TimeFilter: Patch-Specific Spatial-Temporal Graph Filtration for Time Series Forecasting
Authors: Yifan Hu
Guibin Zhang
Peiyuan Liu
Disen Lan
Naiqi Li
Dawei Cheng
Tao Dai
Shu-Tao Xia
Shirui Pan
Venue: ICML 2025 poster (2025)
Keywords: Time Series Forecasting
Primary area: deep_learning->sequential_models_time_series

Abstract: Time series forecasting methods generally fall into two main categories: Channel Independent (CI) and Channel Dependent (CD) strategies. While CI overlooks important covariate relationships, CD captures all dependencies without distinction, introducing noise and reducing generalization. Recent advances in Channel Clustering (CC) aim to refine dependency modeling by grouping channels with similar characteristics and applying tailored modeling techniques. However, coarse-grained clustering struggles to capture complex, time-varying interactions effectively. To address these challenges, we propose TimeFilter, a GNN-based framework for adaptive and fine-grained dependency modeling. After constructing the graph from the input sequence, TimeFilter refines the learned spatial-temporal dependencies by filtering out irrelevant correlations while preserving the most critical ones in a patch-specific manner. Extensive experiments on 13 real-world datasets from diverse application domains demonstrate the state-of-the-art performance of TimeFilter. The code is available at https://github.com/TROUBADOUR000/TimeFilter.

TimeFilter: Patch-Specific Spatial-Temporal Graph Filtration for Time Series Forecasting
Yifan Hu 1* Guibin Zhang 2* Peiyuan Liu 1* Disen Lan Naiqi Li
Dawei Cheng
Tao Dai
Shu-Tao Xia
Shirui Pan
Abstract
Channel Clustering
CD Cluster
Cluster
(CC)
KMAAN
wamma
Time series forecasting methods generally fall into two main categories: Channel Independent (CI) and Channel Dependent (CD) strategies. While CI overlooks important covariate relation- ships, CD captures all dependencies without dis- tinction, introducing noise and reducing gener- alization. Recent advances in Channel Clus- tering (CC) aim to refine dependency model- ing by grouping channels with similar charac- teristics and applying tailored modeling tech- niques. However, coarse-grained clustering strug- gles to capture complex, time-varying interac- tions effectively. To address these challenges, we propose TimeFilter, a GNN-based framework for adaptive and fine-grained dependency mod- eling. After constructing the graph from the input sequence, TimeFilter refines the learned spatial-temporal dependencies by filtering out ir- relevant correlations while preserving the most critical ones in a patch-specific manner. Ex- tensive experiments on 13 real-world datasets from diverse application domains demonstrate the state-of-the-ar performance of TimeFilter. The code is available at https: / /gggtht..b.ccoo com/ PROUBADOURO00/TI wwto//eieeFFiiie.o iler.
DTW 0.091 (lligh Simulaariy
ITW 0JIKS Stiriiarity
Suppoet Case V
KMAM
WV MAA
DTW 6.875 (lligh Similtariyy
0.33 Stevthrriyy
KMMA DTW 0.226 (Low Sinilaaits)
Case
DTW 6.057 (Hligh Sunilazie)
Elletricity
Channe]
Chansel
Chanae 3
Figure I. Analysis of three channels from the Electricity dataset shows the pros and cons of CI, CD, and CC strategies in different cases. Dynamic Time Warping (DTW) is metric for measuring the similarity hetween two sequences, with lower values indicating higher similarity. The CI strategy ignores the highly correlated co- variate information in the Supporting Case (left). The CD strategy fuses all information, including the irrelevant dependencies in the Supporting Case (right). The CC strategy addresses these issues by using global correlation to form CD and CI clusters. However. it cannot model the dynamic interactions of channels different time steps at fine-grained level. For example, in the Failure Case, it still faces the same dilemma CD and CI strategies.
ing (Volkovs et al., 2024), traffic flow analysis (Shu et al., 2021), health monitoring (Morid et al., 2023). and financial investment (7hu et al., 2024a;b). The literature on deep learning approaches for this task can be broadly catego- rized into hhannel-independen (CI) methods and channel- dependent (CD) methods (Han al., 2024b).
1. Introduction
CI methods rely on the individual historical values of each channel for prediction (Nie et al., 2023; Dai et al., 2024; Zeng et al., 2023), while CD methods go a step further by modeling the dependencies between different channels (Yu et al., 2024; Huang et al., 2023; Zhang & Yan, 2023). How- ever, it is important to recognize that data from different domains often exhibit significant variations in underlying distributions and characteristics. For instance, in climate- related data, there are typically natural physical dependen- cies among variables (Karevan & Suykens, 2020), whereas in electricity consumption data, the usage patterns of indi- vidual users can vary drastically, with little to no interde- pendence. This stark contrast suggests that the assumption
Multivariate time series forecasting is challenging task due to the need to capture both temporal dynamics and inter-channel dependencies. This problem arises a wide range real-world applications, including weather forecast-
'Equal contribution 'Tsinghua Shenzhen International Grad- uate School, Tsinghua University "National University of Sin- gapore "Fudan University 'Tongji University "Shenzhen Uni- versity "Griffith University. Correspondence to: Tao Dai <daitao. @ caae@@ggalicoc gmmai..co>>, Shirui Pan <s, @ www.fith..c........ au>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

TimeFilter
of channel independence in CI methods, as well as the in- discriminate modeling of inter-channel dependencies in CD methods, both have inherent limitations.
RiloorMakie
Rempeaa Comdakee
Spuzac Condaaee
Spenalcmpent Cancanas
IN Depadea
Cunderner (6T i-31)
To address this challenge, recent works have introduced the concept of Channel Clustering (CC), where channels with similar patterns are grouped into clusters based on data- dependent criteria. For instance, CCM (Chen et al.. 2024) dynamically clusters channels with inherent similarities, ap- plying CD modeling within each cluster and CI modeling across different clusters. However, CCM only considers relationships within the same cluster. DUET (Qiu et al., 2024b), on the other hand, extends this idea by introducing clustering in the temporal dimension as well. By dynami- cally assigning different modeling strategies for inter-cluster and intra-cluster relationships, these approaches enhance forecasting capabilities by effectively capturing both intra- cluster and inter-cluster dependencies.
4
Complete Separated
Adaptive Trade-off m.m. Complete Collaborative
(c) Filvalas
10 Chasmiigg
Towdftker
CCM INIPE IKDEPES)
(Dun)
Fine-grained
Coane-gralned
The dependency map of different strategies. f, is the time step and z; is one channel. (a) CT strategy preserves only the temporal dependencies. (b) strategy fuses dependencies. (c) Patch-wise Filtration finely selects dependencies for each patch. (d) Channel-wise Clustering coarsely models channel dependencies.
Figure 2.
Despite their success, it is notable that existing CC methods typically rely on a searse-grained approach, where similarity across channels is calculated using data from all time points for clustering. As illustrated in Figure 1, this coarse-graing clustering fails flexibly select appropriate dependency modeling strategies for specific time periods. The complex dependencies between channels evolve over time, and chan- nels that are closely related within a cluster may become completely independent at certain time intervals. In such cases, applying a CD method could lead to overfitting (Zhao & Shen, 2024). Conversely, channels that are unrelated in the overall cluster may become highly correlated at exact times, and using a CI method would risk losing valuable inter-variable information. This limitation underscores the need for more dynamic and adaptive approaches that can capture time-varying dependencies.
TimeFilter begins with Spatial-Tema Temporal Construction Module, which segments the input time series into non- overlapping patches and constructs spatial-temporal graph. After that, Patch-Specific Filtration Module applies Mix- ture of Experts (MoE)-style dynamic router to filter out redundant dependencies, honing in on the most critical spa- tiotemporal relationships. Finally, Adaptive Graph Learn- ing Module leverages the graph structure to aggregate rele- vant information and produce accurate forecasting results. Extensive experiments on several datasets show that Time- Filter achieves consistent state-of-the-art performance in both long- and short-term forecasting. Our contributions can be summarized as follows:
Motivated by the observations above, we transition from pre- vious coarse- grained, channel-wise clustering approaches to a ffnner-grainee patch-wise partitioning strategy, where each channel is divided into non-overlapping segments. With this partitioning, as shown in Figure 2(b), there are three key types of relationships: ! temporal dependen- cies between different patches within the same channel, @ inter-channel (spatial) dependencies at the same time step, and spatial-temporal dependencies between different channels and time periods. These relationships are often in- tricately intertwined in real-world time series data. To more robustly model these complex dependencies, we further design filtering method for each dependency region, effec- tively removing irrelevant noisy relationships and ensuring that only the most significant connections are considered.
Novel Paradigm. We for the first time advocate for fine- grained segmentation of dependencies, capturing spatial- temporal relationships at the patch level while adaptively filtering out irrelevant information.
Technical Contribution. We propose TimeFilter, a novel dependency modeling strategy that employs patch-specific filtration to learn better omni-series representations for time series forecasting.
Holistic Evaluation. Comprehensive experiments demon- strate that TimeFilter achieves state-of-the-arr perfor- mance in both long- and short-term forecasting, with error reductions of 4.48% and 5.34%, respectively.
2. Related Work
Technically, we propose TimeFilter, a novel framework crafted to capture intricate spatiotemporal dependencies across both temporal and spatial dimensions, while dynam- ically filtering out extrancous information. Specifically,
2.1. Multivariate Time Series Forecasting
Multivariate time series forecasting (MTSF) has widespread applications in various domains. In recent years, deep learn-
2

