TimeES: Probabilistic and Deterministic Time Series Forecasting via Evolutionary Spectra

Weiwei Ye; Renhe Jiang; Hangchen Liu; Dongyuan Li; Yoshihide Sekimoto

cs.LGSep 26, 2026

TimeES: Probabilistic and Deterministic Time Series Forecasting via Evolutionary Spectra

Authors: Weiwei Ye, Renhe Jiang, Hangchen Liu, Dongyuan Li, Yoshihide Sekimoto

Organizations: The University of Tokyo

Abstract

Real-world time series are inherently non-stationary, with trends, periodic patterns, and uncertainty evolving over time. While the Fourier domain offers a natural lens to model time series, current deep learning approaches do not explicitly model evolution and randomness in the Fourier spectra, which limits their ability to accurately predict both the expected trajectory and its uncertainty in non-stationary time series. Motivated by Evolutionary Spectra (ES) theory, we propose TimeES, a general framework that enables probabilistic and deterministic forecasting via the evolutionary spectra theory. Specifically, we derive a parameterizable evolutionary spectra formulation, recasting non-stationary random process modeling as learning an evolving representation modulated by random variables. Furthermore, we reduce the complexity of the estimated spectra from O(NM) to O(NK), where K << M/2, by exploiting Hermitian symmetry and spectral energy sparsity for frequency selection. Based on a simple linear backbone, our proposed TimeES achieves consistent state-of-the-art performance across both deterministic and probabilistic forecasting tasks, with high efficiency and interpretability. Code is available at: https://github.com/wwy155/TimeES.

Figures & tables

Figure 1: A case study on the ETTm1 using 7 days samples with daily and trend spectrum (0 and 1 frequency). (a-c): spectra obtained by different methods; (d-f): probabilistic forecasting results of different methods. TimeES produces point-wise spectra via neural network and quantify period-aware uncertainty interval.

Figure 2: The overview of TimeES. TimeES estimates the interpretable amplitude matrix

\mathbf{A}

by a neural network; then the DES is used to synthesize time series with a physical phase matrix

\mathbf{F}

and a probabilistic realization vector

\mathbf{w}

, enabling general time series forecasting.

Interpretation	$\mathbf{y}$	$\mathbf{w}$
Deterministic	$\displaystyle\frac{1}{\sqrt{M}}(\hat{\mathbf{A}}_{\mathbf{y}}\odot\mathbf{F})\mathbf{1}$	$W_{k}=1,\ \forall k$
Probabilistic	$\displaystyle\frac{1}{\sqrt{M}}(\hat{\mathbf{A}}_{\mathbf{y}}\odot\mathbf{F})\mathbf{w}$	$W_{k}\sim\mathcal{CN}(\mu_{W_{k}},\sigma_{W_{k}})$

Table 1: Deterministic vs. Probabilistic Interpretation.

Method	$A(n,\omega_{k})$	$W_{k}$
iDFT	$(\mathbf{F}\mathbf{x})_{k}$	$1$
iSTFT	$(\mathbf{F}\mathbf{x}^{(m)})_{k},\;m=\lfloor n/L\rfloor$	$1$
DES (ours)	$\text{MLP}_{n,k}(\mathbf{x})$	$\mathcal{CN}(\mu_{W_{k}},\sigma_{W_{k}})$

Table 2: Special cases of the

A(n,\omega_{k})

and

W_{k}

Models	Datasets	ETTh1	ETTh2	ETTm1	ETTm2	ECL	EXG	ILI	Solar	Traffic
TimeGrad	CRPS	0.606	1.212	0.647	0.775	0.397	0.826	1.140	0.293	0.407
(2021)	MSE	1.062	3.462	1.218	1.690	0.505	1.567	4.197	0.475	0.983
CSDI	CRPS	0.492	0.647	0.524	0.817	0.577	0.855	1.244	0.432	1.418
(2022)	MSE	0.949	1.226	1.002	1.723	1.007	1.701	4.515	0.763	1.731
TimeDiff	CRPS	0.465	0.471	0.464	0.316	0.750	0.433	1.153	0.700	0.771
(2023)	MSE	0.517	0.456	0.537	0.268	0.879	0.402	3.958	0.821	1.350

Table 3: Experimental results on nine real-world datasets. Bold face indicates the best result, and underline indicates the second-best result.

