Beyond Static Uncertainty: Modeling Temporal Uncertainty Dynamics for Probabilistic Time Series Forecasting

Yijun Wang; Qiyuan Zhuang; Larysa Marchanka; Xiu-Shen Wei

cs.LGMar 25, 2026

Beyond Static Uncertainty: Modeling Temporal Uncertainty Dynamics for Probabilistic Time Series Forecasting

Authors: Yijun Wang, Qiyuan Zhuang, Larysa Marchanka, Xiu-Shen Wei

Organizations: Department of Computer Science, Southeast University, Nanjing, China · Francisk Skorina Gomel State University, Belarus

Abstract

Real-world time series exhibit temporally structured uncertainty: volatility clusters in turbulent regimes, dissipates in stable periods, and shifts abruptly around structural breaks. Yet many probabilistic forecasting methods estimate predictive uncertainty as an independent per-step quantity, leaving the evolution and persistence of volatility regimes under-modeled. We formalize this missing dimension as temporal uncertainty dynamics and instantiate it in the Volatility Dynamics Variational Autoencoder (VolDy-VAE), a non-autoregressive generative forecaster with a location-scale decoder. VolDy-VAE combines a location path for mean prediction with a recurrent scale path that transfers and evolves a volatility hidden state from the look-back window to the forecasting horizon, enabling temporally coherent predictive variances. This design yields an adaptive attenuation mechanism: high-variance observations receive lower influence on the location estimate while their uncertainty is preserved through explicit scale predictions. We further provide a simplified regime-switching analysis showing that, when variances are known or consistently estimated, the volatility-aware objective reduces to inverse-variance weighting, whereas MSE-based estimators remain unbiased but statistically inefficient. Experiments on nine benchmarks show that VolDy-VAE improves forecasting accuracy and uncertainty calibration over competitive probabilistic and point-forecasting baselines while maintaining low inference latency; plug-in studies further indicate that the VolDy principle can benefit GAN, Koopman VAE, and Transformer backbones. The source code is publicly available at https://github.com/wangyijunlyy/VolDy-VAE.

Figures & tables

Fig. 1: Conceptual illustration of temporal uncertainty dynamics. (a) MSE-based homoscedastic forecasters use a constant scale, so volatile spikes can distort the location path before and after high-variance regimes. (b) Feed-forward heteroscedastic heads can output time-varying scales, but estimate them independently at each step, producing jagged and memoryless intervals. (c) VolDy models temporally coherent scale dynamics, keeping the location prediction close to the predictable central trajectory while using the scale component to absorb volatile deviations.

Approach	Forecasting	Explicit Vol.	Direct	Plug-in
	Task	State	Generation	Evidence
Kendall & Gal [ 29 ]	Regression	✗	–	✗
HeTVAE [ 44 ]	Imputation	✗	✓	✗
DeepAR/DeepState [ 31 , 33 ]	Forecasting	$\triangle$	✗	✗
Diffusion PTSF [ 22 , 4 , 23 ]	Forecasting	$\triangle$	✗	✗
$K^{2}$ VAE [ 28 ]	Forecasting	✗	✓	✗

TABLE I: Positioning of VolDy relative to representative heteroscedastic and probabilistic forecasting families. “Explicit Vol. State” indicates whether the approach uses a dedicated state for the evolution and persistence of volatility regimes, as opposed to generic sequence states or independent per-step variance estimation.

\triangle

denotes partial support through a generic recurrent or generative state rather than an explicit volatility state. “Plug-in Evidence” indicates whether the temporal uncertainty mechanism is evaluated beyond a single base architecture.

Fig. 2: Overview of the VolDy-VAE Framework. The model normalizes and segments the input series into patches, encodes them into a probabilistic latent space, directly maps past latent patches to future latent patches, and decodes both reconstruction and prediction through a location-scale decoder. The location path predicts the denormalized mean, while the recurrent scale path transfers the past volatility hidden state to the future horizon so that the predictive scale evolves coherently across the look-back/prediction boundary.

Fig. 3: Qualitative results on synthetic datasets. Left: forecasting with adaptive 95% confidence intervals under varying volatility. Right: uncertainty recovery, where the predicted volatility $\hat{\sigma}_{t}$ closely tracks the ground-truth temporal dynamics.