---

TimeFilter
ing models have shown great promise in MTSF. RNN-based methods (Lin et al., 2023), leveraging the Markov assump- tion, model the sequential nature by considering current and previous time steps. CNN-based methods (Dai et al., 2024; Wu et al., 2023) apply convolution operations along the time dimension to capture local temporal patterns, How- ever, they struggle with long-term dependencies due to van- ishing gradients and limited receptive fields, respectively. Transformer- based methods (Liu et al., 2024b;a) use atten- tion mechanisms to model long-term temporal dependencies, yielding strong performance. However, they tend to over- look the sequential nature of time series, which is crucial for maintaining temporal consistency. MLP-based methods (Hu et al., 2024; Yu et al., 2024), with their dense weights, cap- ture interactions between different time steps in simple but effective way. Their straightforward architecture makes them popular choice for time series forecasting tasks. Ad- ditionally, GNN-based methods (Wu et al., 2020; Huang et al., 2023) are applied to capture the relationships between different variables in a graph structure.
where C denotes the number of channels and L denotes the look-back horizon. X; E IR represents one of the channels. The objective is to construct a model that predicts the fu- ture sequences Y = $c. E RCT, where T denotes the forecasting horizon.
3.2. Dependency Matrix
The dependency matrix M E R describes the depen- dencies within channels at different time patches as the ad- jacency matrix of a apatial-temport graph G = {V.E} with V/ as the node set and & as the edge set, where n = C x N and N denotes the number of patches in the look-back se- quence. M is utilized to depict the inter-node relationship, where M[i.j] indicates the weight of edge eij E E.
Moreover, we decompose G into vatch-specifir ego graphs [G:)]- Each central node of the graph Gi represents the specific patch of input sequence. Then each ego graph Gj is divided into three subgraphs: spatial subgraph Go, temporal subgraph G,T and spatial-temporal subgraph G.STT Each subgraph describes different type of dependency of specific patch. The adjacency matrix of each subgraph can be obtained by applying masks to M.
2.2. Dependency Modeling
Regarding modeling dependencies in MTSF, some mod- els adopt a Channel Independence (CT) strategy (Nie et al., 2023; Zeng et al., 2023; Lin et al., 2024), as in Figure 2(a), preserving only the temporal dependencies. These mod- els use only the historical information of each individual sequence, ignoring the interactions between variables. Al- though the CI approach is robust, it wastes the potentially relevant relationships from inter-channel information. In contrast, the Channel Dependency (CD) strategy (Liu et al., 2024b; Zhang & Yan, 2023; Yu et al., 2024), as in Fig- ure 2(b), models the full spectrum of information, providing information gains but also including redundant dependen- cies, thus reducing robustness. Therefore, designing adaptive dependency modeling strategy remains signif- icant challenge. Recent research (Chen et al., 2024; Qiu et al., 2024b) employs a Channel Clustering (CC) strategy to dynamically seleet modeling strategies based on channel similarity, as in Figure 2(d). For instance, variables æ and ry are chosen to form a CD cluster, where inter-channer rela- tionships are modeled, while variable 22 is assigned to CI cluster, relying only on its own historical values. However, this coarse- coarse-grained approach fails to capture the dynami- cally evolving interactions. Therefore, as in Figure 2(c), we design a robust and ine-grained filtering mechanism to customize the dependencies needed for each time segment.
4. Method
The core of TimeFilter, as illustrated in Figure 3, to capture fine-grained spatial-temporal dependencies through three key components: (i) Spatial-Temporal Construction Module segments the input sequence into non-overlapping patches and constructs the spatial-temporal graph; (ii) Patch-Specific Filtration Module filters out irrelevant in- formation through the Mixture of Experts (MoE) with Dy- namic Routing mechanism to obtain the spatial-temporal de- pendencies for the current sequence; (iii) Adaptive Graph Learning Module leverages the graph structure to aggre- gate pertinent information and derives the forecasting results. The details of each essential module are explained in the following subsections
4.1. Temporall Construction (STC) Module
In this stage, we construct the spatial-temporal graph using the input look-back sequence. First, each channel of the input sequence X E RC is divided into non-overlapping patches X, E RCXNxP where each patch has a length of P and the number of patches N [ Then each patch is mapped to an embedded patch token X, E RxNNxD
3. Preliminaries
Patching(XX, X, = Embedding(X (x))
(I)
3.1. Problem Definition
The Embedding(-) operation transforms each patch from its original length P to a hidden dimension D through a trainable linear layer. In addition, we flatten the first two dimensions to obtain a set of spatial-temporal patches with
In the multivariate time series forecasting task, let X = [x1.X2. xc} E C be the input time series,
3

---

TimeFilter
(i) Spatial- Temporal Construction
(ii) Patck- Patch-Specche Specific Filtration
A
Spatial-temporal
Sponil subgraph
Temporal Filter
joint graph
Adjerenry
(Pl) #1))
f
matnix
Mi PM,
Prunod/mmme momoo//nmmwwlleeo
Tempoeal snbgruph S (V. (e, Ie 11
SdiAboo siges
Temponal odges
Ego-graph graggapphh for patch
Spanial retropaal subgraph
6i" (V. (talts E (E (e we)
Spatial-empprrlll niges
Allocwring single
Spatial Filter
(iii) Adapiive Gruph Learning
OOO O
Routing Network VioD 5 a
M, p(M, I l
Projection
DeNorm
X3
Spatial-temporal patches
& Morrn
Patching
Ego-graph 51-1 for patch Miea
Dynamic
Expert
Graph neural block CNNO
yi
xp Ex (C(CC))
Allocation
Sparse ego graph G (V.E)
92
X1
Allocating! mullipk exports
Spa.-tem. Filter
M p(M, (G7 US?E)
X2
Prediction Y R
X3
L = Lyred
Multivariate time series X E RC
Ego-grape Gn for patch Pn
by Limp +
(V.E)
2 Layn
Figure 3. The overall structure of TimeFilter, which consists of: (i) Spatial- Temporal Construction devised to construct the spatial- temporal graph from the input X; (ii) Patch-Specific Filtration facilitates spatial-tempont dependencies by filtering irrelevant information for each patch; (iii) Adaptive Graph Learning is leveraged to predict the future Y based on GNN.
C' x N patches containing the enhanced local informa- tion X, IR
Here E is the spatial edge set, ET is the temporal edge set, e, is one of the edges and vi E 2..,...,, n.. The adjacency matrix of each subgraph M; can be obtained by applying different masks to M.
Next, we calculate the projection distances in multi-head manner, allowing for richer similarity representation. We first divide Xp into different heads X,, E IR 1R x and then calculate the projection distances Dist(-) by dot product. Here, H is the number of heads and AT is the transpose.
With the above designs, we obtain ego graphs Gi and the corresponding adjacency matrix M, to represent the correlations of each patch.
Dist(XA) Linear(X,)) * Linear X T
R Xxnxn
(2)
4.2. Patch-Specific Filtration (PSF) Module
Subsequently, based on above distances, we construct the graph with the k-Nearest Neighbor (k-NN) method. Each spatial-temporal patch is considered as node, resulting in total of n nodes. For each node, we retain the & nearest neighbors. In practice, we retain the k [a n. nearest neighbors, where a is hand-tuned scaling factor. Formally, the adjacency matrix M can formulated as follows
After constructing the graph, each edge connects patch nodes, representing their dependency relationship. At this stage, design patch-specific MoE module composed of routing network, dynamic expert allocation and filtration to identify effective dependencies, filter out irrelevant ones, and obtain the spatial-tempore relationship.
Routing Network. We observe that time series data from different domains exhibit distinct characteristics and un- derlying structures, necessitating the modeling of different types dependencies. Therefore, for each ego graph G, we design three experts for different dependencies, including Temporal Filter, Spatial Filter, and Spa.-tem. Filter. Based the router's 's decisions, filters with different dependency regions are assigned to different patches. Specifically, the
M = (-NN(Ge
(3)
Instead of using the whole graph G with non-negligible noise (Huang et al., 2023), we decompose G into patch- specific ego graphs {G.) J to effectively resist noise, each corresponding to patch. Each ego graph is further divided into three regions: temporal, spatial and spatial-temporal.
of = (V, (emoee E E
(4) filtration operation can be formulated as follows:
g = (V, (eivleio E E
(5) Temporal Filter M, ~ p(M (T) if the selection in-
GH (V, (eivleio E 8 (8S (E()))
(6) cludes a Temporal Filter, the temporal edges E are re-