Figure 3: A probabilistic forecasting case of ETTh1 testing sample. We present 96% prediction intervals as the purple areas.

Models Autoformer			FEDformer		TimesNet		FITS		FreTS		PatchTST		iTransformer		TimeES (Ours)
Datasets	S.V.	M.V.	S.V.	M.V.	S.V.	M.V.	S.V.	M.V.	S.V.	M.V.	S.V.	M.V.	S.V.	M.V.	S.V.	M.V.
ETTm1	1.176	0.580	0.840	0.448	0.811	0.400	0.793	0.418	0.755	0.414	0.763	0.386	0.778	0.407	0.740	0.390
ETTm2	0.498	0.327	0.495	0.305	0.480	0.291	0.478	0.285	0.452	0.323	0.462	0.281	0.487	0.288	0.434	0.278
ETTh1	0.925	0.496	0.882	0.440	0.929	0.458	1.155	0.534	1.010	0.504	0.970	0.469	0.968	0.454	0.878	0.440
ETTh2	0.700	0.450	0.713	0.437	0.659	0.414	0.656	0.396	0.596	0.500	0.633	0.387	0.682	0.383	0.583	0.379
ECL	0.679	0.227	0.610	0.214	0.591	0.193	0.587	0.328	0.530	0.205	0.543	0.216	0.601	0.178	0.516	0.197

Table 4: Experimental results on forecasting, we report MSE. Bold face and underline indicate the best and second-best results. S.V denotes average MSE results on single forecasting and M.V. denotes average MSE results on multivariate forecasting.

Variants	S.V. (MSE)	M.V. (MSE)	P.F. (CRPS)
w/o DES	0.974±0.014	0.554±0.006	$\times$
w/o E	1.282±0.012	0.604±0.010	0.466±0.009
TimeES	0.929±0.009	0.485±0.003	0.349±0.007

Table 5: Experimental results on different variants of TimeES on ETTh1 dataset.

\times

indicates that the variant lacks the DES and is therefore not generative. P.F. denotes probabilistic forecasting.

Model	PatchTST		+TimeES		Improvement		iTransformer		+TimeES		Improvement
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
96	0.414	0.419	0.385	0.402	6.89%	4.15%	0.386	0.405	0.384	0.402	0.56%	0.78%
192	0.460	0.445	0.434	0.437	5.65%	1.86%	0.441	0.436	0.433	0.428	1.88%	1.77%
336	0.501	0.466	0.451	0.457	9.99%	2.02%	0.487	0.458	0.475	0.451	2.44%	1.51%
720	0.500	0.488	0.482	0.479	3.68%	1.82%	0.503	0.491	0.474	0.480	5.83%	2.18%
Avg.	0.469	0.455	0.438	0.444	6.56%	2.42%	0.454	0.448	0.441	0.440	2.84%	1.59%

Table 6: Backbone enhancement on the ETTh1 dataset, TimeES is integrated as a plug-in module.

Figure 4: Low- and high-frequency components analysis on ETTh1 dataset.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Tasks	Benchmarks	Metrics	Input	Output
Long-term Forecasting (Single variate and Multivariate)	ETT{m1, m2, h1, h2}, ECL, EXG, Traffic, Weather, Solar	MSE, MAE	96	{96, 192, 336, 720}
Probabilistic Forecasting	ETT{m1, m2, h1, h2}, ECL, EXG, Traffic, ILI, Solar	CRPS, QICE, MSE, MAE	168	36 (ILI), 192 (others)
Classification	UEA (5 subsets)	Accuracy	315-2500	-

Appendix

Table 7: Summary of experiment benchmarks.

Figure 5: Energy distribution of different datasets. We use pink to highlight frequencies of top 90% energy.

