DrafTS: Time-Aware Decomposition with Residual Correction for Time Series Modeling

1 INTRODUCTION

• Existing frequency-wise filtering misses temporal evolution, while suppressing whole noisy observations can also discard useful dynamics within them.The paper contrasts both approaches with decomposition that extracts a primary component and recovers missed dynamics from the residual.Source: §1 · p. 1, Figure 1 · p. 2
• Underlying dynamics are structured temporal patterns such as trends and periodic behavior, whereas noise is irregular variation without stable temporal patterns.Source: §1 · p. 1
• The Hilbert-transform example shows the first five IMFs and derives instantaneous amplitude and frequency from the analytic signal of c5(t).The figure marks time t∗ and distinguishes real and imaginary parts.Source: Figure 2 · p. 2
• DrafTS uses time-aware masks to capture evolving dynamics in a primary component and residual correction to exploit useful dynamics left by imperfect decomposition.The residual-aware module processes residuals recurrently with fixed hidden weights and trains only its output layer.Source: §1 · p. 2
• Across four time-series tasks, DrafTS improves diverse task-specific backbones without modifying their internal architectures.The framework combines time-aware decomposition with lightweight residual correction.Source: §1 · p. 2

Deep Learning for Time Series Modeling

• Deep-learning approaches span recurrent, convolutional, MLP-based, and attention-based architectures, with later work also extending pre-training, context length, and sparse experts.Greater modeling capacity alone does not ensure separation of evolving dynamics from noise.Source: §2 · pp. 2–3

Modeling paradigms for noise

• Noise-control methods use spectral filtering or regulate observation use, but global spectral operations miss within-window changes and suppressing noisy observations may limit learning useful dynamics.Examples include fixed cutoffs or Fourier selection, window rejection, loss masking, and instance-adaptive dropout.Source: §2 · p. 3

Signal Decomposition Methods

• Fourier and DCT use fixed-frequency bases, while STFT uses fixed windows; EMD with the Hilbert transform instead yields interpretable instantaneous amplitude and frequency.Threshold-based EMD denoising may remove underlying dynamics; DrafTS learns task-supervised time-aware soft masks from analytic features and retains residuals for correction.Source: §2 · p. 3

3.1 NOTATIONS AND PROBLEM FORMULATIONS

• For each variable, DrafTS represents the input as a T × N series with a task-specific target, then splits it into primary and residual components for modeling.The target is a future sequence for forecasting, class labels for classification, and the next observation for anomaly detection.Source: §3.1 · p. 3

3.2 STAGE I: TIME-AWARE DECOMPOSITION

• Reflection padding addresses EMD boundary effects; the central T samples of the extracted IMFs are retained, and the first two or three IMFs are examined for potential noise.The padding length is set to one quarter of the input length.Source: §3.2 · p. 4
• Four analytic features—frequency variation, amplitude variation, analytic innovation, and IMF order—guide time-aware masks that allocate IMF content between primary and residual components.The local window length is based on the estimated oscillation rate, balancing statistical stability and temporal locality.Source: §3.2 · p. 5

3.3 STAGE II: MODELING WITH CORRECTION

• A fixed recurrent reservoir maps the residual into a task-specific correction, with only its output layer trained to provide expressive temporal representations using few trainable parameters.The correction is added to the backbone output to produce the final task output.Source: §3.3 · p. 6

4.1 EXPERIMENTAL SETTINGS

• Experiments cover short- and long-term forecasting, classification, and anomaly detection, using real-world benchmarks and Noisy Lorenz63 under three noise levels.Six representative backbones are evaluated; Noisy Lorenz63 combines bursts, spikes, and missing blocks with clean future targets.Source: §4 · p. 6, §4.1 · pp. 6–7, Table 1 · p. 6

4.2 MAIN RESULTS

• DRAFTS generally improves all six backbones in long- and short-term forecasting; on M4, Crossformer’s MASE and OWA fall by 2.21% and 2.02%, respectively.For long-term forecasting, TimeMixer’s MSE falls by 14.84% on Weather.Source: Table 2 · p. 7, §4.2 · p. 7
• DRAFTS improves all six backbones on classification and anomaly detection; TimeMixer accuracy rises from 45.41% to 53.12%, while TimeFilter’s average F1 rises 1.12 percentage points.TimeFilter’s average F1 changes from 82.73% to 83.85%.Source: §4.2 · p. 7, Figure 5 · p. 8