---

TimeFilter
tained. For example, user behavior data with low inter- channel correlation benefits more from temporal depen- dencies (Yuan et al., 2022), which leverage the historical information of individual users.
smallest set of filters whose cumulative probability exceeds the threshold Top-p, and the number of selected filters q is calculated by:
((,))
arg min (E( KSJ
Σ
Σr(S.)&
2 Top-p
(9)
Spatial Filter M, (( If the selection includes a Spatial Filter, the spatial edges E are retained. This can be crucial for scenarios where spatial dependencies reflect the synchronization between multiple variables in real-time monitoring.
Next, the calculation of output routing is:
(((())) Filter, E S, . E (1,2, ..., m) (10)
((Gi), 0,
Filter, $ S.
Spa.-tem. Filter: M, P(M If the selec- tion includes Spatial-tempord. Filter, the spatial edges E EV(EE (E?) are retained. For instance, in traffic flow pre- diction, spatial-temporal dependencies are pivotal (Wang al., 2024b), capturing the nuanced interactions between traffic dynamics across different locations and their evolu- tion over time.
where S is the set of selected filters controlled by ((G)) in Eq. 9:
S. = (Filtery Filter,, Filters.re))
(11)
Filtration. Finally, each ego graph Gi, Vi E (1,2, n), based on output dynamic routing g(G)) E R", we perform a filtration operation on M, to obtain the adjacency matrix M, and the corresponding edge set & and the sparse graph T. after filtering out redundant dependencies. For example, if S: = {Temporal Filter, Spa.-tem. Filter), the ego graph after filtering will be:
Following the classic concept of a (sparsely-gater mixture- of-experts (Shazeer et al., 2017b; Zhang et al., 2024a) and graph sparsification paradigms (Wang et al., 2024a; Zhang et al., 2024b;c), we calculate the confidence of the current patch for each filter using a noisy gating mechanism.
(12) It is worth noting that the filtration operation on the ego- graphs can be performed in parallel without introducing additional computational complexity.
G. (V, fewlee EEEEEEE D&DH
r(g)) = Softmax (4) (M,)),
(M.) Linear, g (M:) + e Softplus(Linear, (M. (M.)) (8)
where (MM) E R is the calculated scores of patches for all filters, c E (0, 0,, 1) is denotes the standard Gaussian noise, Linear, and Linear, are clean and noisy scores mapping, respectively. m is the number of filters and in TimeFilter m 3.
4.3. dapttiv Graph Learning (AGL) Module
A ffter obtaining 9. = (V,&,), we restore the n ego graphs into global graph ② and aggregate the M, into correspond- ing M
g (V.
Dynamic Expert Allocation. The classical Top- K rout- ing MoE architecture assumes that the same number of experts is assigned to each input token, but this overlooks the differences between inputs. Specifically, in dependency modeling, certain time patches require a diffcrent number of dependencies. For example, when there a clear lead- lag effect in stock data, both temporal and apatial-temporaa dependencies are needed, while during unexpected events, spatial dependencies are more crucial to capture how stocks from the same sector react.
(13)
where N(..) is the set of neighboring nodes of a given node.
Then, the graph learning layer learns the graph adjacency matrix M adaptively to capture the hidden relationships among time series data.
h COMB (-1), AGGGREE (k-
by e ((x.)) (14)
where 0 S 1 S O, O is the number of GNN layers, h (0) = X, E Mxax RJ and M S bedding of node v; at the 1-th layer. AGGR(.) and COMB(.) are the functions used for aggregating neighborhood infor- mation and combining ego- and neighbor-eeprssentation.. respectively. The AGGR(.) employed in TimeFilter is an additive operation, where the representation of each node updated by summing the features of its neighboring nodes.
L 1 O) is the em-
To adaptively customize the dependency relationships for each patch, we propose Dynamic Expert Allocation module to select filters based on the confidence of the patch itself, following MoE-Dynamic (Huang et al., 2024). Unlike the Top-K routing mechanism, which selects fixed number of experts, our method allows the model to evaluate whether the current selection of dependencies is sufficient. If not, it will continue to allocate more dependencies.
Finally, to perform forecasting, our model utilizes a residual module and linear projection in time dimension. This transformation can be expressed as:
Thus, we rank the confidence r from largest to smallest, resulting in sorted index list T. Afterward, we find the
Y Linear (x, + 10)) E CXT
(15)
5

---

TimeFilter
4.4. Loss Function
CCM (Chen al., 2024) is included as plug-in in the baseline PatchTST+CCM, CCM for short.
The loss function of TimeFilter consists of three compo- nents. For the predictors, the Mean Squared Error (MSE) loss pCeed } || & is used to measure the variance between predicted values and ground truth. Moreover, following dynamic routing (Huang et al., 2024), we introduce an dynamic loss Layn to avoid assigning low confidence to all experts, thereby activating a larger number of experts to achieve better performance.
Implementation Details. All experiments are imple- mented in PyTorch (Paszke al., 2019) and conducted on one NVIDIA A100 40GB GPU. We use the Adam op- timizer (Kingma, 2014) with a learning rate selected from {le-s, [1e-,,ee4,, 5e -)_ The Top-p in dynamic expert alloca- tion is selected from {0.0, 0.5}. We select two common met- rics in time series forecasting: Mean Absolute Error (MAE) and Mean Squared Error (MSE). For additional details on hyperparameters and settings, please refer to Appendix A.3.
1 72
m
dan
(Gi)) * 19g()(9),)) (16) 1=1
Moreover, we introduce an expert importance loss Cimp (Shazeer et al., 2017a) to prevent TimeFilter from converging to trivial solution where only single group of experts is consistently selected:
5.2. Results
Long-term Forecasting. We report the results of the long- term forecasting in Table 1. The input length L is 96 for our method and all baselines. The forecasting horizon T is f96, 192, 336, 720}. The results show that TimeFilter adaptively models various dependencies for each dataset in different domains, and has been proven to yield sigmificant improvements. Compared to the second-best model Led- dam ( YYu et al., 2024), TimeFilter reduces MSE and MAE by 4.48% 2..2% respectively.
1
Std(r(G:))
Cump n Mean(r(G.)) +e
(17)
where e is a small positive constant to prevent numerical instability, Std(-) and Mean(.) calculate the standard devi- ation and the mean, respectively. Therefore, the final loss function is defined as:
Additionally, we perform the Wilcoxon test with Led- dam (Yu et al., 2024) and obtain the p-value of 4.66e 100 indicating a significant improvement at the 99% confidence level. The detailed error bars are shown in Appendix B.I.
L = Cred + X1 + d2 Aipp
(18)
where A and A2 are the scaling factors.
Furthermore, according to the scaling law (Shi al., 2024) of TSF, for given dataset size, as the look-back horizon L increases, so does the noise in the input sequence, which does not necessarily improve the performance of the model. In addition, the dataset size has more influence on the opti- mal horizon, while the model size has less influence. There- fore, we conduct a search for different look-back horizons to determine the optimal horizon for the 4 relatively large datasets and compare TimeFilter with other baselines. As shown in Table 2, TimeFilter demonstrates robust resis- tance to noise and maintains SOTA performance even with longer optimal horizon inputs, outperforming other methods. Specifically, compared to the channel clustering strategy methods DUET (Qiu et al., 2024b) and CCM (Chen et al., 2024), TimeFilter reduces MSE and MAE by 1.34%/1.40% and 6. 59%%3.69%2 respectively. This demonstrates the ef- fectiveness of our fine-graines adaptive filtration strategy.
5. Experiments
5.1. Experimental Setup
For long-term forecasting, we conduct extensive experiments on selected 9 commonly real-world datasets from various application scenarios, including ETT (ETThI, ETTh2, ETTm1, ETTm2), Traffic, Electricity, Weather, Solar-Energy (Miao et al., 2024a;b) and Climate (with 1763 channels) datasets. For short-term forecasting, we adopt 4 benchmarks from PEMS (PEMS03, PEMS04, PEMS07, PEMS08) (Liu et al., 2025b;a). The statistics of the dataset are shown in Appendix A.
Datasets.
Baselines. We compare our method with SOTA repre- sentative methods, including GNN-based methods: MS- GNet (Cai et al., 2024), CrossGNN (Huang et al., 2023);
Transformer-baus based methods: CATS (Kim et al., 2024).
iTransformer (Liu et al., 2024b), PatchTST (Nie et al., 2023).
Short-term Forecasting. We report the results of the
Crossformer (Zhang& & Yan, 2023), FEDformer (Zhou et al., 2022); baaedd methods: Leddam (Yu et al., 2024), SOFTS (Han et al., 2024a), DUET (Qiu et al., 2024b), DLin- ear (Zeng et al., 2023), TiDE (Das et al., 2023); CNN- methods: ModemTCN (Donghao & Xue, 2024), Times- Net (Wu et al., 2023), SCINet (Liu et al., 2022). In addition,
short-term forecasting in Table 3. The input length L is 96 for our method and all baselines. The forecasting horizon T is {12, 24, 48). From the table, TimeFilter consistently out- performs other methods across all PEMS datasets. Specif- ically, on PEMS08, TimeFilter reduces MSE and MAE by 13.54%/ /3.1%% 13% compared to the second- best DUET (Qiu
6

