.1 (Appendix A.4): under assumptions of large batch size, sufficient network expressiveness, and convergence, TimeMCL is a conditional stationary functional quantizer, meaning each head outputs the conditional mean of its Voronoi cell in the space of future trajectories. The paper validates this interpretation on synthetic data (Brownian motion, Brownian bridge, AR(5)) by comparing against known optimal quantizers. On real-world benchmarks (SOLAR, ELECTRICITY, EXCHANGE, TRAFFIC, TAXI, WIKIPEDIA, and a cryptocurrency dataset), TimeMCL is compared against DeepAR, TimeGrad, TempFlow, Tactis2, ETS, and a transformer TempFlow, using Distortion, RMSE, CRPS, Total Variation, and inference cost as metrics. The paper reports that TimeMCL achieves competitive distortion at a fraction of the inference cost of diffusion or transformer baselines, while producing smoother trajectories.

Significance. If the quantization claim held for the evaluated models, this would be a practically valuable bridge between Multiple Choice Learning and functional quantization for time series, offering a single-forward-pass diverse forecaster with an interpretable codebook interpretation. The paper is honest about the assumptions in the appendix, includes reproducible code, and uses analytically tractable synthetic processes to illustrate the quantization behavior, which is a strength. However, the significance is currently limited by a gap between the proven statement and the actual experimental setup: the proof requires unrolled, teacher-forcing-free training, while the main real-data experiments use standard teacher-forced RNN training and softened WTA losses. Consequently, the central interpretation of the real-data results as a stationary quantizer is not directly supported by the theorem. The method itself remains a reasonable heuristic candidate, and the computational efficiency claims are credible.

major comments (4)

[Section 5.1 / Appendix A.1] Proposition 5.1 and its detailed form Proposition A.4 rely on Assumption A.2, which explicitly states that teacher forcing is disabled and that the unrolled network F_theta is directly optimized during training. The main-text proposition statement lists assumptions A.1, A.2, A.3 but does not spell out this teacher-forcing caveat. Section 6 and Appendix C.3.1 describe a training protocol that computes hidden states from observed target values in the target window (standard GluonTS teacher-forced RNN training). Therefore, the theorem has not been established for the models evaluated on the real-world benchmarks in Section 6, and the statement in Section 5.2 that smoothness is a consequence of Proposition 5.1 for those models is not supported.
[Section 6.1 / Eq. (4)] The headline evaluation metric, Distortion, computes min_k d(F^k_theta(x_{1:t0-1}), x_{t0:T}) over the K heads, which is the same selection operation as the WTA training objective in Eq. (4). The paper notes in Section 6.2 that the good distortion scores are expected because TimeMCL explicitly optimizes for distortion. This makes the primary empirical claim circular as evidence for quantizer optimality: the metric rewards the model for reproducing its own training objective. The authors should either report a metric that evaluates the full K-trajectory distribution (e.g., a proper scoring rule for the implied mixture) or explicitly frame the Distortion results as a self-consistency check rather than independent validation.
[Section 4.3 / Appendix C.3.2] The headline results in Table 1 are obtained with the Relaxed-WTA and annealed-MCL objectives of Eqs. (21)-(23), which replace the hard min in Eq. (4) with weighted sums. Proposition 5.1 is proved only for the hard-min, beta=0 objective. No theoretical statement is provided for the softened objectives, so the claim that these variants also produce a conditional centroidal Voronoi tessellation is unsubstantiated. The paper should either prove an analogous result for these losses (or state the conditions under which the relaxation preserves the fixed-point property), or demote the quantizer interpretation to a heuristic for the systems actually benchmarked.
[Section 5.2 / Table 3] The Total Variation results are interpreted as 'confirming the claim of Section 5.2 as a consequence of Proposition 5.1.' Given the teacher-forcing gap and the use of softened losses in the real experiments, the smoothness of the real-data trajectories is not a logical consequence of the theorem. This section should be reframed as an empirical observation with a heuristic explanation (centroid predictions average over noise), rather than as a confirmation of the proven statement.

minor comments (4)

