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RouteTS: Frequency-Time Routing for Time Series Forecasting

Gaofeng Lin, Lei Duan

arXiv:2608.14682v1cs.LGstat.ML

TL;DR

Existing forecasting methods struggle to jointly model global periodicity and localized non-stationarity because static single-domain processing can misalign periodic signals or over-smooth transients. RouteTS routes dominant spectral components to a frequency-domain predictor and residuals to a time-domain MLP, achieving favorable accuracy across most benchmarks with minimal computational overhead.

  • Problem

    Static single-domain or fixed-separation forecasting cannot dynamically balance deterministic periodicities and stochastic residuals across heterogeneous time series.

  • Method

    RouteTS uses amplitude routing to send dominant frequencies to a complex-valued frequency predictor and residual components to a time-domain lightweight MLP.

  • Results

    RouteTS achieves the best average MSE on six of eight benchmarks and competitive MAE across most datasets.

  • Takeaways & Limitations

    Optimal routing is dictated by each dataset’s intrinsic spectral signature, supporting accurate forecasting with minimal computational overhead.

  • Takeaways & Limitations

    RouteTS’s channel-independent design does not exploit inter-channel correlations, motivating future integration of relational modeling.

Abstract

from arXiv · show

Real-world time series inherently intertwine global periodic structures with localized non-stationary variations. Existing approaches process these heterogeneous dynamics within a single computational domain, incurring fundamental limitations: time-domain models suffer from periodic misalignment over long horizons, while frequency-domain models over-smooth transient spikes. We argue that the optimal computational domain is not a property of the model, but of the data itself. Based on this principle, we propose RouteTS, a unified forecasting framework that partitions the frequency spectrum via amplitude routing and delegates components to their mathematically optimal domains. Dominant frequencies are processed by a complex-valued linear predictor in the frequency domain to preserve periodic structure, while residual spectral energy is reverted to the time domain and modeled by a lightweight MLP for local variations. Extensive experiments demonstrate that RouteTS achieves competitive prediction accuracy across diverse real-world datasets, with routing decisions guided by the underlying spectral signature. Furthermore, the lightweight design of RouteTS provides significant computational efficiency advantages, offering a principled solution to the longstanding dilemma between global periodicity and local transience.

1 INTRODUCTION

RouteTS addresses the limitations of single-domain forecasting by routing spectral components according to whether they encode global periodicity or localized transience. It processes dominant frequencies in the frequency domain and residual components in the time domain.

  • Motivation: Real-world time series intertwine global structural patterns with localized non-stationary variations, making heterogeneous-component modeling fundamental to robust forecasting.The passage motivates this challenge across energy planning, financial markets, and traffic management.
  • Single-domain limitations: Single-domain architectures struggle because frequency modeling suits sparse periodic structure, whereas localized spikes spread across frequencies and become over-smoothed by compact spectral representations.Periodic-dominant signals exhibit sparse high-amplitude peaks, while transient-dominant signals contain time-localized variations and sharp spikes.
  • Routing principle: Spectral structure motivates routing: sparse high-amplitude components represent global periodicity, while diffuse low-amplitude spectral energy corresponds to localized variations better preserved in time.This complementary spectral–temporal structure supports assigning different components to different computational domains.
  • RouteTS: RouteTS partitions the spectrum by identifying Top-K dominant frequencies and routes them to a complex-valued linear predictor that preserves periodic structure.The framework first transforms the series into the frequency domain before selecting dominant components.
  • RouteTS: RouteTS routes remaining low-amplitude spectral components back to the time domain for modeling, complementing frequency-domain processing of dominant periodic components.The supplied passage describes the residual-component routing but truncates the specific time-domain model details.

2 RELATED WORK

Prior forecasting research has progressed from RNNs and TCNs toward Transformers, Linear models, and frequency-domain methods, but existing approaches struggle to represent periodic structure and localized non-stationarity simultaneously. These limitations motivate routing heterogeneous dynamics to computational domains suited to their underlying characteristics.