---

TimeFilter
Models
Titefiller
DUET
iTranformmr
Leddam
MSGNet
SOFTS
CressCINN
PatchTST
Crossfomer
TimesNet
DLinear
FtDformer
(Ours)
(2024))
CD
(2024)
(2024)
(2024a)
CD
(200))
CD (2023)
CD
(2
CD (2022)
Metric
MSF
MAR|
MSR
MAR
MSF
MAR
MSF
MAR
MSR
MAR
MSR
MAR
MSR
MAR
MSR
MAR
MSR
MAR
MSE
MAR
MSF
MAR
MSR
MAP.
ETT (Avg.)
0,355
0,385
0.337
0.388
0.383
0.399
0.368
0.388
0.384 0,403
0.339 0.396 00337 0.392 (0.381 0.397 (0.655 0.578 10.39] 0.404
(0.446
0.447 10.408 0.426
Weather [0.239
0.269
(0.25]
0.273
0.223)0.258
0.279
0.242
0.272
00.249 0.278 (0.255 0.278 (0.247 0.289 0.289|0.259 (0.259 0:281 |0.259 0:315 0.315|0.250
0.287 10:265 0.315 [0.309 0.360
Electricity
[0.588
0.256]
0.172
0.258
0.178
0,270
00.169
0.263
00.194
0.300 (0.174
0.264 (0.201 0. 0.300 (0.216 0
0.304|0.244
0.334
0.295
[.295|0.225 0.319
(0.214 0.327
Traffic 00.007
0.268
(0:45]
0.269
0.269|0.428
(0.428
0.282
00:467
0.294
(0.64]
0.370
(0:409
0.267
(0.583 0.523 00.535 0.362
0
0.304
0.336
(0.625 0.383
(0.610 0.376
Solar Energy 0.223 0,250 0.237 0.233
0.2233
0.233 0.262 0.230
0.264 0.262 0.288 0.2299 0.256 00324 0.377 0.377|0.270 0.270 0.307 0.307]0.641 0.639 |0.301 0.319
0.319|0.330 0.401
(0.292 0.381
Table Long-term forecasting results. The input length T. is 96, All results are averaged across four different forecasting horizons: T E (96, 192, 336, 7203. The best and second-hest results are in bold and underlined, respectively. See Table 8 for full results.
Models
TimeFilter (Ours)
DUET CC (2024b)
CCM
PDF (2024)
ModemTCN
iTransformer
CATS (2024)
PatchTST (2023)
Crossformer CD (2023)
TimesNet CD (2023)
DLincar (2023)
(2024)
(2024)
(2024b)
Metric
MAB|
MAE|
MSE
MAE|
MSE
MAE
MSE
MAE
MAB
MSE
MAE
MSE
MAB
MAE
MAE
MSB
MAE
Weather
10.216
0.258
0.2218
0.252 0.225 0.263
| (0.225 0.261 00:224 0,264 |0.232 0.270 0.244 0.272 (0.226 0,264 0,234 0.292 0.255 0,282
00.242
0.293
Electricity| 0.150
0.246
00.158
0.248
0.167
0.261 0.156 0.250 0 156 0,253
[0.163 0.258
0.178
0,264 |0.159 0.253 0,181
0.279
0.192
0.192
0.295
10.166 0.264
Traflic |0.360
0.254
00.394
0.257
0.389
0.259
|0.377
0.256
[0.396
0,270
|0.397
0.281
00.449
0.283
|0.391
0.264 0.523
0.284
0.620
0.336
|0.434
0.295
Climate 1.047 0.490| 1.103 0.509|
1.123 0.512 0.555
1.155 0.542 1,000 0.496 1.095 0.509
| 1.387 0.562 1.170 0.550
1.295
0.569]
1.134
0.541
Table 2. Long-term forecasting results. The input length L is searched from (192, 336, 512, 720] for optimal horizon in the scaling law of TSF (Shi et al., 2024). All results are averaged across four different forecasting horizons: T E {96, 192, 336, 720]. The best and sccond- best results are in bold and underlined, respectively. Sec Table 9 for full results.
Models
TimeFilter (Ours)
DUET
iTransoemer CD (2004b)
Loddam UD (2004
SOFTS
PatchTST (202223)
Crosafemeer (D02))
TimesNot
TIDE
DLincar
SCINet
FEDformer
(200))
120044)
CD CU13)
CD (2023)
(102023
(2122)
Metric
MSE
MAE|
MSE
MAE|
MSE
MAE|
MSE
MAE
MSE
MAE
MSB
MAE
MSE
MAB
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
PEMS03 [0,084
0.191| [0,086
0.192 |0,096 0,204
10.101
0.200
0.192 0.151 0.265| 10.138 0.253 0.119 0.271 0:271 0,380
0.219
0.295 0.093
0,203
10.167
0,291
PEMS04 [0.083
0.186
[0.096
0.203
|0.098
0.207 10.102
0.213
19196 0. 161622 0.273 |0. 145 0.267 0. 109 0.220 0.307
0.405
[0.236
0.350 0.085
0.194
|0.195
0.308
PEMS07 (0.071 0.170
J0.076
0.176
(0.088
0.190
0o.087
0.192
0.192|0.075
0.005
0.173 0.1166
0,270
08111 0.272
0.106
0.208|
0.297
0.394
[0.241 0.343
0.112
0.211
J0.133
0,282
PEMSO8 10.083 0,186 10,096
0.192 (0.127 0,212 0.1002 0.211 10.114 0.208 00,238 0.289|
00232 0,270 0.150 0.244 0.347 0,421 (0.281 0. 0,366
0.133
0,225
J0.234 0,326
Table 3. Short-term forecasting results. The input length L is 96. All results are averaged across three different forecasting horizons: (12, 24, 483. The best and second-best results are in bold and underlined, respectively. See Table 10 for full results.
et al., 2024b). On PEMS07 with 883 channels, TimeFilter reduces MSE and MAE by 5.33% 1.33%/1.73% 1.7%% compared to the second-best SOFTS (Han et al., 2024a).
all patches within the look-back sequence are filtered based on the same type of dependency. The w/o Filter refers to the case where no filtering is applied after graph construction.
The result shows that TimeFilter consistently outperforms other filtering methods. We conclude that higher-weight relationships may be due spurious regression (Liu et al., 2024a) rather than effective dependencies. Furthermore, time series data from different domains require different types of relationships.
5.3. Ablation Study
To validate the effectiveness of TimeFilter, we conduct a comprehensive ablation study on its architectural design.
Comparison of different Filter methods. We carefully design six other filtering methods and the comparison is shown in the Filter Replace section of Table 4. Top-K strat- egy selects the top K edges with the largest weights from the ego graph G; of each patch, while Random-K randomly selects K edges. RegionTop-K strategy selects the top K' edges with the largest weights within the region corre- sponding to three types of dependencies, while Region'Tre learns a threshold for each region through a linear mapping and selects edges with weights greater than the threshold in each region. C-Filter refers to channel-win wise filtering, where
MoE component ablation. We conduct an ablation study on the routing mechanism (Static Routing with only the most
significan significant filter selected) and Cimp & dyns demonstrat- ing the importance of dynamic selection and load balancing.
5.4. Model Analysis
Case study on dependency modeling. As shown in Fig- ure 4, we visualize the dependencies and distribution of
7

