Quantile-Guided Diffusion for Probabilistic Time-Series Forecasting

Shadi Omari; Bashar Khoury; Assaf Schuster

cs.LGOct 14, 2026

Quantile-Guided Diffusion for Probabilistic Time-Series Forecasting

Authors: Shadi Omari, Bashar Khoury, Assaf Schuster

Organizations: Department of Computer Science, Technion – Israel Institute of Technology

Abstract

Quantile forecasts summarize uncertainty at each future time step, while diffusion forecasters represent it through sampled future trajectories. We study how predicted quantiles can serve as distributional controls for diffusion forecasting. Specifically, we investigate how the mechanism of quantile integration affects its effectiveness in guiding a pretrained diffusion forecaster. We find that residual control throughout the denoising network yields the most consistent improvements. We instantiate this approach as Control-TS, which learns a quantile-control pathway while keeping the pretrained backbone frozen. Across eight benchmarks, Control-TS consistently improves the backbone's CRPS, achieves competitive performance against state-of-the-art forecasting models, and retains its gains in capacity-matched comparisons. Further analyses examine how these gains depend on quantile granularity, predictor choice, and forecast quality. Finally, we introduce the Conditional Linear Predictive Score (CLPS), which complements distributional metrics by evaluating the downstream forecasting utility of generated futures. Together, these findings support predicted quantiles as an effective interface between explicit uncertainty estimates and diffusion-based trajectory generation.

Figures & tables

Figure 1: Integrating predicted quantiles with diffusion forecasting. (A) concatenates, (B) FiLM, (C) ensembles. (D) adds residuals before the first 1, 2, or all blocks (Control-TS). All improve backbone CRPS on some datasets; full-depth Control-TS gives the most consistent gains on all eight.

Figure 2: Control-TS, the full-depth residual realization. A dedicated quantile pathway injects depth-aligned control through zero-initialized $1\times 1$ projections into the frozen backbone.

Dataset	Backbone	Output ensembling	Input concat.	FiLM	1-block residual	2-block residual	Control-TS (full-depth)
Electricity	0.049 $\pm$ 0.000	0.049 $\pm$ 0.000	0.062 $\pm$ 0.000	0.054 $\pm$ 0.000	0.052 $\pm$ 0.000	0.052 $\pm$ 0.000	0.047 $\pm$ 0.000
Exchange	0.012 $\pm$ 0.001	0.008 $\pm$ 0.000	0.007 $\pm$ 0.000	0.008 $\pm$ 0.000	0.009 $\pm$ 0.000	0.008 $\pm$ 0.000	0.008 $\pm$ 0.000
KDDCup	0.312 $\pm$ 0.001	0.320 $\pm$ 0.003	0.289 $\pm$ 0.002	0.268 $\pm$ 0.005	0.286 $\pm$ 0.001	0.285 $\pm$ 0.002	0.249 $\pm$ 0.001
M4-Hourly	0.034 $\pm$ 0.000	0.034 $\pm$ 0.000	0.027 $\pm$ 0.001	0.026 $\pm$ 0.001	0.032 $\pm$ 0.000	0.026 $\pm$ 0.000	0.022 $\pm$ 0.000
Solar	0.402 $\pm$ 0.000	0.378 $\pm$ 0.001	0.308 $\pm$ 0.004	0.360 $\pm$ 0.002	0.444 $\pm$ 0.000	0.415 $\pm$ 0.001	0.341 $\pm$ 0.000
Traffic	0.099 $\pm$ 0.000	0.099 $\pm$ 0.000	0.117 $\pm$ 0.000	0.116 $\pm$ 0.001	0.125 $\pm$ 0.000	0.118 $\pm$ 0.000	0.097 $\pm$ 0.000

Table 1: Quantile integration mechanisms. Best values are bold , second-best values are underlined .

