Towards Robust Real-World Multivariate Time Series Forecasting: A Unified Framework for Dependency, Asynchrony, and Missingness
Authors: Jinkwan Jang
Hyungjin Park
Jinmyeong Choi
Taesup Kim
Venue: ICLR 2026 Poster (2026)
Keywords: Time Series Forecasting
Channel Dependence
Asynchronous Sampling
Missing Blocks
Primary area: learning on time series and dynamical systems
TL;DR: To address a trio of real-world challenges involving inter-channel dependencies, asynchronous sampling, and test-time missing blocks, we introduce ChannelTokenFormer that achieves robust performance without requiring explicit temporal alignment.

Abstract: Real-world time series data are inherently multivariate, often exhibiting complex inter-channel dependencies. Each channel is typically sampled at its own period and is prone to missing values due to various practical and operational constraints. These characteristics pose three fundamental challenges involving channel dependency, sampling asynchrony, and missingness, all of which must be addressed simultaneously to enable robust and reliable forecasting in practical settings. However, existing architectures typically address only parts of these challenges in isolation and still rely on simplifying assumptions, leaving unresolved the combined challenges of asynchronous channel sampling, test-time missing blocks, and intricate inter-channel dependencies. To bridge this gap, we propose ChannelTokenFormer, a Transformer-based forecasting framework with a flexible architecture designed to explicitly capture cross-channel interactions, accommodate channel-wise asynchronous sampling, and effectively handle missing values. Extensive experiments on public benchmark datasets reflecting practical settings, along with one private real-world industrial dataset, demonstrate the superior robustness and accuracy of ChannelTokenFormer under challenging real-world conditions.

Published as a conference paper at ICLR 2026
TOWARDS ROBUST REAL-WORLE MULTIVARIATE TIME SERIES FORECASTING:
A UNIFIED FRAMEWORK
FOR DEPENDENCY, ASYNCHRONY, AND MISSINGNESS
Jinkwan Jang" Hyungjin Park Jinmyeong Choi
Taesup Kim
Graduate School of Data Science Seoul National University
ABSTRACT
Real-world time series data are inherently multivariate, often exhibiting complex inter-channel dependencies. Each channel is typically sampled at its own period and is prone to missing values due to various practical and operational constraints. These characteristics pose three fundamental challenges involving channel depen- dency, sampling asynchrony, and missingness, all of which must be addressed si- multaneously enable robust and reliable forecasting in practical settings. How- ever, existing architectures typically address only parts of these challenges in iso- lation and still rely on simplifying assumptions, leaving unresolved the combined challenges of asynchronous channel sampling. test-time missing blocks, and intri- cate inter-channel dependencies. To bridge this gap, we propose ChannelTokeen Former, a Transformer-baset forecasting framework with flexible architecture designed to explicitly capture cross-channel interactions, accommodate channel- wise asynchronous sampling, and effectively handle missing values. Extensive experiments on public benchmark datasets reflecting practical settings, along with private real-world industrial dataset, demonstrate the superior robustness and accuracy of
under challenging real-world conditions.
INTRODUCTION
Accurate time series forecasting is critical in domains such as industrial monitoring (Jean-Picee al. 2024 ZZhao et al. 2024b), energy systems Yao et al. 2025 Nascimento et al. 2023 Hul 2024]. and healthcare (He & Chiang, 2025 Tang et al. 2024), where predictive insights directly influence operational safety, resource efficiency, and long-term outcomes. Yet, real-world time se- ries forecasting faces many practical challenges, including irregular sampling, concept drift within channels, and uncertainty quantification. Among these, complex interdependencies among channels, channel-wine -wiee asynchrony, and block-wise missingness often oversimplified by current modeling approaches, which typically assume channel indepennennd regularly sampled, and complete time series. In this work, explicitly address these three key challenges within unified framework.
One key challenge lies in the complex interdependencies among channels. Signals from differ- sensors subsystems are rarely independent; their interactions often encode latent correlative dynamics crucial for accurate forecasting. Some studies have directly adopted channel-dependent architectures (Zhang & Yan 2023 Wang al. 2024c Liu et al. (024c), exploring inter-channel de- pendencies under specific structural assumptions. As an alternative, many models employ a channel- independent design (Nie et al. 2023 Wang al. 2024b), which has shown competitive robustness to distributional drift (Han et al. 2024b). Nevertheless, carefully designed channel- dependent ap- proaches (Chen et al.) 2024a] Lee et al. (024) can still exploit interdependencies to provide richer predictive signals and deliver performance gains beyond channel-independent baselines.
Equal contribution.
'Corresponding author.

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Conventional Setting: Regular Sampling Rate
Charnel 1 (h-mmaad)
Interpolation-nnduced "Distortion"
Amp.
NosAH b
[FFT Magnitude Spectrum)
Channel N
Newly Proposed
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(FFT Phose Spectrum)
Channal- wasse Anynchronuu Sampling
Inter Pototooaiion (Fonward Fil, Linear, Spiine,,. Regular Patching
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Figure I: Our proposed practical conditions highlight the simultaneous presence of channel- asynchronous sampling, block-wise missingness at test time, and inter-channel dependencies. Inter- polation over coarsely sampled regions leads to signal distortion. See Appendix B for more details.
Another difficulty arises from the hetcrogeneity of data sources (Reiss et al., 2019 Filho et al, 2024 Dong et al. 2025 Ying et al. 2025 Agency 2025). Time series signals commonly origi- nate from diverse sensors, such as those tracking temperature, pressure, actuator positions, bi- ological signals. Due to differing physical properties and application contexts, these signals are frequently sampled varying temporal resolutions, often leading to channel-wise (multi-source) asynchronous sampling in practice. Nonetheless, most existing approaches (Wang et al.) 2024cc Chen et al.) 2025) assume idealized input conditions: fully observed sequences sampled identical intervals and aligned timestamps across channels. Such assumptions ignore variations in sampling periods and sequence lengths. complicating both model design and data preprocessing in real-world settings. Beyond coannel-iddeeendent strategies, the challenge of channel-wis, asynchronous sam- pling remains underexplored.
As with interdependencies and sampling variability, missing data represents another key challenge in time series modeling. In practice, discrete missing values are already present in standard bench- marks such as ETTm, and modern neural time-series models can exhibit certain degree of implicit robustness to such patterns (see Appendix (.6). However, real-world signals often contain long contiguous intervals of missing observations, often arising from maintenance issues, sensor mal- functions, or communication failures (You et al., [025 Nejad et al., 2024). In such cases, naive interpolation, which is commonly applied during preprocessing. can be unreliable or misleading, particularly for dynamic signals. More robust approaches can instead benefit from leveraging cross- channel correlations infer missing dynamics, rather than relying solely on extrapolation from partially observed inputs.
These three core challenges highlight the need for forecasting models that remain robust under real- istic conditions, as illustrated in Figure [ In particular, models should be capable of: (i) leveraging cross-channel dependencies to capture complex interactions and maintain structural consistency; (ii) handling asynchronous, variable-length inputs across channels without strict alignment or re- sampling; and (iii) addressing block-wise missing intervals during test-time inference by directly exploiting information from other channels, rather than relying on imputation methods.
While these requirements are clear, prior approaches have only provided partial solutions. Channel-dependent models capture inter-channel dependencies but overlook asynchrony and miss- ingness; ionunell-iddeeedeeen strategies handle misalignment but lose cross-channel structure; and specialized approaches for missing values do not account for asynchrony, while irregular time series methods focus on within-channel non-uniformity under sparse settings rather than cross-channel heterogeneity in sampling.
To overcome these limitations, we propose Foamner a unified Transformer-bass fore- casting framework designed tackle all three challenges in unified and simultaneous manner: channel-wise asynchronous sampling, test-time missing blocks, and cross-channel dependencies. The key is revisit channel tokens under realistic multivariate conditions, viewing them compact
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representations that aggregate local temporal information within each channel while also captur- ing cross-channel dependencies. Although this idea resembles the channel-level summary tokens in iTransformer (Liu et al., 2024c) and TimeXer (Wang et al., 2024e), our framework distinguishes itself by introducing a mask-guided attention strategy that enables a unified treatment of intra- and cross-channel interactions. In this way, the tokens serve as global attention anchors. Predictions made from compressed channel-wise representations that account for heterogeneous sampling peri- ods and masked tokens from missing blocks, regardless of the number of local tokens. This design preserves natural channel resolutions, remains robust to partially observed inputs, and adaptively captures cross-chamen dependencies.
2 RELATED WORK
Channel-Dependder (CD) Strategies for Multivariate Time Series CD strategies jointly model
all channels to capture inter-channel dependencies, and has been widely adopted across various
neural architectures, including TCN/CNNs (Luo & Wang
2024
Wu et al. 2023 MLPs (Han al.
2024a; Huang et al.] (2024). Transformers (Cheng et al. 2024
Liu et al., 2024b) Shu & Lampos
2024 Wang et al. 2024e c Zhang & Yan
2023] Liu et al. 2024c Yang al.] [2224] Yu 2023),
and GNNs (Huang et al. 2023] Yi et af. 2023a Cao et al. 2020 Liu et al. 2022). Some recent
models (Chen et al. 2024a| Qiu et al. (025) refine CD strategies by learning sparse or clustered
inter-channel structures. Moreover, TimeFilter (Hu et al. 2025) constructs a spatial-temporal graph over patches and filters out spurious dependencies a patch-specific manner. Generative approaches such as COSCI-GAN (Seyñi et al., 2022) also employ CD principles, emphasizing cross-channe coherence in the context of sequence generation. These approaches are particularly effective when channels are correlated, since sparsely observed or low-resolution signals can benefit from denser ones through cross-channel interactions. In practice, however, most CD strategies only partially address real-world challenges: they typically assume aligned sampling across channels, are not designed to handle block-wise missing intervals and can be sensitive to noisy inter-dependencies.. In contrast, our method utilizes channel tokens that explicitly encode cross-channel relationships while remaining robust under asynchronous sampling and block- wise missingness.
Irregularity of Multivariate Time Series (Multi-Source Asynchrony) Irregularly sampled mul- tivariate time series characterized by channels observed at non-uniform and often unpredictable time points, resulting in irregular gaps within individual channels. A number of recent works (Che)
let al., 2018: Shukla & Marlin 2021 Li et al. 2023a Chen et al. [203] Zhang et al. 2023 2024
Mercatali et al.] |024 Liu et al. 2024a 2025 Kim & |02] Klötergens et al. 20025 address this challenge by developing models that operate directly on non uniform time intervals, often using time encodings, interpolation-based alignment, attention mechanisms, or graph-based structures. Models such as Raindrop (Zhang et al. (022b) and Hi-Patch (Luo et al. (2025) are tailored for highly sparse settings, focusing on wise inter-channel relations rather than capturing temporal patterns. As result, they are less suited to structured multi-source asynchrony and do not align with our proposed practical setting, which unifies three core challenges in continuous, long-term forecasting. In con- trast, we target a real-world setting, including fixed but distinct sampling periods across channels, as categorized in the recent Time- IIme-IMM dataset (Chang et al. 2025).
Missing Value Handling for Time Series and Test-time Missing Intervals Models such as BRITS (Cao et al., 2018) and SAITS (Du et al., 2023) reconstruct missing values through impu- tation based on recurrent dynamics or self-attention, while general-purpose frameworks like Times- Net (Wu et al., 2023) and TimeMixer++ (Wang et al. 2024a) treat imputation as a downstream task. While these models can perform imputation. they typically rely on a separate forecasting module to complete the end-to-end pipeline, which increases complexity and may hinder practical deploy- ment in real-world scenarios. BiTGraph (Chen et al., 2024b) and S4M (Jing et al., 2025) integrate missing- Masuung-value handling into forecasting architectures, but still assume regularly sampled time series. Other studies such as TimeXer (Wang et al., 2024c) and TFT (Zhou) 2023) test robustness to missing inputs by simple replacements, offering explicit mechanism for structured sparsity. SERT (Nejad) et al. 2024) highlights the importance of handling block-wise missingness in real-world forecasting scenarios, reinforcing the need for explicit mechanisms beyond simple replacement. In contrast, approach leverages channel tokens to minimize imputation induced distortion and directly handle test-time missing intervals during inference, providing a robust solution for real-world settings.
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3 PROBLEM FORMULATION
Asynchronous Channel-wise Observations We consider a multivariate time series forecasting task with N channels, where each channel E [I....,N} is sampled riod. Let |- denote the relative sampling periods across channels, normalized such that Given an imput length L, the number of valid sampled points available for channel i becomes L; L/ss]]- Similarly. for prediction horizon of length H, the number of future points to be predicted for channel is H, = [H/s]] This adjustment is made in the data- point domain, not the time domain, so that the actual time span covered by the prediction remains consistent across channels despite different temporal resolutions.
distinct fixed pe-
SN.
Our learning objective computes the forecasting enror only at valid (observable) positions for each channel, which is standard in irregular forecasting settings (Luo et al. 2025 Zhang et al. 2024). We refer to this Shannel-aggregaae MSE (CMSE):
N N
N H:
H:
Littal
1
(
)/0²²
N
For evaluation, we adopt CMSE as the primary metric, along with CMAE, its MAE-based variant.
