CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Authors: Haotian Si
Changhua Pei
Jianhui li
Dan Pei
Gaogang Xie
Venue: ICML 2025 poster (2025)
Keywords: Time Series Forecasting; Lightweight Models; Mechine Learning
Primary area: applications->time_series

Abstract: Recent advances in lightweight time series forecasting models suggest the inherent simplicity of time series forecasting tasks. In this paper, we present CMoS, a super-lightweight time series forecasting model. Instead of learning the embedding of the shapes, CMoS directly models the spatial correlations between different time series chunks. Additionally, we introduce a Correlation Mixing technique that enables the model to capture diverse spatial correlations with minimal parameters, and an optional Periodicity Injection technique to ensure faster convergence. Despite utilizing as low as 1% of the lightweight model DLinear's parameters count, experimental results demonstrate that CMoS outperforms existing state-of-the-art models across multiple datasets. Furthermore, the learned weights of CMoS exhibit great interpretability, providing practitioners with valuable insights into temporal structures within specific application scenarios.

CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Haotian Si 12 Changhua Pei 13
Jianhui Li
Dan Pei
Gaogang Xie
Abstract
studies have shown that lightweight models even out- perform their complex counterparts (Zhang et al., 2022; Zeng et al., 2023; Das et al., 2023; Xu et al., 2024). This prompts us to reconsider fundamental question: whether temporal structures inherently possess a simple but efficient representation that we have previously overlooked.
Recent advances in lightweight time series fore- casting models suggest the inherent simplicity of time series forecasting tasks. In this paper, we present CMoS, super-lightweight time se- ries forecasting model. Instead of learning the embedding of the shapes, CMoS directly mod- els the spatial correlations between different time series chunks. Additionally, we introduce Cor- relation Mixing technique that enables the model to capture diverse spatial correlations with min- imal parameters, and an optional Periodicity In- jection technique to ensure faster convergence. Despite utilizing as low as 1% of the lightweight model DLinear's parameters count, experimen- tal results demonstrate that CMoS outperforms existing state-of-the-art models across multiple datasets. Furthermore, the learned weights of CMoS exhibit great interpretability, providing practitioners with valuable insights into temporal structures within specific application scenarios.
Several previous models (Nie al., 2023; Wang et al., 2024) emphasized learning the representation of shapes, which is also called embeddings, to further build abstract dependency via attention or mixing mechanism. However, we argue that directly modeling the relative positional relationships be- tween different time series chunks can be more robust and interpretable than modeling specific patterns. As shown in Fig. 1, for the subsequences of the sliding window, while their shupes vary over time, as long as their relative po- sitions are consistent, their dependencies, referred to as chunk-to-chunk spatial correlations, often maintain stable and reveal the temporal regularity of the system. This phe- nomenon carries both analytical and practical implications: from an analytical perspective, it reveals that there's 's an in- herent property with data, which can be a robust feature used for forecasting; from practical perspective, we can employ relatively sim- ple parametric methods for modeling such property, thereby substantially reducing model complexity.
in time series
1. Introduction
Time series forecasting plays a vital role in many fields including finance, energy, and weather. By accurately pre- dicting future time steps, it helps organizations make in- formed decisions and optimize resource allocation. As data- driven insights become increasingly essential, several fore- casting methods based on deep learning are proposed with the interaction of structures like RNN (Lin et al., 2023), CNN (Hewage al., 2020; LIU al., 2022), and Trans- formers (Nie al., 2023; Liu et al., 2024a). However, recent
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Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Figure I. time advances, the specific patterns the time win- dow change greatly, while the spatial correlations of the time series chunks remain similar.

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Moreover, we find that such spatial correlation has good decomposability, which offers new perspective for build- ing differentiated spatial correlations in multivariate time series with much fewer parameters. Although the spatial correlations across different time series may be different, there could be inherent relationships between these correla- tions. For example, in power demand forecasting, industrial load exhibits long-term dependencies as it reflects stable consumption patterns shaped by long-term economic struc- tural change, while residential demand shows predominantly short-term dependencies due to its sensitivity to immedi- ate factors like weather conditions or public events. As a result, the total power consumption of this region demon- strates hybrid dependency structure. Clearly, the spatial correlations in these three metries are different from each other. However, shown in Fig. 2. by decomposing the cor- relation dependencies into long-tern parts and short-term parts, we can represent the spatial correlations of all three se- ries simultancously with only two sub-correlation matrices. Also, since each matrix is shared by multiple time series, the learned spatial correlation is less sensitive to noise and outliers within individual series.
Editing. Due to the high interpretability of the correla- tion matrices to be learned, we can inject periodicity into CMoS by directly editing the initial weights. This makes CMoS more easier to model the periodic spatial correla- tions, thereby speeding up convergence and enhancing the performance for time series with great periodicity. Empow- ered by the above techniques, CMoS can achieve state-of- the-art prediction performance with as low as 1 1% of the parameter count compared to the lightweight model DLin- ear. Our code is provided in the anonymous repository https: //gi thub. /oom/CCTC1 oudOps/ CMoS.
In summary, our contributions are as follows:
We propose CMoS, a super-lightweight forecasting method with up to 100x parameter efficiency than DLinear.
Benefiting from chunk-wise spatial correlation model- ing, correlation mixing strategy across channels, and Periodicity Injection techniques, CMoS achieves top- tier performance on long-term multivariate time series forecasting tasks.
Indisstria Rower Deszzod
Reginn Power Detauo
Resiential Power Demand
The learned spatial correlation matrices are highly in- terpretable, which can help us to better understand the potential temporal regularity of real-world systems.
long short-feree moott-term epecial correlation
2. Related Works
long-turm spatial commcaation
short-turm corriativv
2.1. Development of lightweight time series forecasting models
Figure 2. The spatial correlations of multiple time series can be represented by the combination of fewer sub-correlations.
Several previous works (Zhou al., 2021; Wu et al., 2021; Zhou et al., 2022: Nie et al., 2023; Liu al., 2024a) have adapted the transformer structure to the time series fore- casting domain due to its capability of capturing long-term dependencies. The parameter count of these models of- ten reaches the scale of tens of millions. However, DI in ear (Zeng et al., 2023) demonstrated that merely using sim- ple linear layers with fewer parameters can lead to competi- tive even superior prediction performance, revealing that complex models with large parameter sizes are not always necessary to achieve high-quality forecasts. This has signifi- cantly accelerated the development of lightweight models, including FITS (Xu et al., 2024), Sparse'TFF (Lin et al., 2024b), and CycleNet (Lin et al., 2024a), to achieve state- oh-the-art prediction performance with fewer parameters.
In this paper, pioneer leveraging the stability and decom- posability of spatial correlations to build super-lightweight forecasting model for multivariate time series. Specifically, we propose CMoS, a Chunk-wise Mixture of Spatial cor- relations architecture to predict multivariate time series. The architecture involves several techniques to investigate the robust and efficient representation of time series de- pendencies: (I) Chunk-wise Spatial Correlation Modeling. Instead of learning the representation of specific patterns, CMoS focuses on directly building the concise spatial cor- relation matrices for time series, and we prove both theo- retically and experimentally that chunk-wise correlations exhibit stronger noise resistance than point-wise ones. (II) Correlation Mixing Strategy. CMoS introduces a mixture of correlation mechanisms to adaptively generate correla- tion structures across different time series. By learning small set of basis correlation matrices and the correspond- ing mixing weights, CMoS can represent varying spatial relationships across channels while maintaining great pa- rameter efficiency. (III) Periodicity Injection by Weight
The success of the lightweight models inspires a crucial insight: there likely exists an elegantly simple yet highly effective formulation for the structure of time series within the existing prediction framework (predicting future time steps based on lookback windows). Thus we try to shift our focus from modeling sophisticated pattern representations to directly learning the inherent and interpretable spatial
2

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
correlations among all channels, and finally design the super- lightweight model CMoS.
of past L observations.
Motivated by the
of chunk-to-
chunk spatial correlations mentioned in Sec. 1, we rede- fine the forecasting task as the following simple formu- lation. A time series { (t--411,::222----. .t) E R with length L can be divided into the chunk-wise se- ries - - ...SS.........s1::) E R xs with length & 8 when the chunk size is S, and we denote as {x,- 1-'+11 +1: x +2:...., (12....,x.) } in brief. For each chunk x₁_i to be predicted, from the perspective of spatial correlations, it can be simply viewed as linear combination of previous sequences:
2.2. Channel Strategy for Lightweight Models
Multivariate time series can be viewed as a multi-channel signal. Whether it is necessary to model the dependen- cies between different channels has been widely discussed. Some works claimed that modeling these cross-channel de- pendencies can enhance the prediction results (Chen et al., 2023; Han et al., 2024; Liu et al., 2024a), while some thought that modeling all channels separately with only one backbone (Nie et al., 2023: Xu et al., 2024). which is known as Channel Independent Strategy, can provide more robustness and further lead to better results. Since the latter does not require the additional overhead associated with modeling cross cross-channd channel dependencies, this greatly de- creases the overall parameter count. As a result, lightweight models (Xu et al., 2024; Lin et al., 2024b) whose parameter counts are comparable to or smaller than DLinear tend to adopt this Channel Independent strategy.
X++i
X + b;
(2)
3=0
where the learnable 8,y denotes the spatial correlation coef- ficient, indicating the extent to which the & - j-th chunk in the historical window influences the prediction of the i-th chunk. The learnable b, can introduce some basic trends like gradual increase.
In this work, we reexamine the drawbacks of the Channel Independent strategy in the design of lightweight models from the perspective of model capacity. For lightweight models, the scarcity of nonlinear relationships inherently restricts their ability to express diverse temporal structures. Therefore, employing the channel-ine -indeeeeind strategy will confine the model to representing only one single temporal structure. In CMoS, we address this issue by introduc- ing Correlation Mixing, which significantly enhances the model's 's capacity to express a variety of temporal structures with an affordable parameter cost. Additionally, we provide a detailed comparison of the modeling overhead associated with different channel strategies and the predictive perfor- mance of CMoS variants in the Appendix B, further high- lighting the superiority of our Correlation Mixing strategy.
In fact, some existing lightweight models, such as DLin- ear and TSMixer, can be viewed as special cases with a chunk size of I, where they model point-to-peint spatial correlations. However, we find that this type of correlation is more susceptible to random noise. We demonstrate that baunk-to-chune spatial correlation can be more robust to random noise than point-to-point spatial correlation.
Definition 3.1. Consider a linear regression model f(x; 0) = Txx. x. Assume the model is subjected to in- put x' vector. We define the noise sensitivity of the model as the variance of the output change: Var(f (x'; 0) (f(x; 0))
5, where 5 ~ NNu, o²) is a Gaussian noise
V (e T 6) = 0² 20100
Theorem 3.2. Within each chunk, perform a weighted av- erage of the point-to tonnttt--g to-point weights {01, 0.....,,. ,n to ob- tain new weights 6" Ei= a;9; (a; 2 0). Consequently, 2
3. CMoS
In this section, we introduce each component employed in CMoS and demonstrate their advantages.
we have 0² Ei= 0 2 0²6²2 iie. the chunk-wise linear model composed of new weights exhibit lower noise sensi- tivity under Definition 3.1.
3.1. Chunk-wise Spatial Correlation Modeling
The proof of Theorem 3.2 is provided in Appendix F.
Time series forecasting is the task of predicting future values of a sequence {a ,., } based on its past observa- tions. Given a sequence of L observations, the goal is to predict the next value(s) 2z+1, Tt+2, ... HI where H is the forecasting horizon. Formally, the task can be defined as:
Chunking v.s. Patching. The Patching technique, proposed by Nie et al. (2023), is widely used in many existing time se- ries forecasting methods (Goswami et al., 2024; Woo et al., 2024: Liu et al., 2024b). Technically, patching splits only the historical series into segments, and batch-based models focus on generating aggregated representation of the cor- relations between these historical segments (similar to the high-level semantic information in LLMs) and then decode the representation to future time points. However, the black- box nature of such representations make it hard to figure out
2 1itH =
21) 0)
(1)
where f is the forecasting function, 0 represents the model parameters, and (It- (It-L+1, It- ; 2. is the window
3