4.3 ABLATIONS AND ANALYSIS

• On Noisy Lorenz63, RC outperforms three residual alternatives with 12.48K trainable parameters, and RC improves both analytic variants at every SNR while full DRAFTS beats Raw + RC.The ablation results indicate that gains do not come solely from adding RC to the raw backbone.Source: §4.3 · p. 8
• Across a shared input-energy reduction range of 1.05%–9.20%, DRAFTS remains accurate at higher reductions, while FFT-high performs best at only 1.28% reduction.The comparison uses TimeFilter and reports test forecasting errors on Noisy Lorenz63.Source: Table 4 · p. 8, §4.3 · p. 8
• On ILI, DRAFTS reduces MSE and MAE by 27.62% and 7.30%, respectively, while increasing parameters, peak GPU memory, and model-side FLOPs by at most 0.66%.It requires 22.81% fewer training iterations under the early-stopping protocol; analytic signals are precomputed offline.Source: §4.3 · p. 8, Table 6 · p. 9
• On Noisy Lorenz63 at SNR 7 dB, DRAFTS better captures sharp forecasting-boundary transitions and subsequent recovery than the backbone’s often overly smooth predictions.The analysis also reports better retention of nonlinear dynamics, temporal variation, and oscillatory patterns under mixed noise.Source: Figure 6 · p. 9, §4.3 · p. 9
• Decomposition is conservative on periodic ETTm1 but stronger and more variable on heterogeneous ILI; sensitivity analysis finds larger J generally raises MSE, with optima varying by SNR.For RC, size 64 gives the largest mean relative MSE drop across tested SNRs, though its optimum also varies by SNR.Source: §4.3 · p. 9

• DRAFTS uses instantaneous-frequency features to extract a primary component for evolving dynamics and residual correction to exploit dynamics missed by decomposition.Source: §5 · p. 9
• Improvements across four time-series modeling tasks and six backbones support the proposed approach.Source: §5 · p. 9
• Future work will explore online multivariate decomposition to capture evolving cross-variable dynamics without offline preprocessing.Source: §5 · p. 9

• Generative AI tools assisted manuscript writing and polishing, including checks of mathematical notation, dimensions, units, and formula definitions.Source: AI USE STATEMENT · p. 10
• AI tools helped identify and organize related literature and provided feedback on experimental methodology and interpretation of results.Source: AI USE STATEMENT · p. 10
• The authors manually checked mathematical expressions, verified cited claims against original papers, and take responsibility for the final content.Source: AI USE STATEMENT · p. 10

A.1.1 LONG-TERM FORECASTING

• Long-term forecasting covers seven real-world datasets and synthetic Noisy Lorenz63; ECG, Laser, and Lorenz63 use task-specific input and prediction horizons.The first five public forecasting benchmarks follow BasicTS splits; dedicated protocols are used for the remaining datasets.Source: §A.1.1 · p. 14

A.1.2 SHORT-TERM FORECASTING

• M4 short-term forecasting reserves the final H training observations for validation because the dataset has no official validation split.Raw and augmented variants share the same setup, and each configuration is evaluated once with seed 42; those comparisons do not establish significance or cross-seed robustness.Source: §A.1.2 · p. 14

A.1.3 CLASSIFICATION AND ANOMALY DETECTION

• Classification uses a deterministic class-stratified 80/20 validation split, while anomaly detection evaluates five datasets with matched baseline and DRAFTS backbone configurations.Classification results average three seeds; anomaly-detection training uses normal data and repeats three seeds.Source: §A.1.3 · p. 15

A.2 DECOMPOSITION ADAPTIVITY METRICS

• Decomposition adaptivity is characterized by residual magnitude, residual-amplitude variance, and temporal-variation alignment.Alignment reflects association between residual energy and local input variation, not noise-identification accuracy; residual-amplitude variance is zero for a temporally constant amplitude, including an all-zero residual.Source: §A.2 · p. 16