Test-time Missing Intervals In many real-world cases, observations exhibit persistent sparsity due to temporary failures, communication losses, or maintenance, leading to block-wise missing in- tervals at test time, as shown Figure This type of missingness is typically easy detect, as most channels rarely show prolonged zero readings unless the device is disconnected or malfunctioning.
Conventional patch- methods cannot handle variable numbers of patches per channel, and thus typically fill missing entries within each patch using zeros or mean values. In contrast, our model adopts a conolaelaapin patch representation that supports variable patch lengths and counts across channels. This enables the removing fully unobserved patches from the input, rather than merely filling in missing values. This formulation supports forecasting under heterogeneous horizons from channel- -eannel-wise asynchronous sampling, while also accommodating channel-specific missingness in the input at test time. The forecast horizon is assumed to be fully observed, since missing intervals in the future cannot be predicted in advance.
CHANNELTOKENFORMIO
Repurposing Channel Tokens for Realistic Settings Pioneering studies have explored the use of auxiliary tokens, for example, query tokens for extracting rorecasting-relevan information from sequences of varying lengths (Kim et al. (2024), and special tokens for mediating information ex- change across heterogeneous entities (Zhao et al. 2024a). Channel-level summary tokens have also been introduced in i'Transfrmerr (Liu al. (222c) and TimeXer (Wang al., (022e).. Building on these insights, we repurpose channel tokens for realistic multivariate forecasting conditions, where patches are inherently unbalanced across channels due to heterogeneous sampling, optionally their own dominant frequencies, and prolonged missing intervals. In our framework, Channel Tokens act as compact abstractions that summarize each channel's 's patches into stable channel-level embed- dings, so that forecasting relies these embeddings rather than uneven patch sequences that would otherwise complicate the decoder input. This reframing allows the model to remain robust under practical conditions where patch imbalance poses fundamental challenge.
Channel-wise Frequency-based Dynamic Patching and Tokenization Prior rrqquency-guide patching (e.g., Moirai (Woo et al., 2024). LightGTS (Wang et al.) 2025)) in time series foundation models targeted variable-length segmentation for univariate streams with heterogeneous granulari- ties. In contrast, we repurpose this frequency-guided granularity selection for a channel-wise asyn- chronous multivariate setting. The detailed procedure is described in Algorithm [ This enables non-aligned per-channel patching while preserving efficiency by sharing projection layers across channels with the same patch length and introducing dedicated channel tokens. For each channel, we estimate a dominant period via FFT to determine the patch length, and use a sampling-awe aware fallback when clear peak exists. We then split each channel into non-overlapping patches and
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Frequency
"Residual Network with Unified Attention Layer"
tine Intervals
[Inference]
Readionly Masking
Channel Tubene
Output
(Pattch Langtr)
Token-count Agnostic
Positional Eribooding Channel Emteddingg
N
Shand Decodirr (Bemppinng periid!
[Training]
++
Local Tokaut
N
LLocal Token
Grose- Channel Attention
Figure 2: Overview of ChannelTokenFFornee (CTF). All tokens across channels pass through a unified attention layer, where local and global information is aggregated into channel tokens. Only the channel tokens are decoded by decoders, each shared among channels with the same sampling period, produce the final prediction.
map each patch to local token. For each channel, we also initialize a small set of channel tokens. Each local token receives fixed positional embedding Ppux and a learnable channel embedding e while the channel token receives only ( Full details on thresholds, datasct-specific examples, and the fallback procedure are provided in Appendix A.2
(i)
Training-TTi Time Proxying for Test-Time Missingness via Channel-wise Patch Masking Build- ing on channel-wise dynamic patching, our framework supports variable input lengths via patch masking and explicitly targets block-wise missingness test time. Unlike prior uses of random training-
patch masking that primarily serve as regularizer to curb overfitting, repurpose inputs may be partially missing,
time proxy for realistic test-time incompleteness: inference- the model is trained to handle such missing blocks by design. We adopt random patch mask- ing, as introduced in PatchDropout (Liu et al., 2023). applied to a subset channel-wise patches at the input stage during training simulate llok-wiiee missingness at test time. As illustrated in Figure 2 if a missing segment spans an entire patch at test time, the corresponding local token is removed at the input stage, thereby being excluded from the attention computation. This prevents fully unobserved patches from contributing spurious signals, minimizing the risk of propagating in- valid information. Despite temporal gaps induced by masking, temporal ordering and context are retained through fixed positional embeddings. This perspective shift improves resilience to test-time missing intervals and, as a side effect, acts as an implicit regularizer that mitigates overfitting in low-resource settings. This procedure is formalized in Algorithm
A Unified Mask-Guider Attention with Channel Tokens for Real-World Challenges jointly address three practical challenges multivariate time series, we design unified, mask maakk-uided self- attention mechanism that integrates local and global representations within single attention oper- ation. As illustrated in Figure 22, this design unifies both token types in one attention step. Each channel i E {I,..., N) contributes two types of tokens: (i) local tokens T( E (TT xd represent- ing patch-level embeddings, and (ii) global channel tokens ((i) E m summarizing high-level contextual information, where ( and C denote the number of local and channel tokens, respec- tively. The full token sequence is constructed as:
N
x T : : C(N) E Txd where T E + m).
(2)
This unified sequence is passed through masked multi-head self-attention layer with residual con- nection:
(OKT (dd + +M V.
X = X Attention CaatiionQQAAr K,K,V) X + softmax
(3)
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Charnel
Takuna
Tokona
Chantel
Masked (1)
Chermel
only
Attended (0)
0
Read-enty
Tokona
Takona
(a) Attention Mask
(b) Attention Map
Figure 3: Our unified attention masking strategy. Local tokens perform intra-temporal attention within the channel. Channel tokens aggregate local and cross-channel information, but visible to local tokens and do not attend to themselves. Optionally, attention among channel tokens from the same channel can be masked encourage inter-channel interaction and reduce redundancy.
Here, M E TxT encodes structural constraints on how tokens can attend to one another, reflect- ing their types, channels, and masking. Our tailored masking scheme constructs M as illustrated in Figure 3] (1) Local tokens attend only to other local tokens within the same channel, enabling intra-temporal modeling. (2) Channel tokens attend to their own local tokens and other channels' tokens, but are not accessible to local tokens due to their read-only role in the attention mecha- nism. (3) Channel tokens do not attend themselves; for each query ((+), the key C is excluded to avoid self-reinforcement and to encourage informative cross-channel interaction. Our unified masking strategy, inspired read-only prompt attention (Lee et al. 2023) and attention control techniques (Kim et al., 2025), is implemented without modifying the standard Transformer archi- tecture. Stacked masked attention blocks with residual connections and layer normalization enable deeper modeling of temporal and cross-channel dependencies. Although channel- wise tokenization introduces additional parameters and our unified attention remains quadratic the total number of tokens, the proposed masking scheme focuses attention on the most relevant token interactions, and empirically does not lead to prohibitive inference latency for realistic channel counts and sequence lengths (see Appendix
EXPERIMENTS
To assess the effectiveness ChannelTokenFoormer (CTF) under our proposed problem settings, we conduct extensive experiments on four widely used multivariate time series benchmarks adapted to practical conditions, and on wolld datasets from air monitoring and the LNG cargo handling system. Additional results under conventional settings are provided in Appendix
Datasets We evaluate forecasting performance under channel-wise asynchronous sampling using four multivariate datasets. TTTT-paactcall and ETT2-practical (ETTI & ETT2) modify ETTml and ETTm2 (Zhou et al., 2021) by resampling channels to domain-specific temporal resolutions (e.g.. 1-hour for load, 1i-minete for temperature). Weather-practical (Weather), adapted from the Weather benchmark (Wu et al., 202I), assigns heterogencous sampling periods to channels based Saaaa--Solaaiindoweer--pr
on distinct physical characteristics. solar and wind power series from the Monash Forecasting Archive (Godahewa et al., (021). with variables resampled to 20-minute and 5-minute intervals, respectively. EPA-Air (EPA), adapted from the U.S. Environmental Protection Agency's air quality monitoring data (Chang et al. 2025, collects measurements from several regions with heterogencous sampling periods across channels (e.g., 1-hour for temperature, 8-hour for PM2s. among others). We choose four regions, Maricopa, Richmond, LA and Hillsborough. LNG Cargo Handling System (CHS) is real-world industrial dataset collected from an LNG carrier, consisting of sensor channels related cargo operations,
(SolarWind) combines

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Published as a conference paper at ICLR 2026
ship navigation, and surrounding weather conditions. To assess robustness under block-wise test- time missing intervals, we additionally conduct experiments on SolarWind by varying the missing ratio. Detailed dataset specifications are provided in Appendix A.I
Baselines We compare CTF against representative state-of-the-art models across architectures
and modeling strategies that target inter-channel dependencies, irregular sampling, or missingness. Transformer baselines include TimeXer (Wang et al.) (024e), i'Transformer (Liu et al. (024c), and PatchTST (Nie et al. 2023). We also cover CNNs, GNNs, and MIPs, including TimesNet eW et a
2023). CrossGNN (Huang al., 2023), TimeMixer+ (Wang al., 2024a). and DLinear (Zeng ct al. 2023). We further include strong recent
methods, DUET (Qiu et al., 2025
and TimeFilter (Hu et al., 2025); smeeular-saapling methods such as t-PatchGNNN (Zhang et al.
2024) and Hi-Patch (Luo et al. 2025); ) and a missingness-robune approach, BiTGrrpp (Chen et al,
2024b).
Setup and Implementation Details Baseline models for regular time series assume uniformly sampled and fully observed inputs. To satisfy this requirement, we employ linear interpolation to re- cover values unobserved at timestamps not covered by channel-specific sampling periods. To isolate architectural effects from interpolation, we further implement interpolation-free variants, including version of TimeXer (Wang et al. 2024e) modified handle temporally non-aligned cross-channel inputs. Implementation details are provided in Appendix C.4 In contrast, irregular or missing- aware baselines do not require interpolation, as they operate solely on observed values. To simulate block-wise missingness at test time, we randomly mask contiguous input regions. For fairness, reg- ular baselines without explicit missing-value handling are also provided with linearly interpolated values for these masked regions, consistent with their training procedure. For each dataset, input and prediction lengths are selected to reflect the channel- -wise temporal resolution. Details for each dataset are in Appendix A.2 Despite heterogeneous sampling periods across channels, it is practi- cal define input and prediction windows by fixed time duration instead of requiring the same counts of sample points across channels. The actual number of input and output points per channel is determined by its sampling period, with shorter periods yielding more points over the same window.
5.1
MAIN RESULTS
Case Channel-wise Asynchronous Forecasting We evaluate forecasting performance using datasets that reflect realistic sampling heterogeneity across channels. Unlike standard baselines that rely on interpolation, our approach preserves the original sampling structure and operates directly on observed values. When we average CMSE and CMAE over all prediction lengths (Table D. CTF delivers strong overall performance, achieving the best or second-best results across datasets. This underscores the benefit of avoiding interpolation when channels are temporally misaligned. In particular, CTF handles asynchrony more effectively than irregular modeling on most datasets.
Detailed results for each prediction length and dataset provided in Appendix [ As CTF uses frequency-domain information for dynamic patching, we also compare it against frequency-based approaches and still consistently achieve superior performance (see Appendix C.5 for more details).
Table 1: Forecasting performance on the channel-wist asynchrony (Case 1).
Approadi
Clarnn---eeeninni
Clasnck-Enkggeninnn
Iaregals modeling
Model
CTF(our))
TineFiterr
DUET
TimeXer
ITrans
CoosCINN
TimeaNer
TimsMiisas
PUCATST
DLiseu
Hi.Puch
(PudiGNN
Meuic
CMSE CHAE SMAB/CNSE CMAE CMSE CMAE CMAE
CMSE CHAE
CHIE CMAE
CMSE CMAE CSSE CMBE CHAE
BUTT
0,599 9:410 (1412 0.118 0.424 0,425
(.422 0.424
0.435 0443
0.428 0.415 (1452 0.444 0.433 0,432
g.111 p.51d 0.425 0,430
0,448 0.454
0.405 0.570
RTT3
0,377
0,83 (n 183 a 388
0.988 0.189
(.880 0 158
o 0.396
0.386 0858
(1 1.999 a 906 0.396 0.400 0,900 a 906
0.455 0.449
0.907
0.401 (1999 0.406
SoluWind
0.403
0,422 (1404 0.44)
0:435 0.491 0.424
0.459
0.485
0.465
0.478
0.429 0.468 0.417 0.467
0..42 0.516
0.431
0.473 0.447
Westee
0,275
0,296
0.309 0.322
0.300
0.310
0.323
0.285
0.315
(1331 0.322
0.276 0.296 0.275
0.331
0,30] 0.320 0.312
EPA
0.776
0.58s (1863
0.599
(.788
0.579
0.856
0.611
0.382
0.611
0.979
0.665
(1937
0.643
0.931
0,637
0.844
0.597
[.047
0.713
0,805
0.629
0.801 0.628
0.285 0.126 (0.304
0.307
0.133
0.298
0.128
0.305
0.132
0.298
0.139
(0.330
0.140
0.296 0.129
0,315 0.132
(0.35) 0.248
00301
0.129
(.299
Case 2: Channel-wise Asynchronous Forecasting with Test-time Missing Blocks We further evaluate all models under more challenging setting, where test-time inputs include bisck-wise miss- ing intervals, beyond discrete missing values, explicitly indicated to the model, on top of channel- wise asynchronous sampling. This configuration encompasses all three core challenges of interest.