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Mixtame of Cecretstione for Chanset
Channet
Chaasel
Weight Mocarst
MACATA
Monir Helgtes
MAAAMAA
Channel
Mixcorn Cacrelationn
Channel
Chunk & Channel Channel-Specificc Specific Weight Allocation
Channol
Spatial Correlation Mixture
Aggregated Prediction Chunks
Recombined Prediction Windows
Lookback Window
Correlation Matrices
Yeristo C R MONSSO
Yet1:t111 E RRNH
Figure 3. CMoS Architecture.
how specific segment influence the final prediction, limiting the interpretability of these methods.
decomposability the spatial correlations, we expand the sole chunk-wise correlations to the combination of basic spatial correlations according to the weight set r (n ..... 2 given chunk-wise lookback window
In contrast, the proposed chunking technique splits both historical and future series, and instead of learning the high- level representations, chunk-based CMoS focus on directly modeling of the spatial correlation between historical and future segments. which is quite interpretable. Our further experiments also demonstrate that chunks enhance both robustness and efficiency.
of the n-th channel {x"
5 ++1** x" 4 , x"):
X+++
e
(0
xi + bb) (3)
Σ
E
k=
j=0
and the correlation matrices 0 , AK are shared by all channels.
3.2. Correlation Mixing
Most recent works either adopt the Channel Independent strategy which is potentially incompatible with multiple temporal structures when parameter count limited, or adopt the channel-mixing strategy while introducing unaf- fordable complexity. To address the above shortcuts, we design the correlation mixing strategy to better model the spatial correlations meanwhile maintaining great parameter efficiency. As a simplified version of Mixture-of-Exper (Ja- cobs et al., 1991) (each expert is just a spatial correlation matrix), this strategy combines the various basic spatial cor- relations with weighted sums obtained by the aggregated information of the lookback windows of each channel, adap- tively generating channel-specific spatial correlations. The strategy includes three components: shared spatial correla- tion matrices, channel-specific information aggregators, and a shared weight allocator.
Two-stage weight allocation. The key building robust spatial correlation matrices well as obtaining the best correlation for each channel is establishing a stable mapping between raw time series data and weights of each matrix. In our formulation, channels with similar temporal structures should have similar weight combinations. However, due to the potential differences in noise levels across time series data from different channels, the weight allocation process may still be affected by the noise level, even if their tem- poral structures are similar. Thus the first stage of weight allocation is to reduce the effect raised by different noise levels. Specifically, for each channel, we design specific information aggregator via a convolution kernel with size c and stride 5. This operation applies the specific degree of smoothing the raw data of certain channel obtain its stable and sparse representation z E R 24_5.
Shared spatial correlation matrices. Inspired by the
z Conv D" D"("
(4)

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
In the second stage, since the effect of the noise level has been mitigated, which means that channels with similar temporal structures have similar representation, we employ linear layer (the parameter matrix R 2=-- xK) shared by all channels weight allocator map the sparse representation zi to the weight set T (76 ................................ 7
3.4. Instance Normalization
We employ the Reversible Instance Normalization (Kim et al., 2022) technique, which is commonly used in many previous works, to address the potential distribution shift issue and improve prediction performance. Specifically, we normalize the original data and add statistical features back to the model's output to obtain the original distribution:
m = Linear(z")
(5)
L+LE 1
which is used for combining correlation matrices in Eq. (3).
L+1
Va
(6)
3.3. Periodicity Injection by Weight Editing
X 14 & x X
Vo+u
(7)
Some recent works (Lin et al., 2024b;a) have shown that explicitly leveraging the periodic features of time series can effectively improve the prediction accuracy of the model. In CMoS, given a periodic time series with the main period p, the time series chunks to be predicted should be more closely related to the corresponding chunks of previous periods, i.., Correlation(x) (x,;x, xt-;) Correlation (x,, Xt-j) for V such that i mod & = and j mod 5 + 0. As result, the spatial correlation matrix will exhibit significant peak every & chunks.
where y and a denote the mean and the variance of the input window.
3.5. Loss Function
We apply the Mean Squared Error (MSE) of the predicted values and the ground truth values as the loss function C:
N Σ |R 1
1
L
According to this assumption, we can directly edit the pa- rameters the first spatial correlation matrix by initializing it with the pre-defined periodic peaks. Specifically, the 0 in the matrix gedtt that satisfy - + mod = 0 is assigned to L. while others are assigned to 0, as shown in Fig For time series data with significant periodicity, since the initialized parameters already incorporate most periodic correlations, the model's learning burden is reduced. By providing a strong inductive bias towards periodicity, this op- eration not only accelerates convergence but also enhances model robustness. Meanwhile, the remaining correlation matrices are freed to focus on capturing non-periodic tem- poral structures, which can lead to an improvement in final prediction performance. In CMoS, we apply this strategy to those datasets with human activities involvement (demon- strating periodicity that aligns with human calendar cycles, e.g., daily or weekly routines), and calculate the period of these datasets by AutoCorrelatin Function (Madsen, 2007). More details about Periodicity Injection are provided in Appendix E.
3.6. Parameter Efficiency Analysis
Suppose the length of the lookback window is L and the forecast horizon is H for N channels. Even with an overly naive strategy of sharing only one temporal structure across all channels, the total parameter count of DLnnear is higher than 2 x L, x H. In CMoS, given chunk size S, convolu- tion kernel size c, and number of basic correlation matrices K (K is a constant and K < NN. The total number of parameters in CMoS is:
H S
2L K W Wiight Allocator Part
K Correlation Part
N Aggregntars Part
For the dataset ETThl with N = 7, S is set to 24, K is set to 4, and c is set to 8, the total number of the CMoS's parameters Rs LxH + L, which is less than 1% of DLinear's 966
parameter count, while CMoS can additionally deal with multiple temporal structures.
4. Experiments
values P
Interval
In this section, we present both the performance and pa- rameter efficiency advantages of CMoS against existing state-of-the-arr baselines, and demonstrate the effectiveness of each design in CMoS. Additional experimental results on hyperparameter sensitivity are listed in Appendix C.2.
Initielized by
Periodicity Injection
Learnable gast
Randomly Initialized
4.1. Setup
Datasets. We conduct experiments on widely used datasets for long-term time series forecasting, including
Figure 4. Illustration of Periodicity Injection.
5

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Table 1. The prediction performance of CMoS and haselines on long-term multivariate time series forecasting task. The results are averaged from all prediction horizons of H E {96, 192, 336, 720}. The best result is highlighted in bold and the second best is highlighted with underline. The results CMoS are averaged over 5 runs on each horizon with standard deviation STD. For detailed results on each prediction horizon, please refer to Appendix C.I.
Datasets
Electricity
Traffic
Weather
ETThl
ETTh2
ETTml
ETTm2
Metrics
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSB
MAB
Complex Models
Informer
0.389
0.406
1,898
0.866
0.292
0.348
0.739
0.589
0.422
0.440
0.584
0.503
0.362
0.387
FedFormer
0.219
0.329
0.62
0.382
0.301
0.345
0.435
0.454
0.406
0.438
0.567
0.519
0.334
0.380
TimesNet
0.190
0,284
0.617
0.327
0.255
0.282
0.468
0.459
0.390
0.417
0.408
0,415
0.292
0.331
PatchTST
0.171
0.270
0.397
0.275
0.224
0.261
0.429
0.436
0.351
0.395
0.349
0,381
0.256
0,314
TimeMixer
0.185
0.284
0.410
0.279
0.226
0.264
0.427
0.441
0.350
0.397
0.356
0.380
0.258
0.318
iTranfformer
0.258
0.397
0.232
0.270
0.439
0.448
0.370
0.403
0.361
0.390
0.269
0.327
Lightweight Models
DE iinnn
0.167
0.264
0.428
0.287
0.242
0.295
0.430
0.443
0.470
0.468
0.356
0,378
0.259
0,324
FITS
0.168
0.265
0.429
0.302
0.244
0.281
0.408
0.427
0.335
0.386
0.357
0.377
0.254
0.313
SparseIFF
0.165
0.258
0.412
0.278
0.240
0.280
0.406
0.419
0.344
0.364
0.361
0,382
0.251
0.312
CycleNet
0.158
0.250
0.421
0.289
0.242
0.278
0.415
0.426
0.355
0.398
0.355
0.379
0.252
0.309
CMuS
0.158
0.250
0.396
0.271
0.220
0.260
0.403
0.416
0.331
0.383
0,354
0.378
0.259
0.316
STD
1.0.011
1.0.01
100001
1.0.011
.00.002
100.002
1.0.004
1.0.003
1.0.003
1000002
1.0.003
1.0.003
1.0.04
10.004
Electricity, , Traffic, Weather, ETTh1, ETTh2, ETTm1, and ETTm2. More details of datasets are listed in Appendix D.1.
of complex models is sufficient enough, while lightweight models are limited to modeling only one temporal struc- ture (e.g., spatial correlation). As a result, they struggle to effectively handle multiple time series with varying tempo- ral structures at the same time. Owing to the mixture of correlations mechanism, CMoS is able to deal with vari- ous temporal structures inherent in numerous time series samples, and even surpasses the prediction performance of complex models.
Baselines. We compare CMoS with several well-known and state-of-the-art methods, including Informer (Zhou et al., 2021), FedFormer (Zhou et al., 2022), TimesNet (Wu et al., 2023), PatchTST (Nie et al., 2023), TimeMixer (Wang et al., 2024), iTransformer (Liu al.. 2024a), and lightweight models like DLinear (Zeng et al., 2023), FITS (Xu et al., 2024), Sparse'TSF (Lin et al., 2024b), and CycleNet (Lin et al., 2024a). More implementation details about baselines are provided in Appendix D.4.
Furthermore, CMoS exhibits remarkable stability in perfor- mance across varying random seed initializations, which can be attributed to the minimal parameter count and multi- ple wowustesss-ehhneeinn strategies, particularly the Chunk- wise Modeling and Periodicity Injection mechanisms. The combined effect of these design choices notably contributes to the model's reproducibility and practical deployment ca- pabilities.
Experimental Settings. Following previous lightweight methods like FITS and SparseTSF, we conduct a grid search on the lookback window of 96, 336, 720. For each set- ting, the results of CMoS are averaged over 5 runs with random seeds. More setting details about CMoS are listed in Appendix D.3.
4.3. Ablation Study
4.2. Main Results
We validate the effectiveness of cach technique in CMoS through multiple ablation studies as follows.
Table presents a comprehensive comparison of prediction results between CMoS and other baseline models across multiple datasets. Overall, CMoS demonstrates superior predictive performance, achieving state-of-the-at results on the majority of benchmarks, ranking first in out of 14 evaluation metrics and second in 3 metrics. It is noteworthy that CMoS consistently achieves optimal results datasets containing more than 20 channels (Eletricity, Traffic, and Weather). Interestingly, on these datasets, for methods em- ploying the channel independence strategy, complex models (STtchTST, TimeMixer, iTransformer) generally outperform the simpler models by a significant margin. The reason lies in that under this strategy, the representation capacity
Effectiveness of Chunk-wise Correlation Modeling. To investigate the effectiveness of chunk- modeling against point-wise modeling, we compare the prediction perfor- mance of CMoS and its variant, CMoS w/o Chunk, imple- mented by setting the chunk size to 1. Even though the parameter size of models using chunk-wise modeling is typ- ically only one-tenth to one-hundredd using point-wise modeling, according to Table 2, the former outperforms the latter in terms of prediction performance on the majority of datasets, and the improvement par- ticularly notable on the ETT-series datasets which exhibit
of that of models
6