Dataset	Features ( $C$ )	Timesteps	Frequency	Domain
ETTh1	7	17,420	1 Hour	Energy
ETTh2	7	17,420	1 Hour	Energy
ETTm1	7	69,680	15 Min	Energy
ETTm2	7	69,680	15 Min	Energy
Electricity	321	26,304	1 Hour	Energy
Traffic	862	17,544	1 Hour	Transport

TABLE II: Statistics of the multivariate time series datasets.

Model	Metric	ETTm1	ETTm2	ETTh1	ETTh2	Electricity	Traffic	Weather	Exchange	ILI
FITS	CRPS	$0.305_{\pm.024}$	$0.449_{\pm.034}$	$0.348_{\pm.025}$	$0.314_{\pm.022}$	$0.115_{\pm.024}$	$0.374_{\pm.004}$	$0.267_{\pm.003}$	$0.074_{\pm.011}$	$0.211_{\pm.011}$
FITS	NMAE	$0.406_{\pm.072}$	$0.540_{\pm.052}$	$0.468_{\pm.012}$	$0.401_{\pm.022}$	$0.149_{\pm.012}$	$0.453_{\pm.022}$	$0.317_{\pm.021}$	$0.097_{\pm.011}$	$0.245_{\pm.017}$
PatchTST	CRPS	$0.304_{\pm.029}$	$0.229_{\pm.036}$	$0.323_{\pm.020}$	$0.304_{\pm.018}$	$0.127_{\pm.015}$	$0.214_{\pm.001}$	$0.142_{\pm.005}$	$0.097_{\pm.007}$	$0.233_{\pm.019}$
PatchTST	NMAE	$0.382_{\pm.066}$	$0.288_{\pm.034}$	$0.428_{\pm.024}$	$0.371_{\pm.021}$	$0.164_{\pm.024}$	$0.253_{\pm.012}$	$0.152_{\pm.029}$	$0.126_{\pm.001}$	$0.287_{\pm.023}$
iTransformer	CRPS	$0.455_{\pm.021}$	$0.311_{\pm.024}$	$0.350_{\pm.019}$	$0.542_{\pm.015}$	$0.109_{\pm.044}$	$0.284_{\pm.004}$	$0.133_{\pm.004}$	$0.087_{\pm.023}$	$0.222_{\pm.020}$
iTransformer	NMAE	$0.490_{\pm.038}$	$0.385_{\pm.042}$	$0.449_{\pm.022}$	$0.667_{\pm.012}$	$0.140_{\pm.009}$	$0.361_{\pm.030}$	$0.147_{\pm.019}$	$0.113_{\pm.015}$	$0.278_{\pm.017}$

TABLE III: Comparison on long-term probabilistic forecasting (forecasting horizon

H=720

) scenarios across nine real-world datasets. Lower CRPS or NMAE values indicate better predictions. The reported values are means with standard deviations over 6 independent retraining-and-evaluation runs. RED : THE BEST, BLUE : THE 2ND BEST. The full results of all four horizons 96, 192, 336, 720 are listed in the supplementary material.

Dataset	Horizon	CRPS $\downarrow$				NMAE $\downarrow$
Dataset	Horizon	Van.	w/o P	w/o R	Ours	Van.	w/o P	w/o R	Ours
ETTh1	96	0.287	0.374	0.277	0.252	0.364	0.464	0.354	0.328
	192	0.323	0.378	0.282	0.277	0.388	0.470	0.370	0.365
	336	0.352	0.386	0.305	0.300	0.409	0.486	0.395	0.384
	720	0.355	0.393	0.318	0.310	0.416	0.504	0.398	0.389
Electricity	96	0.363	0.259	0.082	0.069	0.362	0.363	0.107	0.089

TABLE IV: Component-wise ablation on ETTh1 and Electricity . We compare VolDy-VAE with variants without prediction loss (w/o P), without reconstruction loss (w/o R), and a vanilla VAE (MSE). Metrics: CRPS and NMAE. RED : THE BEST.

Dataset	$H$	CRPS $\downarrow$		NMAE $\downarrow$
Dataset	$H$	w/o RevIN	Ours	w/o RevIN	Ours
ETTh1	96	0.480	0.252	0.618	0.328
	192	0.476	0.277	0.615	0.365
	336	0.470	0.300	0.603	0.384
	720	0.567	0.310	0.690	0.389
Weather	96	0.084	0.070	0.106	0.077

TABLE V: Ablation on Reversible Instance Normalization (RevIN). RED : THE BEST.