  • RNN- and TCN-based models established early forecasting foundations, but limited long-range capacity shifted attention toward Transformer and Linear architectures.
  • Transformers evolved from per-step tokens to patched subsequences, inverted variate tokens, and phase-aligned representations for global dependency modeling.Despite their expressive power, they lack explicit periodic constraints and must learn cycles implicitly through high-dimensional parameterization.
  • DLinear and related methods show that trend-seasonal decomposition with simple linear mappings can outperform sophisticated Transformers, but single projections constrain capacity for abrupt non-stationary changes.Multi-scale feature fusion has further advanced this line of work.
  • Time-domain and Linear paradigms entangle deterministic periodicities with stochastic high-frequency variations, risking either local-noise overfitting or genuine-periodicity blurring.This dilemma motivates processing heterogeneous dynamics in their optimal domains.
  • Pure frequency-domain models such as FITS and FilterNet struggle with localized transients because non-stationary spikes spread across many frequency components and sparse representations over-smooth sharp edges.Later methods incorporate time-domain information to mitigate this limitation.
  • Existing joint and dual-pathway methods use static domain allocation, which can over-smooth signals, amplify noise, or impose fixed time-frequency assignments regardless of dataset characteristics.MixLinear, for example, rigidly assigns local trends to time and global trends to frequency domains.

3 METHOD

RouteTS partitions normalized input spectra by amplitude, routing dominant frequencies to a complex-valued frequency predictor and residual components to a time-domain MLP. The two branch predictions are aggregated in the time domain and inverse-normalized to produce the final forecast.

  • Task Formulation: Given a lookback window X ∈ R^(L×C), RouteTS forecasts a future horizon Y ∈ R^(T×C) for regularly sampled multivariate series.The model learns to predict future values Ŷ solely from the historical input sequence X.
  • Amplitude-Based Routing: RevIN normalization and rFFT transform the input into a frequency representation retaining F = ⌊L/2⌋ + 1 components.The normalized sequence Xnorm is mapped to the frequency domain before routing.
  • Amplitude-Based Routing: For each variable, the Top-K frequencies by amplitude define a binary mask that separates dominant components from the residual spectrum.The dominant stream captures global deterministic periodicities, while the complementary stream retains non-stationary local variations.
  • Frequency-Domain Main Predictor: The Frequency Branch applies a shared complex-valued linear projection from historical frequency dimension F to horizon dimension F′, explicitly learning amplitude scaling and phase shifting.The predicted complex spectrum is converted back to the time domain with irFFT.
  • Time-Domain Residual Predictor and Aggregation: The residual spectrum is reverted with irFFT and forecast by an MLP backbone, after which both branch outputs are aggregated in time space and restored to the original scale with iRevIN.This divide-and-conquer design preserves stable structural cycles and non-stationary local residuals.

4 EXPERIMENTS

RouteTS is evaluated on eight multivariate real-world datasets against ten established forecasting baselines, with results showing competitive accuracy and data-dependent routing. Spectral analyses and efficiency tests further support its adaptive mechanism and lightweight deployment.

  • Experimental Setup: Experiments cover eight multivariate datasets spanning energy, transportation, economics, environment, and healthcare, evaluated against ten established forecasting models.The baselines include PhaseFormer, MixLinear, CFPT, FilterTS, AMD, FBM, iTransformer, FilterNet, FITS, and DLinear.
  • Forecasting Accuracy: RouteTS achieves the best average MSE on six of eight benchmarks and maintains competitive MAE across most datasets.Its routing thresholds are dataset-specific, reflecting differences in spectral structure.
  • Forecasting Accuracy: On COVID-19, pure time-domain processing captures abrupt localized spikes and achieves the lowest MSE and MAE.The dataset lacks prominent periodic patterns, making K = 0 the best-performing routing choice.
  • Spectral Routing Analysis: Traffic and Solar benefit from K = 1, which isolates dominant periodic components while delegating residual non-stationary variations to the Time Branch.Increasing frequency allocation introduces high-frequency noise and degrades performance.
  • Spectral Routing Analysis: ETTh2 and ETTm2 favor K = full for broad multi-frequency periodicity, while Exchange trails lightweight linear baselines despite retaining unified routing robustness.Exchange is dominated by high-frequency stochastic fluctuations and weak periodic structure.
  • Computational Efficiency: 70.55MB of memory on Solar and 4% of CFPT’s parameters with 13% of its memory on ETTm2 demonstrate RouteTS’s efficiency.On Solar, RouteTS maintains competitive MSE; on ETTm2, it matches CFPT’s accuracy.