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In this case, all models receive inputs containing contiguous missing segments filled with zeros. But for a fair comparison, linear interpolation is applied to regular baselines to avoid underestimating their capability. Unlike other approaches, CTF can internally adjust input length by design, enabling the use of zero-patch masking to effectively handle missing regions. As shown in Table our model consistently outperforms most baselines in average CMSE and CMAE for overall prediction lengths across different missing ratios, maintaining high performance even as the severity of missingness increases. This robustness stems from the use of random patch masking during training, which im- proves the model's ability to generalize to incomplete patterns. Unlike other methods, CTF avoids distortion from artificially zero-filled values and preserves signal fidelity under block-wisc missing conditions. Detailed results across datasets are provided in Appendix E The corresponding details are provided in Appendix A.2
Table 2: Channel-wise Asynchronous Forecasting performance on the SolarWind dataset under block-wise test-time missingness with varying missing ratios m (Case 2).
Approads
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0.126 8.40
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10.45 0.459 0.474 1u475 0491 0,432 0.4ss (0.68 0:474 0-45 0.473 04777
(1442 0.523 0.427 0.473 0.450 0.436 (:426
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putt cat 14477
(u4RI 0.555 (0:4 nas nuse 0.512 0.457 0.505 cos 0,405 0u407 u.a9 (0.85 0.500
1.1% 0.538 0.450 0.405 0.667 0,514 0:444 0.523
0.375
9.452
907
a 471 0.40
s69 0.960.955 064 nom nuww 0.546 0,511 0.500 517 0.527 0.511 0:545
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0.6
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0.542
9.487 543
0.470
0.500
9.478
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jo 0.53 (0599
0.620 (.514 assa juso6 ous96 0.553 0.5e2 (1.55 0,554 0u838 0.83 (0.993 0st6
0.52
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0,500 0.542
(.522 0.571
(.495 0.562
5.2 ABLATION STUDY
To assess the effectiveness of our design in address- ing the three key challenges, we conduct an ablation study where each component is selectively removed or replaced. For Dependency, we disable cross- channel attention. For Asynchrony, we remove the dynamic patching mechanism, forcing fixed patch boundaries that ignore dominant frequencies and channel-specific sampling periods. For Missingness, we eliminate patch masking during training and test time, exposing the model to zero-filled inputs in- stead. These experiments show that robust forecast- ing under practical conditions requires all three components, since removing any one degrades per- formance. Additional ablations are provided in Appendix C.1
Table 3: Ablation study under channel-wise asynchronous sampling with block-wise test- time missing (missing ratio = 0.375). Results are averaged across all prediction lengths on the SolarWing Wind dataset.
Ablation Setting
CMSE
CMAE
Full model (CTF)
0.452
0.508
Channel Dependence (CI only)
0.521
wio Dynamic patching
0.494
0.536
wro Patch masking
0.458
0.508
6 ANALYSIS
6.1
UNDERSTANDING THE OPTIMAL NUMBER CHANNEL TOKENS
The number of channel tokens was varied to analyze its impact on asynchronous forecasting, Results in Table 4 show that the optimal choice differs by dataset: two channel tokens work best for ETTI and SolarWind three for EPA, and one for Weather and CHS. These outcomes reflect how channels interact under asynchronous sampling. When channels move in a similar and highly correlated way, single channel token captures the shared pattern and more tokens add redundancy. ETTI and SolarWind, which exhibit two dominant groups of channels, benefit from two tokens. EPA shows stronger heterogeneity, and three tokens better capture its diverse structures. Overall, the optimal number of channel tokens depends on the degree of channel similarity and diversity in each dataset.
6.2 ROBUSTNESS TO INPUT LENGTH VARIABILITY
Random patch masking is primarily designed to make the model robust to incomplete inputs caused by block-wise missing intervals at test time. An additional benefit of this design is robustness to input length variability, which remains limitation for conventional forecasting models. Most exist- ing approaches train and evaluate separate models for each target input length, and even models that technically accept variable lengths often degrade in performance when the sequence length changes.
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Table 4: Effect of varying the number of channel tokens (1, 2. 3) on asynchronous forecasting per- formance. Results are reported in CMSE, averaged over all prediction lengths.
Table 5: Evaluation of robustness to vary- ing input lengths during test time, Our CTF is trained with an input length of 576 using random patch masking, and tested with other shorter input lengths 288, 360, and 432 on the SolarWind dataset.
Dataset
Channel Tokens
ETT2
SolarWind
Weather
RPA
CHS
Test Input Length
288
360
432
576
0.399
0,377
0.274
0.775
0.282
2
0.398
0.377
0,400
0.275
0.777
0.289
Avg. CMSE
0.429
0.461
0.448
0.409
3
0.411
0.375
0.406
0,281
0.774
0.295
Avg. CMAE
0.457
0.495
0.491
0.457
As shown in Table 5 CTF maintains stable performance across all test input lengths on the Solar- Wind dataset without requiring retraining. Since each channel has a different sampling period, we define the input length based on the channel with the smallest sampling interval, which corresponds to 576 time steps over days. Even when this length is reduced to 288 (equivalent to 1 day), the CMSE increases by only 4.9%, demonstrating strong generalization to input length variation, which another essential property for real-world forecasting under inconsistent input conditions.
6.3 FREQUENCY BIAS ANALYSIS IN PREDICTED TIME SERIES
We analyze how interpolation affects the frequency characteristics of predicted time series under practi- cal conditions with channel-wise asynchronous sam- pling. To this end, we focus TimeXer (Wang) eet al. (024e)), which is structurally similar to CTF ours
Low band
Ground Truth Predicted(CTFF) Ours Prodicted
0.40
Amplitude
0.30
Attenuation Prevented (No Ixterpolation
and achieves nearly identical performance in the conventional setting on the ETTml bench- mark. This makes TimeXer a strong reference point for fair comparison. In the practical setting, how- ever, TimeXer can only process asynchronous in- puts by performing interpolation, since its original design cannot directly accommodate non-aligned se- quences without architectural modifications. As a result, any performance gap observed here reflects interpolationninducce distortion rather than inherent modeling differences. We analyze dominant fre- quency difference and amplitude spectrum RMSE across frequency bands in the ETTI dataset (Ta- 6l. In ETTI (Table 6). both models localize dominant cycles similarly, since the main periodic- ities (e.g., 8. 12, and 24 hours) lie in the low- to mid-frequency range where interpolation-inducedd phase delay is minimal. However, amplitude spectrum RMSE shows consistent gaps: as Figure 4 illustrates, both models exhibit some attenuation, but the effect is noticeably stronger for TimeXer, while CTF better preserves spectral energy. Thus, although TimeXer matches CTF in conventional settings, its reliance on interpolation the practical setting introduces frequency bias that propa- gates into predictions. By avoiding interpolation, CTF mitigates this distortion and better retains trend and periodicity under asynchronous sampling.
0.20
band
High bhad
0.10
0.30
Frequency
Figure 4: Frequency-domain comparison be- tween CTF and TimeXer test sample from ETTI, showing that amplitude attenua- tion is prevented across all frequency bands.
Table 6: Frequency-domain comparison between CTF (interpolation-free)) and TimeXer (interpolation-baseel the ETTI for prediction length of 192 (2-day window). Lower values indicate better spectral fidelity. RMSES are computed over three frequency bands under 1-hour sam- pling (fa = 1): Low (0.00-0.10), 10), Mid (0. 10-0.25), and High (0.25-0.50) cycles/hour, corresponding to periodicities longer than 10 hours, between 4 and 10 hours, and shorter than 4 hours, respectively.
Model/ Metric
Dominant Freq. Diff.
Low-band RMSE
Mid-band RMSE
High-band RMSE
CTF (Ours) TimeXer
0.0098
0.1877
0.0735
0.0457
0.0099
0.1901
0.0745
0.0473

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7 CONCLUSION
A practical forecasting setting is introduced where three core real-world challenges oceur simulta- neously: channel-wise asynchronous sampling, block-wise missing intervals, and complex cross- channel dependencies. Instead of interpolation, our approach handles both missing segments and sampling gaps through masking and frequency-based dynamic patching. The unified mask-guided attention leverages channel tokens with read-only masking and cross-channel attention, consistently outperforming prior methods across real-world conditions. Our scalability analysis indicates that the masking guided unified attention design remains computationally manageable for realistic channel counts and input lengths. However, www--hhhhhhdddiiooee regimes with thousands of channels (e.g., Traffic) will require additional mechanisms such as explicit channel sparsification or grouped chan- nel representations, which we leave for future work. Beyond scaling. future work will also consider applying ChannelTokenForree to broader domains, integrating multimodal signals, and improving interpretability of channel interactions, thereby expanding its utility real-world forecasting.
ACKNOWLEDGGENNTS
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2022-00345580, Research on AI Robustness Against Distribu- tion Shift Woald Scenarios; and No. 555-2023-0022266 Center for Optimizing Hyperscale AI Models and Platforms). We sincerely thank HD Korea Shipbuilding & Offshore Engineering Co., Ltd. (HD KSOE), particularly the Thermal Systems Research Lab, for granting access to pro- prietary data under confidentiality agreement and for supporting the secure data-sharing process that enabled this research.
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A IMPLEMENTATION DETAILS
A.1 DATASET CONFIGURATION UNDER MULTI-SOURCE ASYNCHRONY
We conduct long-term forecasting experiments on four modified benchmark datasets adapted to our practical setting, as well as two real-world datasets. In this setting, each channel provides differ- ent number of observations over the same temporal window, reflecting channel-specific sampling periods within the multivariate time series. This configuration more closely mirrors real-world con- ditions, where sensor deployment strategics and data acquisition policies vary across channels.
To further contextualize this setting, note that widely used benchmarks already involve substan- tial preprocessing to obtain complete, regularly sampled grids. For example, the ETT datasets typi- sensor streams for load-related channels into 15-minute or hourly averages. This procedure reduces the statistical complexity of the original processes and introduces its own form of synthetic regularity. Our practical variants do not create any new values; instead, they relax this enforced uniformity by selectively downsampling channels whose variability, as- sessed in the frequency domain, indicates sufficient oversampling, The resulting multi-rate sensing patterns more plausibly reflect real-world conditions while not materially altering the underlying temporal dynamics.
cally aggregate
ETT1-practical & ETT2-practical (ETTI & ETT2) ETT1 and ETT2-practical variants of the commonly used ETTml (Zhou et al. 2021) and ETTm2 datasets. Unlike their original versions, where all channels sampled uniformly at 15-minute intervals, ETT1 -prracccl and ETT2-practical introduce channcl-specific sampling periods to reflect industrial environments, where sensors operate different acquisition frequencies depending on their physical properties and monitoring requirements. Specifically, the channels are grouped into two categories: load-related channels and the oil-temperature (OT) channel. The load-related channels are downsampled to an hourly rate, representing slowly varying electrical load measurements, while the OT channel retains its original I 15-minute resolution to capture higuefryy thermal dynamics.
Monash-Solarrinnn Wind-practia- (SolarWind) Konash-SolarWind-prceee is derived from Aus- tralian Energy Market Operator (AEMO) and the Monash Forecasting Repository (Godahewa al., (021) and built upon high-frequency power generation data originally collected 4-second intervals. In real-world energy monitoring systems, distinct sensors are used for solar and wind power, each exhibiting unique temporal behaviors and operational constraints. To replicate these real-world dynamics, we apply channel-specific downsampling: solar-related channels are resam- pled 20-minute intervals, capturing smooth and gradual irradiance trends, while wind-related channels are resampled at 5-minute intervals to reflect the rapid fluctuations characteristic of wind dynamics. This restructured dataset captures modality-specific asynchrony and sampling hetero- geneity, offering challenging benchmark for assessing model robustness under realistic conditions where different signal types operate on distinct temporal resolutions.
