Bridging Time and Frequency: A Joint Modeling Framework for Irregular Multivariate Time Series Forecasting

1. Introduction

• Irregular sampling makes global periodic patterns and fine-grained local dynamics difficult to model together.Classical DFT and FFT assume uniform sampling, while uneven patch density can make reliable feature aggregation difficult and degrade representation quality.Source: §1 · pp. 1–2
• TFMixer uses a learnable NUDFT for irregular timestamps and query-based patch mixing to aggregate informative local representations.It fuses time- and frequency-domain representations for forecasting and uses inverse NUDFT for explicit seasonal extrapolation.Source: §1 · p. 2
• TFMixer is a unified joint time–frequency framework that explicitly bridges global spectral dependencies and local temporal dynamics.Its Global Frequency Module and Local Time Module process the two representations in parallel.Source: §1 · p. 2
• TFMixer achieves state-of-the-art performance on multiple real-world IMTSF datasets.The introduction reports extensive experiments but gives no quantitative comparison here.Source: §1 · p. 2

2.1. Irregular Multivariate Time Series Forecasting

• Irregular forecasting research includes temporal-decay RNNs, Neural ODEs, set-based models, and graph-based methods for asynchronous sensor relationships.The passage also describes patching approaches for local structures in irregular sequences.Source: §2.1 · pp. 2–3
• Patch-based IMTSF methods capture local structure but can struggle with information-density imbalance and often omit frequency-domain analysis.The passage cites tPatchGNN as an example of segmenting irregular sequences into transformable patches.Source: §2.1 · p. 3

2.2. Frequency Domain Analysis in Time Series

• Frequency-domain methods capture long-term periodic patterns, but FFT requires uniform grids, limiting direct use on irregular multivariate series.KAFNet pre-aligns irregular observations to a shared grid, whereas TFMixer applies learnable NUDFT to raw non-uniform observations.Source: §2.2 · p. 3

3.1. Problem Definition

• An IMTS consists of N univariate variables with timestamped observations, non-uniform intervals, and inter-variable asynchronicity.Source: §3.1 · p. 3
• The forecasting task maps historical observations to estimates for future timestamp queries.The supplied formulation identifies the historical observations O and target prediction queries Q as inputs to the predictive model.Source: §3.1 · p. 3

3.2. Canonical Pre-Alignment Representation for IMTS

• Canonical pre-alignment represents an IMTS with shared chronological timestamps, a value matrix, and an observation mask.The shared timestamp set contains unique observation times; the mask indicates whether each variable is observed at each time.Source: §3.2 · p. 3

4.2. Global Frequency Module

• Learnable NUDFT maps irregular observations directly to spectral representations, and a spectrum encoder extracts global frequency features without requiring interpolation.The module also retains refined spectral coefficients for inverse NUDFT seasonal-bias estimation.Source: §4.2 · p. 4, §4.2.1 · p. 4, §4.2.2 · p. 4

4.2.1. LEARNABLE NUDFT

• An MLP nonlinearly refines raw spectral coefficients while preserving their dimensionality, adapting amplitude and phase before encoding them into the model’s latent space.The encoded frequency representation is used for fusion, while refined coefficients remain available for inverse NUDFT.Source: §4.2.1 · p. 4, §4.2.2 · p. 4

4.3.1. PATCH ENCODER

• Transformable patches use consistent time windows across variables, continuous-time embeddings, and time-aware convolution to encode local semantics and accommodate variable-length sequences.A patch mask marks whether observations are present, including for empty patches.Source: §4.3.1 · p. 5

4.3.2. QUERY-BASED PATCH MIXING

• Learnable query patches attend to encoded patches to aggregate informative temporal features and suppress noise from sparse patches into a fixed-capacity representation.Positional encodings preserve patch order in the attention operation.Source: §4.3.2 · p. 5
• Dual-Mixing Blocks alternate patch mixing within variables and variable mixing across channels, then aggregate the semantic dimensions into a fixed-length temporal context.The variable mixer exchanges information among channels to capture cross-variable dependencies.Source: §4.3.2 · pp. 5–6

4.4. Output Module

• The Output Module fuses time and frequency features, decodes initial predictions, adds a seasonal bias estimated by inverse NUDFT at future timestamps, and applies masked denormalization.The seasonal bias is scaled by a learnable parameter.Source: §4.4 · p. 6

4.5. Loss Function

• Training combines MAE forecasting loss on observed future timestamps with MAE reconstruction loss on historical observations reconstructed from spectral coefficients.The reconstruction objective is intended to encourage the frequency module to capture physical periodicities rather than spurious noise.Source: §4.5 · p. 6

5.1.1. DATASETS

• The evaluation uses PhysioNet, MIMIC, HumanActivity, and USHCN, spanning healthcare, biomechanics, and climate science.The paper directs readers to Appendix A.1 for dataset statistics and further details.Source: §5.1.1 · p. 6

5.1.2. BASELINES

• Fourteen baselines comprise IMTS classification or imputation models and IMTS forecasting models.The forecasting group includes NeuralFlows, CRU, GNeuralFlow, tPatchGNN, GraFITi, Hi-Patch, KAFNet, and APN.Source: §5.1.2 · p. 6

5.1.3. IMPLEMENTATION DETAILS

• Forecast targets are the next three timestamps for most datasets and 300 milliseconds for HumanActivity; experiments use five random seeds and report mean and standard deviation.Lookback periods are 36 hours for MIMIC-III and PhysioNet, 3000 milliseconds for Human Activity, and 3 years for USHCN.Source: §5.1.3 · p. 7

