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 τ∈{12,24,48,96} for PEMS datasets for short-term forecasting, and τ∈{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
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 Zphase and Zamp on Electricity and Solar-Energy datasets under the setting of lookback window T=96 and prediction length τ=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 τ = 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 τ∈{12,24,48,96} for PEMS and τ∈{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 ± 0.001
0.332 ± 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 ± 0.001
0.358 ± 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 ± 0.001
0.381 ± 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 ± 0.001
0.417 ± 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 τ∈{12,24,48,96} for PEMS datasets for short-term forecasting, and τ∈{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∈{48,96,192,336,720} and fixed prediction length τ=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∈{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 τ = 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 τ = 720. Training Time denotes the average time required per epoch for the model.
Figure 7: Sensitivity analysis of the hidden dimension dmodel under the setting of lookback window T=96 and prediction length τ=96 .
Figure 8: Sensitivity analysis of the cycle length W under the setting of lookback window T=96 and prediction length τ=96 . The values in parentheses indicate the inherent periodicity of each dataset.
Figure 9: Visualization of forecasting results on the Solar-Energy dataset with lookback window T=96 and prediction length τ=96 .
Figure 10: Visualization of forecasting results on the ETTm1 dataset with lookback window T=96 and prediction length τ=96 .
Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.
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.
Yingbo Zhou, Yutong Ye, Zhiwei Ling +5
Fudan University · Beihang University · Zhejiang University +1
Deep learning has advanced time series forecasting, but periodicity drift, in which cycle timing and phase vary over time, remains a challenging problem. Existing methods predominantly model these sequences on fixed time grids, suffering from a limited ability to accommodate phase-related variation. To address this limitation, we propose POEM, a phase-aware forecasting framework based on latent feature rotation using the special orthogonal group in two dimensions, denoted by SO(2). POEM aims to reduce the phase-related variability by learning a phase-correction coordinate and applying an invertible SO(2)-based rotation to paired latent features. To extrapolate this correction coordinate, Directional Phase Increment Attention (DPIA) retrieves historical phase increments from similar temporal contexts and integrates them into future phase corrections. Experiments demonstrate that POEM achieves competitive performance, while qualitative visualizations suggest that the learned phase-aware transformation makes latent trajectories more regular.
Jiawen Zhu, Shuhan Liu, Shengxuan Li +2
School of Software Technology, Zhejiang University · State Key Lab of CAD&CG, Zhejiang University
Real-world time series forecasting faces the fundamental challenge of non-stationary statistical properties, including shifts in mean and variance over time. While reversible instance normalization (RevIN) has shown promise by stationarizing inputs and denormalizing outputs, it relies on the strong assumption that historical and future distributions remain identical. We observe that in many practical applications, distribution shifts follow cyclical patterns that correlate with periodic positions (e.g., seasonal and holiday volatility). To this end, we propose PAMod, a lightweight yet powerful framework that models cyclical distribution shifts via Phase-Amplitude Modulation in the normalized feature space. PAMod learns periodic embeddings to modulate representations: phase modulation captures mean shifts, while amplitude modulation adapts to variance changes. Crucially, we prove mathematically that modulating in normalized space is equivalent to applying dynamic denormalization, offering an elegant unification of distribution adaptation and representation learning. Extensive experiments on twelve real-world benchmarks demonstrate that PAMod achieves state-of-the-art performance with fewer computational resources. Furthermore, our modulation mechanism, as a novel plug-and-play technique, can improve existing time-series forecasting methods with simple integration.
Yingbo Zhou, Yutong Ye, Shuhao Li +5
Fudan University · Beihang University · Beijing University of Posts and Telecommunications