Weather-practical (Weather) Weather Srather-practical is constructed from the widely used Weather (Wu et al.) 2021) benchmark. While the original dataset assumes uniform sampling periods across all channels, this assumption rarely holds real-world environmental monitoring systems. In Weather-practical, we reorganize the channels to reflect heterogeneous sampling frequencies, based on the intrinsic temporal dynamics of each channel. For instance, fast-changing environmental channels such as wind velocity, precipitation, and solar radiation are sampled at 10-minute intervals, whereas more stable channels such as air pressure or vapor pressure are sampled at coarser hourly rate. Mid-range channels such as air temperature and specific humidity are sampled at 20-minute or 30-minute intervals in accordance with their moderate temporal variability. This configuration, summarized in Table mimics the asynchronous and heterogeneous acquisition patterns encountered in operational weather stations and provides a more realistic testbed for evaluating forecasting models under non-uniform input conditions.
hhtp://www.emwwch.om.auu
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Table Channel Specification in the Weather-practical Dataset
Channel
Description
Sampling Period
P (mbar)
Air Pressure
hour
(PC)
Air Temperature
20 min
Tpot (K)
Potential Temperature
hour
Tdew (9C)
Dew Point Temperature
hour
rh ($)
Relative Humidity
min
VPmax (mbar)
Saturation Water Vapor Pressure
hour
VPact (mbar)
Actual Water Vapor Pressure
hour
vPdef (mbar)
Water Vapor Pressure Deficit
hour
sh (g/kg)
Specific Humidity
hour
H2OC mmmol/mol)
Water Vapor Concentration
hour
rho (kg/m²)
Air Density
hour
(m/s)
Wind Velocity
10 min
max. wv (m/s)
Max Wind Velocity
min
wd (9)
Wind Direction
10 min
rain (mm)
Precipitation
10 min
raining (5)
Duration of Precipitation
min
SWDR (M/m²)
Short Wave Downward Radiation
10 min
PAR (μmоl/ /mm /s)
Photosynthetically Active Radiation
min
max. PAR (umo 1 / /s)
Max PAR
10 min
Tlog (cC)
Internal Logger Temperature
hour
oT
Operational Timestamp (Offset)
10 min
EPA-Air (EPA) EPA-Air is sourced from air quality monitoring data (Chang al. 2025) of the U.S. Environmental Protection Agency collects measurements from several monitoring regions across the United States. In real-world air quality monitoring, different channels are reported heterogeneous temporal resolutions due to channel-specific measurement practices. To reflect these conditions, we assign channel-see channel-specific sampling periods: temperature- relattd variables are recorded at 1-hour intervals, fine particulate matter (PM2s) is reported at 8 8-hour intervals, air quality indices (AQI) are provided at daily intervals, and ozone measurements are reported at weekly intervals. For our experiments, we selected four representative cities, Maricopa, Richmond, Los Angeles, and Hillsborough, from the original dataset that contains measurements from eight metropolitan areas. This real-world dataset preserves the inherent asynchrony and heterogeneity of environmental sensing, yielding a benchmark that mirrors practical conditions where meteorological variables, pollutant measurements, and derived indices recorded on distinct temporal resolutions.
LNG Cargo Handling System (CHS) CHS is a real-world dataset collected from an LNG (Liq- uefied Natural Gas) carrier during a ballast voyage. The dataset contains time series mcasurements from total of 52 channels, each sampled at distinct frequencies based on their physical character- isties and operational requirements. Specifically, weather and navigation channels, such latitude and wave period, are sourced from external systems and sampled at a coarse hourly rate. In contrast, onboard sensor channels are sampled more frequently to support real-time monitoring and control. These include machinery-relatee channels (e.g.. Low Duty Compressor (LDC) suction pressure, main engine inlet pressure, and LNG consumption), sampled at 10-minute intervals, and cargo tank sensor channels (c.g.. tank temperature at top, mid, and bottom levels, and tank pressure), sampled at 1-minute intervals. For experimental evaluation, we selected 10 key channels from the full set of 52, focusing on those most relevant navigation and weather conditions, cargo tank thermodynamics, and machinery operations.
The LNG carrier is equipped with four independent cargo tanks. During voyage, ambient heat ingress into the cargo tanks causes the LNG stored in these tanks to partially vaporize, producing what known Boil-Off Gas (BOG). Efficient BOG management is critical: the gas can be used as fuel, reliquefied, or combusted via a Gas Combustion Unit (GCU), which converts it into CO, be- fore discharge. The operational objective is to minimize unnecessary combustion while maximizing energy utilization as fuel, all while maintaining tank pressure within safe operational bounds. The
thtpp://www.epa.gvv//ee
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(b) Diagram of LNG Cargo Handling System
(a) LNG Carrier (from Wikimedia) and Route
Figure 5: Overview of the LNG Cargo Handling System (CHS).
CHS dataset captures the dynamic interplay among external conditions. cargo tank behavior, and onboard machinery responses. provides a foundation for predictive modeling of key control chan- nels, including tank gas pressure and LDC suction pressure. Accurate prediction of these channels important for safe and energy-efficient voyage execution under the channel-wise asynchronous sampling conditions found in real-world LNG cargo operations.
A.2 IMPLEMENTATION DETAILS
Model Hyperparameters and GPU Setup All experiments were conducted using single NVIDIA RTX 3090 GPU with 24GB memory. We used the Adam optimizer across all models, with initial learning rates per dataset for fair comparison: 10-4 for ETT1, ETT2, SolarWind and EPA, 10-2 for Weather, and 10-3 3 for the CHS dataset. Training was run for a maximum of 10 epochs with early stopping based on validation performance. For models with stackable encoder block, the number of encoder layers was set to 2. For Transformer-basee models, the model dimen- sion dmooee searched over {128, 256, 512]. and the feedforward expansion ratio du / dmodel was selected from 2, Details of the hyperparameters used for each dataset are fully specified in the provided code repository 3] We adopted most baseline implementations from the TimesNet (Wul et al. 2023) repository. DUET (Qiu et al. 2025). TimeFilter (Hu et al.] 2025). CrossGNN (Fuang al. 2023). t-PatchGNN (Zhang et | 2024). Hi-Patch (Luo al. 2025), BiTGraph (Chen al., 2024b), originally provided in a separate repository, were reimplemented and integrated into the
TimesNet framework for consistency. The hyperparameters of these baselines were aligned with those specified in their original repositories. In our proposed Channe/TokenFormer (CTF), the num- ber of channel tokens was selected from {1, 3), with the optimal value determined separately for
each dataset.
Training and Evaluation under Our Proposed Practical Setting For regular time series baseline models, both training and testing were conducted on linearly interpolated data due to architectural constraints. To ensure a fair and consistent comparison, we preserved the original design of the baselines and trained them on interpolated inputs with MSE loss. contrast, our proposed model, Chamne/TookenFomee (CTF), is explicitly designed to operate on non-interpolated inputs that reflect the original, variable sampling periods of each channel. To ensure uniform input shape compatible with existing architectures, we apply forward-fil fill interpolation during dataset preprocessing: how- ever, this step serves merely as a structural tool to meet input dimensionality requirements, not as a modeling assumption. During the channel-wise patch embedding phase, all forward-filled filled values are explicitly excluded: only the valid indices L,, corresponding to truly observed data points, are used for computation. For irregular and missing handling time series baselines, we followed their original frameworks by providing an input mask, which was used for both data processing and MSE com- putation. This procedure is exactly consistent with the forward-fill interpolation-basee approach. To ensure fair evaluation, both the basclines and our model compute -agnnne--ggregated metrics (CMSE and CMAE, defined in Section only over the valid indices H4, which correspond ac- tually observed target values. Thus, although the input preprocessing methods differ, all models are
*Code:
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evaluated on the same set of ground truth targets (y1..... (.H:). the number of valid test samples is inherently reduced, since the test set is subsampled according to the maximum sampling factor across channels. In summary, each model is trained under settings aligned with its architectural requirements, but all are evaluated uniformly based on their accuracy in predicting truly observed values. Some recent methods adopt "drop-last" trick (Qiu et al. 2024) to improve performance. In our experiments, we do not apply this operation.
). Note that under the practical setting,
Input/Output Length Settings per Dataset As described in Section diction lengths for each dataset to reflect the temporal characteristics of the underlying signals. Instead of fixing the number of time steps across all channels, we define these lengths over a consis- tent time duration to accommodate channel-wise variations in both sampling period and dominant frequency components. For implementation and reporting convenience, however, we represent input and output lengths in terms of the number of steps corresponding to the channel with the smallest sampling period. The input and prediction lengths (in time steps, relative to the finest resolution channel) for each dataset are as follows: ETTI, ETT2: 192 (input) + [192, 336, 768, 1152] (pre- diction), SolarWind: 576 + [288, 576, 864, 1 1152J, Weather: 144 + [144, 288, 576, 864), EPA: 96 + [96, 192, 288, 384]. CHS: 120 - [120, 240, 480, 840]. Additional details such as patch length for each channel in each public dataset are fully specified in the released code files.
5 we set the input and pre-
Details of Channel-wise Frequency-based Dynamic Patching In dominant frequency delection via Fast Fourier Transform (FFT), each input channel is represented as a one- me--imensionnl sequence x(), and its amplitude spectrum A (6) [+ is computed. Instead of relying on predefined set of candidate periods, the dominant period is searched within reasonable range (empirically up to ~ 200), which covers common real-world cycles such as hourly, daily, and weekly periodicities. The frequency index k with the highest amplitude in this range is identified and converted into the corresponding dominant period pf kr If the amplitude A (1) at Kj exceeds fixed threshold
(empirically set to the 70th percentile of A pÊ is regarded length assignment with relative sampling periods, the relative sampling period of channel i is defined as r; Smi.." where s; is the actual sampling period and Smin is the minimum across all channels.
strong dominant period. In patch
The patch length 4. representing the number of points per patch, is then assigned as 4 = Let /ra], ensuring consistency with both the detected periodicity and the sampling resolution. In the fall- back mechanism, if no dominant frequency passes the 70th percentile threshold, a sampling-awe aware patching rule is applied that directly determines patch lengths based on each channel's sampling rate, without enforcing periodicity-based constraints. This approach is designed for cases where periodicity is too weak or noisy to guide patching, and it prioritizes maintaining consistency with the intrinsic resolution of each channel.
Example of Channel-wise Frequency- Dynamic Patching For example, in the Sola dataset, wind power is recorded every 5 minutes and solar power every 20 minutes. Over 2-day period, this corresponds to 576 time steps for wind power (resulting patch length: 48 steps) and 144 for solar power (resulting patch length: 18 steps). Assuming patching is performed based on each channel's 's dominant period, this yields 576 / 48 = 12 local tokens for wind power and 144 18 = for solar power. Detailed calculations are provided below.
Relative sampling periods: r1 1, = FFT-detected dominant periods: pi 48, P5 72
Resulting patch lengths: 1 = [ [48/1] = 48 1p5/r2] [72/4] 18
Number of patches:
-Pos1 = [L4]] = [576/48] = 12
- Pakz = L/ r2) = /576 4) =
Algorithmic Details of Frequeney-base Dynamic Patching and Tokenization Aim plements the frequency-based dynamic patching and tokenization procedure described below. Given that all channels are observed over the same time interval, we first compute the relative sampling pe- riod of each channel with respect the smallest sampling period across channels. This determines how many samples (i.c., its effective sequence length) fall into this interval. For each channel, we then set the patch size primarily based on its dominant period. In practice, if the dominant period
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Algorithm 1 Frequency-based Dynamic Patching and Tokenization
Require: Multivariatc series RR*K; sampling periods {sx N dominant periods ( (D. horizon H;; mask prob. q; #channel tokens y
Ensure: Token sequence S
I: Smin <- E[N
2: --] []
final token list
3: for
to N do
4:
si - S./Smin
P relative sampling period valid length of channel i patch length in samples number of patches
5:
Li + L/s;
6: P. Da/84
7: K; I./p. = L/D,
8:
Partition X( th into (1/)) Jo=n where p5 R
T(
local tokens kept after masking
10:
for j = to K, do
p( fully unobserved (test time) Bernoulli(g)
(training) then
> patch masking
12:
continue
13:
end if 7, Wp. P pos
14:
6 Append 7,(6) to T
W ERexP, To ERd
15:
16: end for
17:
Initialize learnable channel tokens ( 1k cha¹ E Rd
18: 5( [T((, c ..... Cm] e,thn
c is broadcast to all tokens
19:
Append 5(1) to S
20:
end for
21:
return
cannot be reliably estimated for channel, one can instead define its patch size using only the rel- ative sampling period, i.e., a sampling-aware patching rule. The resulting series for each channel then partitioned into patches, and we discard patches that are either entirely unobserved during test time or randomly skipped with probability q during training, so that the token sequence reflects only informative segments. Fach remaining patch linearly projected and enriched with positional embeddings, after which a small set of learnable channel tokens is appended and shared channel embedding is added all tokens from the same channel. This yields unified token sequence that encodes heterogeneous sampling periods and channel-wise frequency characteristics without relying on interpolation.
Test-time Missing Conditions For controlled missingness injection, vary the overall missing ratio and apply block-wise masking by independently sampling contiguous blocks for each channel and each sample. This procedure mimics naturally occurring block-wise gaps caused by sensor mal- functions or maintenance events in real deployments. It produces a range of challenging, randomly located missing blocks that are applied identically to all compared methods at test time.