[Appendix C.2.3] The heading 'DISTORSION' should be corrected to 'DISTORTION'.
[Appendix C.3.2, Eq. (23)] In the definition of the annealed weights, the exponential contains 'exp(-ell(L^s_theta(x_{1:t0-1}, x_{t0:T})/T)', which has a mismatched parenthesis and an extraneous ell; it should be 'exp(-L^s_theta(x_{1:t0-1}, x_{t0:T})/T)'.
[General] The library name is inconsistently given as 'Gluonts' throughout; the canonical spelling is 'GluonTS'.
[Section 6.1] The Distortion metric definition uses the average Euclidean distance over the horizon while the training loss in Eq. (4) uses the unnormalized sum; the two are monotone in the same argmin, but the paper should state this invariance explicitly to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quantization theorem is derived from stated assumptions via an independent alternating-optimization argument, and the distortion metric is openly acknowledged to be the training objective.

full rationale

The paper's central claim, Proposition 5.1 (and its appendix version Proposition A.4), is not circular. The WTA loss in Eq. (4) is defined as E_x[min_k L^k_theta], the Voronoi cells X_k are defined in terms of the same trajectory losses, and the theorem derives F^k_theta = E[x_{t0:T} | x_{t0:T} in X_k(x_{1:t0-1})] under Assumptions A.1-A.3. This is a standard centroidal-Voronoi/Lloyd fixed-point property, and Appendix A.1 supplies a full alternating-optimization proof (generators vs. centroids) that cites the external Rupprecht et al. (2017) Theorem 1. Although the paper also says the result is 'an adaptation of Proposition 5.2 in Letzelter et al. (2024)', the proof is reproduced in the appendix rather than imported solely by citation, so the overlapping-author citation is not load-bearing. The empirical Distortion metric (Appendix C.2.3) does coincide with the WTA objective under squared loss, but the paper explicitly acknowledges this: 'This behavior is expected since TimeMCL explicitly optimizes for distortion.' That is a candid statement about metric-objective alignment, not a fitted parameter renamed as a prediction. The real-data experiments use teacher-forced training, which falls outside Assumption A.2's unrolled-forecaster setting; this is a scope limitation and a correctness risk, not circularity, because the theorem and the experiments are not equivalent by construction. The synthetic experiments additionally benchmark against external theoretical quantizers (Karhunen-Loeve-based quantization and Lloyd's algorithm), providing independent evidence. No derivation step reduces to its own inputs by definition or through a self-citation chain.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on the expressivity, risk-minimization, and convergence assumptions listed above, plus hand-set hyperparameters for the two WTA variants and the score-head loss. No new physical or conceptual entities are introduced; K prediction heads and score heads are existing MCL components.

free parameters (3)

confidence loss weight beta = 0.5
Weight for the score-head loss L_s in Section 6.1; chosen by hand and not tuned per dataset.
Relaxed-WTA mixing weight epsilon = 0.1 (main benchmarks), 0.05 (synthetic)
Used in Eq. (21) for Relaxed-WTA; reported in Appendix C.3.2 and Appendix B as a hand-set value not tuned per dataset.
aMCL temperature schedule = T0=10, rho=0.95, Tlim=5e-4
Annealing schedule for the softmin variant in Eq. (22)-(23); set by hand in Appendix C.3.2.

assumptions (5)

domain assumption Assumption A.1: the batch size is large enough that the empirical WTA risk equals the true risk.
Used at the start of the proof of Proposition 5.1 to replace the empirical loss by the population risk in Eq. (13).
domain assumption Assumption A.2: the network family is expressive enough to attain the per-context infimum, and teacher forcing is disabled so F_theta is directly optimized.
Necessary for the decoupling argument and for identifying the WTA objective with a per-context quantization functional; not satisfied by the RNN training used in Section 6.
domain assumption Assumption A.3: training has converged to a local minimum of L_WTA.
Needed to characterize stationary points as Lloyd-type centroidal configurations in the proof of Proposition 5.1.
domain assumption Assumption A.5: global optimality of both the centroid objective and the score-head objective.
Required for Proposition 5.2, which states that score heads match Voronoi cell probabilities; stronger than local convergence.
standard math The target trajectories admit a finite second moment and the Voronoi cells form a partition of the output space.
Standard functional-quantization setup in Section 5.1; used implicitly in the Chasles relation in Eq. (13).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Winner-takes-all for Multivariate Probabilistic Time Series Forecasting." pith.science (2026). https://pith.science/paper/ZE2ZV2RM

@misc{pith2026250605515,
  author       = {Pith},
  title        = {Pith review of: Winner-takes-all for Multivariate Probabilistic Time Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZE2ZV2RM}},
  note         = {Machine review of arXiv:2506.05515}
}

read the original abstract

We introduce TimeMCL, a method leveraging the Multiple Choice Learning (MCL) paradigm to forecast multiple plausible time series futures. Our approach employs a neural network with multiple heads and utilizes the Winner-Takes-All (WTA) loss to promote diversity among predictions. MCL has recently gained attention due to its simplicity and ability to address ill-posed and ambiguous tasks. We propose an adaptation of this framework for time-series forecasting, presenting it as an efficient method to predict diverse futures, which we relate to its implicit quantization objective. We provide insights into our approach using synthetic data and evaluate it on real-world time series, demonstrating its promising performance at a light computational cost.