---

TimeFilter
Catagories
Filter Replace
MoE
Cases
TimeFitter
Top-K
Random- K
RogionTop-R K
RegionThre
C-Filter
w/o Filter
Static Routing
w/o Cimp drs
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
Weahter
0.239
0.269
0.243
0,272
0.243
0,272
0.241
0,271
0.241
0.270
0.242
0,271
0.244
0,273
0.242
0,271
0.243
0,272
Electrictty
0.158
0.256
0.
0,263
0.168
0,264
0.
0.168
0,263
0.172
0,267
| 0.117
0,265
0. 167
0,263
0 0,164
0,260
0.163
0.250
Traffic
0.407
0.268
0.413
0.271
0.412
0.271
0.411
0.270
0.412
0.271
0.413
0.271
0.414
0.273
0.410
0.271
0.410
0.271
Solar
0.223
0.250
0.231
0.258
0,231
0.258
0.233
0.257
0.235
0.259
0.234
0.258
0.234
0.262
0.228
0.258
0.227
0.258
ETThl
00.466
0.431
00.41
0.438
0.441
0.437
0.441
0.438
0.437
0.436
0.431
0.436
0.442
0.438
0.440
0.434
0.430
0.432
ETTm2
0.272
0.321
0.276
0.324
0.276
0.323
0.273
0.323
0.276
0.324
0.277
0.325
0.277
0.324
0.274
0.322
0.273
0.322
PEMS08
0.086
0.191
0.190
0.086
0.191
0.084
0.188
0.084
Table 4. Ablation study of TimeFilter. The The input length I. is 96. See Table 11 for full results.
ETTb2 (a5)=0.51)
Weather (co)
Electricity (c00=00.00)
Floctriiity
Traffic
Diizzer SOFTS
Diaam SCETS
0.104
02:00
0.10-
Tiea
MSE
TieuFibe
375
0.35
51000
210000
Nam of
Num of
Nam of
Electricity
Look Pack Hortiom
Looko Back Hox boon
9.06
0.07
0.27
8:39
0,092
8.072
0.678
0.03
6.06
8.000
Figure 5. Influence of look-hack horizon. TimeFilter consistently outperforms other models under different look-back horizons.
Sputta
Temperal Spm.7
Npatal
Vempora Spu- Fem.
Syuttal
Tempend Ipa.Ien.
Figure 4. Above are the dependencies learned by TimeFilter from the ETTH2, Weather and Electricity datasets. Below is the distribu- tion of selected filters, where the x-axis represents the dependency types and the y-axis represents the number of diffcrent filters.
BTTme
Tofffe
12W0-4 Tik Luceve lohn
11A4-
Menoey U'AAE
TT SOFTS
MIL
Paick
0.100
CHIME Then
selected filters of one batch in the ETTh2, Weather, and Electricity datasets. Indeed, it shows that different datasets require different types of dependency and that TimeFilter can capture these dependencies powerfully. Furthermore, as the correlation (Qiu et al., 2024a) within datasets increases, TimeFilter tends to utilize more effective dependencies to capture future variations. TimeFilter customizes the filtering of redundant dependencies for each dataset, enhancing its ability to express omni-series representations.
10000 I0ea
0.70
Monssy Uiage
0.133
1000
Training Time (maa/azr)
Traiting Time (mo-1190
Figure 6. Model efficiency comparison under input- 1uput-96-predicee predict-96 of ETTm2 and Traffic datasets.
model configurations and the same batch size. As shown in Figure 6, the efficiency of TimeFilter exceeds other Transformer-basee based and GNN-based methods.
Influence of look-back horizon. To find the optimal hori- zon (Shi et al., 2024), we explore how varying the length of look-back horizons from (48, 96, 192, 336, 512, 720) im- pacts the forecasting performance. As shown in Figure 5, TimeFilter effectively resists noise and exploits vital infor- mation in longer sequences, outperforming other models over different look-back horizons. Furthermore, as the hy- perparameter length of the patch P increases proportionally with the look-back horizon L, the memory usage and infer- ence efficiency of TimeFilter remain almost unchanged, thus preventing memory explosion in extremely long sequences.
6. Conclusion
Considering the complex dynamic interactions inherent in real-world multivariate time series, we propose TimeFilter to adaptively, finely, and robustly model the dependencies. Technically, we design patch-specific filtration to effectively remove redundant relationships, thereby exploiting the most important dependencies for accurate forecasting. Exten- sive experiments show that TimeFilter consistently achieves SOTA performance in both long- and short-term forecasting tasks. Our work provides an in-depth exploration of depen- dency modeling in TSF, which we believe may pave the way for further research into dependency representations.
Efficiency Analysis. We comprehensively compare the forecasting performance, training speed, and memory foot- print of TimeFilter and other baselines, using the official
8

---

TimeFilter
Acknowledgements
Hu, Y., Liu, P., Zhu, P., Cheng, D., and Dai, T. Adap- tive multi-scale decomposition framework for time series forecasting, arXiv preprint Axix:22006.03711, 2024.
Dai Tao supported in part by the National Natural Science Foundation of China, under Grant (62302309, 62171248), Shenzhen Science and Technology Program
Huang. Q. Shen, L.,, Zhang, R., Ding, S., Wang, B., Zhou, Z., and Wang, Y. CrossGNN: Confronting noisy multi- variate time series via cross interaction refinement. In Ad- vances in Neural Information Processing Systems, 2023.
Impact Statement
Huang, Q. An, Z., Zhuang, N., Tao, M., Zhang, C... Jin, Y., Xu, K., Xu, K., Chen, I.,, Huang, S., and Feng, Y. Harder task needs more experts: Dynamic routing in MoE models. In Proceedings of the 62nd Anmual Meeting of the Association for Computational Linguistics (Volume I: Long Papers). Association for Computational Linguistics, August 2024.
Time series forecasting plays a crucial role in a variety of real-world domains, such as finance, weather forecasting, and traffic control. Our study presents a novel method for enhancing the dependency modeling. The datasets utilized publicly accessible, promoting transparency and repro- ducibility. We do not anticipate any ethical issues arising from this work. The primary contributions of this research advance the field of time series forecasting, with potential applications in multiple sectors.
Karevan, Z. and Suykens, J. A. Transductive Istm for time- series prediction: An application to weather forecasting. Neural Networks, 125:1-9, 2020.
References
Kim, D., Park, J., Lee, J., and Kim, H. Are self-attentions effective for time series forecasting? Advances in Neural Information Processing Systems, 2024.
Cai, W., Liang, Y., Liu, X., Feng, J., and Wu, Y. Msgnet: Learning multi-scale inter-series correlations for multi- variate time series forecasting. Proceedings of the AAAI Conference on Artificial Intelligence, 2024.
Kingma, D. P. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
Chen, J., Lenssen, J. E., Feng. A., Hu, W., Fey, M., Tassi- ulas, I.,, Leskovec, J., and Ying, R. From similarity to superiority: Channel clustering for time series forecast- ing. Advances in Neural Information Processing Systems, 2024.
Lai, G., Chang, W.- Yang, Y., and Liu, H. Modeling long-and short-term temporal patterns with deep neural networks. In International ACM SIGIR conference on research & development in information retrieval, pp. 95- 104, 2018.
Dai, T., Wu, B., Liu, P., N., Bao, J., Jiang, Y., and Xia, S.-T. Periodicity decoupling framework for long-term series forecasting, International Conference on Learning Representations, 2024.
Lin, S., Lin, W., Wu, W., Zhao, F., Mo, R., and Zhang, H. Segmn: Segment recurrent neural network for long-term time series forecasting. arXiv preprint arXiv:2308. / 1200, 2023.
Das, A., Kong, W., Leach, A., Sen, R., and Yu, R. Long- term forecasting with tiDE: Time-series dense encoder. arXiv preprint arX1v:2304.08424, 2023.
Lin, S., Lin, W., Wu, W., Chen, H., and Yang. J. Sparsetsf: Modeling long-term time series forecasting with 1k pa- rameters. International Conference on Machine Learning, 2024.
Donghao, L. and Xue, W. ModernTNN: A modern pure con- volution structure for general time series analysis. Inter- national Conference on Learning Representations, 2024.
Liu, C., Miao, H., Xu, Q., Zhou, S., Long, C., Zhao, Y., Li, Z.., and Zhao, R. Efficient multivariate time series fore- casting via calibrated language models with privileged knowledge distillation. In ICDE, 2025a.
Godahewa, R., Bergmeir, C., Webb, G. I., Hyndman, R. J., and Montero-Manso, P. Monash time series forecasting archive. arXiv preprint arXiv:2105.066/4 066443,.66666..0 2021.
Liu, C., Xu, Q. Miao, H., Yang. S., Zhang, L., Long, C., Li, Z.,, and Zhao, R. Timecma: Towards Ilm empoweed multivariate time series forecasting via cross-my alignment. In AAAI, volume 39, pp. 18780-18788. 18788, 2025b.
Han, L., Chen, X.-Y., Ye, H.-J., and Zhan, D.-. SOFTS: Efficient multivariate time series forecasting with series- core fusion. In Advances in Neural Information Process- ing Systems, 2024a.
Liu, M., Zeng, A., Chen, M., Xu, Z., Lai, Q. Ma, L., and Xu, Q. SCInet: Time series modeling and forecasting with sample convolution and interaction. Advances in Neural Information Processing Systems, 15:5816-5225 2022.
Han, L., Ye, H.-J., and Zhan, D.C. The capacity and ro- bustness trade-off: Revisiting the channel independent strategy for multivariate time series forecasting. IEEE Trans. on Knowl, and Data Eng., pp. 7122-7142, 2024b.