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CMeS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
higher noise levels. This suggests that chunk-wise correla- tion modeling is approach with both greater parameter efficiency and robustness.
Test Loe with Ft
0.40
Test Loo with FI
viu
Pt
Effectiveness of Correlation Mixing, To evaluate the im- pact of the Correlation Mixing mechanism, we implement a variant CMoS by restricting the number of spatial corre- lation matrices to one (CMoS w/o Correlation Mixing). In this simplified configuration, all channels are constrained to share single spatial correlation, which is equivalent to adopting the channel independence strategy. Our experi- mental results demonstrate that this restriction significantly impacts model performance. Consistent with our compre- hensive comparison against other lightweight models pre- sented in Sec. 4.2, CMoS with Correlation Mixing exhibits substantial performance advantages over its variants, and this superiority is especially pronounced in datasets contain- ing more channels (achieving remarkable improvements of up to 17.5% on the Weather dataset). These results provide compelling evidence for the effectiveness of our correlation mixing strategy and its crucial role in representing various temporal structures while maintaining model efficiency.
016
014
0.30
1000
2000
3000
20000
6000
Decealions
Ticaakkus
(a) Electricity.
(b) ETTh2.
Figure 5. Test loss during training process on different datasets with horizon= 96. PI is the abbreviation of Periodicity Injection. The model with Periodicity Injection exhibits faster loss reduction.
nomena and is influenced by more random and complex factors. As result, the periodicity of most metries in this dataset is either not imprecise or not significant. Therefore, for such datasets, Periodicity Injection may have negative impact. In practice, we recommend using this strategy only when significant and stable periodicity exists in the majority of the time series.
Table 3. The MSE results of CMoS with Periodictiy Injection and w/o Periodicity Injection.
Table The MSE and MAE results of CMoS and its variants 7 datasets with horizon = 96. CorMix is the abbreviation of Correlation Mixing. The best results highlighted in bold, and the worst results are marked with *
Electricity
Truffic
Weather
ETTh1
ETTh2
ETTml
ETTm2
with
0.129
0.367
0.148
0.361
0,274
0,292
0.367
w/o PI
0.130
0.369
0.144
0.366
0.292
0.293
0.170
Variants
CMoS
w/o CorMix
w/o Chunk
Metries
MSE
MAR
MSF
MAE
MSR
MAF
4.4. Effectiveness and Efficiency
Electricity
0.129
0.223
0.137
0
0.132
0.226
Traffic
0.367
0.256
0.385
0.266
0.368
0,259
FITS
Weather
0.144
0.193
0.171
0.226
0,147
0,203
Paschris
ETTh1
0.361
0,383
0,362
0.385
0.368
0.393*
Dilisur
ETTh2
0.274
0.341
0.276
0.338
0.280*
0.344
0.44
Space TSP
ETTml
0.292
0.345
0.308
0.349
0.288
0.340
Nunber Paninottss
50000 190300 199360
Nunber Pacatietess xMi
CycleNer
Chiss
ETTm2
0.167
0.257
0.168
0.257
0.170"
0.262
Figure 6. Comparison of the prediction performance and parameter count between CMoS and other baselines on Electricity dataset with horizon= 192.
Effectiveness of Periodicity Injection. The implemen- tation of Periodicity Injection enables the model to more efficiently learn spatial correlations associated with primary periodic patterns, resulting in substantial improvements in the speed of loss reduction, as shown in Fig.5.
A key advantage of CMoS lies in its ability to achieve su- perior predictive performance with remarkably few param- eters. As demonstrated in our experimental results on the Electricity dataset in Fig. 6, CMoS achieves state-of-the- art prediction accuracy while utilizing only a fraction of the parameters required by most existing lightweight mod- els even one to two orders of magnitude fewer. This demonstrates that our modeling approach, combining spa- tial correlation modeling and correlation mixing, though concise, effectively captures temporal structures in forecast- ing tasks. This reveals the potentially inherent low-rank nature of time series prediction. Furthermore, such a com- pact parametrization enables CMoS to be readily deployed
Furthermore, this mechanism provides strong prior knowl- edge about periodic structures, allowing the remaining pa- rameters to concentrate on capturing secondary periodici- ties and non-periodic components of the time series. This, as shown in Table 3, can potentially enhance overall per- formance in most cases. However, from the experimental results, we observe an exception in the Weather dataset. Unlike other datasets that are dominated by human actiy- ities and are highly correlated with human-made cyclical patterns, the Weather dataset is derived from natural phe-
7

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
on edge devices, enabling high-quality time series forecast- ing even in essorcceconstrainee environments. Meanwhile, to further verify the efficiency of CMoS in practice, we provide the inference FLOPs, GPU memory footprint, and inference time of CMoS and other baselines on Electricity dataset on our platform in Appendix C.3.
Mapping 3. In contrast, the larger weights in Mapping 3 are almost entirely concentrated at the very end of the matrix. This indicates that the mapping models very short- term dependencies, relying heavily on observations from the past hour or even the last few minutes during prediction. This is a more effective forecasting strategy for time series where long-term trends are unpredictable but short-term trends remain relatively stable.
5. Interpretability
Mapping 4. Mapping 4, on the other hand, exhibits sev- eral noticeable diagonal stripes near the beginning of the matrix. suggesting that the prediction of a particular chunk depends more on data from other chunks observed a long time ago. Therefore, Mapping models long- long-term depen- dencies, which are likely to be more prominent in time series with strong and stable periodic features.
In the CMoS model, each weight 8i, in the spatial correla- tion matrix 0 directly reflects the strength of the relationship between the corresponding historical chunk and the pre- dicted chunk. A larger value of 0t,5 in the matrix indicates that the model places greater emphasis on the i-th histori- cal chunk when predicting the j-th future chunk. To gain deeper insights into the model's 's ability to learn and capture these spatial correlations, as shown in Fig. 7, we visualize four spatial correlation matrices, which are also referred to as correlation mappings, that are learned from the weather dataset without periodicity injection.
Mapping 1. The weights in Mapping I are distributed re relatively evenly, without showing very strong or specific dependencies. We hypothesize that the model uses Mapping to capture finer grained dependencies that other mappings overlook, namely residual dependencies.
Mapping Weights
To further explore how the model combines these founda- tional correlations to perform prediction on each channel, we visualize the original time series along with the average mapping allocation situation corresponding to each time series as shown in Fig. 8 (for all cases please refer to Ap- pendix H). As previously discussed, Channel cxhibits slow and unpredictable trend variations, prompting the model to exclusively utilize Mapping 3 to capture very short-term dependencies. Channel also exhibits some slow trends, but a peak occurs every other day, and the shape of the peak is more similar to that of nearby peaks. Consequently, the model primarily combines Mapping 2 and Mapping 3 to jointly model the very short-term and cross-day depen- dencies. Channel demonstrates more distinct periodic characteristics, leading the model to additionally leverage the long-term dependencies captured in Mapping en- hance prediction performance and robustness.
Residual Correlation
Mapping Weights
Short-term
Correlation (Previous Day)
Mapping Weights
Very Short-serm
Correlation
(Previous Hours)
Mapping Wrights
Long-term Correlation
Figure 7. Visualization of the learned spatial correlation matrices (mappings) on the Weather dataset. The chunk size is set to 4.
The visualization analysis reveals that each spatial correla- tion mapping captures and emphasizes distinct patterns of temporal dependencies, which can help understand the in- herent patterns of the whole system. We explain the features of each mapping one by one:
Claassi 1
20
300
300
Caard
Under the setting with a chunk size of 4, the weights in Mapping exhibit a very distinct diagonal stripe pattern at the tail end, where each weight 0.,, on this stripe roughly satisfies the condition 180-i+j 36. Considering that the weather dataset is sampled every ten minutes (6 x 24/4- 36 chunks per day), this stripe indicates that the predicted chunk is highly dependent on the historical data from the same chunk one day prior. In other words, Mapping 2 primarily models the short-term dependency between the predicted value and its observation from the previous day.
Mapping 2.
270
Claaneel
40
Figure 8. Raw time series and their corresponding averaged map- ping allocation proportion.
8