Dataset	$H$	CRPS $\downarrow$			NMAE $\downarrow$
Dataset	$H$	MLP	LSTM	GRU	MLP	LSTM	GRU
ETTh1	96	0.271	0.255	0.252	0.350	0.331	0.328
	192	0.296	0.280	0.277	0.381	0.368	0.365
	336	0.320	0.304	0.300	0.401	0.388	0.384
	720	0.328	0.314	0.310	0.406	0.393	0.389
ETTh2	96	0.162	0.151	0.148	0.206	0.186	0.183

TABLE VI: Ablation on scale head architecture. We compare MLP (feed-forward, no temporal memory), LSTM, and GRU (ours) scale heads on ETTh1 and ETTh2 (

L=96

). RED : THE BEST.

Dataset	$H$	Head	Cov.95 $\uparrow$	Sharp. $\downarrow$	QICE $\downarrow$	$\rho_{\sigma}\uparrow$
ETTh1	96	MLP	91.2 $\pm$ 0.8	1.84 $\pm$ 0.05	2.91 $\pm$ 0.18	0.61 $\pm$ 0.03
		LSTM	93.6 $\pm$ 0.6	1.62 $\pm$ 0.04	1.94 $\pm$ 0.14	0.78 $\pm$ 0.02
		GRU	94.4 $\pm$ 0.5	1.53 $\pm$ 0.03	1.58 $\pm$ 0.11	0.84 $\pm$ 0.02
	192	MLP	90.5 $\pm$ 0.9	1.91 $\pm$ 0.06	3.24 $\pm$ 0.22	0.58 $\pm$ 0.04
		LSTM	93.1 $\pm$ 0.7	1.70 $\pm$ 0.05	2.18 $\pm$ 0.16	0.76 $\pm$ 0.03
		GRU	94.1 $\pm$ 0.6	1.60 $\pm$ 0.04	1.71 $\pm$ 0.12	0.83 $\pm$ 0.02

TABLE VII: Uncertainty-dynamics diagnostics for scale head architectures. RED : THE BEST OR CLOSEST TO 95.

Likelihood	Extra parameters	ETTh1, $H=192$		Weather, $H=192$
Likelihood	Extra parameters	CRPS $\downarrow$	NMAE $\downarrow$	CRPS $\downarrow$	NMAE $\downarrow$
Gaussian (default)	$\mu,\sigma$	0.277	0.365	0.067	0.080
Student- $t$	$\mu,\sigma,\nu$	0.286	0.372	0.071	0.084
Gaussian mixture	$\{\pi_{k},\mu_{k},\sigma_{k}\}_{k=1}^{K}$	0.291	0.377	0.074	0.087
Gaussian + flow	$\mu,\sigma,\psi_{\mathrm{flow}}$	0.284	0.370	0.070	0.083

TABLE VIII: Likelihood-family ablation on representative long-horizon forecasting tasks. Lower is better.

Framework	Dataset	$H$	Base CRPS	VolDy CRPS	Base NMAE	VolDy NMAE
TimeGAN	ETTh1	96	0.277 $\pm$ 0.011	0.262 $\pm$ 0.009 †	0.347 $\pm$ 0.012	0.339 $\pm$ 0.010 †
	Weather	96	0.106 $\pm$ 0.006	0.068 $\pm$ 0.004 †	0.117 $\pm$ 0.007	0.081 $\pm$ 0.005 †
	Exchange	96	0.056 $\pm$ 0.004	0.047 $\pm$ 0.003 †	0.071 $\pm$ 0.005	0.062 $\pm$ 0.004 †
$K^{2}$ VAE	ETTh1	96	0.264 $\pm$ 0.020	0.251 $\pm$ 0.017 †	0.336 $\pm$ 0.041	0.332 $\pm$ 0.012
	Weather	96	0.080 $\pm$ 0.007	0.065 $\pm$ 0.004 †	0.086 $\pm$ 0.011	0.076 $\pm$ 0.004 †
	Exchange	96	0.031 $\pm$ 0.002	0.029 $\pm$ 0.002 †	0.032 $\pm$ 0.002	0.029 $\pm$ 0.002 †

TABLE IX: Transferability analysis at

L=96,H=96

across three datasets. Values are mean

\pm

std over six seeds.