5 CONCLUSION

RouteTS addresses the trade-off between periodic misalignment and over-smoothing by routing spectral energy according to amplitude and the data’s intrinsic spectral signature. It delivers favorable accuracy with minimal computational overhead, while remaining limited by its channel-independent design.

  • Core contribution: RouteTS dynamically routes spectral energy by amplitude to address the conflict between preserving global periodicities and modeling non-stationary variations.The framework uses a complex-valued Frequency Branch to preserve the amplitude-phase structure of dominant spectral components.
  • Core contribution: Optimal routing is strictly dictated by the data’s intrinsic spectral signature.This conclusion is supported by the paper’s extensive evaluations.
  • Empirical findings: RouteTS achieves favorable accuracy across most benchmarks and remains resilient to long-range phase drift and mixed-regime instability.Mechanism analyses confirm these robustness properties.
  • Empirical findings: The predictive gains require minimal computational overhead.The conclusion emphasizes computational efficiency alongside accuracy improvements.
  • Limitations and future work: A current limitation is RouteTS’s channel-independent design, motivating future integration of inter-channel relational modeling.The paper identifies inter-channel relational modeling as a promising future direction.

AI USE STATEMENT

Generative AI was used only to draft and edit portions of the paper and refine LaTeX formatting, while the authors independently developed and verified the research content.

  • Generative AI tools assisted with drafting and editing portions of the paper and refining LaTeX code for table and figure formatting.
  • The authors did not use generative AI to generate datasets, develop models, formulate mathematical claims, design methods or experiments, or interpret results.
  • All AI-assisted work was reviewed and verified against the authors’ research contributions, experimental records, and manually implemented code.The authors independently developed the core methodology, mathematical formulations, experimental design, and software implementation.

REPRODUCIBILITY STATEMENT · A EXPERIMENTAL DETAILS

The paper provides a reproducibility framework covering RouteTS architecture, data flow, datasets, preprocessing, baselines, and training hyperparameters. These details are distributed across specified sections, appendices, and tables.

  • REPRODUCIBILITY STATEMENT: RouteTS architecture and data flow are fully specified in Sections 3.1–3.3.These sections are identified as the primary source for reproducing the proposed framework.
  • A EXPERIMENTAL DETAILS: Dataset statistics and preprocessing steps are documented in Appendix A.1.The appendix provides dataset-level information needed to reproduce the experiments.
  • A EXPERIMENTAL DETAILS: Baseline descriptions are documented in Appendix A.2.The baseline configurations are included as part of the experimental documentation.
  • A EXPERIMENTAL DETAILS: Training hyperparameters are reported in Table 1 and Appendix A.3.The reported hyperparameters include dataset-specific batch sizes, learning rates, and routing thresholds K.
  • A EXPERIMENTAL DETAILS: The reproducibility materials include dataset-specific batch sizes and learning rates.These settings are explicitly identified among the training hyperparameters reported in the paper.
  • REPRODUCIBILITY STATEMENT: The paper states that its experimental details are provided comprehensively to facilitate reproduction of the reported results.This reproducibility aim motivates the documentation across the architecture sections, appendices, and Table 1.

A.1 DATASETS · A.2 BASELINES · A.3 IMPLEMENTATION DETAILS

The study evaluates RouteTS on eight diverse benchmark datasets, compares it with ten representative baselines, and uses dataset-specific training settings with early stopping. The benchmarks span periodic, non-stationary, high-dimensional, and financial time series.

  • A.1 DATASETS: Eight benchmark datasets are characterized by their input channels, temporal sampling rates, and total available timesteps.Table 4 provides the statistical overview of these datasets.
  • A.1 DATASETS: The dataset suite spans periodic and non-stationary settings, including traffic congestion, solar irradiance, currency volatility, and transformer-temperature forecasting.PM2.5 further introduces high dimensionality and spatial-temporal correlations, while Exchange lacks explicit temporal periodicities.
  • A.2 BASELINES: Ten representative baselines cover Transformer-based, Linear-based, MLP-based, and frequency-domain forecasting models.The comparison set includes PhaseFormer, MixLinear, CFPT, FilterTS, AMD, FBM, iTransformer, FilterNet, FITS, and DLinear.
  • A.3 IMPLEMENTATION DETAILS: Training was capped at 100 epochs with early stopping using a validation patience of 12 epochs to prevent overfitting.The optimization configuration was adapted to dataset scale, channel dimensionality, and convergence behavior.
  • A.3 IMPLEMENTATION DETAILS: Batch sizes were 64 for PM2.5, 32 for ETTh2, ETTm2, ILI, and COVID-19, 16 for Traffic and Solar, and 8 for Exchange.Initial learning rates ranged from 1 × 10−2 for ILI to 5 × 10−4 for Exchange and PM2.5.