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
6. Conclusion
ing Research, 2023. ISSN 2835-8856. URL https: / openrevieewww.o.mmmmeee.e net http://frrrumiiiccccaaaaaa.....
In this paper, we propose a super-lighteiight time series forecasting model CMoS, which directly builds the spatial correlation within different time series chunks, To enhance the model's capacity to model varying temporal structures, we propose the Correlation Mixing strategy to adaptively combine the foundational correlations for one specific chan- nel. Also, we introduce the Periodicity Injection trick to accelerate the convergence. The experimental results show that CMoS achieves top-tier prediction performance with an extremely limited number of parameters. Additionally, our further analysis demonstrates that the model has excel- lent interpretability, which helps us better understand the underlying patterns of the system.
Goswami, M., Szafer, K., Choudhry, A., Cai, Y., Li, S., and Dubrawski, A. Moment: A family of open time- series foundation models. In International Conference Machine Learning. 2024.
Han, L., Chen, X.-Y,, Ye, H.-J., and Zhan, D.-C. SOFTS: Efficient multivariate time series forecasting with series-core fusion. In The Tiihth- Annual Conference Neural Information Processing Systems, 2024. httpttss:/ hwppp/eeeeeeeeeeeeeeeeeeeeeeeeee net forum?
id=8 9AU15L1U
Hewage, P., Behera, A., Trovati, M., Pereira, E., Ghahre- mani, M., Palmieri, F., and Liu, Y. Temporal convo- lutional neural (ten) network for an effective weather forecasting using time-series data from the local weather station. Soft Computing, 24:1 14:16433-16482 2020.
Acknowledgements
This work was partially funded by the National Natural Science Foundation of China (Grant No. W2412136), the National Key Research and Development Program of China (No.2022YFB2901 800), the National Natural Science Foun- dation of China (62202445), and the National Natural Sci- ence Foundation of China-Research Grants Council (RGC) Joint Research Scheme (62321166622)
Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. Adaptive mixtures of local experts. Neural Computation, 3(1):79-87, 1991. doi: 10. (166//eoo.199 1991.3.1.79.
Kim, T., Kim, J., Tae, Y., Park, C., Choi, J.-H., and Choo, J. Reversible instance normalization for accu- rate time-series forecasting against distribution shift. In International Conference on Learning Representations, 2022. URL https: http///pppnreeeeeeeeeeeeeeoeeeer net fortm? i id=cGDAkOolCOp.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Lin, S., Lin, W., Wu, W., Zhao, F., Mo, R., and Zhang, H. Segrnn: Segment recurrent neural network for long-term time series forecasting. arXiv preprint arXiv:2308. 12000 2023.
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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
A. CMOS Circuit & Our CMoS Model
The design of our model, which coincidentally shares the CMoS nomenclature, partly draws some inspiration from CMOS (Complementary Senal--xii-SSemmoooee
circuits. There exists an interesting parallel between the two: just as CMOS circuits dynamically switch their outputs between VDD (high voltage level) and GND (ground level) response to varying input signals, model adaptively combines different spatial correlations in response various channels within the lookback window, as shown in Fig. 9.
VDD
Cornelation
Correlation
Input Signal
VDD or GND according to
Time Series of channel
Neural Network
Combination of
Correlations according
GND
(a) The logic of CMOS circuits (take the NOT gate as example).
(b) The logic of our CMoS model.
Figure 9, The logical similarity between CMOS circuits and CMoS model.
B. More Information about Channel Strategy
B.1. Classification of Channel Strategies
We summarize the channel strategies that existing methods applied as follows (and shown in Fig. 10):
Private Line. This strategy is adopted by some methods that decompose multivariate time series forecasting task to multiple univariate time series forecasting tasks (Box & Pierce, 1970; Oreshkin et al., 2020). In this mode, each channel is modeled completely separately, with each channel having its own dedicated network. For a dataset with N channels, this results in model complexity of O(N), and may suffer from overfitting since the training data of each network is not enough.
One Bus. This strategy is also called Channel-Independern in several works (Nie et al., 2023; Liu et al., 2024b; Goswami et al., 2024; Lin et al., 2024b). Each channel can only receive past information from its own, and all channels share only one temporal structure. Although this strategy is claimed to be able to avoid overfitting since the network is trained with more data, unified structure may struggle to handle the diverse temporal structures when faced with different time series from the same system.
Data-Mixing. This strategy, adopted by several recent works (Zhang & Yan, 2023; Chen et al., 2023; Liu et al., 2024a; Han et al., 2024), assumes that data from the other channels can benefit the predictions for one specific channel. However, this will result the model complexity of O(N2) for building cross-channel dependencies, making it a cumbersome solution that is not suitable for many cases, especially for edge devices.
This strategy is proposed by CMoS. We posit that channel dependencies manifest through similar spatial correlations across different channels. By mixing K (K < N) fundamental correlations, we can derive the unique temporal structure for each channel while maintaining great parameter efficiency.
Correlation-Mixing
Table shows the model complexity when using different channel strategies with the same backbone (assuming the complexity of the backbone is constant C") on a dataset containing N channels.
B.2. Experimental Results on Different Channel Strategy
We further compare CMoS with its variants under these channel strategies by removing or rewriting the correlation mixing module. For the data- data-mixing strategy, since the model with complexity of O(CN2) exceeds the GPU memory limit when the number of channels large (more than 100), for comprehensive analysis, we chose TSMixer, model based on Multilayer

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
(a) Private Line
(b) One Bus
(c) Data Mixing
(d) Correlation Mixing
Figure 10. Illustration of different channel strategies.
Table 4. The model complexity under different channel strategies. channel is N.
The complexity of the backhone is constant C, and the numher of
Strategy
Private Line
One Bus
Data Mixing
Correlation Mixing
Complexity
O(CN)
O(C)
O(CN3)
O(CK), K <
Perceptron (MLP) architecture that incorporates simplified data mixing techniques (the model's parameter count is still several hundred times that of CMoS), our benchmark model under data-mixing strategy. Table 5 includes the forecasting results under all channel strategies. One Bus strategy, mentioned above, will limit the capacity of the model, while the Private Line strategy will limit the number of training samples for each separate network, both would degrade the performance. Although the Data Mixing strategy brings heavy complexity, its performance still does not surpass that of CMoS. We will discuss this intriguing phenomenon in the next section.
Table 5. Predicition performance under different channel strategics. All datasets include 20 channels.
Strategy
Private Line
One Bus
Data Mixing
Correlation Mixing
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Electricity
0.135
0.229
0.137
0.231
0.131
0.229
0.129
0.223
Traffic
0.399
0.281
0.385
0.266
0.376
0.264
0.367
0.256
Weather
0.148
0.206
0.171
0.226
0.147
0.203
0.144
0.193
B.3. Cross-channel Dependency from the Perspective of Information Theory
Contrary to the assumption of Data Mixing strategy that building cross-channel needs additional modules compared with models using Private Line strategy, from the perspective of information theory, we believe that cross-channel dependencies should not lead to increase in model complexity, but rather provide greater opportunities for model compression. Consider the Shannon Entropy for the i-th channel ((+)
Σ P(x log P(x*). the Shannon Entropy for the j-th channel H (X²) = E P(x)) log P. and their joint entropy H(X X = E' x²) log P(x', x²). The existence of cross-channel dependency means that X provides some information for X Under the information theory the two channels is greater than 0. Since I(X X') = (X*) + H(X)) H( X²), we have H(X: X²) H(X+) + H(X') if the cross-channel independency exists. The Private Line strategy can be seen modeling of Σ (X').. Therefore, in the presence of cross-channel dependencies, assume that fewer parameters required to model the mapping from the historical window to the prediction window, as the joint information entropy of the data H H(X) .x......"" XN) is smaller than Σ (X*)..
framework, this means the mutual information I I(XiX
X
C. More Results
C.1. Full Results Compared with Baselines
The full results compared with baselines with horizons E (96, 192, 336, 720] are shown in Table Compared to all late-offtthe-an methods, CMoS maintains top-2 prediction accuracy in most settings.
12

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Table 6. The full prediction performance of CMoS and baselines on long-term multivariate time series forecasting task with horizons (96, 192, 336, 720]. The best result is highlighted in bold and the second best is highlighted with underline. The results of CMoS are averaged over 5 runs on each horizon.
Datasets
Electricity
Traffic
Weather
ETTh1
ETTh2
ETTml
ETTm2
Metrics
MSE
MAE
MSE
MAE
MSE
MAE
MSB
MAE
MSE
MAB
MSE
MAE
MSE
MAE
Informer
0.215
0.321
0.682
0.391
0.206
0.253
0.715
0.571
0.362
0.394
0,419
0.422
0.216
0.302
FedFormer
0.191
0.305
0.593
0.365
0.175
0.242
0.379
0.419
0.337
0.380
0,463
0.463
0.216
0.309
TimesNet
0.169
0.271
0.595
0.312
0.168
0.214
0.389
0.412
0.334
0.370
0.340
0.378
0.189
0.265
Horison = 96
PatchTST
0.143
0.247
0.370
0.262
0.149
0.196
0.377
0.397
0.274
0.337
0.289
0.343
0.165
0.255
TimeMixer
0.153
0.256
0.369
0.256
0.142
0.198
0.372
0.401
0.281
0.351
0.293
0.345
0.165
0.256
iTransformer
0.134
0.230
0.363
0.265
0.157
0.207
0.386
0.405
0.297
0.348
0.300
0.353
0.175
0.266
DLinear
0.140
0.237
0.395
0.275
0.170
0.230
0.379
0.403
0.300
0.364
0.300
0.345
0.164
0.255
FITS
0.139
0.237
0.400
0.280
0.172
0.225
0.376
0.396
0.277
0.345
0.303
0.345
0.165
0.254
SparseTSF
0.138
0.233
0.389
0.268
0.169
0.223
0.362
0.388
0.294
0.346
0.312
0.354
0.163
0.252
CycleNet
0.129
0.223
0.397
0.278
0,167
0.221
0.374
0.396
0,279
0.341
0.299
0.348
0.160
0.247
CMoS
0.129
0.223
0,367
0.256
0.144
0.193
0.361
0.383
0.274
0.341
0.292
0.345
0.1
0.257
Informer
0.263
0.362
2,802
1.275
0.261
0.300
0.726
0.574
0.460
0.448
0.547
0.480
0.320
0.365
FedFormer
0.203
0.316
0.614
0.381
0.274
0.344
0.420
0.444
0.415
0.428
0.575
0.516
0.297
0.360
TimesNet
0.180
0,280
0.613
0.322
0.219
0.262
0.440
0.443
0.404
0.413
0.392
0.404
0.254
0.310
Horison sontson = 192
PatehTST
0.158
0.260
0.386
0.269
0.191
0.239
0.409
0.425
0.348
0.384
0.329
0.368
0.221
0.293
TimeMixer
0.168
0.269
0,400
0.271
0.192
0.243
0.413
0.430
0.349
0.387
0.335
0.372
0.225
0.298
iTransformer
0.154
0,250
0.384
0.273
0.200
0.248
0.424
0.440
0.372
0,403
0.341
0,380
0.242
0.312
DLinear
0.154
0.250
0.407
0.280
0.216
0.275
0.427
0.435
0.387
0.423
0.336
0.366
0.224
0.304
FITS
0.149
0.248
0.412
0.288
0.215
0.261
0.400
0.418
0.331
0.379
0.337
0.365
0.219
0.291
SparseTSF
0.151
0.244
0.398
0.270
0.214
0.262
0.403
0.411
0.339
0.377
0.347
0.376
0.217
0.290
CycleNet
0.144
0.237
0.411
0.283
0.212
0.258
0.406
0.415
0.342
0.385
0.334
0.367
0.214
0,286
CMoS
0.142
0.236
0.379
0.261
0.186
0.237
0.405
0.409
0.333
0.383
0.334
0.366
0.228
0.299
Informer
0.334
0.416
0.881
0.496
0.309
0.332
0.741
0.588
0.454
0.464
0.654
0.531
0.400
0.414
FedFormer
0.221
0.333
0.627
0.389
0.331
0.374
0.458
0.466
0.389
0.457
0.618
0.544
0.366
0.400
TimesNet
0.204
0.293
0.626
0.332
0.278
0.302
0.523
0.487
0.389
0.435
0.423
0.426
0.313
0.345
Horizon = 336
PatchTST
0.168
0.267
0.396
0.275
0.242
0.279
0.431
0.444
0.377
0,416
0.362
0.390
0.276
0.327
TimeMixer
0.9
0.291
0.407
0.272
0.247
0.284
0.438
0.450
0.367
0.413
0,368
0.386
0.277
0.332
iTransformer
0.169
0.265
0.396
0.277
0.252
0.287
0.449
0.460
0.388
0.417
0.374
0.396
0,282
0.337
DLinear
0.169
0.268
0.417
0.286
0.258
0.307
0.440
0.440
0.490
0.487
0.367
0,386
0.277
0.337
FITS
0.170
0.268
0.426
0.301
0.261
0.295
0.419
0.435
0.350
0.396
0.368
0.384
0.272
0.326
TSa
0.166
0.260
0.411
0.275
0.257
0.293
0.434
0.428
0.359
0.307
0.367
0.386
0.270
0.327
CycleNet
0.161
0.254
0.424
0.289
0.260
0.293
0.431
0.430
0.371
0.413
0.368
0.386
0.269
0.322
CMoS
0.161
0.254
0.397
0.270
0.240
0.281
0.412
0.420
0.342
0.384
0.366
0.386
0.273
0.325
Informer
0.502
0.525
3.225
1.302
0.390
0.388
0.772
0.623
0.410
0.454
0.715
0.578
0.512
0.468
FedFormer
0.259
0.364
0.646
0.394
0.423
0.418
0.474
0.488
0.483
0.488
0.612
0.551
0.459
0.450
TimesNet
0.206
0.293
0.635
0.340
0.353
0.351
0.521
0.495
0.434
0.448
0.475
0.453
0.413
0.402
Horison = 720
PatchTST
0.214
0.307
0.435
0.295
0.312
0.330
0.457
0.477
0.406
0.441
0.416
0.423
0.362
0.381
TimeMixer
0.228
0.320
0.462
0.316
0.318
0.330
0.486
0.484
0.401
0.436
0.426
0.417
0.360
0.387
iTransformer
0.194
0.288
0.445
0.308
0.320
0.336
0.495
0.487
0.424
0.444
0.429
0.430
0.375
0.394
DLinear
0.203
0.300
0.454
0.308
0.324
0.367
0.473
0.494
0.704
0.597
0.419
0.416
0,371
0.401
FITS
0.212
0.304
0.478
0.339
0.326
0.341
0.435
0.458
0.382
0.425
0.420
0.413
0.359
0.381
SSa
0.205
0.293
0.448
0.297
0.321
0.340
0.426
0.447
0.383
0.424
0.419
0.413
0.352
0.379
CycleNet
0.198
0.287
0.450
0.305
0.328
0.339
0.450
0.464
0.426
0.451
0.417
0.414
0.363
0.382
CMoS
0.200
0.288
0.442
0.295
0.311
0.332
0.433
0.451
0.374
0.423
0.425
0.417
0.367
0.385
13