Method	Electricity	Exchange	KDDCup	M4-Hourly	Solar	Traffic	UberTLC	Wikipedia
ARIMA	0.344 $\pm$ 0.000	0.008 $\pm$ 0.000	0.514 $\pm$ 0.000	0.031 $\pm$ 0.000	0.558 $\pm$ 0.003	0.486 $\pm$ 0.000	0.478 $\pm$ 0.000	0.654 $\pm$ 0.000
DLinear	0.058 $\pm$ 0.001	0.015 $\pm$ 0.004	0.318 $\pm$ 0.015	0.055 $\pm$ 0.007	0.794 $\pm$ 0.027	0.131 $\pm$ 0.000	0.250 $\pm$ 0.006	0.259 $\pm$ 0.002
DeepAR	0.051 $\pm$ 0.000	0.013 $\pm$ 0.004	0.362 $\pm$ 0.017	0.046 $\pm$ 0.006	0.429 $\pm$ 0.055	0.103 $\pm$ 0.002	0.169 $\pm$ 0.002	0.215 $\pm$ 0.003
TFT	0.060 $\pm$ 0.001	0.011 $\pm$ 0.000	0.543 $\pm$ 0.048	0.038 $\pm$ 0.002	0.371 $\pm$ 0.006	0.128 $\pm$ 0.005	0.202 $\pm$ 0.009	0.219 $\pm$ 0.004
PatchTST	0.055 $\pm$ 0.001	0.010 $\pm$ 0.001	0.420 $\pm$ 0.011	0.034 $\pm$ 0.004	0.728 $\pm$ 0.015	0.151 $\pm$ 0.007	0.219 $\pm$ 0.004	0.209 $\pm$ 0.001
iTransformer	0.074 $\pm$ 0.000	0.010 $\pm$ 0.000	-	-	0.496 $\pm$ 0.000	0.158 $\pm$ 0.000	-	0.262 $\pm$ 0.000

Table 2: Forecasting results on eight benchmarks. CRPS; lower is better. Best/second-best values are bold / underlined , including ties. Our evaluations report mean

\pm

SD over three independent runs with different random seeds.

Figure 3: Qualitative predictive distributions. Control-TS and TSDiff medians and 90% intervals on shared M4-Hourly and Electricity contexts. Shaded regions denote the forecast horizon.

Backbone	Metric	Improved datasets	Range $\Delta$ (%)
CSDI	CRPS	8/8	$[+0.34,+11.05]$
MG-TSD	CRPS-sum	5/8	$[-7.23,+38.27]$

Table 3: Quantile control on additional diffusion backbones. CRPS for univariate CSDI; CRPS-sum for multivariate MG-TSD. Results are based on three independent runs with different random seeds. Positive

\Delta

denotes relative improvement over the corresponding backbone. Scores are not pooled across backbones or with the main benchmark comparison. Appendix B.17 provides detailed results and protocols.

Panel of Figure 4 — Figure 4: Robustness to degraded control signals. Left: CRPS improvement over TSDiff as Gaussian corruption increases; positive values indicate retained gains. Right: selected M4-Hourly and Electricity examples with inaccurate quantile scaffolds; Control-TS stays closer to the realized continuation than the scaffold. Full corruption results and failure cases are in Appendices B.13 , B.14 .

		Electricity	Exchange	KDDCup	M4-Hourly	Solar	Traffic	UberTLC	Wikipedia
Linear (CLPS)	TSDiff	0.070 $\pm$ 0.001	0.096 $\pm$ 0.000	0.900 $\pm$ 0.001	0.041 $\pm$ 0.000	0.543 $\pm$ 0.002	0.157 $\pm$ 0.001	0.303 $\pm$ 0.003	0.392 $\pm$ 0.006
Linear (CLPS)	Control-TS	0.072 $\pm$ 0.004	0.028 $\pm$ 0.000	0.699 $\pm$ 0.006	0.038 $\pm$ 0.001	0.526 $\pm$ 0.003	0.154 $\pm$ 0.001	0.306 $\pm$ 0.001	0.374 $\pm$ 0.007
DeepAR	TSDiff	0.054 $\pm$ 0.002	0.011 $\pm$ 0.003	0.480 $\pm$ 0.027	0.025 $\pm$ 0.001	0.380 $\pm$ 0.016	0.116 $\pm$ 0.002	0.174 $\pm$ 0.002	0.224 $\pm$ 0.006
DeepAR	Control-TS	0.054 $\pm$ 0.001	0.012 $\pm$ 0.008	0.411 $\pm$ 0.033	0.035 $\pm$ 0.007	0.387 $\pm$ 0.009	0.110 $\pm$ 0.002	0.173 $\pm$ 0.006	0.223 $\pm$ 0.005
TFT	TSDiff	0.058 $\pm$ 0.001	0.015 $\pm$ 0.002	0.818 $\pm$ 0.099	0.032 $\pm$ 0.002	0.374 $\pm$ 0.007	0.137 $\pm$ 0.007	0.238 $\pm$ 0.011	0.232 $\pm$ 0.003
TFT	Control-TS	0.060 $\pm$ 0.001	0.007 $\pm$ 0.000	0.632 $\pm$ 0.001	0.032 $\pm$ 0.001	0.364 $\pm$ 0.007	0.124 $\pm$ 0.014	0.221 $\pm$ 0.010	0.230 $\pm$ 0.001

Table 4: Train-on-Hybrid, Test-on-Real CRPS. Mean

\pm

SD over three independent runs with different random seeds. The better synthetic source is bold , including ties.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Wikipedia denoiser L2 loss over 1000 epochs.