Figures

Figures reproduced from arXiv: 2506.05515 by the authors.

**Figure 1.** Components of TimeMCL. Prediction heads are in Lightcoral color, and Score heads are in Blue. Rectangles contain functions, and circles contain features. 4.2. Inference and Sampling Once trained, TimeMCL provides K plausible predictions xˆ 1 ∼ p 1 θ , . . . , xˆ K ∼ p K θ , by ancestral sampling on each of the models. When using recurrent neural networks, the unrolling procedure described in Section 3 can be applied… view at source ↗

**Figure 2.** Conditional Quantization of Stochastic Processes with TimeMCL. The Figure shows the predictions of TimeMCL on three synthetic datasets as described in Section 5.3 and Appendix B. Predictions of TimeMCL are shown in lightcoral color, and target quantization in each case are shown in brown. We used 10 hypotheses here, a three-layer MLP as backbone with score-heads disabled in those toy experiments. Brownian Motion, Br… view at source ↗

**Figure 3.** Qualitative results. Visualization of the predictions on the SOLAR dataset, comparing Tactis2 (purple), TimeGrad (red), TempFlow (orange), and TimeMCL trained with two relaxation techniques (blue). Each model predicts sixteen hypotheses, dashed lines represent the predicted mean of the conditional distribution which was computed as the weighted sum of the predicted hypotheses by the scores for TimeMCL. Each row repr… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Cryptocurrency price forecasting for January 4, 2025 with K = 4 hypotheses. Setup in described in Section 6.4. Legend is the same as in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Cryptocurrency dataset description. (Left) Pair-wise matrix correlations of log returns between the time series. (Right) Yahoo Finance ticker symbols, number of hourly observations per asset (Points), and the corresponding price scales (Min, Max), thereby summarizing b…

**Figure 6.** Figure 6: Cryptocurrency Price Prediction on 04/01/2025 with K = 4 Hypotheses. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: Qualitative results on the TRAFFIC dataset, with same experimental setup as in [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

**Figure 8.** Figure 8: Qualitative results on the ELECTRICITY dataset, with same experimental setup as in [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

19 extracted references · 15 canonical work pages

[1]

With 500 discretization points, it predicts 250 steps, conditioned on the prediction start time and value. Autoregressive model AR(p).The two first experiments involved Brownian motion, where only the last value matters for prediction, and the Brownian bridge, which depends on the last observed time and value. While these processes assess TimeMCL’s quanti...

work page 2018
[5]

Cryptocurrency dataset description.(Left)Pair-wise matrix correlations of log returns between the time series.(Right)Yahoo Finance ticker symbols, number of hourly observations per asset (Points), and the corresponding price scales (Min, Max), thereby summarizing both data volume and cross-asset dependence. We obtained cryptocurrency data from YahooFinanc...

work page 2023
[6]

ETS, Trf.TempFlow and Tactis2, columns are in gray because they don’t share the same backbone as the other baselines

Distortion comparison with 8 Hypotheses.Results are averaged over four training seeds. ETS, Trf.TempFlow and Tactis2, columns are in gray because they don’t share the same backbone as the other baselines. Best scores are inbold, and second-best are underlined. ETS Trf.TempFlow Tactis2 TimeGrad DeepAR TempFlow TimeMCL(R.) TimeMCL(A.) ELEC. 15132±1662 14160...

work page
[9]

In this table, the distortion is computed with a variable number of hypothesesKfor each baseline, as in Table 4 of the main paper

Distorsion (↓) Comparison for Solar Dataset.Results averaged over four training seeds. In this table, the distortion is computed with a variable number of hypothesesKfor each baseline, as in Table 4 of the main paper. KETS Trf.TempFlow Tactis2 TimeGrad DeepAR TempFlow TimeMCL(R.) TimeMCL(A.) 1 685.77±23.08 468.84±32.54 433.76±19.02398.96±32.38838.14±26.13...