---

TimeFilter
Liu, P., Wu, B., Hu, Y., Li, N., Dai, T., Bao, J., and Xia, S.-T. Timebridge: Non-stationarity matters for long-term time series forecasting, arXiv preprint arXiv:2410.04442, 2024a.
Shu, W., Cai, K., and Xiong, N. N. A short-term traffic flow prediction model based on an improved gate recurrent unit neural network. IEEE Transactions Intelligent Transportation Systems,
16665, 2021.
Liu, Y., Hu, T., Zhang. H., Wu, H., Wang, S., Ma, L.,, and Long, M. iTransformer: Inverted transformers are effec- tive for time series forecasting. International Conference on Learning Representations, 2024b.
Volkovs, K., Urtans, E.,, and Caune, V. Primed unet-lstm for weather forecasting. In Proceedings of the International Conference on Advances in Artificial Intelligence, ICAAI '23, pp. 13-17, New York, NY, USA, 2024. Association for Computing Machinery. ISBN 9798400708985. doi: 10. 1..11446333888363366660
Miao, H., Liu, Z., Zhao, Y., Guo, C., Yang, B., Zheng, K., and Jensen, C. S. Less is more: Efficient time se- ries dataset condensation via two-fold modal matching. PVLDB, 18(2):22-2338 2024a.
Wang, K., Li, G., Wang. S., Zhang, G., Wang. K., You, Y., Fang, J., Peng. X., Liang, Y., and Wang, Y. The snowflake hypothesis: Training and powering gnn with one node one receptive field. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Dis- covery and Data Mining, KDD '24, pp. 3152-3163, New York, NY, USA, 2024a. Association for Com- puting Machinery. ISBN 9798400704901. doi: 10. 1 1145/3637528.6677766 URL https: http:s///doooooeo /rr// 10. N0.1145/3637522 3671766.
Miao, H., Zhao, Y., Guo, C., Yang, B., Zheng, K., Huang, F., Xie, J., and Jensen, C. A unified replay- based contin- prediction on streaming data. In ICDE, pp. 1050-1000 2024b.
uous learning framework for spatio-
Morid, M. A., Sheng, O. R. L., and Dunbar, J. Time series prediction using deep learning methods in healthcare. ACM Trans. Manage. Inf. Syst., 14, 2023.
Nie, Y., H. Nguyen, N., Sinthong, P., and Kalagnanam, J. A time series is worth 64 words: Long-term forecasting with transformers. In International Conference on Learning Representations, 2023.
Wang, Y., Xu, Y., Yang. J., Wu, M., Li, X., Xie, L., and Chen, Z. Fully-comnected spatial-tempores graph for multivariate time-series data. Proceedings of the AAAI Conference on Artificial Intelligence, 2024b.
Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al. Pytorch: An imperative style, high-performance deep learning library. Advances in Neural Information Processing Systems, 32, 2019.
Wu, H., Hu, T., Liu, Y., Zhou, H., Wang, J., and Long, M. TimesNet: Temporal 2d-variation modeling for gen- eral time series analysis. In International Conference Learning Representations, 2023.
Qiu, X., Hu, J., Zhou, L., Wu, X., Ju, Zhang, B., Guo, C., Zhou, A., Jensen, C. S., Sheng, Z., and Yang, B. Tfb: Towards comprehensive and fair benchmarking of time series forecasting methods. Proc. VLDB Endow, 2024a.
Wu, Z., Pan, S., Long, G., Jiang. J., Chang, X., and Zhang, C. Connecting the dots: Multivariate time series forecasting with graph neural networks. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2020.
Qiu, X., Wu, X., Lin, Y., Guo, C., Hu, J., and Yang, B. Duet: Dual clustering enhanced multivariate time series forecasting. arXiv preprint arXiv:24 12. 10859, 2024b.
Yu, G., Zou, J., Hu, X., Aviles-Rivero, A. I., Qin, J., and Wang, S. Revitalizing multivariate time series forecasting: Learnable decomposition with inter-series dependencies and intra-series variations modeling. In Forty-firsi Inter- national Conference on Machine Learning, 2024.
Shazeer, N.. Mirhoseini, A., Maziarz, K., Davis, A., Le, Q. Hinton, G., and Dean, Outrageously large neural networks: The sparsely-gated mixture-of-expert layer. In International Conference on Learning Representations, 2017a.
Yuan, E., Guo, W., He, Z., Guo, H., Liu, C., and Tang, R. Multi-behavior sequential transformer recommender. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Informa- tion Reirieval, SIGIR '22. Association for Computing Machinery, 2022.
Shazeer, N., Mirhoseini, A., Maziarz, K., Davis, A., Le, Q. Hinton, G., and Dean, J. Outrageously large neural networks: The sparsely-gate mixture-of-experts layer. In International Conference on Learning Representations, 2017b.
Zeng, A., Chen, M., Zhang, L., and Xu, Q. Are transformers effective for time series forecasting? In Proceedings of the AAAI conference on artificial intelligence, volume 37, pp. 11121-11 128, 2023.
Shi, J., Ma, Q., Ma, H., and Li, L. Scaling law for time series forecasting. Advances in Neural Information Processing Systems, 2024.
10

---

TimeFiter
Zhang, G., Sun, X., Yue, Y., Jiang, C., Wang, K., Chen, T., and Pan, S. Graph sparsification via mixture of graphs. arXiv preprint
0et.:2205.114460 2024a.
Zhang, G., Wang, K., Huang, W., Yue, Y., Wang, Y., Zimmermann, R., Zhou, A., Cheng, D., Zeng, and Liang, Y. Graph lottery ticket automated. In The Twelfth International Conference Learning Representations, 2024b. URL https: http:::///ppeeeeeeeeee net Eorum? id=nmB JBzooSX.
Zhang, G., Yue, Y., Wang, K., Fang, J., Sui, Y., Wang. K., Liang Y., Cheng, D., Pan, S., and Chen, T. Two heads are better than one: Boosting graph sparse training via semantic and topological awareness. arXiv preprint arXiv:2402.01222. 2024c.
Zhang, Y. and Yan, J. Crossformer: Transformer utilizing cross-dimension dependency for multivariate time series forecasting. In International Conference on Learming Representations, 2023.
Zhao, L. and Shen, Y. Rethinking channel dependence for multivariate time series forecasting: Learning from lead- ing indicators. In The Twelfth International Conference on Learning Representations, 2024.
Zhou, H., Zhang, S., Peng, J., Zhang, S., Li, J., Xiong, H., and Zhang, W. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 11106-1115,,, 2021.
Zhou, T., Ma, Z., Wen, Q. Wang. X., Sun, L., and Jin, R. FEDformer: Frequency enhanced decomposed trans- former for long-term series forecasting. In International Conference on Machine Learning, pp. 27268-2728 PMLR, 2022.
Zhu, P., Li, Y., Hu, Y., Liu, Q., Cheng, D., and Liang, Y. Lsr-igru: Stock trend prediction based on long short-term relationships and improved gru. International Conference on Information and Knowledge Management, 2024a.
Zhu, P., Li, Y., Hu, Y., Xiang, S., Liu, Q. Cheng, D., and Liang, Y. Mci-gru: Stock prediction model based on multi-head cross-attention and improved gru. arXiv preprint arXiv:2410.20679, 2024b.