Single-Variate Long-term Forecasting
Datasets	ETTm1		ETTm2		ETTh1		ETTh2
Variants	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
TimeES	0.893	0.619	0.590	0.553	0.929	0.670	0.618	0.585
w/o E	1.306	0.792	0.673	0.587	1.282	0.840	0.785	0.676
w/o DES	0.966	0.662	0.639	0.569	0.974	0.699	0.744	0.646
Multi-Variate Long-term Forecasting

Appendix

Table 8: Ablation results on ETTs datasets. We report MSE, MAE for 720-steps single- and multi-variate long-term forecasting and report CRPS, MSE for 192 steps multivariate probabilistic forecasting. The best result is denoted in bold characters.

Dataset	Horizon	KNF	Koopa	SKOLR	TimeBridge	TimeMixer	TimeXer	TimeES
ETTm1	96	0.691	0.331	0.342	0.324	0.344	0.387	0.328
	192	0.707	0.377	0.397	0.365	0.368	0.441	0.364
	336	0.718	0.402	0.445	0.395	0.394	0.514	0.402
	720	0.742	0.466	0.536	0.457	0.455	0.592	0.465
ETTm2	96	0.230	0.182	0.194	0.178	0.178	0.364	0.176
	192	0.274	0.245	0.256	0.245	0.246	0.526	0.240

Appendix

Table 9: Mean Squared Error (MSE) results on deterministic multivariate forecasting tasks across different datasets and prediction horizons. Bold values indicate the best performance for each setting. The last row shows the number of times each model achieves the best performance.

Figure 6: (a-f) Predictive uncertainty quantification by TimeES across different datasets. We plot the last 500 predicted time steps of the first four variables in each dataset. The orange line shows the ground truth, the blue line represents the predicted mean, and the gray shaded region indicates the 96% prediction interval.

Figure 7: Average accuracy across datasets.

	Autoformer	FEDformer	TimesNet	FITS	FreTS	PatchTST	iTransformer	TimeES
EthanolConcentration	31.60	31.20	35.70	25.10	24.79	36.42	31.60	36.81
MotorImagery	48.40	45.00	50.40	49.60	50.40	52.80	55.60	52.00
SelfRegulationSCP1	84.00	88.70	91.80	72.70	89.56	81.78	88.39	92.83
StandWalkJump	29.33	30.67	37.33	38.67	45.33	41.33	37.33	46.67
UWaveGestureLibrary	85.90	85.30	85.30	61.81	85.10	80.19	87.50	86.50
Avg.	55.85	56.17	60.11	49.58	59.04	58.50	60.08	62.96

Appendix

Table 10: Classification accuracy on five datasets. Bold and underline indicate best and second-best results respectively.

Figure 8: Efficiency analysis on ETTh1 dataset.

Figure 9: We conduct a sensitivity analysis of the hyper-parameter

r

in multivariate long-term forecasting and probabilistic forecasting tasks. Specifically, we set the prediction horizon to 720 time steps. We report the Mean Squared Error (MSE) for deterministic forecasting and the Continuous Ranked Probability Score (CRPS) for probabilistic forecasting, evaluated on the ETT (ETTm1, ETTm2, ETTh1, ETTh2), ECL, EXG, Traffic, Weather, and Solar datasets.

Models	Datasets	ETTh1	ETTh2	ETTm1	ETTm2	ECL	EXG	ILI	Solar	Traffic
TimeGrad	CRPS	0.606	1.212	0.647	0.775	0.397	0.826	1.140	0.293	0.407
(2021)	QICE	6.731	9.488	6.693	6.962	7.118	9.464	6.519	7.378	4.581
CSDI	CRPS	0.492	0.647	0.524	0.817	0.577	0.855	1.244	0.432	1.418
(2022)	QICE	3.107	5.331	2.828	8.106	7.506	7.864	7.693	9.957	13.613
TimeDiff	CRPS	0.465	0.471	0.464	0.316	0.750	0.433	1.153	0.700	0.771
(2023)	QICE	14.931	14.813	14.795	13.385	15.466	14.556	14.942	14.914	15.439

Appendix

Table 11: Probabilistic forecasting performance (CRPS, QICE) across nine datasets.