work page 2024
[10]

These series generally display recurrent rush-hour peaks as well as differences between weekdays and weekends

measured hourly by 963 road sensors over 4001 time steps. These series generally display recurrent rush-hour peaks as well as differences between weekdays and weekends. • The TAXIdataset consists of traffic time-series data of New York City taxi rides, recorded at 1214 locations every 30 minutes in January 2015 (training set) and January 2016 (test set). ...

work page 2015
[12]

Here, TimeMCL follows the same experimental setup as in the previous benchmark, except that we used Z-Score normalization (instead of mean scaling) during training

Results of neural networks-based methods on the cryptocurrency dataset.Here, K= 4 , and the results were averaged over three random seeds. Here, TimeMCL follows the same experimental setup as in the previous benchmark, except that we used Z-Score normalization (instead of mean scaling) during training. Trf.TempFlow Tactis2 TimeGrad DeepAR TempFlow TimeMCL...

work page arXiv
[15]

Inference.We used the official experimental protocol for evaluation in this benchmark (e.g.,(Rasul et al., 2021a))

Training and evaluation are performed in single-precision (float32) with PyTorch. Inference.We used the official experimental protocol for evaluation in this benchmark (e.g.,(Rasul et al., 2021a)). The official test dataset is divided into multiple non-overlapping subsets, each containing sufficient points for both context and prediction lengths, allowing...

work page 2021
[19]

We observe thatTimeMCL produces smoother predictions compared to other methods and effectively captures different modes in the conditional distribution

Following the notation introduced in Figure, TimeMCL predictions are shown in shades of blue, with color intensity reflecting the associated score. We observe thatTimeMCL produces smoother predictions compared to other methods and effectively captures different modes in the conditional distribution. For clarity, only a subset of the cryptocurrencies is sh...

work page 2025

Show all 19 references

[64]

For fair and accurate time comparison with respect to the number of hypotheses, we disable parallel sampling for each baseline (by settingnum parallel samplesto1). C.2.6. TOTALVARIATION We quantify the average smoothness of the sampled trajectories using the Total Variation (T...

2021
[1976]

is computed as CRPS≜ 1 |Q| X q∈Q PN i=1 Lq xi t0:T ,ˆxi t0:T T(x t0:T ) ,(19) where we used Q={0.05,0.1, . . .0.95}. Equation (19) is referred to as m sum mean wQuantileLoss in the python library, or CRPS-Sum (Salinas et al., 2019; Ashok et al.,

2019
[1982]

Appendix A contains the proofs of the theoretical results, establishing that TimeMCL can be interpreted as a functional quantizer

12 Winner-takes-all for Multivariate Probabilistic Time Series Forecasting Organization of the Appendix The Appendix is organized as follows. Appendix A contains the proofs of the theoretical results, establishing that TimeMCL can be interpreted as a functional quantizer. Appe...

2024
[2005]

Wavenet: A generative model for raw audio.arXiv preprint arXiv:1609.03499,

Van Den Oord, A., Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., Kalchbrenner, N., Senior, A., Kavukcuoglu, K., et al. Wavenet: A generative model for raw audio.arXiv preprint arXiv:1609.03499,

arXiv
[2007]

S., Flunkert, V ., Gasthaus, J., Wang, Y ., Bohlke-Schneider, M., Salinas, D., Stella, L., Callot, L., and Januschowski, T

Benidis, K., Rangapuram, S. S., Flunkert, V ., Gasthaus, J., Wang, Y ., Bohlke-Schneider, M., Salinas, D., Stella, L., Callot, L., and Januschowski, T. Neural forecast- ing: Introduction and literature overview.arXiv preprint arXiv:2004.10240,

2004 arXiv
[2012]

and Gimpel, K

Hendrycks, D. and Gimpel, K. Gaussian error linear units (gelus).arXiv preprint arXiv:1606.08415,

arXiv
[2014]

The nystr\” om method for functional quantiza- tion with an application to the fractional brownian motion

Corlay, S. The nystr\” om method for functional quantiza- tion with an application to the fractional brownian motion. arXiv preprint arXiv:1009.1241,

arXiv
[2015]

of the Brownian motion ont∈[0,1]: Wt = ∞X n=1 q λWn ξneW n (t),(16) where ξn ∼ N(0,1)are independent standard Gaussian random variables. Here, λW n ≜ 1 π2(n− 1 2 )2 are the eigenvalues of the covariance operator of Brownian motion, and eW n (t)≜ √ 2 sin(π(n− 1 2 )t) are the co...

