Paloma: Phase-Conditioned Residual Modulation for Time Series Forecasting

Jingru Fei; Wei Fan; Kun Yi; Qi Zhang; Zhendong Niu

cs.LGOct 18, 2026

Paloma: Phase-Conditioned Residual Modulation for Time Series Forecasting

Authors: Jingru Fei, Wei Fan, Kun Yi, Qi Zhang, Zhendong Niu

Organizations: Beijing Institute of Technology · University of Auckland · State Information Center · Tongji University

Abstract

Explicitly modeling periodicity has proven to be an efficient approach for time series forecasting, where the residual, obtained after removing the periodic component from the time series, is typically treated as unstructured noise. However, in this work, we show that such residuals still exhibit structured, phase-dependent deviations, as supported by both empirical observations and theoretical analysis. This suggests that residuals should not be treated merely as unstructured noise, but instead as phase-conditioned signals whose variations arise in both direction and magnitude. Motivated by this perspective and inspired by phase and amplitude modulation in classical signal processing, we propose Paloma, a simple yet effective framework for modeling residual deviations. Specifically, Paloma comprises two complementary modules: a Phase Rotation module, which performs phase-conditioned rotations to capture directional changes in the feature space, and an Amplitude Scaling module, which applies phase-conditioned affine transformations to model magnitude variations. We further abstract the two modules into a general-purpose plugin, termed the Paloma technique, which can be readily integrated into existing forecasting architectures to modulate input sequences in a phase-aware manner, enabling backbone models to capture phase-dependent variations in both direction and magnitude. Extensive experiments on twelve datasets demonstrate that Paloma achieves state-of-the-art and robust forecasting performance. Additional results show that the Paloma technique consistently improves a wide range of baseline models, validating its effectiveness as a general-purpose component. Our Code is available at https://github.com/TSTS13/Paloma.

Figures & tables

Figure 1: Visualization of structured residual deviations. (a) and (b) present two representative channels from the Electricity dataset. From top to bottom, the rows show sampled sequences, learned cycles from CycleNet, and the residual mean and standard deviation visualized as one-dimensional heatmaps under a cycle length of 168. (c) is constructed by combining the learned cycle from (b) with Gaussian noise, with the row-wise definitions consistent with those in (a) and (b). Additional experiments for (c) and visualizations on the Weather dataset are provided in Appendix A .

Figure 2: Overall architecture of Paloma. Paloma is a lightweight framework for modeling residual deviations, centered on (i) the Phase Rotation module, which captures phase-conditioned directional variations via rotation in the feature space, where G-S ortho. denotes Gram–Schmidt orthogonalization Golub and Van Loan (2013) ; Greub (2012) , and (ii) the Amplitude Scaling module, which models phase-dependent magnitude variations through affine modulation.

Models	Paloma		DistDF		GTR		RAFT		FreDF		TimeXer		CycleNet		FilterNet		iTransformer		PatchTST		DLinear
	[ ours ]		[ICLR’26]		[ICLR’26]		[ICML’25]		[ICLR’25]		[NeurIPS’24]		[NeurIPS’24]		[NeurIPS’24]		[ICLR’24]		[ICLR’23]		[AAAI’23]
Metrics	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	0.363	0.372	0.378	0.394	0.367	0.389	0.381	0.400	0.392	0.399	0.382	0.397	0.379	0.396	0.392	0.401	0.407	0.410	0.387	0.400	0.403	0.407
ETTm2	0.267	0.308	0.277	0.321	0.268	0.315	0.281	0.330	0.278	0.319	0.274	0.322	0.266	0.314	0.285	0.328	0.288	0.332	0.281	0.326	0.350	0.401
ETTh1	0.421	0.416	0.430	0.429	0.439	0.434	0.428	0.433	0.437	0.435	0.437	0.437	0.457	0.441	0.441	0.439	0.454	0.448	0.469	0.455	0.456	0.452
ETTh2	0.361	0.386	0.367	0.393	0.372	0.400	0.382	0.410	0.371	0.396	0.368	0.396	0.388	0.409	0.383	0.407	0.383	0.407	0.387	0.407	0.559	0.515

Table 1: Performance comparison with a fixed lookback window

T=96

. The prediction lengths are set as

\tau\in\{12,24,48,96\}

for PEMS datasets for short-term forecasting, and

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

for the others for long-term forecasting. Results are averaged over the four prediction lengths. The best and second-best results are highlighted in red and blue , respectively. Full results are provided in Table 6 , and additional baseline results are reported in Table 7 .