To specifically evaluate robustness under structured contiguous missing conditions, we additionally introduce block-wise missing intervals into the test-time inputs of the SolarWind dataset. Each channel is assigned randomly positioned missing blocks, with each block spanning exactly one patch length. This ensures that at least one patch per channel is entirely unobserved, enabling direct assessment of our model's patch-level masking strategy at test time. During training, our model applies random patch masking, as shown in Table training with a 0.4 random masking ratio consistently improved model robustness across different missing ratios.
B SPECTRAL DISTORTION INDUCED BY INTERPOLATION
B. BACKGROUND
In practical time series forecasting scenarios, multivariate signals often feature channel-wise asyn- chronous sampling periods across channels, stemming from both inter-system and intra-system heterogeneity in sensor design and operation. However, most existing forecasting models assume regular and complete time grid, necessitating the use of interpolation to align all channels
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Table 8: Average forecasting performance of our CTF model on the SolarWind dataset, evaluated under block-wise missing at test time with varying missing ratios m, and trained with different patch masking ratios E.
Missing ratio (Train)
0.1
0.2
0.3
1 0.4
0.5
Metric
CMSE CMAE
CMSB CMAE
CMSE CMAE
CMSE CMAE
CMSE CMAE
m 0.125
0.420 0.472
0.414
0.466
00.413
0.466
0.408
0.463
0.412
0.466
I 0.250
0.458 0.504
0.444
0.491
0.439
0.488
0.428
0.482
0.427
0,481
0.375
0.490
0.537
0.474
0.521
0,464
0.515
0.452
0.508
0.452
0.509
m 0.500
0.527
0.573
0.498
0.549
0.488
0.541
0.474
0.533
0.474
0.536
common temporal resolution prior to model ingestion. While this preprocessing step ensures input compatibility, it inevitably introduces spectral distortion (Feng et al.) 2004] Fitzgerald & Ander- Ison] [992] by artificially synthesizing intermediate values, thereby altering the original frequency characteristics of the signals. Interpolation imposes artificial continuity by estimating unobserved values from nearby observations, which can fundamentally reshape both the temporal dynamics and spectral content of the signal. This appendix provides theoretical explanation of how such interpolation-inde induuedd distortion arises in the frequency domain, as observed through the Fast Fourier Transform (FFT).
B.2 LINEAR INTERPOLATIIN-NNDUCE FREQUENCY DOMAIN DISTORTION
Let x() denote a real-valued time series obtained via uniform but sparse sampling relative to other channels. Let ztt) denote the version of xtt) with intermediate values filled in via linear interpo- lation between uniformly sampled points. This interpolation process can be interpreted as a volution in the time domain, as linear interpolation is equivalent to applying a piecewise linear (triangular) filter to the sampled signal. By the Convolution Theorem, this results in multiplication in the frequency domain:
(() = x(f) H(f),
where H(f) is the frequency response of the triangular interpolation kernel, given by sine sinc?(f). This imposes low-pass filtering effect that attenuates high-frequency components of the original signal, and constitutes the primary source of spectral distortion introduced by linear interpolation. Such distortion manifests in multiple forms: (i) amplitude attenuation within the Nyquist limit, due to low-pass filtering that suppresses mid-to-hight frequency energy: (ii) amplitude gaps, due to the non- uniform and oscillatory nature of the interpolation kernel's 's frequency response, which causes broad spectral suppression in the mid-to-high frequency range; (iii) spectral leakage beyond the Nyquist frequency, caused by the non-ideal frequency response of finite-support interpolation kernels; and (iv) phase delays, introduced by the filtering process, especially in the mid-to-high frequency range, and observable in the FFT phase spectrum.
(1) Amplitude Attenuation Under linear interpolation, the reconstructed signal exhibits reduced overall spectral energy, particularly in the mid-to-high frequency range:
{ < [X(
This occurs because each frequency component is scaled by a factor [/( < 1 for f > 0, resulting in suppressed spectral magnitude:
= [H()) < (X()).
Attenuation becomes stronger higher frequencies, leading to net reduction in total spectral en- ergy and smoother appearance in the time domain. This loss of impair the model's s ability capture mrniigr temporal patterns in forecasting tasks.
information can
(2) Amplitude Gaps The frequency response of linear interpolation exhibits oscillatory behavior due to its squared sine form. resulting in repeated dips and low-gain regions across the spectrum, particularly in the mid-to-high frequency range. When these regions coincide with frequencies that
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contain substantial energy in the original signal, the interpolated spectrum displays significant local suppression observable as amplitude gaps:
[x()) < (x() for f [fmia= fhigh]-
Such non- non-uniform attenuation alters the spectral envelope and can adversely impact the performance of forecasting models.
(3) Spectral Leakage (High-Frequenc) Artifacts) The frequency response of linear interpola- tion, given by H(f) sinc"(f) is not band-limited and extends well beyond the Nyquist frequency, with decaying side lobes across the spectrum. As result, interpolation artificially introduces high- frequency components into the signal, even when the original data is strictly band-limited. This is particularly evident in the FFT magnitude spectrum(see Figure 6a), where the interpolated signal exhibits spectral energy in regions beyond the Nyquist limit. These high-frequeng, artifacts are not part of the true signal but emerge from the spectral spreading effect caused by the interpolation ker- nel. They distort the original spectral structure by injecting spurious out-of-band energy, which can mislead models.
(4) Phase Delay Although linear interpolation produces smooth and continuous waveform from discretely sampled data, it introduces a frequency-dependent shift in the phase spectrum. Let ZX()) and ZX (f) denote the phase spectra of the original and interpolated signals, respectively. Then the phase difference is given by:
2$(1= - 5.
which remains negative across most frequencies, with the delay becoming notably apparent from the mid-frequency band onward (see Figure 6b). This indicates that the interpolated signal lags behind the original in phase. The delay is negligible at low frequencies but grows rapidly in the high- frequency band, due to the squared sine frequency response of linear interpolation. When the signal's dominant spectral components lie in the high-frequeng, region, this phase delay leads to substantial misalignment of key waveform features such as peaks and transitions. Even if the interpolated waveform appears visually smooth in the time domain, the underlying temporal displacement of eigh-frequency content can significantly degrade forecasting accuracy, especially in tasks that rely on precise timing of local patterns.
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(a) FFT Magnitude Spectrum and Distortion
(b) FFT Phase Spectrum and Delay
Figure 6: Spectral distortion caused by linear interpolation in the ETTh1 dataset. The left panel shows amplitude attenuation and spectral leakage the magnitude spectrum, while the right panel illustrates phase delay in the FFT phase spectrum.
B.3 SUMMARY AND IMPLICATIONS FOR FORECASTING PERFORMANCE
The four types of distortion (amplitude attenuation, amplitude gaps, spectral leakage, and phase delay) collectively degrade the spectral integrity of interpolated signals. These artifacts can mis- lead forecasting models into learning spurious temporal or frequency patterns. The issue becomes more severe in multivariate settings with channel-wist asynchronous sampling, where aligning all channels to a uniform time grid via interpolation introduces artificial synchrony and continuity. To mitigate such distortion, advocate for interpolation-fre modeling. where models are designed
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to operate directly asynchronously sampled and non-uniformly aligned signals, without relying on synthesized values that risk obscuring the true structure of the original signals.
Input-level distortions act as primary source of frequency bias and, in turn, forecasting emror. Thus, the key priority is not to design ever more claborate froquency-doman heads on top of al- ready interpolated inputs, but rather adopt interpolation-ffer architecture that can ingest asyn- chronously sampled, partially missing multivariate series without forcing them onto an artificially regular grid. In practice, we observe that frequency-based approaches such as FreTS and FITS vulnerable to such frequency biases, showing significant performance degradation on real-world datasets with strong channel-wise asynchrony (see Table 18 for detailed results). Moreover, the frequency-bing analysis comparing CTF and TimeXer, which are fundamentally similar except for CTF's interpolation-fies module, indicates that removing interpolation yields more faithful spec- tral characteristics relative to the original time series and, consequently, improved forecasts (see CTF designed with exactly this objective in mind.
Section 6.3
C FURTHER ANALYSIS
C.I ADDITIONAL ABLATIONS
To complement the main experiments, we provide additional ablation studies highlighting the con- tribution of key architectural components in ChammerTokenFoomer (CTF). shown in Table [ incorporating learnable channel-specifee embeddings consistently improves performance across all datasets. This suggests that explicitly encoding channel identity through channel embeddings pro- vides a strong inductive bias, particularly in heterogeneous multivariate settings. Furthermore, as shown in Table [0] setting patch lengths to match each channel's periodic characteristics, rather than using fixed patch length, has clear impact forecasting performance. This underscores the importance of reflecting channel-specificc sampling periodicities when handling asynchronous time series.
Table Effect of channel embedding on forecasting performance. Results are reported CMSE, averaged over all prediction lengths. Overall, channel embedding improves performance most datasets, indicating effectiveness modeling channel-specific dynamics.
Setting / Dataset
ETTI
ETT2
SolarWind
Weather
EPA
CHS
CTF with channel embedding
0.398
0.377
0.400
0.269
0.775
0.282
CTF w/o channel embedding
0.404
0.379
0.399
0.283
0.781
0.285
Impact of patch length forecasting performance. Results reported in CMSE, aver- aged over all prediction lengths. In general, frequency- guidedd patching improves performance on most datasets, suggesting that adapting patch lengths to each channel's periodic structure benefi- cial.
Table 10:
Patch length Dataset
ETTI
ETT2
SolarWind
Weather
EPA
CHS
FFT-based
0.398
0.377
0.400
0.274
0.775
0.282
0.404
0.378
0.405
0.289
0.817
0.300
0.401
0.378
0.400
0.289
0.807
0.301
32
0.424
0.383
0.399
0.306
0.799
0.307
To further examine the effectiveness of our design beyond what was demonstrated Section 5.2 we conduct an extended ablation study focusing on the three practical challenges of multivariate forecasting: Dependency, Dependency.Asyyncinvn Asynchrony, and Missingness. As summarized in Table II we selectively disable each architectural component corresponding the three core challenges addressed by design, and evaluate all ablation settings under varying levels test-time missingness. To remove Channel Dependence, eliminate attention between channel tokens from different channels. To
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Charnet
Losel
Tokama
Channel
Masked (1)
Attended (0)
0
2
Local
Key
Tasana
Local
Channel
Key
Figure 7: Visualization of the attention masks used in Channel-Independer (CI) masking strate- gies. CI-ReadOnly (left) allows only unidirectional attention from channel tokens to local tokens within the same channel, while CI-Mutual (right) permits bidirectional attention between local and channel tokens. In both cases, no cross-channel attention allowed.
assess the impact of Dynamic Patching. we prevent the model from incorporating dominant fre- quency information extracted from FFT well as channel-specific sampling periods. For Patch Masking, we disable the masking mechanism for block-wise missing intervals during both training and testing. Across all settings, the full model consistently achieves the best results, confirming that each architectural element contributes complementarily to the model's overall robustness. Among these, patch masking emerges as critical for handling contiguous missing blocks, while channel dependency and dynamic patching offer additional gains under asynchronous and heterogeneous conditions.
Table 11: Results of the ablation study under channel-wist asynchronous sampling with block- alsck-wise test-time missing. Results are shown for different missing ratios m, with metrics averaged across all prediction lengths on the SolarWind dataset.
Test-time Missing Ratio
0.125
0.250
0.375
I/
0.500
Ablation Setting
CMSE
CMAE
CMSE
CMAE
CMSE
CMAE
CMSE
CMAE
Full model (CTF)
0.408
0.463
0.428
0.482
0.452
0.508
0.474
0.533
w/o Channel Dependence
0.408
0.463
0.429
0.484
0.474
0.521
0.522
0.551
w/o Dynamic patching
0.452
0.496
0.474
0.515
0.494
0.536
0.517
0.562
w/o masking
0.412
0.465
0.434
0.484
0.458
0.508
0.482
C.2 COMPARISON OF ATTENTION MASKING STRATEGIES
We conduct an ablation study comparing six variants of attention masking strategies within our unified attention layer. To avoid analytical redundancy, all strategies share common constraint: channel tokens are prohibited from attending to any other tokens within the same channel, including themselves. This restriction eliminates redundant intra-channel interactions and encourages richer cross-channel communication.
The six masking strategies fall into two categories: Channel-Independerr (CI) and Channel- Dependent (CD). The key distinction lies whether cross-channel attention among channel to- kens is permitted. CD strategies further subdivided based on the scope and indexing of allowed inter-channel connections. Table presents comparative results for the masking strategies on the SolarWind dataset (Case 2).
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Charnel
Locel
Loogl Tekens
Channel
Masked
Quary
Attendad (0)
2
0 0
0|
Takana
Local
Channel
Key
Tosana
Local
Channel
Key
Figure 8: Visualization of the attention masks used in Channel-Dependent (CD) masking strate- gies with global cross-channel token attention. CD-ReadOnly (left) extends CI-ReadOnly by al- lowing channel tokens to attend to channel tokens from other channels, while CD-Mutual (right) permits bidirectional attention within channels and additionally enables global cross-channel atten- tion among channel tokens.