Models	Datasets	ETTh1	ETTh2	ETTm1	ETTm2	ECL	EXG	ILI	Solar	Traffic
TimeGrad	MAE	0.813	1.496	0.831	0.967	0.504	1.058	1.414	0.446	0.535
(2021)	MSE	1.062	3.462	1.218	1.690	0.505	1.567	4.197	0.475	0.983
CSDI	MAE	0.708	0.900	0.752	1.069	0.822	1.081	1.481	0.675	0.925
(2022)	MSE	0.949	1.226	1.002	1.723	1.007	1.701	4.515	0.763	1.731
TimeDiff	MAE	0.479	0.485	0.477	0.333	0.764	0.446	1.169	0.713	0.784
(2023)	MSE	0.517	0.456	0.537	0.268	0.879	0.402	3.958	0.821	1.350

Appendix

Table 12: Deterministic forecasting performance (MSE, MAE) across nine datasets.

Model		Autoformer		FEDformer		TimesNet		FITS		FreTS		PatchTST		iTransformer		TimeES
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.886	0.636	0.713	0.598	0.682	0.551	0.643	0.511	0.600	0.508	0.624	0.508	0.622	0.517	0.607	0.505
	192	1.252	0.798	0.770	0.614	0.748	0.568	0.745	0.545	0.710	0.553	0.710	0.545	0.730	0.554	0.690	0.534
	336	0.974	0.681	0.886	0.664	0.810	0.592	0.811	0.573	0.788	0.592	0.786	0.585	0.829	0.609	0.770	0.568
	720	1.594	0.884	0.992	0.705	1.006	0.689	0.974	0.633	0.921	0.649	0.931	0.646	0.931	0.642	0.893	0.619
	Avg	1.177	0.750	0.840	0.645	0.812	0.600	0.793	0.566	0.755	0.576	0.763	0.571	0.778	0.581	0.740	0.556

Appendix

Table 13: Full results of the single variate forecasting task. We compare extensive competitive models under different prediction lengths following the setting of iTransformer [ 39 ] . The input sequence length is set to 96 for all baselines. The statistical significance is within 1%.

Model		Autoformer		FEDformer		TimesNet		FITS		FreTS		PatchTST		iTransformer		TimeES
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.475	0.505	0.379	0.419	0.338	0.375	0.353	0.378	0.352	0.385	0.329	0.367	0.334	0.368	0.328	0.366
	192	0.553	0.496	0.426	0.441	0.374	0.387	0.389	0.400	0.390	0.407	0.367	0.385	0.377	0.391	0.364	0.384
	336	0.621	0.537	0.445	0.459	0.410	0.411	0.447	0.441	0.421	0.423	0.410	0.399	0.426	0.420	0.402	0.402
	720	0.671	0.561	0.543	0.490	0.478	0.450	0.482	0.445	0.495	0.475	0.439	0.454	0.491	0.459	0.465	0.445
	Avg	0.580	0.525	0.448	0.452	0.400	0.406	0.418	0.416	0.414	0.423	0.386	0.401	0.407	0.410	0.390	0.400

Appendix

Table 14: Full results of the multivariate forecasting task. We compare extensive competitive models under different prediction lengths following the setting of iTransformer [ 39 ] . The input sequence length is set to 96 for all baselines. The statistical significance is within 1%.

Single-Variates
Model		PatchTST		+TimeES		Improvements		iTransformer		+TimeES		Improvements
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.624	0.508	0.603	0.505	3.37%	0.63%	0.622	0.517	0.612	0.496	1.56%	4.02%
	192	0.710	0.545	0.704	0.533	0.88%	2.35%	0.730	0.554	0.721	0.544	1.35%	1.77%
	336	0.786	0.585	0.780	0.574	0.70%	1.80%	0.829	0.609	0.787	0.576	5.07%	5.39%
	720	0.931	0.646	0.921	0.635	1.15%	1.63%	0.931	0.642	0.923	0.634	0.86%	1.23%

Appendix

Table 15: We evaluate TimeES as a decomposition framework on long-term single variate and multivariate forecasting tasks in ETTs datasets, the best results are denote in bold characters.

Figure 10: (a-f) Predictive uncertainty quantification by TimeES across different datasets. We plot the last 500 predicted time steps of the first four variables in each dataset. The orange line shows the ground truth, the blue line represents the predicted mean, and the gray shaded region indicates the 96% prediction interval.

Figure 11: Spectral decomposition analysis on four dataset. We visualize the high and low frequencies.