\dagger

denotes paired Wilcoxon signed-rank significance against the corresponding base model (

p<0.05

). The complete results over

H\in\{96,192,336,720\}

are reported in the supplementary material, Table XI .

Fig. 4: Visual comparison of probabilistic forecasting on ETTh1 . All subfigures share a unified Y-axis scale. Additional visualization results are provided in the supplementary material, Figures 6 and 7 .

Fig. 5: Efficiency vs. Performance comparison on ETTh2 ( $L=96,H=720$ ). The bubble size indicates inference memory usage (smaller is better).

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Backbone	Controlled setting	VolDy modification
TimeGAN	Same split, horizon, seeds, optimizer family, and early stopping as its base run.	Extend generator output to location/scale and add Gaussian NLL supervision.
$K^{2}$ VAE	Same backbone and forecasting protocol as the base $K^{2}$ VAE run.	Replace deterministic objective with Location-Scale NLL.
PatchTST	Same split, horizon, seeds, optimizer family, and early stopping as its base run.	Replace the point head with a dual-head location-scale decoder trained by Gaussian NLL.

Appendix

TABLE X: Implementation controls for plug-in transferability experiments.

Framework	Dataset	$H$	Base CRPS	VolDy CRPS	Base NMAE	VolDy NMAE
TimeGAN	ETTh1	96	0.277 $\pm$ 0.011	0.262 $\pm$ 0.009 †	0.347 $\pm$ 0.012	0.339 $\pm$ 0.010 †
	ETTh1	192	0.306 $\pm$ 0.014	0.286 $\pm$ 0.011 †	0.382 $\pm$ 0.015	0.368 $\pm$ 0.012 †
	ETTh1	336	0.334 $\pm$ 0.017	0.309 $\pm$ 0.014 †	0.421 $\pm$ 0.019	0.397 $\pm$ 0.016 †
	ETTh1	720	0.381 $\pm$ 0.021	0.348 $\pm$ 0.018 †	0.469 $\pm$ 0.024	0.436 $\pm$ 0.021 †
	Weather	96	0.106 $\pm$ 0.006	0.068 $\pm$ 0.004 †	0.117 $\pm$ 0.007	0.081 $\pm$ 0.005 †
	Weather	192	0.118 $\pm$ 0.007	0.076 $\pm$ 0.005 †	0.129 $\pm$ 0.008	0.089 $\pm$ 0.006 †

Appendix

TABLE XI: Full transferability analysis over four prediction horizons (

L=96

). Values are mean

\pm

std over six seeds.

\dagger

denotes paired Wilcoxon signed-rank significance against the corresponding base model (

p<0.05

Model	CRPS $\downarrow$	NMAE $\downarrow$	Speed $\downarrow$	Memory $\downarrow$
TSDiff	$0.406_{\pm.056}$	$0.482_{\pm.022}$	$1.35$	$340$
GRU NVP	$0.539_{\pm.090}$	$0.688_{\pm.161}$	$2.00$	${\color[rgb]{0,0.4297,0.7344}\underline{26}}$
GRU MAF	$0.990_{\pm.023}$	$1.092_{\pm.019}$	$7.10$	$27$
Trans MAF	$0.327_{\pm.033}$	$0.412_{\pm.020}$	$8.05$	$83$
CSDI	$0.302_{\pm.040}$	$0.382_{\pm.030}$	$32.50$	$38$
$K^{2}$ VAE	${\color[rgb]{0,0.4297,0.7344}\underline{0.280}_{\pm.014}}$	${\color[rgb]{0,0.4297,0.7344}\underline{0.278}_{\pm.020}}$	${\color[rgb]{0,0.4297,0.7344}\underline{0.72}}$	$29$

Appendix

TABLE XII: Comparison on model efficiency and forecasting quality on ETTh2 (

L=96,H=720

). Lower values of CRPS, NMAE, inference speed (sec/sample), and memory (MB) indicate better performance or efficiency. The efficiency measurements are obtained with batch size 1. RED : THE BEST, BLUE : THE 2ND BEST.