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
C.2. Parameter Sensitivity
We conduct experiments to investigate the impact of the hyperparameters chunk size and the number of spatial correlation matrices the final prediction performance. Fig. 1la demonstrates the prediction results with different chunk sizes. Since the main period in the Electricity dataset is 24 x 7 168, when the chunk size is set to 16 which is not common factor of the period, the MSE increases greatly. In practice, setting the chunk size as common divisor of the potential periods would be a better strategy for CMoS.
Fig. 1lb demonstrates the prediction results with different numbers of spatial correlation matrices. For the Electricity dataset which contains over 100 channels, when the number is small (1 and 2), the model fails to capture sufficient temporal dependencies, leading suboptimal performance. However, more matrices bring more parameter count and more computational cost, and may lead to the network becoming overly sparse, which might also slightly affect the overall performance. Therefore, for datasets with medium or large number of channels, it would be a better choice when the number of spatial correlation matrices is set to an integer between 4 and 8.
0.140
0.145
0.140
0.135
MSE
0.135
MSE
0.130
0.130
0.125
0.120
0.125
8
12
16
24
1
4
6
8
12
Chunk Size
Number of Spatial Correlation Matrices
(a) MSE results with varied chunk size.
(b) MSE results with the varied number of spatial correlation matrices.
Figure 11. MSE results on Electricity dataset with multiple hyperparameter settings. The horizon is set to 96.
C.3. Inference Efficiency
We additionally provide the inference FLOPs, GPU memory footprint, and inference time of CMoS and other baselines Electricity dataset using 3090 GPU follows. The batch size of all methods is set to 64 for fair comparison.
Table inference FLOPs, GPU memory footprint, and inference time CMoS other baselines on Electricity dataset using one
3090 GPU,
Method
DLinear
CycleNet
SparseTSF
FITS
iTransformer
PatchTST
TimeMixer
CMoS
FLOPS
5.31G
5.68G
1.02G
5.33G
249.51G
1196.08G
10.58G
2.96G
GPU Memory
245MB
267MB
262MB
691MB
2271MB
22014MB
18642MB
252MB
Inference Time
1.81s
1.83s
1.49s
4.71s
1.92s
2.90s
2.85s
1.58s
D. Detailed Experimental Settings
D.1. Datasets
We train and test all methods on well-known datasets that are widely used for long-ten forecasting: Electricity, Traffic, Weather, ETTh1, ETTh2, ETTml, and ETTm2. Since Christoph Bergmeir (2024) strongly questioned the reasonability of
14

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
another dataset called Exchange raae on the 24 workshop, we excluded this dataset from our experiments. The number of time steps, number of channels, sample rate, the period obtained by AutoCorrelation Function, and the description of each dataset are listed in Table 8. Following most previous works (Zhou et al., 2022; Nie et al., 2023; Xu et al., 2024; Lin et al., 2024a), the ETT-series datasets are divided into training, validation, and test sets with 6:2:2 ratio, while others are divided into training, validation, and test sets with a 7:1:2 ratio.
Table 8. The detailed information of datasets.
Period obtained by ACF
Datascts
Time steps
Channels
Sample rate
Description
Electricity (2021)
26304
321
1 hour
168
Electricity consumption data of 321 clients
Traffic (2021)
17544
862
1 hour
168
Road occupancy rates measured by 862 sensors
Weather (2021)
52696
21
10 min
144
21 meteorological factors
ETThl (2021)
17420
7
1 hour
24
7 factors of electricity transformer
ETTh2 (2021)
17420
hour
24
7 factors of electricity transformer
ETTml (2021)
69680
15 min
96
7 factors of electricity transformer
ETTm2 (2021)
69680
15 min
96
7 factors of electricity transformer
D.2. Environment
All model instances in the paper are implemented with Py'Torch (Paszke et al., 2019). We conducted all experiments on the server equipped with 12 Intel(R) Xeon(R) Gold 5317 CPUs @ 3.00GHz and 4 NVIDIA GA102 GeForce RTX 3090 GPUs. Each experiment is conducted with one GPU.
D.3. Implementation Details of CMoS
Due to the simplicity of CMoS, we only need to set a few hyperparameters. In our experiment, one set of hyperparameters is searched for each dataset. The chunk size searched from (2,4.8.24), the number of spatial correlation matrices searched from {2,4,8), and the lookback window is searched from (66,33,,200). We use the AdamW (Loshchilov & Hutter, 2019) optimizer and the learning rate is searched from {2 x 10-5,,7 10-5, 9,8 10-5, 8 x 10-4). We additionally adopt the StepLR scheduler with slepsize 20 and gammaa 0.75. Each model is trained for 200 epochs with batch size= 64.
D.4. Experimental Settings other baselines
We reimplement CycleNet and SparseTSF according their source code in https: //////gttttt/tt cwm/iss- 1138/ SPa and https: / /ttpttttttt/ttt// window and other byperparameters for all baselines, providing comprehensive and reliable benchmark for these baselines. Therefore, for other baselines included in the latest version of TFB, we directly refer to the prediction performance provided by TFB.
TFB (Qiu et al., 2024) searches the lookback
E. Details of Periodicity Injection
E.1. Implementation Details of Periodicity Injection
For each dataset, firstly, use the AutoCorrelation Function (ACF) to caleulate the dominant period of this dataset (the period of all datasets calculated by ACF are listed in the 5th column in Table 8 in Appendix D.I). Next, during the grid search process for the byperparameter chunk size, for each experiment setting, we input the the calculated and the hyperparameter chunk size into Algorithm I provided in Appendix E.2, to obtain a modified matrix initialized via Periodicity Injection. This matrix is then used to replace the first basic correlation matrix the initialized model, thereby completing the Periodicity Injection operation.
E.2, Pseudocode of Periodicity Injection
15

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Algorithm 1 Pseudocode of Periodicity Injection
Input: Zero-Initialized Spatial Correlation Matrix 0 E IR x period p. chunk size S for i = 1 to !! do
for E 2 to with step - do
if +
then
0 0
end if end for end for Return: Modified Matrix
F. Theoretical Proof of Theorem 3.2
Proof. To prove Theorem 3.2, we need to demonstrate that:
Σ i= anly
)
Σ (as 2 0)
(9)
E as
Since a; 2 0, we have (E (E==);) E
af. Thus, have:
Di ag
i=1 a;0,)²
(10)
E Oi
1= o
According to the Cauchy- Scuwha-Schwazz inequality, we have:
a,0,)²
0
(11)
Thus, we finally prove that:
Σ 0:0s D ai
EE 1=
a,0,)²
n
0 (a: 2 0)
(12)
E of
1=1
The equality holds if and only at most one a; is non-zero (i E (0....
n - 1)). Therefore, the original theorem is proven.
G. Further Discussion
Although CMoS has demonstrate its superior prediction performance and efficiency on existing datasets, we further discuss some theoretical aspects that are not yet fully developed.
Spatial Correlation Modeling, The premise of modeling spatial correlation is that the time series data exhibit certain relatively stable patterns, including but not limited to local smoothness, periodicity, and long-term trends. For those irregular time series, such as in the case time series generated by random walks, the effectiveness this modeling approach has yet to be fully validated. Nevertheless, it notable that periodicity or stationary trend constitutes fundamental indicator of time series predictability and forecasting potential. Long-term forecasting can be very challenging if time series exhibit no periodicity stationary trend.
Extension of Theorem 3.2 Theorem 3.2 considers Gaussian noise, which is one of the most common and widely adopted assumptions. For those Non-Gaussian noise, take the burst noise as an example, it can be viewed as extreme deviations that occur the tail of a noise distribution. So we can define burst noise B()) mathematically as follows based on extreme value theory. Let X (t) be a noise process. According to the -Baundds-al Haan theorem, the conditional distribution of the exceedances events exceeding sufficiently high threshold follow a Generalized Pareto Distribution (GPD):
GPD(y:,, (() S xl > 1 - (1+ Ey
-1/8
16

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
where y = 0, is the scale parameter, and g 0 is the shape parameter.
Next, since the burst noise can be regarded discrete events, we define their occurence times T; follow Poisson process with intensity A(u). So we can finally define burst noise (t) as
B(I) y if (T:), otherwise
where y is i..d. GPD(o,E).
However, when we attempt to replace 5 with B in Definition 3.1, we find it difficult to theoratically analyze Var(#" B) since B is sparse and thus Var(0" B) relies heavily on specific samples of GPD.
As an alternative, we choose to experimentally compare the performance chunk/point-level modeling on the time series with random burst noise. Specifically, we construct several time series using sine functions, and following the above formulation, additionally inject some burst noises into all time series. The prediction results are listed in Table 9, and conclude that chunk-level modeling is more robust than point-level modeling when facing burst noise, which can be seen an empirical extension of Theorem 3.2 on other non-Gaussian noise like burst noise.
Table 9. Prediction performance on time series with burst noisc.
MSE
MAE
Chunk-level
0.0246
0. 166
Point-level
0.0249
0.1176
H. Extended Visualization the Weather Dataset
We provide more visualization on the Weather dataset to showcase more relationships between the raw time series and their corresponding mapping allocation proportions. Fig. 12 provides visualizations of all the raw time series channels within long time window in the Weather dataset, and Fig. 13 provides visualizations of mapping allocation situation on all channels.
17