Channel	Trend (DC)	Weekly (168h)	Daily (24h)	12h	8h
HUFL (high useful load)	1.95	0.46	1.60	0.80	0.95
HULL (high useless load)	2.55	0.83	1.34	0.71	0.78
MUFL (middle useful load)	2.02	0.44	1.57	0.79	0.94
MULL (middle useless load)	2.56	0.84	1.32	0.70	0.81
LUFL (low useful load)	1.63	0.84	1.81	0.94	0.96
LULL (low useless load)	2.07	1.05	1.34	0.83	1.10

Appendix

Table 16: Share (%) of the predictive variance contributed by selected frequencies on ETTh1, averaged over 672 test windows. A uniform share would be 1.18%. The largest value in each column is in bold .

Figure 12: Share of the predictive variance contributed by each frequency on ETTh1 for each channel ( $M=168$ , 85 frequencies). Red and blue cells indicate shares above and below the uniform share of 1.18%, respectively; $k=1$ corresponds to the weekly period.

Dataset	Fixed grid	Learnable
ETTh1	0.363 $\pm$ 0.006	0.372 $\pm$ 0.008
Off-grid synthetic	0.156 $\pm$ 0.018	0.157 $\pm$ 0.017

Appendix

Table 17: CRPS (mean

\pm

std over 3 seeds) of TimeES with frequencies fixed on the DFT grid or learned jointly with the amplitudes.

Setting	Model	ETTh1	ETTh2	ETTm1	ETTm2
$L=10$ , $H=24$	KooNPro	0.527	0.351	0.395	0.277
$L=10$ , $H=24$	TimeES	0.355	0.202	0.375	0.175
$L=168$ , $H=24$	KooNPro	0.398	0.380	0.352	0.290
$L=168$ , $H=24$	TimeES	0.292	0.206	0.175	0.132
$L=168$ , $H=192$	KooNPro	3.079	2.922	3.465	3.656
$L=168$ , $H=192$	TimeES	0.349	0.320	0.310	0.230

Appendix

Table 18: CRPS comparison with KooNPro on the ETT datasets (3 seeds per setting).

L

denotes the history length and

H

the forecast horizon; the original setting of KooNPro is

L=10

and

H=24

, and the TimeES results for

L=168

H=192

are from Table 3 . Bold indicates the better result.

Dataset	ETTh1	ETTh2	ETTm1	ETTm2	ECL	EXG	Traffic	Weather	Solar	ILI	Avg.
Coverage	0.885	0.862	0.914	0.900	0.915	0.992	0.904	0.965	0.928	0.963	0.923

Appendix

Table 19: Fraction of the spectral energy of the test split captured by the frequencies selected on the training split (

M=96

r=0.9

Input length	24	72	96	168 (default)	336
ETTh1	0.356	0.336	0.339	0.349	0.369
ETTm1	0.398	0.321	0.298	0.310	0.299

Appendix

Table 20: CRPS of TimeES with different input lengths (horizon 192, mean over 3 seeds). The default input length is 168. Bold indicates the best result.

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Time series forecasting under non-stationarity faces a fundamental tension between capturing stable representations and adapting to distribution shifts. Existing methods implicitly rely on static historical assumptions, leading to a critical failure mode we term Phase Amnesia, where models become blind to the evolving global context. To resolve this, we formalize non-stationary dynamics through three physical hypotheses: wold decomposition, dynamical phase evolution, and heteroscedastic manifold generation. These principles inspire PULSE, a physics-informed, plug-and-play framework adopting a Disentangle--Evolve--Simulate design philosophy. Specifically, PULSE utilizes phase-anchored disentanglement to resolve optimization interference caused by dominant trends, employs a Phase Router to actively generate future trajectories, and introduces Statistic-Aware Mixup (SAM) to ensure robustness against out-of-distribution volatility. Empirically, PULSE enables a simple MLP backbone to achieve state-of-the-art or highly competitive performance across 12 real-world benchmarks. This validates that a correct physics-informed inductive bias is far more critical than raw architectural complexity for non-stationary forecasting. The code is available at: https://github.com/Gemost/PULSE.

Abstract

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