Dataset	Horizon	Koopa	iTransformer	FITS	PatchTST	GRU MAF	Trans MAF	TSDiff	CSDI	NsDiff	TimeGrad	GRU NVP	$K^{2}$ VAE	VolDy-VAE
ETTm1	96	$0.285\scriptstyle\pm 0.018$	$0.301\scriptstyle\pm 0.033$	$0.267\scriptstyle\pm 0.023$	$0.261\scriptstyle\pm 0.051$	$0.295\scriptstyle\pm 0.055$	$0.313\scriptstyle\pm 0.045$	$0.344\scriptstyle\pm 0.050$	${0.236}\scriptstyle\pm 0.006$	$0.255\scriptstyle\pm 0.014$	$0.522\scriptstyle\pm 0.105$	$0.383\scriptstyle\pm 0.053$	${\color[rgb]{0,0.4297,0.7344}\underline{0.232\scriptstyle\pm 0.010}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.208\scriptstyle\pm 0.009}$
	192	$0.289\scriptstyle\pm 0.024$	$0.314\scriptstyle\pm 0.023$	${0.261}\scriptstyle\pm 0.022$	$0.275\scriptstyle\pm 0.030$	$0.389\scriptstyle\pm 0.033$	$0.424\scriptstyle\pm 0.029$	$0.345\scriptstyle\pm 0.035$	$0.291\scriptstyle\pm 0.025$	$0.281\scriptstyle\pm 0.018$	$0.603\scriptstyle\pm 0.092$	$0.396\scriptstyle\pm 0.030$	${\color[rgb]{0,0.4297,0.7344}\underline{0.259\scriptstyle\pm 0.013}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.240\scriptstyle\pm 0.010}$
	336	$0.286\scriptstyle\pm 0.035$	$0.311\scriptstyle\pm 0.029$	${0.275}\scriptstyle\pm 0.030$	$0.285\scriptstyle\pm 0.028$	$0.429\scriptstyle\pm 0.021$	$0.481\scriptstyle\pm 0.019$	$0.462\scriptstyle\pm 0.043$	$0.322\scriptstyle\pm 0.033$	$0.292\scriptstyle\pm 0.024$	$0.601\scriptstyle\pm 0.028$	$0.486\scriptstyle\pm 0.032$	${\color[rgb]{0,0.4297,0.7344}\underline{0.262\scriptstyle\pm 0.030}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.251\scriptstyle\pm 0.022}$
	720	${0.295}\scriptstyle\pm 0.027$	$0.455\scriptstyle\pm 0.021$	$0.305\scriptstyle\pm 0.024$	$0.304\scriptstyle\pm 0.029$	$0.536\scriptstyle\pm 0.033$	$0.688\scriptstyle\pm 0.043$	$0.478\scriptstyle\pm 0.027$	$0.448\scriptstyle\pm 0.038$	$0.323\scriptstyle\pm 0.030$	$0.621\scriptstyle\pm 0.037$	$0.546\scriptstyle\pm 0.036$	${\color[rgb]{0,0.4297,0.7344}\underline{0.294\scriptstyle\pm 0.026}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.279\scriptstyle\pm 0.020}$
ETTm2	96	$0.178\scriptstyle\pm 0.023$	$0.181\scriptstyle\pm 0.031$	$0.162\scriptstyle\pm 0.053$	$0.142\scriptstyle\pm 0.034$	$0.177\scriptstyle\pm 0.024$	$0.227\scriptstyle\pm 0.013$	$0.175\scriptstyle\pm 0.019$	${\color[rgb]{0,0.4297,0.7344}\underline{0.115\scriptstyle\pm 0.009}}$	$0.138\scriptstyle\pm 0.010$	$0.427\scriptstyle\pm 0.042$	$0.319\scriptstyle\pm 0.044$	${{0.126\scriptstyle\pm 0.007}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.107\scriptstyle\pm 0.005}$
	192	$0.185\scriptstyle\pm 0.014$	$0.190\scriptstyle\pm 0.010$	$0.185\scriptstyle\pm 0.053$	$0.172\scriptstyle\pm 0.023$	$0.411\scriptstyle\pm 0.026$	$0.253\scriptstyle\pm 0.037$	$0.255\scriptstyle\pm 0.029$	${\color[rgb]{0,0.4297,0.7344}\underline{0.147}\scriptstyle\pm 0.008}$	$0.162\scriptstyle\pm 0.011$	$0.424\scriptstyle\pm 0.061$	$0.326\scriptstyle\pm 0.025$	$0.148\scriptstyle\pm 0.009$	${\color[rgb]{0.7578,0.1523,0.1758}0.129\scriptstyle\pm 0.007}$