Local
Local
Tolens
Masked (1)
Qpary
0
011
0
1 2
Looal
Chanset
Kay
Local
Charnel
Figure 9: Visualization of the attention masks used in Indexed Channel-Dependent (CD) masking strategies. CD-RRadOOnly-ndeeeel (left) and CD-Mutual-Indexed (right) restrict cross-channel attention to channel tokens with the same index across different channels. This design promotes role-specific specialization while maintaining intra-channel locality for local tokens.
Masking (2 variants): All attention operations are confined to within each channel. No cross-channel interaction allowed (see Figure 7)..
CI-ReadOnly: Channel tokens can attend to local tokens (read-only summarization), not vice versa.
CI-Mutual: Local and channel tokens within channel can attend to each other bidi- rectionally.
Channel-Dependees Masking (4 variants): Local tokens remain strictly within- channel,,,, attending only to tokens from the same channel. In contrast, channel tokens are permitted interact across channels, subject specific masking rules.
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Global Choss-haannel Token Attention: Each channel token attend to all chan- nel tokens from other channels, but not to those from its own channel. This facilitates rich cross-channel communication while maintaining strict inter-channel exclusivity (see Figure
* CD-ReadOnly: Extends CI-ReadOnly by allowing channel tokens to also attend channel tokens from other channels. This allows each channel token sum- marize both intra- and inter-channel information without influencing local token representations directly.
* CD-Mutual: Extends CI-Mutual with additional global attention among chan- nel tokens across different channels. enables two-way communication between channel tokens, potentially enhancing global coordination and information ex- change across channels.
Indexed Cross-Channel Token Attention: When multiple channel tokens signed per channel, attention is restricted to those with matching indices across chan- nels (e-g., index-0 tokens attend only other index-0 tokens). This design encourages wise specialization and alignment (see Figure
* SleadOnly-Indexetl Read-only variant allowing same-index channel tokens attend to each other across channels. This selective communication enables controlled abstraction across semantically aligned channel positions.
CD-Mutual-Indexee Mutual variant enabling bidirectional intra-channel and same-index inter-channel attention. It balances local coherence and cross-channel consistency by unifying both local and indexed global interactions.
Table 12: Comparison of attention masking strategies on the SolarWind dataset under the block-wise test-time missing scenario. Results reported in CMSE and CMAE, averaged over all prediction lengths with a missing ratio m = 0.375. Among the methods, CD-ReadOnly and CD-ReadOnly. Indexed achieve the best performance under our proposed practical setting.
Strategy
CI-ReadOnly
CI-Mutual
CD-ReadOnly
CD-Mutual
CD-ReadOOnly-nneexed No-Muutal-Indexxee
CMSE
0.474
0.457
0.452
0.456
0.452
0.457
CMAF
0.521
0.512
0.508
0.510
0.508
0.512
C.3 SCALABILITY ANALYSIS
We further investigate the scalability of Chamn/Toocnnoormer (CTF) with respect to both the num- ber of channels and the input sequence length.
Channel scalability. We measured runtime per training iteration and maximum memory usage as the number of input channels increases. Results in Table 13 show that CTF scales reliably up 275 channels on a single 24GB GPU. Out-of-memory (OOM) is observed only at 280 channels. Training runtime grows proportionally with channel count (c.g.. 2.69s/iter C' 200), while inference latency remains comparable to TimeXer (0.903s vs. 0.917s).
Table 13: Channel scalability of CTF on single GPU with 24GB VRAM. We report training runtime (s/iter) and peak memory usage (MB) as the number of input channels increases. CTF scales reliably up to 275 channels, with out-of-memory (OOM) occurring at 280 channels.
Metric / # of Channels
5
30
50
100
200
275
280
Runtime (s/iter)
0.0186
0.1193
0.2307
0.7173
2.6912
5.4175
00M
Memory (MB)
108.9
460.3
911.2
2843.9
10386.1
17834.9
Input length scalability. We also tested longer input horizons. As shown in Table 14 inference runtime remains stable (0.014s - 0.016s) as sequence length grows from 96 to 2048. Memory usage scales linearly with input length.
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Table 14: Input length scalability of CTF. We report training runtime (s/iter), peak memory usage (MB), and inference latency as input length grows. Inference runtime remains stable (0.014s - 0.016s), while memory usage scales linearly with length.
Metric/ / Input Length
96
512
1024
2048
Training Runtime (s/iter)
0.028
0.048
0.074
0.147
Training Memory (MB)
126.
403.3
945.8
2921.4
Inference Runtime (s/iter)
0.0140
0.0147
0.0147
0.0159
Inference Memory (MB)
126.
404.8
945.8
2921.4
Summary. These results demonstrate that the masking- based attention design CTF can handle high-density multivariate forecasting and long-horizon inputs with stable inference latency. While memory grows linearly and limits appear at very high channel counts, the model remains efficient practical regimes.
C.4 COMPARISON WITH MODIFIED TIMEXER FOR ASYNCHRONOUS SAMPLING
For TimeXer (Wang et al., 2024c). we incorporated channel-wist asynchronously sampled inputs, along with their corresponding sampling information, as exogenous inputs, following the same strat- egy (shared tokenizers) used our model. To ensure consistency and reduce redundancy, channels with identical sampling periods shared the same modeling configuration when each was used as the forecasting target. We conducted experiments on the ETTI, ETT2, and SolarWind datasets for Case (Section 5.1), and on the SolarWind dataset for Case 2 (Section [.I.p Results are reported as CMSE and CMAE averaged over four prediction horizons. As summarized in Table [3] our CTF achieves consistently better comparable performance compared to the modified TimeXer variants.
Tabble 15: Comparison of CTF with modified TimeXer under asynchronous sampling. Results are reported in CMSE and CMAE averaged over four prediction horizons.
Model
CTF (ours)
TimeXer (mod.)
TimeXer
Dataset / Metrie
CMSE
CMAE
CMSE
CMAE
CMSE
CMAE
ETT1
0.398
0.411
0.404
0.416
0.420
0.424
Case
ETT2
0.377
0,383
0.374
0.382
0.381
0.389
SolarWind
0.400
0.448
0.417
0.450
0.420
0.468
Case
SolarWind
0.440
0.496
0.474
0.504
0.462
0.504
C.5 COMPARISON WITH FREQUENCY-BASED APPROACHES
As representative frequency-based approaches, we compare FreTS Yi et al. 2023b) and FITS (Xu al. 2024) against CTF range of datasets exhibiting realistic channel-wise asynchrony. Because these methods operate primarily in the frequency domain, they are expected to be vulner- able to frequency biases introduced by interpolation-base results (Table 18) show significant performance degradation under strong inter-channel asynchrony, particularly on EPA and CHS, whereas CTF remains substantially more robust.
preprocessing; indeed, the experimental
C.6 ANALYSIS OF HANDLING DISCRETE MISSINGNESS
First, the real-world benchmarks we use already contain discrete missingness. For example, in ETTm (15-minutely) data, one can observe that missing entries are implicitly encoded as zeros. Not only our model but also other baselines rely on a small neural tokenizer (patch embedding network) to process these imputs. We empirically observe that this stage already provides reasonable degree of robustness to discrete missing values. Figure I0 illlsstras this effect. Even when 30% of entries converted into discrete missing values, the resulting patch-embedding cloud in t-SNE space
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remains closely aligned with that of the clean inputs. This behavior is consistent with the fact that the patch tokenizer is a shallow neural network (e.g., a small MLP-based projector) that implements a relatively smooth, approximately Lipschitz-continuous mapping from input patches to embeddings. Under such mappings, localized perturbations that affect only small fraction of time steps such as isolated missing entries induce only bounded changes in the embedding space, rather than causing drastic geometric distortions (Fazlyab et al. 2019;; Zhhag et al. 2022a). Consequently, discrete missingness tends be absorbed as small, localized noise at the tokenizer level, which helps explain why downstream forecasting performance is largely preserved even when nontrivial fraction of discrete missing values is present.
To evaluate robustness more precisely under discrete missing conditions at test time, we introduce discrete missing values into the test-time inputs of two datasets, ETTI and EPA. both datasets, we randomly insert zero-filled gaps of short, varying lengths (5 to 20 time steps), which are substantially smaller than the patch size, into each channel. These settings are designed test model robustness to scattered missing values. For ETTI with discrete missingness, we compare CTF against standard time-series baselines, including strong CD and CI approaches. For EPA with discrete missingness, select two representative baselines, Hi-Patch (Luo et al. 2025) and ContiFormer (Chen et al.) 2023). Since ContiFormer is an -DE-based model with prohubitively long training times, we reim- plement within the NetesNet framework by approximating its ODE solver, following the structure of XemeXer (Wang et al., 2024e), to ensure both efficiency and consistency. However, because this reimplemented ContiFormer does not natively conform the pvegular-sampliin frameworks con- sidered and operates on interpolated inputs, we treat it under the same procedure as the regular baselines. We focus on two challenging monitoring sites, Los Angeles (LA) and Hillsborough, to evaluate robustness under more difficult real- wallwwolld conditions. Experimental results under this set- ting are reported in Table [6] Table [7] Overall, we observe that most models exhibit only minor degradation under discrete missingness for small missing ratios 10.1-..3). Across all tested missing ratios, however, CTF consistently maintains stronger performance than the compared methods on both datasets.
Table 16: Average CMSE and CMAE for all prediction lengths of ETTI dataset under short-range test-time missing intervals with varying missing ratios m. CTF is trained with random patch masking ratio of 0.4 to enhance robustness against missing inputs.
Approach
Channel Dependent
Channel- Inddeeedend
Midel
CTF(uur))
TimeFilter
DUET
TimeXer
iTrans.
C
TimesNet
TimeMisares
PatchT
DI iinerr
Metric
CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE CMSE CMAE
0.415
0.424
0.412
0.418
0.445
0.441
0.455
0.434 0.490 0.449 0.448
0.439
0.453 0.442
m 0.2
0.434
0.438
0.433
0.438
0.475
0.465
0.466
0.461
0.477 0.467 0.471 0.452 0,508 0.486 0.474 0.469
0.465
0.458
0.483 0.465
m
0.3
0.453
0.451
0.455
0.453
0.506
0.489
0.495
0.481
0,500
0.483
0.495
0.470
0.527 0.502
0.499
0.486
0.491 0.478
0,514 0.486
0.1
0.471
0.463
0.474
0.466
0.572
0,530
0.526
0.502
0.523
0.499
0.520
0.487
0,550 0,519 0.522 0.503
0.517
0.497
0.544 0,506
0.493
0.181
0.595
0.563
0.526
0.548
0.506
0,576
0.518
0.547
0.577 0.526
D FORECASTING RESULTS IN A CONVENTIONAL MULTIVARIATE SETTING
To demonstrate that our model remains effective even under standard multivariate cases, we addi- tionally evaluate it on regularly sampled multivariate time series without any missing values. By aligning the sampling period and patch length across all channels, our model can be directly ap- plied to this conventional setting. This evaluation shows that our model remains applicable beyond
the proposed practical settings. For comparison, baseline results (Hu et al. 2025 Qiu et al., 2025 Chen et al.) 2025 Wang et al. 2024e Liu et al. 2024c; Huang et al. 2023 Li et al. [023b Nie jet al., 2023 Zhang & Yan 2023 Das et al. 2023 Wu et al. 2023 Zeng et al. 2023) are obtained
either from the original publications or by reproducing them using the TSLib codebase (Wang al.)
2024d). We compare forecasting performance on five widely used benchmark datasets (Zhou et al.
2021 Wu et al. 202I): ETThl. ETTh2, ETTm1, ETTm2, and Weather. As shown in Table 19 our model achieves performance comparable to or better than sfate-of the basclines across all benchmark datasets in the conventional forecasting setting.
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ETT1 patch embeddings (missing, ratio= 3)
Table 17: Average CMSE and CMAE for all prediction lengths of EPA dataset (LA, Hillsborough) under short-range test-time missing intervals with varying missing ratios m. CTF is trained with a random patch masking ratio of 0.4 to enhance robustness against missing in- puts.
no missing
missing (ratiom0.3) 31
0-
Model CTF (ours)
Hi-Patch
ContiFommer
Metric
CMSE CMAF|CMME
CMAE
0.1 1,074 0.702
1.084 0.742
1,219 0.744
-20-
m 0.2 1.146 0.731
1.121
0.754
1,237
0.757
m 0.3
1,226
0.756
1.204
0.781
1,254
0.773
m 0.4 1.290 0.788
1.383
0.839
1.298
0.801
-40
Table 18: Forecasting performance on the thanmel-wise asynchrony compared with frequency-based approaches.
-60
-40 -20
20 40 60
Figure 10: t-SNE visualization of patch embeddings produced by CTF's patch tokenizer for a batch of ETTI test samples under two conditions: with 30% discrete missing values injected and with no injected missing- ness. The near-overlap between the two embedding distributions suggests that this patch embedding stage already provides a reasonable degree of robustness to discrete missing values.