Panel of Figure 6 — Figure 6: Validation CRPS evaluated every 50 epochs during 1000-epoch Controller training (two example datasets).

Dataset	Train	Test	Domain	Freq.	Median Seq. Len.	Pred. Len.
Electricity a	370	2590	$\mathbb{R}^{+}$	H	5833	24
Exchange b	8	40	$\mathbb{R}^{+}$	D	6071	30
KDDCup c	270	270	N	H	10850	48
M4-Hourly d	414	414	N	H	960	48
Solar e	137	959	$\mathbb{R}^{+}$	H	7009	24
Traffic f	963	6741	$(0,1)$	H	4001	24

Appendix

Table 5: Overview of the datasets and statistics used in our experiments.

Dataset	GluonTS name	Source / provider	License or terms of use
Solar	solar_nips	NREL / Monash Time Series Forecasting Repository	CC BY 4.0 via Monash/Zenodo.
Electricity	electricity_nips	UCI Machine Learning Repository	CC BY 4.0.
Traffic	traffic_nips	Caltrans PeMS / Monash Time Series Forecasting Repository	CC BY 4.0 via Monash/Zenodo.
Exchange	exchange_rate_nips	LSTNet multivariate time-series repository	Research use stated by the source repository; no formal named license identified.
M4-Hourly	m4_hourly	M4 Competition / Monash Time Series Forecasting Repository	CC BY 4.0 via Monash/Zenodo.
KDDCup	kdd_cup_2018_without_missing	KDD Cup 2018 / Monash Time Series Forecasting Repository	CC BY 4.0 via Monash/Zenodo.

Appendix

Table 6: Existing dataset assets used in this work. Licenses and terms are reported according to the corresponding official dataset pages, repositories, or documentation pages accessed on 2026-05-07.

Component	Setting
Hardware	NVIDIA GeForce RTX 2080 Ti
Diffusion steps	$T_{\mathrm{diff}}=100$
$\beta$ schedule	linear from $10^{-4}$ to $10^{-1}$
Backbone block	S4 residual blocks (bidirectional), $d_{\mathrm{state}}=128$
Residual depth	$K=3$ (small configuration)
Hidden width	64

Appendix

Table 7: Summary of key hyperparameters used by our implementation and configs.

	TSDiff-Wide		Control-TS
Dataset	Params [k]	CRPS	Params [k]	CRPS
Electricity	501.4	0.048 $\pm$ 0.000	505.6	0.047 $\pm$ 0.000
Exchange	501.5	0.012 $\pm$ 0.000	493.1	0.008 $\pm$ 0.000
KDDCup	501.4	0.345 $\pm$ 0.002	504.2	0.249 $\pm$ 0.001
M4-Hourly	498.8	0.035 $\pm$ 0.000	407.9	0.022 $\pm$ 0.000
Solar	501.4	0.409 $\pm$ 0.001	412.4	0.341 $\pm$ 0.000

Appendix

Table 8: Complete-system capacity comparison: mean

\pm

SD. Control-TS includes the backbone, Controller, and external forecaster; its results are from Table 2 . Best mean CRPS is bold .

	DeepAR-Large		TFT-Large		Control-TS
Dataset	Params [k]	CRPS	Params [k]	CRPS	Params [k]	CRPS
Electricity	502.7	0.047 $\pm$ 0.001	493.5	0.048 $\pm$ 0.000	505.6	0.047 $\pm$ 0.000
Exchange	496.5	0.022 $\pm$ 0.006	500.3	0.010 $\pm$ 0.001	493.1	0.008 $\pm$ 0.000
KDDCup	502.7	0.380 $\pm$ 0.019	493.5	0.354 $\pm$ 0.044	504.2	0.249 $\pm$ 0.001
M4-Hourly	502.7	0.047 $\pm$ 0.006	493.5	0.030 $\pm$ 0.003	407.9	0.022 $\pm$ 0.000
Solar	502.7	0.371 $\pm$ 0.020	493.5	0.552 $\pm$ 0.045	412.4	0.341 $\pm$ 0.000

Appendix

Table 9: Capacity-comparable explicit forecasters: mean

\pm

SD. Control-TS results are from Table 2 . Best values are bold .

Mechanism	Better than the backbone
Ensembling	4/8
Input concatenation	4/8
FiLM	4/8
1-block residual	3/8
2-block residual	3/8
Control-TS	8/8

Appendix

Table 10: Datasets on which each interaction mechanism improves the backbone.