Design	ETTm1		ETTh1		Electricity		Weather		Traffic		Solar-Energy
Design	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Paloma	0.363	0.372	0.421	0.416	0.166	0.251	0.231	0.262	0.461	0.260	0.197	0.234
w/o PR	0.368	0.374	0.427	0.420	0.170	0.258	0.238	0.270	0.471	0.267	0.206	0.247
w/o AS	0.369	0.374	0.430	0.424	0.170	0.259	0.238	0.270	0.468	0.269	0.201	0.243
w/o PR,AS	0.376	0.380	0.428	0.425	0.174	0.262	0.243	0.280	0.474	0.283	0.209	0.248

Table 2: Ablation study of the Phase Rotation (PR) and Amplitude Scaling (AS) modules on six datasets. The reported results are averaged over all prediction horizons. Detailed results are provided in Table 9 .

LookBack		iTransformer						PatchTST						DLinear						FilterNet
Setup		Original		+ Paloma Tech.		Imporve		Original		+ Paloma Tech.		Imporve		Original		+ Paloma Tech.		Imporve		Original		+ Paloma Tech.		Imporve
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Electricity	96	0.150	0.241	0.137	0.233	8.67%	3.32%	0.164	0.254	0.137	0.234	16.46%	7.87%	0.195	0.277	0.139	0.239	28.72%	13.72%	0.147	0.245	0.137	0.239	6.80%	2.45%
	192	0.164	0.255	0.157	0.250	4.27%	1.96%	0.174	0.265	0.155	0.250	10.92%	5.66%	0.194	0.279	0.156	0.256	19.59%	8.24%	0.160	0.250	0.151	0.245	5.63%	2.00%
	336	0.178	0.272	0.172	0.267	3.37%	1.84%	0.193	0.285	0.176	0.272	8.81%	4.56%	0.207	0.295	0.172	0.275	16.91%	6.78%	0.173	0.267	0.164	0.261	5.20%	2.25%
	720	0.210	0.300	0.192	0.290	8.57%	3.33%	0.232	0.320	0.220	0.309	5.17%	3.44%	0.244	0.330	0.207	0.309	15.16%	6.36%	0.210	0.309	0.201	0.302	4.29%	2.27%

Table 3: Generalization performance of Paloma Tech. on different models. The lookback window is fixed at 96 for all the experiments.

Figure 3: t-SNE visualizations of $\mathbf{Z}_{\mathrm{phase}}$ and $\mathbf{Z}_{\mathrm{amp}}$ on Electricity and Solar-Energy datasets under the setting of lookback window $T=96$ and prediction length $\tau=96$ . The inherent periodicities of the Electricity and Solar-Energy datasets are 168 and 144, respectively, and the numbers in parentheses denote the corresponding ranges of phase indices.

Model	Parameters	MACs	Training Time	Memory Usage	MSE	MAE
GTR	150.71K	3.21M	9.4s	1754MB	0.341	0.342
CycleNet	108.32K	2.22M	8.2s	1414MB	0.344	0.344
DLinear	139.68K	2.94M	7.4s	1420MB	0.345	0.381
FilterNet	369.40K	7.19M	12.1s	1434MB	0.342	0.340
iTransformer	1.29M	27.11M	16.6s	1858MB	0.358	0.349
PatchTST	2.30M	326.33M	21.2s	2130MB	0.354	0.340

Table 4: Efficiency comparison between Paloma and other models on the Weather dataset with lookback window T = 96 and forecast horizon

\tau

= 720. Training Time denotes the average time required per epoch for the model.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Comparison between real-world residuals and synthetic noise-based counterparts under different noise types. Subfigure (a) shows the real-world case from the Electricity dataset (318th channel). Subfigures (b)–(f) present synthetic sequences constructed by combining the learned cycle in (a) with different noise types, including Gaussian, Gaussian mixture, Laplace, Poisson, and pink noise. The meaning of each row is consistent with that in Figure 1 . Unlike real-world data, where residual statistics exhibit clear phase-dependent patterns, all noise-based counterparts fail to produce consistent phase-aligned structure, regardless of distributional complexity or variability characteristics.

Figure 5: Visualization of structured residual deviations on the Weather dataset. The row definitions are consistent with those in Figure 1 .