5.2. Main Results

• TFMixer reduces MSE by 3.0% and MAE by 8.1% versus the second-best baseline, APN, across four diverse IMTS datasets.The results cover healthcare, biomechanics, and climate science.Source: §5.2 · p. 7, Table 1 · p. 7

5.3.1. ABLATION STUDIES

• Removing either branch, the reconstruction loss, spectrum refinement, or query-based patch mixing degrades forecasting performance.The ablations support complementary global and local branches, reconstruction regularization, refined spectral extraction, and query-based aggregation.Source: §5.3.1 · pp. 7–8, Table 2 · p. 8

5.3.2. PARAMETER SENSITIVITY

• Smaller datasets tend to favor dimensions 32 or 64, while larger datasets tend to favor 128 or 256; query-patch counts of 8 or 16 are typically optimal.Too few query patches miss semantic information, whereas too many fragment and couple it.Source: §5.3.2 · p. 8, Figure 4 · p. 8, Figure 5 · p. 8

• TFMixer is designed for irregular multivariate time series forecasting and jointly models time and frequency.Its dual-branch approach targets global dependencies and fine-grained local dynamics.Source: §6 · p. 8
• The framework combines a Global Frequency Module based on learnable NUDFT with a Local Time Module using query-based patch mixing.The conclusion says these modules address non-uniform sampling and information density imbalance.Source: §6 · p. 8
• Evaluations on real-world datasets show TFMixer consistently outperforms existing state-of-the-art methods.Source: §6 · p. 8

A.1. Datasets

• The four benchmarks include clinical datasets, irregular human-activity sensor data, and climate records from selected U.S. stations.The study uses 1,114 USHCN stations from 1996 to 2000, yielding 26,736 instances.Source: §A.1 · p. 11
• PhysioNet contains 12,000 patient series with 41 sporadically recorded clinical signals, while MIMIC-III covers 96 variables for 23,457 ICU patients.Both descriptions concern patients' first 48 hours of ICU hospitalization.Source: §A.1 · p. 11

A.2. Baselines

• PrimeNet is an IMTS pretraining model; SeFT (Horn et al., 2020) uses attention over unordered observations, and GRU-D (Che et al., 2016) uses trainable decay terms.mTAN (Shukla & Marlin, 2021) converts IMTS to reference points for classification.Source: §A.2 · p. 11
• NeuralFlows (Bilos et al., 2021b) models trajectory density with continuous-time normalizing flows, while CRU (Schirmer et al., 2022) models hidden-state transitions as a continuous-time process.GNeuralFlow (Mercatali et al., 2024) adds graph neural networks to NeuralFlows.Source: §A.2 · p. 11
• Forecasting baselines include graph-based tPatchGNN (Zhang et al., 2024a) and GraFITi, hierarchical Hi-Patch, frequency-attention KAFNet (Zhou et al., 2026), and patch aggregation APN (Liu et al., 2026b).Source: §A.2 · p. 11

• Evaluation uses Mean Squared Error and Mean Absolute Error, with dataset-specific lookback periods and forecast horizons.Most datasets forecast the next three timestamps; Human Activity uses a 300-millisecond forecast length.Source: §A.3 · p. 11
• The extended-horizon setting uses 12 hours for MIMIC-III and PhysioNet, 1000 milliseconds for Human Activity, and 1 year for USHCN.Lookback lengths follow Table 1.Source: Table 4 · p. 12
• Each experiment runs independently with five seeds, from 2024 to 2028, and reports mean and standard deviation.Training uses L1 loss and the ADAM optimizer.Source: §A.3 · p. 12

B.1. Masked RevIN Module

• Masked RevIN is designed for irregular forecasting to mitigate distribution shift and non-stationarity.It uses masked normalization before the encoder and masked denormalization afterward.Source: §B.1 · p. 12
• Normalization estimates each input instance’s mean and standard deviation using only visible, unmasked time points.This avoids masked zero paddings or mask tokens biasing the estimates, including under high masking ratios.Source: §B.1 · p. 12
• Denormalization restores the output’s original magnitude and scale using statistics stored during normalization.Source: §B.1 · p. 12

B.2. Varying Lookback Lengths and Forecast Horizons

• TFMixer achieves the best overall performance across most datasets when forecast horizons vary.The evaluation uses 12-hour horizons for MIMIC and PhysioNet’12, 1000 milliseconds for Human Activity, and 1 year for USHCN.Source: §B.2 · p. 13
• Forecasting performance generally improves with longer lookbacks, and TFMixer’s improvement trend is more stable than tPatchGNN’s.For MIMIC, gains plateau or slightly diminish at 36 hours; the authors suggest excessively long windows may introduce distribution shifts or irrelevant dependencies.Source: §B.2 · p. 13

B.3. Parameter Sensitivity

• Smaller datasets tend to perform better with dimensions 32 or 64, while larger datasets require dimensions 128 or 256.Excessive dimensionality can degrade performance on smaller datasets through overfitting.Source: §B.3 · p. 13, Figure 7 · p. 14
• Query-patch counts that are too small or too large hurt performance; the reported optimal W is typically 8 or 16.The cited sensitivity plot concerns the number of learnable query patches.Source: §B.3 · p. 13, Figure 8 · p. 14
• Moderate patch sizes such as 16 or 24 balance local-detail preservation and computational efficiency, while K preferences vary with periodicity and noise.For prominent, complex periodicity, larger K values such as 32 or 64 can help; beyond 16, increasing K may not help simpler or noisier datasets and may introduce spectral noise.Source: §B.3 · p. 13, Figure 9 · p. 14, Figure 10 · p. 14