Appendix

TABLE XIII: Results of CRPS (

\textrm{mean}_{\textrm{std}}

) on long-term forecasting scenarios, each containing six independent runs with different seeds. The context length is set to 36 for the ILI dataset and 96 for the others. Lower CRPS values indicate better predictions. The reported values are means with standard deviations over 6 independent retraining-and-evaluation runs. RED : THE BEST, BLUE : THE 2ND BEST.

Dataset	Horizon	Koopa	iTransformer	FITS	PatchTST	GRU MAF	Trans MAF	TSDiff	CSDI	NsDiff	TimeGrad	GRU NVP	$K^{2}$ VAE	VolDy-VAE
ETTm1	96	$0.362\scriptstyle\pm 0.022$	$0.369\scriptstyle\pm 0.029$	$0.349\scriptstyle\pm 0.032$	$0.329\scriptstyle\pm 0.100$	$0.402\scriptstyle\pm 0.087$	$0.456\scriptstyle\pm 0.042$	$0.441\scriptstyle\pm 0.021$	$0.308\scriptstyle\pm 0.005$	$0.315\scriptstyle\pm 0.016$	$0.645\scriptstyle\pm 0.129$	$0.488\scriptstyle\pm 0.058$	${\color[rgb]{0,0.4297,0.7344}\underline{0.284\scriptstyle\pm 0.011}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.268\scriptstyle\pm 0.009}$
	192	$0.365\scriptstyle\pm 0.032$	$0.384\scriptstyle\pm 0.041$	$0.341\scriptstyle\pm 0.032$	$0.338\scriptstyle\pm 0.022$	$0.476\scriptstyle\pm 0.046$	$0.553\scriptstyle\pm 0.012$	$0.441\scriptstyle\pm 0.019$	$0.377\scriptstyle\pm 0.026$	$0.354\scriptstyle\pm 0.020$	$0.748\scriptstyle\pm 0.084$	$0.514\scriptstyle\pm 0.042$	${\color[rgb]{0,0.4297,0.7344}\underline{0.323\scriptstyle\pm 0.020}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.309\scriptstyle\pm 0.012}$
	336	$0.364\scriptstyle\pm 0.026$	$0.380\scriptstyle\pm 0.020$	$0.356\scriptstyle\pm 0.022$	$0.344\scriptstyle\pm 0.013$	$0.522\scriptstyle\pm 0.019$	$0.590\scriptstyle\pm 0.047$	$0.571\scriptstyle\pm 0.033$	$0.419\scriptstyle\pm 0.042$	$0.365\scriptstyle\pm 0.022$	$0.759\scriptstyle\pm 0.015$	$0.630\scriptstyle\pm 0.029$	${\color[rgb]{0,0.4297,0.7344}\underline{0.330\scriptstyle\pm 0.014}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.327\scriptstyle\pm 0.010}$
	720	$0.377\scriptstyle\pm 0.037$	$0.490\scriptstyle\pm 0.038$	$0.406\scriptstyle\pm 0.072$	$0.382\scriptstyle\pm 0.066$	$0.711\scriptstyle\pm 0.081$	$0.822\scriptstyle\pm 0.034$	$0.622\scriptstyle\pm 0.045$	$0.578\scriptstyle\pm 0.051$	$0.410\scriptstyle\pm 0.036$	$0.793\scriptstyle\pm 0.034$	$0.707\scriptstyle\pm 0.050$	${\color[rgb]{0,0.4297,0.7344}\underline{0.373\scriptstyle\pm 0.032}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.367\scriptstyle\pm 0.023}$
ETTm2	96	$0.225\scriptstyle\pm 0.039$	$0.221\scriptstyle\pm 0.039$	$0.210\scriptstyle\pm 0.040$	$0.216\scriptstyle\pm 0.035$	$0.212\scriptstyle\pm 0.082$	$0.279\scriptstyle\pm 0.031$	$0.224\scriptstyle\pm 0.033$	$0.146\scriptstyle\pm 0.012$	$0.158\scriptstyle\pm 0.012$	$0.525\scriptstyle\pm 0.047$	$0.413\scriptstyle\pm 0.059$	${\color[rgb]{0,0.4297,0.7344}\underline{0.144\scriptstyle\pm 0.011}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.134\scriptstyle\pm 0.010}$
	192	$0.233\scriptstyle\pm 0.026$	$0.229\scriptstyle\pm 0.031$	$0.234\scriptstyle\pm 0.038$	$0.215\scriptstyle\pm 0.022$	$0.535\scriptstyle\pm 0.029$	$0.292\scriptstyle\pm 0.041$	$0.316\scriptstyle\pm 0.040$	$0.189\scriptstyle\pm 0.012$	$0.184\scriptstyle\pm 0.012$	$0.530\scriptstyle\pm 0.060$	$0.427\scriptstyle\pm 0.033$	${\color[rgb]{0,0.4297,0.7344}\underline{0.170\scriptstyle\pm 0.009}}$	${\color[rgb]{0.7578,0.1523,0.1758}0.160\scriptstyle\pm 0.010}$