---

TimeFilter
A. Implementation Details
A.1. Datasets
We conduct extensive experiments on cight widely-used time series datasets for long-term forecasting and PEMS datasets for short-term forecasting. We report the statistics in Table 5. Detailed descriptions of these datasets are as follows:
(1) ETT (Electricity Transformer Temperature) dataset (Zhou al., 2021) encompasses temperature and power load data from electricity transformers in two regions of China, spanning from 2016 to 2018. This dataset has two granularity levels: ETTh (hourly) and ETTm (15 minutes).
(2) Weather dataset (Wu et al., 2023) captures 21 distinct meteorological indicators in Germany, meticulously recorded 10-minute intervals throughout 2020. Key indicators in this dataset include air temperature, visibility, among others, offering comprehensive view of the weather dynamics.
(3) Electricity dataset (Wu et al., 2023) features hourly electricity consumption records in kilowatt-hours (kWh) for 321 clients. Sourced from the UCL Machine Learning Repository, this dataset covers the period from 2012 to 2014, providing valuable insights into consumer electricity usage patterns.
(4) Traffic dataset (Wu et al., 2023) includes data on hourly road occupancy rates, gathered by 862 detectors across the freeways of the San Francisco Bay area. This dataset, covering the years 2015 to 2016, offers a detailed snapshot of traffic flow and congestion.
(5) Solar-Energy dataset (Lai al., 2018) contains solar power production data recorded every 10 minutes throughout 2006 from 137 photovoltaic (PV) plants in Alabama.
(6) Climate dataset (Godahewa et al., 2021) records regional precipitation data across 1,763 stations in Australia.
(7) PEMS dataset (Liu al., 2022) comprises four public traffic network datasets (PEMS03, PEMS04. PEMS07, and PEMS08), constructed from the Caltrans Performance Measurement System (PeMS) across four districts in California.
The data is aggregated into 5-minute intervals, resulting in 12 data points per hour and 288 data points per day.
Tasks
Datasct
Dim
Prediction Length
Dataset Size
Frequency
ETTm1
7
[96, 192, 336, 720)
(34465, 1 11521, 11521)
15 min
ETTm2
(96, 192,336, 720}
(34465, I |1521, I [1521)
15 min
ETTh1
7
[96, 336, 720}
(8545, 2881, 2881)
15 min
Long-term
ETTh2
7
[96, 192, 720}
(8545, 2881, 2881)
15 min
Forecasting
Weather
21
[96, 192, 336, 720}
(36792, 5271, 10540)
10 min
Electricity
321
(96, 192, 336, 720}
(18317, 2633, 5261)
hour
Traffic
862
{96, 192, 336, 720}
(12185, 1757, 3509)
1 hour
Solar-Energe
137
(96, 136, 720)
(36601, 5161, 10417)
10 min
Climate
1763
{96, 192, 336, 720}
(6763, 989, 2070)
10 min
PEMS03
358
12
(15617, 5135, 5135)
5 min
Short-term
PEMS04
307
12
(10172, 3375, 3375)
5 min
Forecasting
PEMS07
883
12
(16911, 5622, 5622)
5 min
PEMS08
170
12
(10690, 3548, 263)
5 min
Table Dataset detailed deseriptions. "Dataset Size" denotes the total number of time points (Train, Validation, Test) split respectively. "Prediction Length" denotes the future time points to be predicted. "Frequency" denotes the sampling interval of time points.
12

---

TimeFilter
A.2. Metric Details
Regarding metries, utilize the mean square error (MSE) and mean absolute error (MAE) for long-term forecasting. The calculations of these metrics are:
T v:)
1 MSE
T Σ
Un
1 MAE
Σ T
V/l
A.3. Experiment details
All experiments implemented in PyTorch (Paszke et al., 2019) and conducted on NVIDIA A100 40GB GPU. We the Adam optimizer (Kingma, 2014). The batch size is set to 16 for the Electricity and Traffic datasets, and 32 for all other datasets. Table 6 provides detailed hyperparameter settings for each dataset.
Tasks
Dataset
Length of Patch
layers
Ir
d d_model
d.ff
Num. of Epochs
ETTml
8
le-4
256
256
10
ETTm2
16
2
1e-4
128
128
10
ETThl
2
2
le-4
128
256
10
Long-term
ETTh2
4
1
le-4
128
256
10
Forecasting
Weather
48
2
5e-4
128
256
10
Electricity
32
2
le-3
512
512
15
Traffic
96
1e-3
512
2048
30
Solar-Energy
48
5e-4
256
512
10
PEMS03
48
2
le-3
512
1024
20
Short-term
PEMS04
48
2
5e-4
512
1024
20
Forecasting
PEMS07
96
2
1e-3
512
1024
20
PEMS08
48
2
1e-3
512
512
20
Table Hyperparameter settings for different datasets. "c layers" denotes the number of graph block. "Ir" denotes the lcarning rate. "d_model" and "d_ff" denote the model dimension of attention layers and feed-forward layers, respectively.
B. Full Results
B.1. Error Bars
In this paper, we repeat all the experiments three times. Here we report the standard deviation of our proposed TimeFilter and the second best model Leddam (Yu al., 2024), as well as the statistical significance in Table 7. We perform the Wilcoxon with Leddam (Yu al., 2024) and obtain the p-value of -1 indicating significant improvement the 99% confidence level.
B.2. Long-term Forecasting
Table and Table present the full results for long-term forecasting, including both the results for the fixed look-back horizon L 96 results and the results obtained through hyperparameter search for the optimal horizon according to the scaling law (Shi et al., 2024) of TSF. The hyperparameter search process involved exploring look-back horizons L E {192, 336, 512, 720). In both settings, TimeFilter consistently achieves the best performance, demonstrating its effectiveness and robustness. In particular, under the optimal horizon condition, TimeFilter can still adaptively model effective dependencies in long sequences.
B.3. Short-term Forecasting
Table 10 present the full results for short-term forecasting. The look-back horizon is set to 96 and the forecasting horizon 24, 48). Across all four PEMS datasets, TimeFilter consistently delivers the highest performance, particularly
E 21,.24,44 with significant improvement on PEMS08, showcasing its effectiveness and reliability.
13

---

TimeFilte
Model
Timeer
Leddam (2024)
Confidence
Dataset
MSE
MAE
MSE
MAE
Interval
ETTml
0.377 + 0.004
0.393 = 0.006
0.386 = 0.008
0.397 + 0.006
99%
ETTm2
0.272 0.004
0.321 0.002
0.281 + 0.003
0.325 0.003
99%
ETThl
0.420 + 0.005
0.428 0.003
0.431 = 0.005
0.429 + 0.004
99%
ETTh2
0.364 + 0.003
0.397 = 0.002
0.373 = 0.009
0.399 + 0.007
99%
Weather
0.239 + 0.006
0.269 + 0.004
0.242 + 0.009
0.272 + 0.006
99%
Electricity
0.158 + 0.005
0.256 = 0.006
0.169 = 0.005
0.263 + 0.005
99%
Traffic
0.407 0.008
0.268 + 0.004
0.467 + 0.006
0.294 + 0.008
99%
Solar
0.223 0.006
0.250 + 0.003
0.230 + 0.008
0.264 + 0.005
99%
Table Standard deviation and statistical tests TimeFilter and second-best method (Ieddamm) on ETT, Weather, Electricity, Traffic, and Solar datasets.
B.4. Ablation Study
We present the full results of the ablation studies discussed in the main text in Table 11. We carefully design six other filtering methods to validate the effectiveness of TimeFilter.
Top-K strategy selects the top K edges with the largest weights from the ego graph G. of each patch.
Random-K randomly selects edges from the ego graph 9, of each patch.
(3) RegionTop-KK strategy selects the top K edges with the largest weights within the region corresponding to three types of dependencies.
(4) RegionThre learns threshold for each region through linear mapping and selects edges with weights greater than the threshold in each region.
(5) C-Filter refers to channel-wise filtering, where all patches within the look-back sequence are filtered based on the same type of dependency.
(6) w/o Filter refers to the case where no filtering is applied after graph construction.
C. Visualization
In order to facilitate a clear comparison between different models, we present supplementary prediction examples for four representative datasets, as Electricity Figure 7, Traffic in Figure 8, Weather in Figure and PEMS08 Figure 10, respectively. The look-back horizon L is 96 for four datasets. The forecasting horizon T 96 for Eleltricity, Traffic and Weather, while T is 48 for PEMS08. Among the different models, TimeFilter delivers the most accurate predictions of future series variations and demonstrates superior performance.
14