	Input concatenation		FiLM
Dataset	Fine-tuned	Frozen backbone	Fine-tuned	Frozen backbone
Electricity	0.062 $\pm$ 0.000	0.065 $\pm$ 0.000	0.054 $\pm$ 0.000	0.060 $\pm$ 0.000
Exchange	0.007 $\pm$ 0.000	0.009 $\pm$ 0.000	0.008 $\pm$ 0.000	0.010 $\pm$ 0.000
KDDCup	0.289 $\pm$ 0.002	0.332 $\pm$ 0.001	0.268 $\pm$ 0.005	0.292 $\pm$ 0.001
M4-Hourly	0.027 $\pm$ 0.001	0.034 $\pm$ 0.001	0.026 $\pm$ 0.001	0.040 $\pm$ 0.001
Solar	0.308 $\pm$ 0.004	0.390 $\pm$ 0.005	0.360 $\pm$ 0.002	0.423 $\pm$ 0.003

Appendix

Table 11: CRPS for fine-tuned and frozen-backbone input concatenation and FiLM. Fine-tuned entries match Table 1 . Best values are bold .

External forecaster	Variant	Electricity	Exchange	KDDCup	M4-Hourly	Solar	Traffic	UberTLC	Wikipedia
Baseline		0.049	0.012	0.312	0.034	0.402	0.099	0.172	0.211
DeepAR	Q1	0.054	0.009	0.303	0.025	0.419	0.098	0.177	0.203
	Q3	0.057	0.009	0.267	0.026	0.397	0.097	0.171	0.205
	Q5	0.056	0.009	0.245	0.022	0.341	0.100	0.173	0.200
TFT	Q1	0.049	0.008	0.282	0.028	0.408	0.107	0.170	0.204
	Q3	0.048	0.008	0.249	0.026	0.392	0.102	0.169	0.202

Appendix

Table 12: Per-variant CRPS on the eight benchmark datasets for each external forecaster (rows) and quantile variant Q1/Q3/Q5 (Section 2.4 ), with the TSDiff baseline for reference. The best entry per column is shown in bold and the second-best is underlined . Tied entries are marked jointly.

	Q1		Q3		Q5
Dataset	Control-TS	Context $\times 1$	Control-TS	Context $\times 3$	Control-TS	Context $\times 5$
Electricity	0.049 $\pm$ 0.000	0.047 $\pm$ 0.000	0.048 $\pm$ 0.000	0.049 $\pm$ 0.000	0.047 $\pm$ 0.000	0.048 $\pm$ 0.000
Exchange	0.008 $\pm$ 0.000	0.010 $\pm$ 0.000	0.008 $\pm$ 0.000	0.011 $\pm$ 0.000	0.008 $\pm$ 0.000	0.012 $\pm$ 0.000
KDDCup	0.282 $\pm$ 0.001	0.311 $\pm$ 0.001	0.249 $\pm$ 0.001	0.319 $\pm$ 0.001	0.256 $\pm$ 0.001	0.282 $\pm$ 0.001
M4-Hourly	0.025 $\pm$ 0.000	0.045 $\pm$ 0.000	0.026 $\pm$ 0.000	0.033 $\pm$ 0.000	0.022 $\pm$ 0.000	0.030 $\pm$ 0.000
Solar	0.408 $\pm$ 0.000	0.375 $\pm$ 0.000	0.392 $\pm$ 0.000	0.449 $\pm$ 0.000	0.341 $\pm$ 0.000	0.440 $\pm$ 0.001

Appendix

Table 13: Observed context as control. CRPS (mean

\pm

SD) using the same full-depth residual architecture. Control-TS values are from Table 2 . The better value in each pair is bold , including ties.

Dataset	Control-TS	No zero-init
Electricity	0.047	0.051
Exchange	0.008	0.009
KDDCup	0.249	0.302
M4-Hourly	0.022	0.027
Solar	0.341	0.433
Traffic	0.097	0.113

Appendix

Table 14: Projection-initialization ablation for full-depth Control-TS. Best values are bold .

Dataset	Pretrained initialization	Control-TS
Electricity	0.050 $\pm$ 0.000	0.047 $\pm$ 0.000
Exchange	0.009 $\pm$ 0.000	0.008 $\pm$ 0.000
KDDCup	0.305 $\pm$ 0.000	0.249 $\pm$ 0.001
M4-Hourly	0.030 $\pm$ 0.000	0.022 $\pm$ 0.000
Solar	0.401 $\pm$ 0.000	0.341 $\pm$ 0.000
Traffic	0.107 $\pm$ 0.001	0.097 $\pm$ 0.000

Appendix

Table 15: Controller-initialization ablation for full-depth Control-TS (CRPS; lower is better). Control-TS initializes its controller residual blocks randomly. Pretrained initialization instead starts each controller block from a copy of the corresponding residual block of the pretrained diffusion backbone; the copies are trained while the backbone remains frozen, and all other settings are unchanged.