B FULL RESULTS · C MECHANISM ANALYSIS SUPPLEMENT

Table 5 presents complete multivariate forecasting results for every prediction length and dataset, complementing the averaged metrics in Table 2 with a granular view across forecast horizons.

  • B FULL RESULTS: Table 5 reports the complete multivariate forecasting results.It covers the full result set rather than averaged metrics alone.
  • B FULL RESULTS: The results span all prediction lengths.This enables performance inspection under different forecast horizons.
  • B FULL RESULTS: The table reports results for each dataset.Dataset-specific performance is included in the complete results.
  • B FULL RESULTS: Table 5 complements the averaged metrics reported in Table 2.Its purpose is to add detail to the averaged summary.
  • B FULL RESULTS: The complete results provide a granular view of model performance.The granularity comes from reporting results across datasets and prediction lengths.
  • B FULL RESULTS: The analysis covers model performance under varying forecast horizons.This is the horizon-level perspective supplied by the complete results.

C.1 AUTOCORRELATION FUNCTION ANALYSIS · C.2 VISUALIZATION OF FREQUENCY-DOMAIN OVER-SMOOTHING · D ABLATION STUDY

The ACF analysis links routing choices to datasets’ temporal dynamics, distinguishing trend-dominated, single-period, and complex-periodic behaviors. Visualization further shows that frequency-domain over-smoothing removes localized fluctuations, whereas RouteTS preserves both periodic structure and fine-grained variation.

  • C.1 AUTOCORRELATION FUNCTION ANALYSIS: ACF measures cyclical predictability and memory retention by quantifying linear dependence between a series and its lagged values.The analysis uses ACF decay profiles and cyclical peaks to diagnose the temporal dynamics underlying routing decisions.
  • C.1 AUTOCORRELATION FUNCTION ANALYSIS: ILI and COVID19 show gradual monotonic ACF decline, while PM2.5 rapidly collapses toward zero; all lack stable deterministic periodicity and select K = 0.These profiles justify bypassing the frequency domain.
  • C.1 AUTOCORRELATION FUNCTION ANALYSIS: Traffic and Solar exhibit undamped, high-amplitude sinusoidal ACF oscillations, supporting K = 1 to isolate their primary harmonic.The routing prevents high-frequency noise from interfering with the time domain.
  • C.1 AUTOCORRELATION FUNCTION ANALYSIS: ETTh2 and ETTm2 display multi-scale ACF periodicity, requiring the full frequency spectrum, whereas Exchange has a smooth random-walk-like decay and is allocated K = full.The contrasting ACF profiles motivate different full-spectrum routing behavior.
  • C.2 VISUALIZATION OF FREQUENCY-DOMAIN OVER-SMOOTHING: FBM over-smooths FBM? No: FBM flattens sharp ground-truth peaks and valleys into low-amplitude oscillations, while RouteTS tracks periodic structure and local fluctuations.The comparison illustrates why pure frequency-domain models fail when localized transients coexist with dominant periodicities.
  • C.2 VISUALIZATION OF FREQUENCY-DOMAIN OVER-SMOOTHING: FBM concentrates spectral energy in a sparse subset of low-frequency components, unlike the ground truth’s broad distribution including non-negligible mid-tohigh-frequency contributions.This excessive concentration is identified as the spectral origin of FBM’s over-smoothing.

D.1 PREDICTOR DESIGN IN TIME BRANCH · D.2 FREQUENCY PREDICTOR DESIGN

RouteTS adopts an MLP for time-domain residuals because it consistently outperforms Linear predictors, especially under non-stationarity, while complex-valued frequency prediction preserves amplitude–phase coupling and improves hybrid and long-horizon forecasting. Together, these ablations support dataset-specific routing and domain-appropriate predictor design.