A.3 PROTOCOL FOR THE DECOMPOSITION COMPARISON

• The decomposition comparison uses validation-calibrated energy-reduction points and compares test-error curves over a shared interval without extrapolation.The common interval is [1.05%, 9.20%]; normalized area under the curve is primary because curve minima depend on how many configurations are evaluated.Source: §A.3 · p. 17

B COMPLETE RESULTS

• Complete experiment results are reported in separate tables for long-term forecasting, short-term forecasting, classification, and anomaly detection.The main text presents aggregated results and selected comparisons.Source: B COMPLETE RESULTS · p. 18

B.1 LONG-TERM FORECASTING RESULTS

• 80 of 84 average comparisons improve across seven real-world datasets, and all 36 average comparisons improve on Noisy Lorenz63 across six models and three noise levels.Gains hold at SNR = 3 dB and are also substantial at higher SNRs.Source: §B.1 · p. 18
• TimeFilter–ILI MSE changes from 2.387 ± 0.402 to 2.000 ± 0.173, a mean paired reduction of 15.01% ± 12.73% across three seeds.MSE improves in all three runs, while MAE improves in two; PatchTST–ETTh1 shows much smaller paired changes.Source: §B.1 · p. 18

B.2 SHORT-TERM FORECASTING RESULTS

• Overall M4 results aggregate six frequency subsets using weights proportional to each subset’s number of series.Lower sMAPE, MASE, and OWA indicate better forecasting accuracy; OWA is relative to the official Naive2 baseline.Source: §B.2 · p. 18
• For Crossformer, weighted-average sMAPE, MASE, and OWA decrease by 1.84%, 2.21%, and 2.02%, respectively.On the Daily subset, all three metrics decrease by more than 10%.Source: §B.2 · p. 18, Table 14 · p. 19

B.3 CLASSIFICATION RESULTS

• Raw and DRAFTS-augmented pairs use the same classification-head design, and results average three runs with seeds 42, 43, and 44.Source: §B.3 · p. 19
• Classification accuracy improves in 54 of 60 backbone–dataset combinations, with one tie, and average accuracy rises for all six backbones.Gains range from 0.57 to 7.71 percentage points; the largest is for TimeMixer, with gains also reported for iTransformer and TimeFilter.Source: Table 15 · p. 20, §B.3 · p. 20

B.4 ANOMALY DETECTION RESULTS

• The unsupervised protocol forecasts the next observation, thresholds scores at the 99.5th percentile of normal validation scores, and uses test labels only for final evaluation.Precision, Recall, and F1-score are averaged over three runs.Source: §B.4 · p. 20
• DRAFTS raises every backbone’s average F1; TimeFilter has the largest gain, from 82.73% to 83.85%.Table 16 reports full anomaly-detection results on five datasets, averaged over three seeds.Source: §B.4 · p. 20, Table 16 · p. 20
• On SMD, Crossformer and TimeFilter gain 5.50 and 4.56 percentage points in F1, respectively, as precision rises while recall is largely preserved.The authors describe this pattern as reducing false alarms without substantially compromising anomaly sensitivity.Source: §B.4 · p. 20

B.5 ABLATION STUDY

• The ablation uses TimeFilter on Noisy Lorenz63 at SNR ∈ {3, 5, 7}, with input length 96 and prediction horizons 16 and 64.Variants share data splits, training protocol, and seed; checkpoints are selected by validation MAE without test-set retuning.Source: §B.5 · p. 21
• DRAFTS achieves the lowest MSE and MAE in all six SNR–horizon settings, and adding RC to the ϕ-based variant improves both metrics in every setting.Table 17 reports the detailed ablation results for TimeFilter at seed 42.Source: Table 17 · p. 21, §B.5 · p. 21
• Raw + RC is not consistently beneficial, and neither no-RC analytic variants nor Raw + RC match the full model across the six evaluated settings.Raw + RC improves some short-horizon cases but degrades both SNR=3 settings and the long-horizon SNR=5 case.Source: §B.5 · p. 21
• The full model outperforms Raw + RC and direct-attribute RC controls at both horizons for every SNR, consistent with complementary analytic encoding and residual correction.The authors report that ϕ provides a stronger basis for correction than direct attributes.Source: §B.5 · p. 21