Dataset	1 block	2 blocks	Control-TS
Electricity	0.052	0.052	0.047
Exchange	0.009	0.008	0.008
KDDCup	0.286	0.285	0.249
M4-Hourly	0.032	0.026	0.022
Solar	0.444	0.415	0.341
Traffic	0.125	0.118	0.097

Appendix

Table 16: Controller-depth ablations. bold identifies the full-depth reference, which is lowest or tied-lowest at displayed precision.

Figure 7: Examples of corrupted quantile-control signals under additive Gaussian perturbations in normalized control space. Solid curves show the original quantile trajectories and dashed curves show the perturbed controls for representative datasets and increasing noise levels.

Dataset	Clean	$\sigma=0.01$	$\sigma=0.02$	$\sigma=0.05$	$\sigma=0.1$	Backbone
Electricity	0.047	0.048	0.049	0.058	0.080	0.049
Exchange	0.008	0.010	0.014	0.031	0.065	0.012
KDDCup	0.249	0.249	0.251	0.252	0.261	0.312
M4	0.022	0.023	0.024	0.037	0.053	0.034
Solar	0.341	0.374	0.377	0.389	0.424	0.402
Traffic	0.097	0.098	0.099	0.108	0.131	0.099

Appendix

Table 17: CRPS under Gaussian degradation of the control signal at inference time. Noise is added in normalized control space. Entries better than the unguided diffusion-backbone baseline are shown in bold , while ties are underlined .

Figure 8: Failure cases of Control-TS: misaligned quantile scaffolds where the forecast follows the unreliable signal.

Dataset	Backbone (frozen) [k]	Controller (trainable) [k]	Controller / Backbone
Electricity	193.4	191.4	0.99
Exchange	193.4	190.1	0.98
KDDCup	193.4	190.0	0.98
M4	192.1	188.2	0.98
Solar	193.4	191.4	0.99
Traffic	193.4	190.0	0.98

Appendix

Table 18: Parameter budget of Control-TS per dataset. The diffusion backbone is held fixed during Controller training; only the Controller branch receives gradient updates. Counts in thousands of parameters (k); ratio is Controller / Backbone.

Dataset	$\Delta$ Time (%)	GFLOPs	$\Delta$ VRAM (%)	$\Delta$ Time (%)	GFLOPs	$\Delta$ VRAM (%)
	Inference (sampling)			Training (one step)
Electricity	$+19.1$	21.5 / 26.8	$+6.1$	$+13.5$	0.20 / 0.24	$+66.2$
Exchange	$+61.1$	24.2 / 28.7	$+4.0$	$+65.6$	0.23 / 0.24	$+66.0$
KDDCup	$+17.1$	21.5 / 25.2	$+5.9$	$+13.2$	0.20 / 0.21	$+64.8$
M4	$+39.2$	17.0 / 18.2	$+3.5$	$+6.5$	0.17 / 0.14	$+63.9$
Solar	$+21.4$	21.5 / 26.4	$+4.5$	$+14.9$	0.20 / 0.24	$+65.0$

Appendix

Table 19: Computational overhead of Control-TS relative to the frozen backbone for all eight benchmarks (

B{=}16

T_{\mathrm{diff}}{=}100

, with quantile observation self-guidance). Each

\Delta

column is the relative increase (%) of Control-TS over the diffusion backbone; GFLOPs columns report

{\color[rgb]{0.1211,0.4648,0.707}\text{Backbone}}\,/\,{\color[rgb]{0.8398,0.1523,0.1563}\text{Control-TS}}

	CSDI: univariate	MG-TSD: multivariate
Dataset	CRPS	CRPS-sum
Electricity	$+7.90$	$+6.35$
Exchange	$+9.98$	$-4.16$
KDDCup	$+3.90$	$-7.23$
M4-Hourly	$+9.73$	$+3.98$
Solar	$+0.34$	$+38.27$

Appendix

Table 20: Paired test-score reductions (%) from Control-TS, using 100 forecast paths. Positive values indicate improvement; negative values indicate regression.

Figure 9: Additional qualitative comparisons between Control-TS and the TSDiff baseline across selected benchmark series.

Figure 10: Examples with accurate quantile scaffolds: Control-TS tightens its predictive distribution around the realized future.

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