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CMoS: Rethinking Time Series Prediction Through the Lens of Chunk-wise Spatial Correlations
Chanoe
Chooel
Choosel
102
Channe
Chooel
Chams
Chomse
100
50
Chaannd 135
Channel 54
Chasaed 15
Chanad
Chataed
100
Choud 10
Chascel
Chared
Pigure 12. Visualizations all channels within long time window in the Weather dataset.
Channel
Channel
Channel 3
Channel
Channel 5
Channel
Channel 7
1.00
1.00
1.00
1.60
1.00
1.00
1.00
Proportion
0.75
Proporice
0.75
Propariicn
0.75
0.75
Prsportion
0.75
0.75
Proportion
0.75
0.00
0.50
0.50
0,50
0.00
3
2
2
3
2
Map
Map
Map Index
Map
Channel
Channel
Channel 10
Channel
Channel 12
Channel
Channel 14
1.00
1.0
1.00
1.00
1.00
1.00
1.00
Proportio
0.75
Pooprrten
075
Propartkc
0.75
075
Proportion
0.75
0.75
Proportion
0.75
8.60
0.50
0.50
0.50
0.50
6.00
3
2
2
5
3
2
3
Indle
Map Inilex
Indes
Map Indea
Channel 15
Channel
Channel 17
Channel
Channel 19
Changel
Channel 21
1.00
1.00
1.00
1.00
1.00
1.00
1.00
Proportko
6.75
0.75
Proparikio
6.75
0.75
Proportke
0.75
0.75
0.75
8.00
0.60
0.50
0.60
0.50
0,50
0.00
0.25
0.25
2
2
2
2
3
Map
Indox
Map
Indox
Map
ladex
Maap Ludox
Figure 13, Visualizations of mapping allocation situation on all channels.
18