Model Metric
CTF (ours)
FreTS
FITS
CMSE CMAE CMAE CMSE CMAE
ETT1
0,399 0.410 (0:436 0.432 0.443 0.430
ETT2
0.377 0.383
0.440 0,428
0.391 0.389
SolarWind
0.403
0.452
0.416
0.502 0.485 0.481
Weather
0.275
0.296
0.285
0.328 0.297 0.304
EPA
0.776
0.586
1.018
0.684
1.245 0.762
CHS
0.285 0.126
0.485 0.344
0.289 0.121
Table 19: Full results of the long-term forecasting task in the conventional setting. The best results highlighted in bold, and the second-best results are underlined.
Model CTF( Tinehhserss TimeFiller
DUET
SingleTM
TincXer
iTeas.
CoosCNNN
Rlincar
PackTST
Crosformz
TIDE
TinesNei
Dlitere
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owe
cver
lowe: owel ove coee
per owdr
bet our
pest oue
oyst our
cusr
os p.771 0.80713361 0.40
(,.300 0.394 0.395
la 556 wol0 852 cu403a 5860.455o S82
0.448 01 (464/n 554 4,02/0 086004000
1:412 0.2002220041 0.441
0.415
0,420
0.429 0.455
11z2 a 429 ( 0 42700.22
0.340 44000.00 0471 nara u.525 u oagguu 436 42940 437 0.632
hast
440 sntn
568
46460 0.499
1200 0.2.0-0440000440
0.451
0.448
0.457
0.495 4.400 as3 4443
0.46 05300.00
04720.46
0.500048800005
0.621 5558 $11 a: 5190516
0:425 0,225.441441 0.432
0.422
0.429
0.420
0.428 p.445 0.300.370.00 (4 0...0.4.0.330.000 10..0.3330.4444
0.8.0s50...00..0....... 0.4540.299
0.522 (0:54) 0,5070 459 044500.45004555
95 po:00 o 5401:.76 D.
(:263 0.597)
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2:1 a s1a0 246 wida 2010
wo0 140. o. wos...00..
554 in 0040000040 c4teo San 534j0 915- SaV
192 0.3766366.30000 0.306 0.342 0.379
0,362
0.392
0.365
0.389 3.355 a 363 a a 43000403
550004005
2.57403900
0.40
0.656 0 a 0.509/ 402 1-4140 0776
0.413
02000-438.4440000 0.415
0.407
0.433
0.412
0434
40 0.43
0.40
04300.4400044
0.491 0.400
1:044
0.763
194 0 0-46
IM 0.691
0.g00.39000000 0.30000000000033 0.330 0.350
003364
0,307
0.307/0.372
0.372 0.397 pu353 0.39 33370
0.407
0.303
0.410 0.1.0........0000 w0t.000.....00.0..3 0.80 0.942
0.a0n
0.654 (166) 0.550/0 414 0.4270.5599.55
95 11117 u 151 3.3
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0.554
0.304 0:356 ia 531 a 96 1 518 o Mal 1340. tost
17
1555 o 170
120 ton 0.40
0-426 in 30d a Serio 318 575 545 173
192 p.359 0.351 0.348 0 352
0.35
0.180
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a 350 a sen 962 s4 i SSt 19
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0,390
0.355
0.451 908 00-00/0 374 a 3870 3 Sao
0.386
0,402
0.404
0.002
2900 and
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1200 0:4404220 0 4.000-440000
0.425
0.452 0.437
0.463 0.377
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0.560
(K4 (.666 an8 c5000
0.44
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0.303|0.30
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0.466 0.419 0.419/0 400 0.406/ 0L407
99 1 1.1 U.245
[0.tow
0,255
0.174
0.255 in 133 a aus 1711 o250 1000 260 1766
0.75
0.2560.207
1.356 i 207 U. 30000 18T 0.20 193-0.293
PRI 10 0 200000 10000.200 222 0.201
00255
00200
0:52 nnod I Eote i 2o0n 209 soa 2000 un
1200
k1460 0. tu
4414
0.400 200 a 954 (n 249 v000 284/0 1od.
01295 0.336
0,313
0.293
0.53000
00310 1:340 220 0 0330 tie o 330
0.04
0.30
0.345
0.307
0.3430.557
0.542 0.337 D. 11-23
0.35 1 309 0.427
720 1o 1922
(1390
0 M99 0.300 of soo i tieee t0e wid 011 o tn
0.000
1 4u www.
401 10020.000000 0.000 1:700
1004 (1 0 536 (n a08 01.00 co 0.522
0.274 0.321 0,260
0.330
( 22222
0:321 o 02200 a 0.334 275 i ol2 274 0.22 258 0 so2
282
0.330
0.3260.757
0.610 ( 355 D 404400 201 0.335jo..800 04401
99 10.59 U 1.155 1.205
00155 o 0.199 0.103 0.202 i1 162 o 0.337 157 0.315
174
0 2111
0.17
0.115
0.233
1 1 2.33 172 2,220 196-0.25
10200 0 251 112011
0:245
0.200
o.rs 0190 sow 2000
1221
0.20
2400..77
1:22
0.250
2000
0.277
1 2299 (1 219 201 20110/2170 0.26
0.263 0 0.37 0.205
0.250
0,289
0.274 0.294
255 0.20 261
0229
0278
0.20
0.310
0.292 0.307
0.278
0.297
0.272
0.335
0.335 (2 3,300 283 0.385
720 0.34
11334
0.342
0.541
0.309
0.345 i 340 (l51 3:0
547
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0533
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0.418 01251 033860 385 01355/0.545 0.381
Aug 0 n.262
p.230
0.269
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(.275/ 243
0.27
2710
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0.315 (77 a 520/0 259 2.25// 0.31
E
FULL RESULTS OF OUR PROPOSED PRACTICAL SETTING
Table 20 shows the results for Case using asynchronous practical datasets, where datasets with
channel-wise asynchronous sampling were linearly interpolated onto regular sampling grid for
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the regular time series baselines. Table 21 presents the results for Case 2, which adds block-wise test-time missing inputs on top of asynchronous sampling. All results include error bars based on 5 random seeds to reflect performance variability. Across both cases, model consistently demonstrates stronger performance than baseline methods.
F FORECAST VISUALIZATION
Figure 11a illustrates the forecasts four models including our CTF on channel 3 of the ETTI dataset under Case 1 in Section[5] The prediction horizon shows periodic sharp drops and recoveries, which are critical for accurate forecasting. Our model closely tracks these rapid transitions, while baseline models tend underpredict their magnitude. We attribute this the interpolation-free nature of our model, which avoids the frequency biases introduced by linear interpolation.
We also examine Case in Section 5 which involves both asynchronous sampling and block-wise test time missing. Figure [1bb] presents visualization of model predictions for the solar power channel from the SolarWind dataset. In this sample, a missing block of length 144 is present in the middle of the test input. This causes TimeMixer++ and TimeXer to produce noticeably noisy forecasts. Although CrossGNN shows relatively higher robustness, its predictions exhibit an un- derestimation of the periodic transitions. In contrast, our model accurately identifies the missing region and applies masking, effectively preventing the injection unreliable inputs. As result, it successfully captures both the timing and magnitude of the periodic transitions, across the missing block.
G DISCUSSION
To jointly capture temporal dynamics and cross-channel dependencies under practical settings, we proposed unified attention masking strategy that merges intra- and inter-channel attention into a single attention layer. This design avoids the need for separate attention modules, reducing the number of parameters and simplifying the overall model architecture. This parameter efficiency is particularly beneficial for datasets with limited data, as it reduces the risk of overfitting while maintaining the model's ability to capture essential temporal and cross-channel patterns. However, this unified structure introduces scalability challenges when applied to nigh-dimensional datasets. As the number of channels increases, the attention layer must process quadratic number of token interactions within single computation step, leading to substantial computational overhead and slower training. promising direction mitigate this issue is to optimize the attention computation, for example, by explicit channel sparsification or grouped channel representations.
Beyond scalability, another limitation lies the objective function. Although our model demon- strates strong predictive performance across diverse practical settings, it still suffers from amplitude attenuation, especially over longer forecasting horizons, due to the use of a standard mean squared error loss. To address this, it is crucial to explore alternative loss functions that better preserve spectral characteristics and periodicity, rather than relying solely on point-wise accuracy. This issue also extends to evaluation. Conventional metries such MSE may fail penalize overly smoothed forecasts, often favoring models that suppress meaningful high-frequency or transient patterns. As result, structurally inaccurate predictions may still appear favorable under MSE, particularly in long-term forecasting scenarios dominated by coarse trends. Refining evaluation metrics to better capture structural fidelity is therefore essential. Designing both loss functions and evaluation pro- tocols that align with the characteristics of asynchronous, partially observed time series remains an important and open research challenge.
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Table 20: Full error bars for the channel-wii -wise asynchronous long- long-term term multivariate forecasting task.
CMSE
Chandd Indepsicci
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Published as a conference paper at ICLR 2026
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Published as a conference paper at ICLR 2026
USE OF LARGE LANGUAGE MODELS (LLMS) IN PAPER WRITING
LLMs were used exclusively for light copy-editing (grammar, clarity, concision) and small LaTeX phrasing/formating suggestions. We did not employ LLMs for retrieval or discovery (e.g., related- work search) or for research ideation. LLMs did not draft technical content or citations and did contribute level comparable coauthor. All text and claims were written, checked, and finalized by the authors; any LLM-suggested edits were adopted only after manual verification to prevent hallucinations or unsupported statements.
ETHICS STATEMENT
This work focuses on methodological contributions for multivariate time-series forecasting. Our experiments rely primarily on publicly available datasets from established academic and institutional sources. In addition, we include one private dataset provided under confidentiality agreements. This dataset contains personally identifiable information, and followed all necessary protocols to ensure secure handling, access control, and compliance with data-sharing restrictions. No raw private data will be released; only aggregated statistics and trained model weights are reported in the paper.
Our proposed model, CTF, is designed general forecasting framework and does not target spe- cific individuals, groups, or sensitive applications. Potential downstream uses of time-series fore- casting models include socially beneficial applications (e-g., energy demand planning. environmen- tal monitoring) as well as sensitive ones (e.g., financial or policy decision- vecision-making)... We encourage responsible usage, with attention fairness, transparency, and possible societal impacts, All experi- ments were conducted under standard computational settings, with environmental costs comparable typical machine learning research practice.
REPRODUCIBILITY STATEMENT
We have taken multiple steps ensure the reproducibility of our work. All model architectures, training procedures, and evaluation protocols are described in detail in the main text and Appendix. Hyperparameters, preprocessing steps, optimization settings, and model configurations are provided in Appendix A.2 For publicly available datasets, we specify their sources in Appendix [A.] to facilitate exact replication. For the private dataset used under confidentiality agreements, raw data cannot be released, but provide aggregate statistics, evaluation splits, and implementation details sufficient for reproducing the reported results.
Our codebase (including model implementation, training scripts, and evaluation pipeline) will be made available upon publication. All experiments were conducted on standard hardware (NVIDIA repeated across
RTX 3090 GPUs) with fixed random seeds to reduce variance; each experiment three random seeds, and average results are reported. We also provide details on software dependen- cies (Python version, PyTorch version, and major libraries) ensure compatibility with common research environments.
33

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Published as a conference paper at ICLR 2026
ChannelTokenFormer (Ours)
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(a) Channel in the ETTI dataset under asynchronous sampling. Our model accurately cap- tures periodic sharp drops and recoveries, whereas the baselines tend underpredict their magnitude.
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Solar power channel in the SolarWind dataset under asynchronous sampling with block- wise test-time missing. Despite a long missing block, our model robustly captures periodic tramsitions, while the baselines exhibit noise or trend drift.
Figure 11: Visualization of forecasting results on (a) the ETTI and (b) the SolarWind datasets.