Dataset	Dim	Prediction Length	Dataset Size	Frequency	Cycle Length	Domain
ETTh1, ETTh2	7	{96, 192, 336, 720}	(8545, 2881, 2881)	Hourly	24	Electricity
ETTm1, ETTm2	7	{96, 192, 336, 720}	(34465, 11521, 11521)	15min	96	Electricity
Weather	21	{96, 192, 336, 720}	(36792, 5271, 10540)	10min	144	Weather
Electricity	321	{96, 192, 336, 720}	(18317, 2633, 5261)	Hourly	168	Electricity
Traffic	862	{96, 192, 336, 720}	(12185, 1757, 3509)	Hourly	168	Transportation
Solar-Energy	137	{96, 192, 336, 720}	(36601, 5161, 10417)	10min	144	Energy

Appendix

Table 5: Detailed descriptions of the datasets. Dim indicates the number of variates in each dataset. Prediction Length refers to the number of future time steps to be predicted. Dataset Size represents the total number of time points in the (Train, Validation, Test) splits, respectively. Frequency indicates the sampling interval of the time series. Cycle Length denotes the length of the global cycle.

Models		Paloma		DistDF		GTR		RAFT		FreDF		TimeXer		CycleNet		FilterNet		iTransformer		PatchTST		DLinear
		[ ours ]		[ICLR’26]		[ICLR’26]		[ICML’25]		[ICLR’25]		[NeurIPS’24]		[NeurIPS’24]		[NeurIPS’24]		[ICLR’24]		[ICLR’23]		[AAAI’23]
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.295	0.332	0.316	0.357	0.305	0.349	0.325	0.369	0.324	0.362	0.318	0.356	0.319	0.360	0.321	0.361	0.334	0.368	0.329	0.367	0.345	0.372
	192	0.343	0.358	0.358	0.380	0.349	0.375	0.364	0.388	0.373	0.385	0.362	0.383	0.360	0.381	0.367	0.387	0.377	0.391	0.367	0.385	0.380	0.389
	336	0.376	0.381	0.392	0.404	0.380	0.398	0.391	0.407	0.402	0.404	0.395	0.407	0.389	0.403	0.401	0.409	0.426	0.420	0.399	0.410	0.413	0.413
	720	0.439	0.417	0.448	0.437	0.435	0.436	0.444	0.438	0.469	0.444	0.452	0.441	0.447	0.441	0.477	0.448	0.491	0.459	0.454	0.439	0.474	0.453

Appendix

Table 6: Performance comparison with prediction lengths

\tau\in\{12,24,48,96\}

for PEMS and

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

for others and fixed lookback window

T=96

. Results are averaged over four prediction lengths. Highlight the best results in red and the second-best results in blue .

Models		Paloma		OLinear		TimeMixer		FITS		Fredformer		FreTS		TimesNet
		[ ours ]		[NeurIPS’25]		[ICLR’24]		[ICLR’24]		[KDD’24]		[NeurIPS’23]		[NeurIPS’23]
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.295 $\pm$ 0.001	0.332 $\pm$ 0.001	0.302	0.334	0.320	0.357	0.353	0.375	0.326	0.361	0.339	0.374	0.338	0.379
	192	0.343 $\pm$ 0.001	0.358 $\pm$ 0.001	0.357	0.363	0.361	0.381	0.486	0.445	0.363	0.380	0.382	0.397	0.389	0.400
	336	0.376 $\pm$ 0.001	0.381 $\pm$ 0.001	0.387	0.385	0.390	0.404	0.531	0.475	0.395	0.403	0.421	0.426	0.429	0.428
	720	0.439 $\pm$ 0.001	0.417 $\pm$ 0.002	0.452	0.426	0.454	0.441	0.600	0.513	0.453	0.438	0.485	0.462	0.495	0.464

Appendix

Table 7: Additional performance comparison with a fixed lookback window

T=96

. The prediction lengths are set as

\tau\in\{12,24,48,96\}

for PEMS datasets for short-term forecasting, and

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

for the others for long-term forecasting. The best and second-best results are highlighted in red and blue , respectively. The reported results with standard deviation of Paloma are averaged from 5 runs (with different random seeds of 2024, 2025, 2026, 2027, 2028).

Figure 6: Forecasting performance with the lookback window

T\in\{48,96,192,336,720\}

and fixed prediction length

\tau=96

LookBack		$T=336$								$T=720$
Models		PatchTST		CycleNet		GTR		Paloma		SparseTSF		CycleNet		GTR		Paloma
		[ICLR’23]		[NeurIPS’24]		[ICLR’26]		[ ours ]		[ICML’25]		[NeurIPS’24]		[ICLR’26]		[ ours ]
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.385	0.405	0.374	0.396	0.369	0.395	0.355	0.380	0.362	0.388	0.379	0.403	0.368	0.400	0.359	0.389
	192	0.414	0.421	0.406	0.415	0.409	0.420	0.399	0.408	0.403	0.411	0.416	0.425	0.409	0.424	0.406	0.418
	336	0.440	0.440	0.431	0.430	0.432	0.434	0.428	0.425	0.434	0.428	0.447	0.445	0.443	0.449	0.439	0.438

Appendix

Table 8: Performance comparison with longer lookback windows

T\in\{336,720\}

. Highlight the best results in red and the second-best results in blue . The baseline results follow Table 11 in GTR [ 4 ] .