Reviews:
[{"review_id": "xOQQJnbmYx", "reviewer": "Reviewer_aVfa", "summary": "There's a recent line on making small architectures that match the performance of large deep learning models for TS forecasting, which raises the question of the relevancy of DL for time series forecasting. The authors propose CMoS, a novel architecture for time series forecasting that is very lightweight. The main contributions of the architecture include two elements: (1) chunk-wise spatial correlation modelling, which models the prediction of each chunk as a linear combination of previous chunks, and (2) correlation mixing, which uses cross-channel aggregation to get channel-specific spatial correlation mixtures with a low parameter count. The authors also introduce periodicity injection due to the structure of the correlation matrix, allows them to introduce pre-defined periodic peaks into the spatial correlation matrix's initialization. The authors use standard MSE loss and RevIN. Experiments are run on typical TSLib datasets against a few existing baselines. The authors ablate the different components of their proposed architecture, and offer analyses of their architecture's efficiency and interpretability.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your detailed and thoughtful review! We will address your concerns point by point.**\n\n> Q1: Novelty of some components\n\nYour example is quite helpful! So we explain the two aspects you mentioned following the format you provided:\n- The use of ConvNet. \n\nExisting methods like DeepTCN or TimesNet employs one ConvNet backbone for all channels. Since different channels may exhibit different noise levels, only one backbone may struggle to resist interference from various levels of noise. In contrast, **CMoS allocate a specific lightweight ConvNet for each channel to eliminate the channel-specific noise variations**, thus enhancing the model's robustness. \n\n**It is also notable that compared to this specific component, the whole Correlation-Mixing framework (including ConvNet) is a more important innovation.** As shown in Appendix B, **compared with existing channel strategies, our Correlation-Mixing can effectively model diverse temporal and even cross-channel dependencies with great efficiency**. ConvNet plays an important role in reducing the effect of noise in this framework.\n- Chunk v.s. Patching. \n\nTechnically, patching **only splits the historical series into segments**, and patch-based models like PatchTST focus on **generating aggregated representation of the correlations between these historical segments** (similar to the high-level semantic information in LLMs) and then decode the representation to future time points. The black-box nature of such representations make it hard to figure out how specific segment influence the final prediction, limiting the interpretability of these methods. \n\nIn contrast, chunk **splits both historical and future series**, and instead of learning the high-level representations, chunk-based CMoS focus on **directly modeling of the spatial correlation between historical and future segments**. which is quite interpretable. Also, we proved that chunks bring benefits in robustness and efficiency.\n\n*To better understand their differences in interpretability*, we visualized the patch-based representation (att1&2.png, based on the attention score of PatchTST's 2 layers) and chunk-based correlation (cmos.png) on weather dataset in anonymous repo https://anonymous.4open.science/r/kjUH. We can easily figure out how historical segments contribute to future segments for each time series in cmos.png ,while it's hard to obtain similar or other interpretable information from att1&2.png.\n\n**We will provide more details and recent works in the next version according to your valuable suggestions!** Also, reviews about works before 2023 will be added.\n> Q2: Inject incorrect periodicity\n\nThat's an interesting question. To simulate this case, we randomly select an integer x!=period as interval in injection phase. Comparing the below results (MSE) with Table 3 in the paper,  it can be seen that **using incorrect period performs barely better than random initialization**.  So we suggest injecting the correct period only when most time series in the system passed the ACF test—a scenario that commonly occurs in real-world applications.\n||Ele.|Tra.|Wea.|ETTh1|ETTh2|ETTm1|ETTm2|\n|-|-|-|-|-|-|-|-|\n|wrong period|0.130|0.372|0.151|0.371|0.297|0.294|0.174|\n> Q3: More Baselines\n\nWe follow your suggestion to include more baselines. The prediciton results (MSE) of naive, seasonal naive, Time-LLM and Crossformer are listed in the below. *Since Time-LLM leverage very large LLM as part of the model, the trainging time is quite long (over 2 days for single setting). Therefore, we currently have only obtained partial results. The full results are on in its way.* \n\nFrom the results, CMoS outperforms these methods, indicating that CMoS is a quite effective model. \n||Ele.|Tra.|Wea.|ETTh1|ETTh2|ETTm1|ETTm2|\n|-|-|-|-|-|-|-|-|\n|Naive|1.611|2.770|0.353|1.319|0.533|1.271|0.385|\n|Naive Season|0.230|0.630|0.371|0.598|0.477|0.489|0.358|\n|Crossformer|0.181|0.523|0.235|0.452|0.861|0.465|0.589|\n|Time-LLM|-|-|-|0.414|0.340|0.360|0.262|\n|CMoS|0.158|0.396|0.220|0.403|0.331|0.354|0.259|\n> Q4: Other concerns\n\nVery sorry that the response here cannot cover every point you have mentioned due to the text length limitation, and **we can further discuss any uncovered points during the discussion phase**. Here are the brief replies for some points:\n- Parameter concerns: We determined the grid sets through extensive experiments. \n- More visualizations and ablations: These are valuable suggestions. We will perform more experiments and  analysis.\n- K matrices are shared: As mentioned in *Sec. Introduction*, the K matrices are designed to learn some basic temporal structures in the system, and each channel finds a specific way to combines these basic correlations. This design brings both efficiency and robustness benefits.\n- Nonlinear injection: If certain modules strongly correlate with periodicity, we can generalize Periodicity Injection to these modules. We believe this would be an interesting direction to explore.", "claims_and_evidence": "- Novel architecture CMoS\n  - chunk-wise Spatial correlation modelling\n  - Correlation mixing strategy\n\nUsing convnets for forecasting is not a new idea. However, the related work lacks any reference to these works. As such, it is difficult to determine how novel the introduced architecture actually is, and what components of it are novel. As the authors state, patching and chunking are quite similar, but the discussion on this topic is quite brief. As is stands, it is difficult to evaluate the novelty of the claims without a proper literature review on comparable architectures. Part of the purpose of the Related Work section is to help differentiate works that are similar in nature, and so should discuss the use of patching more extensively, any differences it has with chunking (other than purpose), the use of convolutions for forecasting. For example, see https://arxiv.org/abs/1906.04397, section 2, paragraph 2 which discusses many other examples of architectures that leverage similar convolutional biases.\n\n- Novel weight init strategy for periodicity injection\n\nThis is an interesting strategy for smart initialization. One obvious limitation to this method is not only that you must know what the periodicity looks like, that periodicity must be fixed (cannot vary over time). Furthermore, it is unclear what happens if you inject a periodicity that is actually incorrect into a model, if e.g. you make an incorrect assumption around the periodicity. I would like to see experiments around this. Nevertheless, I find this idea quite interesting, albeit hard to generalize to other architectures.\n\n- CMoS is SOTA \n\nThe baselines that the authors compare to are those available in the tslib library. Therefore, if their results are better than those of the best model in that library, they are SOTA. Looking at the best model on there currently, CMoS is SOTA on those datasets. However, Tslib seems to be missing many models, including Time-LLM, which performs better (see Table 1 of https://arxiv.org/abs/2310.01728). It would also be useful to add some non-DL baselines to compare, e.g. naive, naive with drift, seasonal naive. Also, if you're using TFB, why not have included crossformer in your table? PatchTST is often second-best, and crossformer is competitive with patchTST in the TFB paper.\n\n- Interpretable learned spatial correlation matrices\n\nThe authors visualize the spatial correlation matrices and interpret them for the weather dataset, which is informative. \n\n\n\n- Difference between chunk and patch: really underexplored. Describe in an appendix or something?", "methods_and_evaluation_criteria": "See above re: CMoS being SOTA.", "theoretical_claims": "Theorem 3.2 Has a proof in Appendix F, that I did not check.", "experimental_designs_or_analyses": "The datasets are widely used and typical within the field.\n\n- It's unclear how you chose to specify the grids for the chunk size and spatial correlation search. It's also unclear is what is meant by \"lookback window search\".\n\n- The authors conducted ablations on the components of the method, showing they are all important for the method's success.\n\n- Interpretability analysis through visualizations. Interesting discussion of the mappings. A way to quantify this approach would be better, especially if you could tie these mappings to the actual time series. Figure 8 is a start, and it would be interesting to expand this analysis to other datasets, or to compare datasets where some transfer might be expected and see if the mappings have similarities across datasets, e.g. in between the ETT datasets.", "relation_to_broader_scientific_literature": "This paper relates to a line of work around deep learning for time series forecasting. Autoformer introduced the Time Series Library https://github.com/thuml/Time-Series-Library that has been used regularly by the community to benchmark TSF models on a set of common datasets. Among these models, some are extremely parameter-efficient and question whether transformers are useful/necessary for time series forecasting, such as DLinear https://arxiv.org/abs/2205.13504. This work continues that line of work, using the same TSLib to benchmark their model and comparing against baselines in that repo.", "essential_references_not_discussed": "- The review needs to go back further than 2023, especially considering that this is a more traditional paper (see \"Claims And Evidence\", the first claim, the paper on temporal convolutions, the related work section, paragraph 2, as a place to start).", "other_strengths_and_weaknesses": "Originality: This work further elucidates the importance of architectural choices for efficient ts forecasting.\n\nSignificance: The results of this work are significant, in that they show solid performance with less parameters than typical methods.\n\nClarity: The paper structure is standard and clear. It's well-written overall, with a few typos.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Did you ablate for overlapping chunks? \n- How about multiple layers of convolution?\n- Why are the K correlation matrices shared by all channels?\n- how do you select the degree of smoothing to apply during the two-stage weight allocation?\n- Are the train/test splits also across time?\n- Can the periodicity injection be applied to other architectures, for example, those with nonlinearities?", "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": "bwPCJdvnOn", "reviewer": "Reviewer_VtAP", "summary": "This paper presents CMoS, a super-lightweight time series forecasting model that utilizes chunk-wise spatial correlations to achieve parameter-efficient and interpretable predictions. The key innovation lies in directly modeling the spatial dependencies between time blocks of fixed size, rather than point-oriented patterns, as theory and experience suggest, to enhance noise robustness. CMoS introduces a correlation hybrid strategy that combines a small group of shared basis correlation matrices (e.g., long-term, short-term, periodic) with channel-specific adaptive weights to achieve diversified time structure modeling while maintaining minimal parameters. In addition, periodic injection through weight initialization accelerates the convergence of periodic data. The experiment demonstrated state-of-the-art performance across seven benchmarks, with interpretable correlation matrices revealing different time dependencies (e.g., daily periodicity, residual trends). This work highlights the inherent simplicity of time structures and provides a framework for resource efficiency forecasting.\n\n## update after rebuttal\nI thank the authors for their thorough response to reviewers' feedback and the improvements made to the paper. After examining the rebuttal and considering the other reviews, I've updated my overall recommendation to \"3 - Weak accept\" as the authors have sufficiently addressed the primary questions and concerns I raised.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your detailed and thoughtful review! We will address your concerns point by point.**\n\n> Q1: How could Theorem 3.2 be extended to nonlinear dependent or non-Gaussian noise (e.g., burst noise), and what empirical evidence supports its robustness under such conditions?\n\n- **Extended to Non-Gaussian noise like burst noise**\n\nTake the burst noise as an example, it can be viewed as extreme deviations that occur in the tail of a noise distribution. So we can define burst noise $B(t)$ mathematically as follows based on extreme value theory. Let $X(t)$ be a noise process. According to the Pickands–Balkema–de Haan theorem, the conditional distribution of the exceedances events exceeding a sufficiently high threshold $u$ follow a Generalized Pareto Distribution (GPD): $GPD(y ; \\sigma, \\xi)= P(Y(t)\\le y|X(t)>u) \\approx\n1-\\left(1+\\frac{\\xi y}{\\sigma}\\right)^{-1 / \\xi}$,where $y=x-u>0$, $\\sigma>0$ is the scale parameter, and $\\xi > 0$ is the shape parameter. Next, since the burst noise can be regarded as discrete events, we define their occurence times ${T_i}$ follow a Poisson process with intensity $\\lambda(u)$. So we can finally define burst noise $B(t)$ as $B(t) = y+u,\\text{if}\\ t\\in\\{T_i\\},\\ \\text{otherwise}\\ 0$, where $y$ is i.i.d. $GPD(\\sigma, \\xi)$.\n\nHowever, when we attempt to replace $\\delta$ with $B$ in Definition 3.1, we find it difficult to theoratically analyze $Var(\\theta^TB)$ since **B is sparse and thus $Var(\\theta^TB)$ relies heavily on specific samples of GPD**. \n\nAs an alternative, we choose to **experimentally compare the performance** of chunk/point-level modeling on the time series with random burst noise. Specifically, we construct several time series using sine functions, and following the above formulation, we additionally inject some burst noises into all time series. Some segments of these time series are visualized in *burst.png* in the anonymous repo https://anonymous.4open.science/r/kjUH. The prediction results are listed in the following table, and **we can conclude that chunk-level modeling is more robust than point-level modeling when facing burst noise, which can be seen as an empirical extension of Theorem 3.2 on other non-Gaussian noise like burst noise**. \n||MSE|MAE|\n|-|-|-|\n|Chunk-level|0.0246|0.1166|\n|Point-level|0.0249|0.1176|\n\n- **Extended to nonlinear dependency**\n\nWith regard to **modeling nonlinear dependency**, the presence of nonlinear interactions makes it difficult to derive closed-form solutions or establish rigorous mathematical proofs. Specifically, in Definition 3.1, if $f(\\theta)$ is a nonlinear function, it's hard to obtain a parametric mathematical expression for $Var(f(x';\\theta)-f(x;\\theta))$, which hinders the following theoretical derivations. So we design a 3-layer MLP with **nonlinear activation** to validate the effectiveness of chunk-level modeling. From the results, **we can find that chunk-level modeling outperforms point-level modeling in most cases, indicating its robustness on multiple datasets**.\n||Ele.|Tra.|Wea.|ETTh1|ETTh2|ETTm1|ETTm2|\n|-|-|-|-|-|-|-|-|\n|Chunk NonLinear|0.166|0.414|0.237|0.410|0.362|0.357|0.263|\n|Point NonLinear|0.170|0.431|0.245|0.413|0.381|0.359|0.260|\n\n> Q2: Inference speed and memory footprint\n> \n\nThat's a quite good suggestion! We provide the inference FLOPs, GPU memory footprint, and inference time of CMoS and other baselines on Electricity dataset using a 3090 GPU as follows. The batch size of all methods is set to 64.\n\n||FLOPS|Memory|Infer. Time|\n|-|-|-|-|\n|Dlinear|5.31G|245MB|1.81s|\n|CycleNet|5.68G|267MB|1.83s|\n|SparseTSF|1.02G|262MB|1.49s|\n|FITS|5.33G|691MB|4.71s|\n|iTransformer|249.51G|2271MB|1.92s|\n|PatchTST|1196.08G|22014MB|2.90s|\n|TimeMixer|10.58G|18642MB|2.85s|\n|CMoS|2.96G|252MB|1.58s|\n\nAlthough the memory allocation strategy and powerful computational performance of the 3090 may narrow the gap in computational overhead among models, it can still be seen that, **CMoS consistently maintains an advantage in computational overhead** except for SparseTSF. It is also notable that the prediction performance of CMoS greatly outperforms SparseTSF, especially on those datasets with more channels. **This means CMoS can achieve the best effectiveness-efficiency balance among all methods**.\n\n> Q3: Automatically select the chunk size of the dataset.\n\nThat's quite a valuable question. As there is no enough information about period, using adhoc chunk size might not be a good choice. The good news is that **since CMoS is a super-lightweight model, it is entirely affordable performing hyperparameter search within a certain range in most cases. Therefore, in practice, we can take advantage of advanced hyperparameter optimization algorithms (such as Bayesian optimization) or frameworks (such as Optuna) to automatically find the optimal chunk size for best performance** on these aperiodic or irregularly sampled series. \n\n*Thank you again for your valuable review, and we hope our response can address your concerns.