  • D.1 PREDICTOR DESIGN IN TIME BRANCH: The MLP outperforms the Linear baseline across all evaluated datasets and prediction horizons, with gains governed by non-stationarity and residual complexity.The comparison covers Traffic and Solar (K = 1) and PM2.5 and COVID-19 (K = 0).
  • D.1 PREDICTOR DESIGN IN TIME BRANCH: On COVID-19, Linear MSE escalates from 0.28 to 1.35 as the horizon extends, whereas MLP MSE remains below 0.30.The result is attributed to nonlinear activation handling chaotic, high-frequency residuals and abrupt distributional shifts.
  • D.1 PREDICTOR DESIGN IN TIME BRANCH: On Traffic and Solar (K = 1), the MLP achieves lower MSE than Linear across all horizons despite frequency-branch isolation of dominant periodic structure.Frequency routing captures dominant periodicity, but localized non-stationary deviations retain nonlinear temporal dependencies.
  • D.1 PREDICTOR DESIGN IN TIME BRANCH: On PM2.5 (K = 0), the MLP–Linear performance gap stays within 0.01 MSE while the MLP remains consistently better without measurable degradation.The MLP’s nonlinear inductive bias extracts weak local structures in this noise-dominated regime.
  • D.1 PREDICTOR DESIGN IN TIME BRANCH: These results establish the MLP as RouteTS’s default time-domain predictor, while K = 0 or K = 1 routing allocates capacity between local dynamics and frequency computation.The routing mechanism avoids unnecessary frequency-domain computation when local modeling is sufficient.
  • D.2 FREQUENCY PREDICTOR DESIGN: The complex frequency predictor couples amplitude and phase through cross-terms, unlike the equal-parameter real-valued ablation that independently projects real and imaginary components.The complex mapping uses W = WR + jWI and preserves joint scaling and rotation in the complex plane.
  • D.2 FREQUENCY PREDICTOR DESIGN: On Solar, the complex predictor’s MSE advantage over the real variant widens from 2.2% at H=96 to 4.0% at H=192.This hybrid periodic–transient regime benefits from phase consistency during dual-branch aggregation.
  • D.2 FREQUENCY PREDICTOR DESIGN: At H=720 on Exchange, the real-valued decomposition reaches MSE 1.147 versus 0.910 for the complex predictor, a +26.0% degradation.The complex formulation prevents accumulated phase drift during long-horizon spectral extrapolation; on ETTh2 and Traffic, its average MSE remains strictly lower with a gap below 4%.

E ROBUSTNESS ANALYSIS · E.1 GAUSSIAN NOISE INJECTION · E.2 VARYING INPUT LENGTH

RouteTS is evaluated for robustness to Gaussian noise and varying look-back lengths against representative Transformer-, Linear-, and frequency-based baselines. It shows graceful noise degradation and improves monotonically with longer historical context, while matching FBM at the longest ETTh2 window.

  • E ROBUSTNESS ANALYSIS: RouteTS is tested under high-frequency noise injection and varying input lengths alongside PhaseFormer, DLinear, and FBM.The evaluation covers Solar (K = 1), ETTm2 (K = full), and look-back lengths L ∈ {96, 192, 336, 720}.
  • E.1 GAUSSIAN NOISE INJECTION: RouteTS’s Solar performance degrades gracefully from 0.186 to 0.324 across increasing noise levels.The model selectively extracts dominant periodicities while isolating high-frequency noise, effectively functioning as an adaptive low-pass filter.
  • E.1 GAUSSIAN NOISE INJECTION: RouteTS demonstrates superior noise immunity across all tested Gaussian noise levels on Solar and ETTm2.Solar represents a single-frequency regime, while ETTm2 represents a complex periodic regime.
  • E.2 VARYING INPUT LENGTH: RouteTS improves monotonically as the look-back window expands, leveraging longer context to capture multi-scale periodic structures.The tested lengths are L ∈ {96, 192, 336, 720} on ETTh2 and ETTm2.
  • E.2 VARYING INPUT LENGTH: At L = 720 on ETTh2, RouteTS achieves comparable performance to FBM at 0.273.Both models maintain stable improvement as the look-back window extends.
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