Design		ETTm1		ETTh1		Electricity		Weather		Traffic		Solar-Energy
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Paloma	96	0.295	0.332	0.360	0.378	0.134	0.222	0.154	0.190	0.425	0.243	0.175	0.224
	192	0.343	0.358	0.414	0.409	0.151	0.238	0.195	0.238	0.442	0.254	0.192	0.240
	336	0.376	0.381	0.452	0.428	0.170	0.256	0.249	0.284	0.462	0.262	0.204	0.235
	720	0.439	0.417	0.458	0.449	0.208	0.289	0.324	0.336	0.514	0.279	0.215	0.238
	Avg	0.363	0.372	0.421	0.416	0.166	0.251	0.231	0.262	0.461	0.260	0.197	0.234

Appendix

Table 9: Full ablation study of the Phase Rotation (PR) and Amplitude Scaling (AS) modules on six datasets. The lookback window is fixed at 96, and results are averaged over four prediction lengths.

Model	Parameters	MACs	Training Time	Memory Usage	MSE	MAE
GTR	350.99K	2.5M	2.4s	1738MB	0.493	0.476
CycleNet	210.04K	1.47M	2.0s	1400MB	0.520	0.484
DLinear	139.68K	979.44K	1.8s	1398MB	0.519	0.516
FilterNet	369.37K	2.4M	2.6s	1412MB	0.481	0.473
iTransformer	737.49K	5.16M	2.7s	2016MB	0.503	0.491
PatchTST	2.75M	60.74M	2.9s	1600MB	0.500	0.488

Appendix

Table 10: Efficiency comparison between Paloma and other models on the ETTh1 dataset with lookback window T = 96 and forecast horizon

\tau

= 720. Training Time denotes the average time required per epoch for the model.

Model	Parameters	MACs	Training Time	Memory Usage	MSE	MAE
GTR	1.1M	827.95M	30.3s	2336MB	0.514	0.301
CycleNet	563.84K	362.08M	25.5s	1764MB	0.502	0.314
DLinear	139.7K	120.61M	22.6s	1696MB	0.645	0.394
FilterNet	502.41K	408.64M	28.5s	2078MB	0.498	0.323
iTransformer	6.72M	5.79G	42.3s	5860MB	0.467	0.302
PatchTST	10.74M	69.46G	205.8s	16614MB	0.514	0.320

Appendix

Table 11: Efficiency comparison between Paloma and other models on the Traffic dataset with lookback window T = 96 and forecast horizon

\tau

= 720. Training Time denotes the average time required per epoch for the model.

Panel of Figure 7 — Figure 7: Sensitivity analysis of the hidden dimension $d_{\mathrm{model}}$ under the setting of lookback window $T=96$ and prediction length $\tau=96$ .

Panel of Figure 8 — Figure 8: Sensitivity analysis of the cycle length $W$ under the setting of lookback window $T=96$ and prediction length $\tau=96$ . The values in parentheses indicate the inherent periodicity of each dataset.

Panel of Figure 9 — Figure 9: Visualization of forecasting results on the Solar-Energy dataset with lookback window $T=96$ and prediction length $\tau=96$ .

Panel of Figure 10 — Figure 10: Visualization of forecasting results on the ETTm1 dataset with lookback window $T=96$ and prediction length $\tau=96$ .

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Reliable periodic patterns serve as a fundamental basis for accurate multivariate time series forecasting. However, existing methods either implicitly extract periodicity through complex model architectures (e.g., Transformers) with high computational overhead or overlook the intrinsic phase-amplitude coupling when modeling periodic components explicitly. To address these issues, we propose a novel Cycle-aware Phase-Amplitude Modulation Network (PAMNet) that explicitly decomposes periodic patterns into complementary phase and amplitude components. The core innovation lies in its dual-branch modulator, featuring dedicated learnable embeddings for phase positioning and amplitude modulation. The phase branch employs cyclical embeddings to capture phase-dependent mean shifts, while the amplitude branch models intensity variations to adapt to changes in variance. A lightweight modulator with element-wise fusion efficiently combines these components, enabling explicit modeling of their interactions without complex attention mechanisms. Extensive experiments on twelve real-world datasets demonstrate that our method achieves state-of-the-art performance through its novel phase-amplitude decoupling mechanism, offering a new perspective for cyclical modeling in time series forecasting.

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