*", "claims_and_evidence": "Yes, I believe the paper's claims are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed method is reasonable for time series forecasting. With the support of theoretical analysis and ablation research, Chunk-wise spatial correlation modeling solves the problem of noise robustness and parametric efficiency. The correlated mixing strategy effectively balances model capacity and lightweight design through  shared basis matrices, while allowing for channel-specific adaptations. Periodicity injection provides a practical inductive bias for cyclical patterns without overcomplicating the architecture. Standard Indicators (MSE/MAE) are used to assess seven established benchmarks to ensure fair comparisons.", "theoretical_claims": "The paper’s theoretical claim (Theorem 3.2) asserts that chunk-wise spatial correlations reduce noise sensitivity compared to point-wise modeling. The proof in Appendix F correctly applies the Cauchy-Schwarz inequality, showing that averaging the point-wise weights into chunk-wise weights reduces the L2 norm of the parameter and thus the noise variance. However, the theorem assumes linear and Gaussian noise, which may not be exactly consistent with real-world time series dynamics (e.g., non-Gaussian noise, nonlinear dependence). While the proof is mathematically sound under these assumptions, its practical significance depends on how closely the linear chunk-wise model approximates the real data.", "experimental_designs_or_analyses": "Experiments are valid using standard benchmarks and metric. Ablation studies support key designs (chunk-wise modeling, correlation mixing), and parameter efficiency is well quantified.", "relation_to_broader_scientific_literature": "CMoS builds on recent advances in lightweight time series models (e.g., DLinear, FITS) but uniquely addresses their limitations in handling diverse temporal structures. While channel-independent strategies (PatchTST, SparseTSF) reduce parameters but restrict model capacity, and channel-mixing methods (iTransformer) incur high complexity, CMoS bridges this gap via correlation mixing—sharing basis matrices while adapting to channel-specific patterns, akin to \"mixture of experts\" but tailored for spatial correlations. Its chunk-wise modeling aligns with patching in PatchTST but prioritizes robustness over semantic embeddings. Periodicity injection extends CycleNet’s explicit cyclical modeling but integrates it into interpretable correlation weights. The work advances the paradigm of \"simple yet effective\" models, demonstrating that lightweight designs can achieve both efficiency and expressiveness by rethinking temporal dependencies.", "essential_references_not_discussed": "I believe there is no essential reference missing from the discussion.", "other_strengths_and_weaknesses": "Strengths:\n1.\tThe paper is well-structured, with clear technical exposition (e.g., chunk-wise formulation, correlation mixing pseudocode) and intuitive visualizations (Fig. 7-8) that enhance interpretability.\n2.\tThe paper compellingly critiques the limitations of channel-independent strategies for lightweight models and introduces correlation mixing as a novel middle ground between parameter efficiency and multi-pattern modeling.\n3.\tThe extreme parameter efficiency (1% of DLinear) and interpretable correlation matrices offer direct value for edge deployment and domain analysis (e.g., energy systems).\n\nWeaknesses:\n1.\tTheorem 3.2 assumes linear and Gaussian noise, ignoring nonlinear dependencies and real-world noise types (e.g., burst noise), limiting its practical relevance.\n2.\tDespite the emphasis on lightweight design, key metrics such as inference speed, and memory footprint are ignored, leaving the utility of the deployment unproven.\n3.\tChunk size selection depends on prior periodic knowledge (e.g., divisor of 24/168), requiring manual tuning of the new data set, and reducing the availability of aperiodic or irregularly sampled series.", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "1.\tHow could Theorem 3.2 be extended to nonlinear dependent or non-Gaussian noise (e.g., burst noise), and what empirical evidence supports its robustness under such conditions?\n2.\tDespite the emphasis on lightweight design, why are inference speed and memory footprint excluded from the evaluation? Can you provide some benchmarks to verify the usefulness of your deployment?\n3.\tWhat strategies can automatically select the chunk size of the dataset without explicit periodic or irregular sampling, thereby reducing the reliance on manual tuning?", "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": "4ZVtbWogNl", "reviewer": "Reviewer_fxdG", "summary": "This paper works on time series forecasting task, and the main idea is to split time series into chunks, and build up chunk-chunk spatial correlations to achieve robust time series forecasting. The paper is clearly written, the proposed modules are accompanied with good motivations and solid proofs.\n\n## Update After Rebuttal\n\nI've read the rebuttal and other reviewers' comments, my final rating is weak accept. The reasons why I cannot give higher ratings are: (1) The performance gap in Table 1 is minor; (2) Although the proposed method can greatly reduce the parameter number, the FLOPs and inference time remain comparable to other methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your detailed and thoughtful review! We will address your concerns point by point.**\n\n> Q1: Details about Fig. 1\n\nAs an illustrative example, we use the MSE of each pair of chunks as the correlation value (labeled in the figure), and a lower MSE meaning stronger correlation. **Only when the shapes of two chunks are relatively similar can the MSE between them be close to zero, indicating a strong correlation between the two chunks**. In other cases, there is no significant relationship between them since their MSE is usually much greater than zero. We would be pleased if the above description can address your conerns!\n\n> Q2: The differences between your method and other chunk-based techniques.\n\nThanks for your valuable question. Despite all three approaches employing chunk operations on raw time series, \n\n**both Chunkformer and REWTS  emphasize modeling intra-chunk relationships within individual chunks**. However, this design suffers from poor interpretability, as the internal mechanisms (such as attention weights of intra-chunk correlation) are often opaque. **It's hard for us to understand how these intra-chunk relationships contribute to the future predictions**. Moreover, since intra-chunk modeling is highly sensitive to each value within a chunk, **the model's performance is often greatly affected when the data contains high levels of noise or outliers**.\n\n**In contrast, our CMoS focuses on direclty modeling the relationships between historical chunks and prediction chunks (inter-chunk relationships)**. This approach offers several key advantages:\n- **Enhanced Interpretability**: Our method provides clear insights into how historical chunks influence future predictions (as shown in Sec.5).\n- **Improved Robustness**: The focus on broader temporal relationships rather than fine-grained intra-chunk patterns makes our method more resilient to noise in the data, which is theoretically proved in Sec. 3.1 and experimentally proved in Sec. 4.3.\n\n**We will add this point to our related works to help readers better understand the differences between CMoS and other methods.**\n\n> Q3: Can you compare your aggregated spatial correlations with attention-based methods?\n\nSure! We visualized the attention-based representation (att1&2.png, based on the attention score of PatchTST's 2 layers) and the correlation of CMoS (cmos.png) on weather dataset in anonymous repo https://anonymous.4open.science/r/kjUH. \n\nFrom the picture, we find that there is a significant difference between the two. The attention-based correlation represents the attention relationships **among historical data points, but it does not reveal how these relationships contribute to the prediction of future time series**. As a result, it is difficult to derive an intuitive explanation of the forecasting process from the attention-based correlation. In contrast, our correlation **directly captures the mapping from historical data to the future time series**. This allows us to **clearly see which historical chunks contribute more to the prediction of future chunks**, making the model’s decision process more interpretable.\n\n> Q4: Can you further include time cost and flops? \n\nThat's a quite good suggestion! We provide the inference FLOPs, GPU memory footprint, and inference time of CMoS and other baselines on Electricity dataset using a 3090 GPU as follows. The batch size of all methods is set to 64.\n\n||FLOPS|Memory|Infer. Time|\n|-|-|-|-|\n|Dlinear|5.31G|245MB|1.81s|\n|CycleNet|5.68G|267MB|1.83s|\n|SparseTSF|1.02G|262MB|1.49s|\n|FITS|5.33G|691MB|4.71s|\n|iTransformer|249.51G|2271MB|1.92s|\n|PatchTST|1196.08G|22014MB|2.90s|\n|TimeMixer|10.58G|18642MB|2.85s|\n|CMoS|2.96G|252MB|1.58s|\n\nAlthough the memory allocation strategy and powerful computational performance of the 3090 may narrow the gap in computational overhead among models, it can still be seen that, **CMoS consistently maintains an advantage in computational overhead** except for SparseTSF. It is also notable that the prediction performance of CMoS greatly outperforms SparseTSF, especially on those datasets with more channels. **This means CMoS can achieve the best effectiveness-efficiency balance among all methods**.\n\n> Q5: Some failure cases. \n\nWhen the underlying data distribution shifts significantly over time (i.e. concept drift), such as sudden changes in market behavior or consumer patterns, the basic correlations may not include the new time strutures, so these rapid distribution shifts can affect the prediction accuracy of CMoS. \n\nHowever, it's also important to note that this is a fundamental difficulty in time series forecasting that the entire field is actively working to address. A possible solution is to quickly update the model when facing concept drifts, and owing to the efficiency advantage, CMoS can be updated more rapidly and at a higher frequency. **This rapid adaptation capability allows CMoS to mitigate the impact of concept drift more effectively than other methods**.", "claims_and_evidence": "1. In Figure 1, which specific techniques are used to obtain the correlation value? It's interesting yet a bit confusing why the obtained correlation values are similar, though the time series in each chunk looks very different. Please give more details.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes, no issues are found.", "experimental_designs_or_analyses": "Yes, the conducted experiments are reasonable. A few experiments are missing, and I mention them in \"Questions for Authors\" below.", "relation_to_broader_scientific_literature": "The key contributions of this paper are reducing parameter sizes specially for time series data, I think this technique would be general to all time series related tasks.", "essential_references_not_discussed": "Can you discuss the differences between your method and the following chunk-based techniques?\n\n[1] Ju, Yue, Alka Isac, and Yimin Nie. \"Chunkformer: learning long time series with multi-stage chunked transformer.\" arXiv preprint arXiv:2112.15087 (2021).\n\n[2] Johnsen, Pål V., et al. \"Recency-Weighted Temporally-Segmented Ensemble for Time-Series Modeling.\" arXiv preprint arXiv:2403.02150 (2024).", "other_strengths_and_weaknesses": "Strengths:\n1. This paper can effectively reduce the number of model parameters to achieve robust time series forecasting.\n\nWeaknesses:\n1. The performance gap is quite minor.", "other_comments_or_suggestions": "N.A.", "questions_for_authors": "1. Can you compare your aggregated spatial correlations (from basic correlations) with attention-based methods? I'm interested if your correlations can be similar or not.\n2. In Section 4.4, for the efficiency-related experiments, can you further include time cost and flops? It's possible that you use less parameters but have more computations.\n3. Can you provide some failure cases when your method cannot have good performance? Theoretically, I think aggregation-based correlations cannot well cover all cases.", "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": "dgfDPgmjks", "reviewer": "Reviewer_SVxE", "summary": "This paper proposes  the CMoS, a highly  lightweight model for time series forecasting tasks. Unlike previous studies, CMoS capture the temporal patterns in a chunk-level manner. The Correlation Mixing mechanism builds robust correlationmatrices, and Periodicity Injection technique help to leverage the periodicity information.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your detailed and thoughtful review! We will address your concerns point by point.**\n\n> Q1: The statement may not hold when time series data shows no periodicity.\n\nWe appreciate your careful examination of this matter. **Indeed, as you pointed out, the formulation becomes less rigorous for some irregular time series, such as in the case of a time series generated by random walks. Nevertheless, it is notable that periodicity or stationary trend constitutes a fundamental indicator of time series predictability and forecasting potential**. Long-term forecasting can be very challenging if time series exhibit no periodicity or stationary trend. For example, DLinear demonstrates that a naive baseline that simply copies the most recent observation can outperform almost all SOTA deep learning models that on the aperiod financial dataset [1]. We also reach a similar conclusion by analyzing the 1st channel of Weather dataset described in Sec. 5. So, **our statement holds true for time series that are predictable and have strong forecasting potential**. And through extensive experiments, **we demonstrate both the soundness and superiority of our approach, which is designed based on the principles of the statement, across the majority of real-world scenarios**.\n\n*Also, motivated by your concerns, we will include an additional discussion section in the revised paper to elaborate on this issue, as well as other potential limitations.*\n\n> Q2: Some equations exists errors.\n\nQuite thanks for your reminder! We will carefully review all the formulas and correct the errors in the revised paper.\n\n> Q3: The details of Periodicity Injection for each dataset should be described in the experimental part.\n\nThanks for your valuable suggestions. We will add the below detailed description of Periodicity Injection in our revised paper according to your suggestion:\n\nFor each dataset, **firstly, we use the AutoCorrelation Function (ACF) to calculate the dominant period $p$ of this dataset** (the period of all datasets calculated by ACF are listed in the 5th column in Table 7 in Appendix D). Next, during the grid search process for the hyperparameter chunk size, **for each experiment setting, we input the the calculated $p$ and the hyperparameter chunk size $S$ into Algorithm 1 provided in Appendix E, to obtain a modified matrix initialized via Periodicity Injection**. This matrix is then used to replace the first basic correlation matrix of the initialized model, thereby completing the Periodicity Injection operation.\n\n> Q4: Please specify the reason and the advantages against SparseTSF\n\nSparseTSF significantly reduces the number of parameters by downsampling the original time series. Additionally, it adopts a Channel-Independent strategy (Described as One Bus in Appendix B)—modeling a single shared temporal structure for all time series—to minimize the overall number of model parameters. However, the oversimplified modeling of the time series forecasting task results in severely limited representational capacity. **Since it can only model a simple and singular temporal structure for a given time series system, this method fails to accurately capture the diverse temporal patterns that may exist across various time series within the system**. As a result, its forecasting performance tends to be suboptimal.\n\nIn contrast, with the help of Correlation Mixing strategy, **CMoS focuses on building several fundamental basic correlations that represent the diverse and inherent patterns of the whole system, and apply a specific mixing strategy for each channel to capture the channel-specific and more accurate temporal structure, thereby greatly improving the prediction performance.** According to the experimental results, CMoS consistently outperforms SparseTSF with a significant margin on datasets containing more than 20 channels (more channels often contain a greater diversity of temporal structures). Therefore, **when comprehensively considering the trade-off between prediction performance and parameter efficiency, our method stands out as the optimal choice among existing approaches including SparseTSF**.\n\n[1] Zeng, Ailing, et al. \"Are transformers effective for time series forecasting?.\" Proceedings of the AAAI conference on artificial intelligence. Vol. 37. No. 9. 2023.\n\n**Thank you again for your kind review, and we hope our response can address your concerns!**", "claims_and_evidence": "- In figure 1, thios paper claims that \"the specific patterns in the time window change greatly, while the spatial correlations of the time series \nchunks remain similar.\" This may be a specific case for time series data with obvious periodicity. This may not hold when time series data shows no periodicity.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. There are no problems for proofs.", "experimental_designs_or_analyses": "Yes. There are no problems.", "relation_to_broader_scientific_literature": "Previous methods model the temporal relationship in point-level or learn feeature from patches. This paper proposes learn the relationship in the chunk-level. It provides a new perspective to leverage the relationship between channels, instead of simply employing Channel Independent strategy.", "essential_references_not_discussed": "There are no problems.", "other_strengths_and_weaknesses": "Strengths:\n- The writing is clear and fluent.\n- The method is effective and easy to implement.\n\nWeaknesses:\n- Some equations exists errors. For example, in the Theorem3.2, the superscript of the summation symbol should be n-1. Similar problem exists in equation 3. \n- The details of Periodicity Injection for each dataset should be described in experimental part.\n- This is a lightweight model, but SparseTSF seems much more lightweight than this model. Please specify the reason and the advantages against SparseTSF.", "other_comments_or_suggestions": "No more comments.", "questions_for_authors": "Please see the weakness and problems above.", "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:

Meta-review: This paper presents CMoS, a lightweight time series forecasting model that introduces chunk-wise spatial correlation modeling and correlation mixing to achieve strong performance with minimal parameters. The idea of modeling inter-chunk correlations, along with a shared-and-adapted correlation strategy across channels, is both elegant and practical—offering a strong balance between efficiency, interpretability, and predictive accuracy. Reviewers appreciate the thoughtful design, clear writing, and the extensive experimental evidence demonstrating consistent performance across multiple benchmarks. While some concerns were raised about the marginal performance gains over recent baselines and the reliance on prior knowledge for periodicity injection, the authors addressed these issues reasonably well in the rebuttal. Overall, this paper makes a valuable and well-executed contribution to the growing body of work on efficient and interpretable time series forecasting. I lean toward acceptance.
Decision: Accept (poster)