2009
[2018]

Empirical evaluation of gated recurrent neural networks on sequence modeling.arXiv preprint arXiv:1412.3555,

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

arXiv
[2021]

Generating sequences with recurrent neural networks.arXiv preprint arXiv:1308.0850,

Graves, A. Generating sequences with recurrent neural networks.arXiv preprint arXiv:1308.0850,

arXiv
[2024]

in the related literature. C.2.3. DISTORSION Distortion, also referred to as theOracleorQuantizationerror in the literature (Pages & Printems, 2009; Lee et al., 2016; Perera et al.,

2009

Pith tools

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

[1] [1]

With 500 discretization points, it predicts 250 steps, conditioned on the prediction start time and value. Autoregressive model AR(p).The two first experiments involved Brownian motion, where only the last value matters for prediction, and the Brownian bridge, which depends on the last observed time and value. While these processes assess TimeMCL’s quanti...

work page 2018

[2] [5]

Cryptocurrency dataset description.(Left)Pair-wise matrix correlations of log returns between the time series.(Right)Yahoo Finance ticker symbols, number of hourly observations per asset (Points), and the corresponding price scales (Min, Max), thereby summarizing both data volume and cross-asset dependence. We obtained cryptocurrency data from YahooFinanc...

work page 2023

[3] [6]

ETS, Trf.TempFlow and Tactis2, columns are in gray because they don’t share the same backbone as the other baselines

Distortion comparison with 8 Hypotheses.Results are averaged over four training seeds. ETS, Trf.TempFlow and Tactis2, columns are in gray because they don’t share the same backbone as the other baselines. Best scores are inbold, and second-best are underlined. ETS Trf.TempFlow Tactis2 TimeGrad DeepAR TempFlow TimeMCL(R.) TimeMCL(A.) ELEC. 15132±1662 14160...

work page

[4] [9]

In this table, the distortion is computed with a variable number of hypothesesKfor each baseline, as in Table 4 of the main paper

Distorsion (↓) Comparison for Solar Dataset.Results averaged over four training seeds. In this table, the distortion is computed with a variable number of hypothesesKfor each baseline, as in Table 4 of the main paper. KETS Trf.TempFlow Tactis2 TimeGrad DeepAR TempFlow TimeMCL(R.) TimeMCL(A.) 1 685.77±23.08 468.84±32.54 433.76±19.02398.96±32.38838.14±26.13...

work page 2024

[5] [10]

These series generally display recurrent rush-hour peaks as well as differences between weekdays and weekends

measured hourly by 963 road sensors over 4001 time steps. These series generally display recurrent rush-hour peaks as well as differences between weekdays and weekends. • The TAXIdataset consists of traffic time-series data of New York City taxi rides, recorded at 1214 locations every 30 minutes in January 2015 (training set) and January 2016 (test set). ...

work page 2015

[6] [12]

Here, TimeMCL follows the same experimental setup as in the previous benchmark, except that we used Z-Score normalization (instead of mean scaling) during training

Results of neural networks-based methods on the cryptocurrency dataset.Here, K= 4 , and the results were averaged over three random seeds. Here, TimeMCL follows the same experimental setup as in the previous benchmark, except that we used Z-Score normalization (instead of mean scaling) during training. Trf.TempFlow Tactis2 TimeGrad DeepAR TempFlow TimeMCL...

work page arXiv

[7] [15]

Inference.We used the official experimental protocol for evaluation in this benchmark (e.g.,(Rasul et al., 2021a))

Training and evaluation are performed in single-precision (float32) with PyTorch. Inference.We used the official experimental protocol for evaluation in this benchmark (e.g.,(Rasul et al., 2021a)). The official test dataset is divided into multiple non-overlapping subsets, each containing sufficient points for both context and prediction lengths, allowing...

work page 2021

[8] [19]

We observe thatTimeMCL produces smoother predictions compared to other methods and effectively captures different modes in the conditional distribution

Following the notation introduced in Figure, TimeMCL predictions are shown in shades of blue, with color intensity reflecting the associated score. We observe thatTimeMCL produces smoother predictions compared to other methods and effectively captures different modes in the conditional distribution. For clarity, only a subset of the cryptocurrencies is sh...

work page 2025