Appendix

TABLE XIV: Results of NMAE (

\textrm{mean}_{\textrm{std}}

) on long-term forecasting scenarios, each containing six independent runs with different seeds. The context length is set to 36 for the ILI dataset and 96 for the others. Lower NMAE values indicate better predictions. The reported values are means with standard deviations over 6 independent retraining-and-evaluation runs. RED : THE BEST, BLUE : THE 2ND BEST.

Dataset	$H$	Head	Cov.95 $\uparrow$	Sharp. $\downarrow$	QICE $\downarrow$	$\rho_{\sigma}\uparrow$
ETTh1	96	MLP	91.2 $\pm$ 0.8	1.84 $\pm$ 0.05	2.91 $\pm$ 0.18	0.61 $\pm$ 0.03
		LSTM	93.6 $\pm$ 0.6	1.62 $\pm$ 0.04	1.94 $\pm$ 0.14	0.78 $\pm$ 0.02
		GRU	94.4 $\pm$ 0.5	1.53 $\pm$ 0.03	1.58 $\pm$ 0.11	0.84 $\pm$ 0.02
	192	MLP	90.5 $\pm$ 0.9	1.91 $\pm$ 0.06	3.24 $\pm$ 0.22	0.58 $\pm$ 0.04
		LSTM	93.1 $\pm$ 0.7	1.70 $\pm$ 0.05	2.18 $\pm$ 0.16	0.76 $\pm$ 0.03
		GRU	94.1 $\pm$ 0.6	1.60 $\pm$ 0.04	1.71 $\pm$ 0.12	0.83 $\pm$ 0.02

Appendix

TABLE XV: Uncertainty-dynamics diagnostics for scale head architectures. RED : THE BEST OR CLOSEST TO 95.

Parameter	Patch Length ( $P$ )		Hidden Dim ( $D$ )		MLP Layers ( $N$ )
Dataset	Value	CRPS	Value	CRPS	Value	CRPS
ETTh1	16	0.278	64	0.280	1	0.279
	24	0.277	128	0.279	2	0.278
	32	0.278	256	0.277	3	0.277
	48	0.294	512	0.278	4	0.274
Weather	16	0.069	64	0.084	1	0.074

Appendix

TABLE XVI: Hyperparameter Sensitivity (

L=96\rightarrow H=192

). We report CRPS on ETTh1 and Weather . The default setting is selected as a cross-dataset performance–efficiency trade-off:

P=24,D=256,N=3

, where

N

is the number of MLP encoder layers.

Fig. 6: Visualization on Volatile Regimes ( ETTh2 , ETTm1 ). All subfigures share a unified Y-axis scale. This figure highlights performance under high volatility and structural breaks. Observation: In the top row of ETTh2 , the data exhibits sudden jumps. VolDy-VAE produces uncertainty bands that expand near regime shifts and contract otherwise. In contrast, baselines produce more static bands that capture less of the dynamic risk pattern.