34


Reviews:
[{"review_id": "DYKXczcuQO", "reviewer": "Reviewer_657H", "summary": "This paper introduces ChannelTokenFormer, a unified Transformer-based framework targeting robust multivariate time series forecasting under three often neglected but crucial real-world conditions: channel-wise asynchronous sampling, block-wise missingness at test time, and complex cross-channel dependencies. The model employs a mask-guided unified attention mechanism with channel tokens, frequency-driven dynamic patching per channel, and explicit patch masking for missingness during training and inference. The approach is evaluated on several benchmark and real-world datasets (including channel-heterogeneous resamplings and a large-scale industrial application), consistently demonstrating improved performance versus state-of-the-art baselines, both under idealized and challenging practical setups.", "questions": "1. Can the authors provide more insights or theoretical grounding for why the proposed mask-guided attention scheme is robust to a wide variety of missingness patterns and channel asynchrony, beyond heuristic and experimental support?\n2. How sensitive is the model to hyperparameters governing patch size selection—especially when the dominant period is weak or ambiguous? Could a poorly tuned FFT threshold degrade performance?\n3. Regarding dataset construction: can the authors clarify to what extent the practical benchmark setups (modified from canonical datasets) reflect organic missingness and sampling asynchrony found in the wild, rather than synthetically controlled structures? Can the robustness of CTF be validated on additional non-synthetic real-world datasets?\n4. Could the authors expand on which failure modes arise with their approach (e.g., with highly bursty, uncorrelated missingness, or in scenarios with low cross-channel redundancy)?\n5. As shown in Table 15, the model does not achieve state-of-the-art performance on long-horizon forecasting tasks under conventional multivariate settings. Does this indicate that its practical effectiveness may be limited when the data are fully observed and regularly sampled?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper formalizes a comprehensive and realistic forecasting challenge: simultaneous handling of dependency, asynchrony, and missingness, as illustrated clearly in Figure 1, moving beyond isolated treatment of these aspects in prior works.\n2. The paper introduces a unified mask-guided attention mechanism for channel tokens (see Figure 2 and Figure 3) that elegantly separates local intra-channel modeling from cross-channel global aggregation, supporting asynchronous and missing inputs naturally.\n3. The model integrates frequency analysis (FFT-driven patch length selection) to respect per-channel periodicities and sampling densities, supported by theorized and empirical evidence (Appendix, Table 9).\n4.  Patch masking for both training and test scenarios is well-motivated and shown to bolster model robustness. The ablation studies reinforce the necessity and effectiveness of each mechanism, especially for structured block-wise missingness.\n5.  Detailed quantitative experiments across a rich collection of public and industrial datasets, as summarized in Tables 1 and 2, show state-of-the-art performance. Ablations in the appendix clarify the interplay among model components.\n6.  The paper provides in-depth analysis of interpolation-induced spectral distortion justifying the benefits of the interpolation-free approach and thoughtfully connecting theory with practical modeling.\n7. The method's scalability to high channel counts and long sequences is empirically demonstrated (**Table 12, Table 13**), supporting claims of practical deployability.", "weaknesses": "1.  While practical motivation for the mask-guided attention and frequency-based patching is strong, the formal theoretical analysis is somewhat lacking.  For example, the impact of frequency-based patching on generalization is empirically supported but not mathematically justified.\n2. The bulk of the validation is empirical, with most arguments for effectiveness given by ablations and performance gains. The absence of principled theoretical results or proofs (e.g., why the particular masking scheme is optimal or robust under certain classes of missingness) is noticeable. For example, the construction of attention masks (as shown in **Figures 7, 8, 9**) is justified via heuristic motivation rather than analytical optimality or formal properties.\n3. The manuscript provides a detailed exposition of the proposed method; however, certain sections are densely structured and may be difficult for readers to follow. In addition, the notation conventions—such as the definitions of the mask matrix, channel tokens, and sampling factors (see Sections 3 and 4)—are somewhat cumbersome and could benefit from simplification or clearer explanations. Moreover, some formulas are embedded directly within the main text, which further hinders readability and could be better presented as numbered equations or in a dedicated formula environment.\n4. The modification of canonical datasets into \"practical\" versions (e.g., Weather-practical, SolarWind) relies on synthetic resampling and missingness injection, which may bias the comparison to baselines. While the intention and utility of these datasets are strong, it is unclear how they reflect real-world statistical complexity versus controlled synthetic heterogeneity, and possible overfitting to constructed artifacts cannot be ruled out.\n5.  While the main architecture is described at a reasonably high level, some key hyperparameters are not specified in detail and are only provided as approximate ranges. Given the relatively complex pipeline and variable-dependent model configuration, this may hinder independent replication.\n6. Appendix B discusses the spectral effects of interpolation (with supporting figures), but the argument remains somewhat qualitative. Providing a more comprehensive, formal, and comparative mathematical analysis, including its impact on prediction accuracy and data outcomes, could enhance the scientific value and make the frequency-related discussion more broadly applicable.\n7.  The paper acknowledges some scalability and generalization challenges in the conclusion but otherwise does not engage deeply with possible failure modes or cases where ChannelTokenFormer might underperform (e.g., non-stationary environments, highly irregular or bursty missingness patterns, or settings with weak cross-channel structure).\n8. Limited Superiority in Conventional Settings: Although the model is evaluated on regularly sampled multivariate time series without missing values to demonstrate its general applicability, the results shown in Table 15 indicate that it does not consistently outperform existing state-of-the-art baselines in this conventional forecasting setting. This suggests that the architectural advantages of ChannelTokenFormer—designed primarily for handling asynchrony and missingness—may not translate into clear gains when data are fully observed and regularly sampled.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qH5iv8XMHL", "reviewer": "Reviewer_9xPZ", "summary": "A practical forecasting setting is introduced where three core real-world challenges occur simultaneously: channel-wise asynchronous sampling, block-wise missing intervals, and complex crosschannel dependencies. Instead of interpolation, this approach handles both missing segments and sampling gaps through masking and frequency-based dynamic patching.", "questions": "1、The proposed algorithm appears to have notable limitations. Can the authors clarify whether their method can effectively handle missing values during training, especially when the missing values occur in a discrete rather than continuous manner? How does the model ensure robustness and applicability under such conditions?\n\n2、Furthermore, while the authors claim that their method addresses variable modeling, multi-source asynchrony, and missing values, have they considered other equally important challenges in time series analysis, such as non-stationarity, concept drift, and probabilistic forecasting? Could the omission of these aspects make the overall logic of the paper appear fragmented and lacking a coherent research focus?\n\n3、In the comparison section, the authors only include Channel-Dependent methods from 2023 and 2024. Why are more recent representative works from 2025, such as **TimeFilter** and **DUET**, not included in the comparison? Would the absence of these up-to-date baselines weaken the credibility and timeliness of the experimental results?\n\n4、Lastly, in lines 295–296, the authors claim that their method ensures computational efficiency. However, does introducing a mask mechanism actually reduce computational costs? When the dataset contains a large number of variables, wouldn’t the computational overhead instead increase significantly, thereby affecting the overall efficiency?\n\nIf my problem is solved, I am willing to improve my score.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1、Multivariate time series forecasting is important to various domains.\n\n2、There are quite a few nice illustrations.\n\n3、This work focuses on an important problem that could have real-world applications.\n\n4、The figures and tables used in this work are clear and easy to read.", "weaknesses": "1、The proposed algorithm has notable limitations. The authors should further clarify whether their method can effectively handle missing values during training, especially in scenarios where missing values appear in a discrete rather than continuous manner. It remains unclear how the model ensures robustness and applicability under such conditions.\n\n2、In addition, although the authors claim that their method addresses three key challenges in real-world scenarios—variable modeling, multi-source asynchrony, and missing values—many other critical issues in time series analysis are not discussed, such as non-stationarity, concept drift, and probabilistic forecasting. As a result, the paper’s overall logic appears somewhat fragmented, lacking a coherent and systematic research focus.\n\n3、In the comparison section, the authors only include several Channel-Dependent methods from 2023 and 2024. However, as a paper submitted to ICLR 2026, it fails to compare against more recent representative works published in 2025, which weakens the experimental credibility and timeliness.\n\n4、In lines 295–296, the authors claim that their method ensures computational efficiency. However, from an implementation perspective, introducing a mask mechanism does not actually reduce computational costs. On the contrary, when the dataset contains a large number of variables, the computational overhead of the proposed method may increase significantly, potentially reducing overall efficiency.\n\n5、In the NIPS 2024 workshop[1], some researchers pointed out that current methods sometimes use the \"drop-last\" trick [2] to improve performance. Therefore, It is recommended that you clarify whether the \"drop - last\" operation was used in your paper in the implementation details section of your paper for transparency.\n\nIf my problem is solved, I am willing to improve my score.\n\n[1] Fundamental limitations of foundational forecasting models: The need for multimodality and rigorous evaluation\n\n[2] TFB: Towards Comprehensive and Fair Benchmarking of Time Series Forecasting Methods", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gMlLYogkSh", "reviewer": "Reviewer_e4Zq", "summary": "This paper introduces ChannelTokenFormer (CTF), a novel Transformer-based framework for multivariate time series forecasting designed to be robust to real-world data challenges. The authors identify three key challenges that often co-occur in practice but are typically addressed in isolation by existing methods: (1) complex inter-channel dependencies, (2) channel-wise asynchronous sampling (different channels having different sampling rates), and (3) block-wise missing values at test time.\n\nThe core contribution is a unified model that handles these three challenges simultaneously without relying on interpolation, which can introduce spectral distortion. The key components of CTF are: 1.Channel Tokens: Repurposed from prior work, these tokens act as global summaries or \"attention anchors\" for each channel. 2. Frequency-Based Dynamic Patching: Each channel is patched non-uniformly based on its dominant frequency (detected via FFT) and sampling rate, naturally accommodating asynchrony. 3. Unified Mask-Guided Attention: A single attention mechanism processes both local (patch) tokens and global (channel) tokens. A carefully designed attention mask controls information flow, allowing local tokens to attend only within their channel, while channel tokens can aggregate information from their own local tokens and other channel tokens, enabling cross-channel communication. 4. Patch Masking: Missing blocks are handled by simply removing the corresponding local tokens from the input, a strategy for which the model is prepared via random patch masking during training.\n\nThe authors conduct extensive experiments on \"practical\" versions of four standard benchmarks (ETT, Weather, SolarWind) and two real-world datasets (EPA-Air, LNG Cargo Handling). The results demonstrate that CTF consistently outperforms a wide range of state-of-the-art models in settings with asynchronous sampling and test-time missingness.", "questions": "1. The paper mentions the FREQUENCY BIAS. How does the bias relate to the forecasting performance? Can it be alleviated by a more specific design in the frequency domain?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper tackles a crucial, real-world problem by addressing channel dependency, asynchrony, and missingness in a unified manner, moving beyond the idealized assumptions common in much of the literature.\n2. The proposed ChannelTokenFormer, with its unified mask-guided attention, offers an elegant solution that avoids signal-distorting interpolation. The integration of frequency-based dynamic patching is a smart way to handle heterogeneous sampling rates.\n3. The paper is exceptionally clear, well-written, and illustrated, making the contributions easy to understand and appreciate. The detailed appendix further underscores the quality and care put into the research.", "weaknesses": "1. The authors compare their method with mainly channel-dependent transformer-based methods. However, many methods have achieved SOTA performance with non-transformer architectures [1] [2]. Comparison with these methods is necessary for a comprehensive evaluation.\n\n2. The section introducing the research methodology lacks essential mathematical formulas and specific descriptive details, which creates certain difficulties for readers to fully understand the implementation logic and operational steps of the proposed method. It is recommended to supplement the core formulas and elaborate on key technical parameters or procedural details to enhance the clarity and reproducibility of the methodology.\n\n3. The unified attention mechanism has a computational complexity that is quadratic in the total number of tokens (patches + channel tokens). The authors' own analysis shows OOM at 280 channels on a 24GB GPU. While sufficient for many applications, this could be a limitation for very high-dimensional problems with thousands of channels, such as the Traffic prediction problems [3]\n\n4. This article uses frequency-domain information for assistance, but it does not conduct a comparison of frequency-domain methods [4][5].\n\n[1]  Si-An Chen, Chun-Liang Li, Sercan Ö. Arik, Nathanael C. Yoder, Tomas Pfister: TSMixer: An All-MLP Architecture for Time Series Forecast-ing. Trans. Mach. Learn. Res. 2023 (2023)\n[2] Han Lu, Xu-Yang Chen, Han-Jia Ye, De-Chuan Zhan: SOFTS: Efficient Multivariate Time Series Forecasting with Series-Core Fusion. NeurIPS 2024\n[3] http://pems.dot.ca.gov/\n[4] Zhijian Xu, Ailing Zeng, Qiang Xu: FITS: Modeling Time Series with 10k Parameters. ICLR 2024\n[5] Kun Yi, Qi Zhang, Wei Fan, Shoujin Wang, Pengyang Wang, Hui He, Ning An, Defu Lian, Longbing Cao, Zhendong Niu: Frequency-domain MLPs are More Effective Learners in Time Series Forecasting. NeurIPS 2023", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3dyTyVKU7Z", "reviewer": "Reviewer_zsWm", "summary": "This paper studies the multivariate time series forecasting problem and focus on modeling cross-channel correlations, handling misalignment in sampling frequencies and periods, and dealing with missing data. Based on the challenge that no existing methods properly considered all these three aspects together, this paper proposed a new unified framework for doing that simultaneously. Experiments demonstrate the effectiveness of the proposed framework against existing methods.", "questions": "Could the authors elaborate on the motivation of their proposed framework, so that it is clearer how their proposed framework can effectively tackle the listed challenges and introduce technical contribution?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "1. Comprehensive experiments provide evidence that the proposed framework achieves its design goal, and provide insights into the performance behavior of the proposed framework under ablation settings.\n2. The proposed framework is straightforward and should be easy to replicate.", "weaknesses": "1. While the presentation of the existing challenges is quite detailed and straightforward, the paper could improve on the presentation of the design choices of the proposed method. Right now it is unintuitive how the proposed framework and its core design choices is effective at tackling the aforementioned challenges.\n2. The paper could elaborate further on the technical contribution of the proposed method, how it advances on top of existing methods and techniques.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)