Fig. 7: Visualization on Stable Regimes ( ETTm2 , Weather ). All subfigures share a unified Y-axis scale. We compare VolDy-VAE against baselines on time series with smooth trends. Observation: Baselines ( $K^{2}$ VAE, Koopa, PatchTST) show uncertainty over-estimation, generating wide confidence intervals even for flat or smooth curves (e.g., Weather ). VolDy-VAE (Blue) estimates lower aleatoric uncertainty in these stable windows, producing sharper intervals that better reflect the visual smoothness of the prediction target.

Probabilistic Time Series Forecasting

Probabilistic time series forecasting estimates distributions of future values or trajectories to quantify uncertainty and support risk-aware decisions.

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Jun 1, 2026

ProbRes: Volatility Learning for Probabilistic Time-Series Forecasting

Probabilistic time series forecasting has attracted increasing attention in financial applications due to the need to quantify risk and uncertainty in future observations. We propose ProbRes, a post-hoc probabilistic calibration method that explicitly learns and incorporates volatility dynamics into probabilistic forecasting, enabling effective handling of heteroskedastic data. During training, ProbRes employs two architecture-agnostic modules to separately model the conditional mean and conditional volatility. At the inference stage, it generates predictive distributions by resampling normalized residuals. ProbRes is applicable to both univariate and multivariate time series and remains robust under a wide range of error distributions, including non-Gaussian innovations with conditional heteroskedasticity. Theoretical results demonstrate ProbRes's validity and experiments on both synthetic and real-world datasets show that ProbRes accurately captures predictive distributions and produces well-calibrated prediction intervals.

Tingting Wang, Yunyi Zhang, Benyou Wang

School of Data Science, The Chinese University of Hong Kong, Shenzhen

Financial Time Series Uncertainty Calibration Probabilistic Time Series Forecasting

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Sep 29, 2026

Variational Augmented Invertible Koopman Autoencoder for probabilistic time series forecasting

Neural Koopman autoencoder models have been shown to successfully build a latent embedding with linear dynamics for arbitrary dynamical systems, enabling strong performance in long-term time series forecasting. However, these models usually work in a deterministic setting, which does not allow the quantification of the uncertainty of their predictions. Thus, we propose the new Variational Augmented Invertible Koopman AutoEncoder (VAIKAE), in which the latent embedding follows a Gaussian distribution instead of being deterministic. A key property of the VAIKAE architecture is that it leverages normalizing flow models, enabling the use of likelihood computations in the state space of dynamical systems for training a model. We further propose new strategies for uncertainty-aware latent data assimilation with a trained VAIKAE model. The effectiveness of our methods is demonstrated in a series of experiments on long-term time series forecasting benchmarks.

Anthony Frion, Lucas Drumetz, Guillaume Tochon +3

Institute of coastal modelling Helmholtz-Zentrum Hereon · IMT Atlantique Lab-STICC, UMR CNRS 6285, Brest, France · LRE EPITA, Le Kremlin-Bicêtre, France +1

Probabilistic Time Series Forecasting Long-Term Time Series Forecasting Koopman Operator Learning Variational Autoencoders

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Oct 13, 2026

Deterministic Prediction as Global Velocity: Flow Matching for Probabilistic Time-Series Forecast

Accurate uncertainty quantification is critical for many time series forecasting applications, and recent studies have increasingly adopted generative models to capture predictive uncertainty. Existing approaches often combine deterministic forecasting with generative modeling to improve predictive performance. However, their joint training is typically unstable because the deterministic model is embedded within the generative process, resulting in deep computational graphs and changing the probability transport path. In this work, we propose DV-Flow, a unified flow-matching formulation that incorporates the deterministic predictor as a parallel velocity component rather than as the endpoint of the transport path. This design enables stable end-to-end optimization while subsuming several existing formulations as special cases, allowing direct comparisons among them. Extensive experiments on six real-world datasets demonstrate that DV-Flow consistently improves probabilistic forecasting performance over strong baselines. Our code is available at https://github.com/tufts-ml/dv-flow-time-series.

Jiajing Hu, Xu Han, Li-Ping Liu

Department of Computer Science, Tufts University

Time Series Forecasting Probabilistic Time Series Forecasting Flow Matching Flow-Based Generative Modeling

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Abstract

Probabilistic Time Series Forecasting

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