---

TimeFilter
Models
TimeFilter (Ours)
DUET
Tiaasformer (20246)
Leddam (2024)
MSGNNet (2024)
SOFTS
CrosGNN (2023)
PatchTST (2023)
Cinssormerr (2023)
TimesNet (2023)
DI (2023)
FFDformer
(2024b)
(20244)
(2022)
Metric
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE|
MSE
MAE
MSE
MAB
MAE
MSE
MAE
MSE
MAB
MSE
96
0,313
0.354
0:324
0,354
0.334
0.368
0.319
0 0.35910
0.319
0,366
0.325
0.361
00.35
0:373
0.329
0.367
0.404
0.426
0.338
0:375
0.346
0.374
0.379
0.419
192
0,356
0.380
0.369
0.379
0.377
0.391
0.369
0.383
0.376
0.397
0.375
0.389
0.372
0.390
0.367
0.385
0.450
0.451
0.387
0.382
0.391
0.426
0,441
336
0.386
0.402
0.404
0.402
0.426
0.420
0.394
0.402
0.417
0.422
0.405
0.412
0.403
0.411
0.399
0.410
0.532
0,515
0.410
0.411
0.415
0.415
0.445
0.459
720
0.452
0.437
0.463
0.437
0.491
0.459
0.460
0.442
0:481
0.458
0.466
0.447
0..461
0.442
0.454
0.479
0.666
0.589
0.478
0.450
0.473
0.451
0.543
0.490
Ang.
00.377
0.393
0..90
0.393
0.400
0.410
00.386
0.397
00.398
0.411
0.393
0.403
0..393
0.404
00.387
0.400
0.513
0.496
|0.400
0.406
00.404
0.408
0.448
0.452
00.169
0.255
0.174
0.255
0.180
0.264
0..176
0.257
00.177
0.262
0.261
0.116
0.266
00.175
0.259
0.287
0,366
0.1187
0.267
0..19
0.293
0.203
0.287
E17m2
192
0.235
0.299
0.243
0.302
0.250
0.309
0,243
0.303
0.247
0.307
0.246
0.306
0.240
0.307
01.241
0.302
0.414
0.492
0.249
0.309
0.284
0.361
0.269
0.328
336
0,299
0.304
0.341
0.311
0.303
0.341
0.312
0.346
0.304
0.345
0.305
0.343
0.597
0.351
0.382
0.429
0,325
0,366
720
0.390
0.393
0.399
0.397
0.412
0.407
0.400
0.398
0:414
0.403
0.405
0,401
0.406
0,400
0.402
0,400
1.730
1.042
0.408
0.403
0.558
0.525
0.421
0,415
Aug:
00.272
0.321
0.280
0.324
00.288
0.332
|0.281
0:325
[0.288
0.330
0.287
0.330
[0.282
0.330
[0.281
0.326
0.757
0.610
[0.291
0.333
|0.354
0.402
0.305
0.349
96
00.370
0.394
0.377
0.393
0..86
0.405
00.377
0.394|
00,390
0.411 0.381
0.399
0,382
0.398
[0.414
0.419
0.423
0.448
00.384
0.402
00.397
0.412
0.576
0.419
192
0,413
0.420
0.429
0,425
0..41
0.436
0.424
0.422
0.442
0.442
0.435
0.431
0.427
0.425
0.460
0.445
0.471
0.474
0.436
0.429
0.446
0.441
0:420
0.448
336
0.450
0.440
0.471
0.446
0.487
0.458
0.459
0:442
0.480
0.468
0.480
0.452
0.465
0.445
0.501
0.466
0.570
0.546
0.499
0.469
0.489
0.467
0:459
0.465
720
0.448
0.457
0.496
0,480
00.503
0.491
0.4463
0.459
0.494
0.488
0.499
0.488
0.442
0.468
00.500
0.488
0.653
0.621
0.521
0.500
0.513
0.510
0.506
0.507
Aug.
00.420
0.428
0.443
0.436
00.454
0.447
00.431
0.429|
00.452
0.452
0.449
0.442
[0.437
0.434
00.469
0.454
0.529
0.522
[0.458
0.450
(0.461
0.457
0.440
0.460
96
0,288
0.337
0.296
0.345
0.297
0.349
0.299
0.343
0..328
0.371
0.297
0.347
0.309
0.359
00.302
0.348
0.745
0.584
0.340
0.374
0..340
0.394
0.358
0.397
192
0.362
0. 399
0.368
0.389
0,380
0.400
0.2622
0.389
0.402
0.414
0.373
0.394
0.390
0.406
00.888
0.400
0.877
0.656
0.402
0.414
0.482
0.479
0.429
0.439
336
0.404
0.424
0.411
0.422
0.428
0.432
0.412
0.424
01:435
0.443
0.410
0.426
0.426
0.444
01.426
0.433
1.043
0.731
0.452
0.452
0.591
0.54]
0.496 0.487
720
0.447
0.433
0.412
0:434
0.427
0.445
0.419
0.438
0:417
0.441
0:411
0.433
0.445
0.464
00:431
0.446
1.104
0.763
0.462
0.468
0..839
0.661
0.463
0.474
Arg.
|0.364
0.397
0.372
0.397
00.383
0.407
00.373
0. 0,399|
00.396
0.417
0.385
0.408
00.373
0.400
00.387
0.407
0.942
0.684
[0.414
0.427
|0.563
0.519
0.437
0.449
96
10.153
0.199
0.163
0.202
0.174
0.214
0.1156
0.202
00.163
0.212
0.166
0.208
0.159
0.218
00.177
0.218
0.158
0.230
0.172
0.220
0..195
0.252
0.217
0.296
Weather
192
0.202
0.246
0.218
0.252
0.231
0.254
0.207
0.
0.250
0.212
0.254
0.217
0.253
0.211
0.266
0.225
0.259
0.206
0.277
0.219
0.261
0.237
0.295
0.276
0.336
336
0.260
0.289
0.274
0.294
0.2778
0.296
0,262
0.291
0:272
0.299
0.282
0.300
0.267
0.310
0.278
0.297
0.272
0.335
0.280
0.306
0.282
0.331
0.339
0,380
720
0,342
0.341
0.349
0.343
0.358
0.349
0.345
0.343
0.350
0.348
0.356
0.351
0.352
0.362
0.354
0.348
0.398
0.418
0.365
0.359
0.345
0.382
0.403
0.428
Aug.
00,239
0.269
0:251
0.273
0.258
0.279
0.2442
0.272
0.249
0.278
0.255
0,278
0.2247
0.289
[0.259
0.281
0.259
0.315
0,255
0.287
(0.265
0.315
0.309
0,360
96
00.133
0.230
0.1
0,233
0.148
0.240
0.114
0.235
00.165
0.274
0.143 0,233
0.173
0.275
00.195
0.285
0.219
0,314
10.168
0.272
00.210
0.302
0.193
0,308
Electticity
192
0.154
0.248
0.163
0.248
0.162
0.253
0.159
0. 0:252
0.184
0.292
0.158
0,248
0.195
0.288
0.1199
0.289
0.231
0.322
0.184
0.289
0.211
0.305
0.201
0.315
336
0.162
0.261
0.5
0.262
0.178
0.269
0.173
0.268
0.1
0,302
0.178
0.269
0,206
0.300
0.:115
0.305
0.246
0,337
0.198
0.300
0.223
0.319
0.214
0,329
720
0.184
0.284
00.204
0.291
0.225
0.317
0.201
0.295
00.231
0.332
0.218
0.305
0.233
0.335
(0.256
0.337
0.280
0.363
0.220
0.320
0.2258
0.350
0.246
0.355
Arg.
00.158
0.256
00.122
0.258
0.178
0.270
0..199
0.263
0..194
0.300
0.174
0.264
00.201
0.300|
00.216
0.304
0.244
0.334
|0.193
0.295
00.225
0.319
0.214 0.327
96
10.375
0.407
0.252
0.395
0.268
0.,426
0.276
006005
0.344
0.376
0.251
0.570
0.310
(0.544
0.359
0.522
0.290
0.593
0. 0.321
(0.600
0.396
0.587
0.366
Ttaffic
192
0.395
0.262
0.431
0.262
0.417
0.276
0.458
0.289
0.613
0.359
0.398
0.261
0.577
0.321
0.540
0.354
0.530
0.293
0.617
0.336
0.599
0.370
0.604
0.373
336
0.456
0.269
0.433
0.283
0,486
0.297
0.692
0.376
0.415
0.588
0.324
0.551
0.358
0.558
0.629
0.336
0.605
0.373
0.621
0,385
720
01.445
0.289
0.509
0.292
0.467
0.302
0.449
0.313
0.702
0.401
0.447
0.287
0.597
0.337
00.586
0.375
0.589
0.328
0.640
0,350
0.665
0.394
0.626
0,382
Aug. 00.407
0.268
0.451
0,269
00.428
0.282
00.467
0.294
[0.641
0.370
0.409
0.267
00.583
0.323
[0.555
0.362
0.550
0.304
0.620
0.336
[0.625
0.383
0.610
0.376
Solae-Enagy
96
0.1193
0.223
0.200
0.207
0.208
0.237
0.1197
0.241
00.008
0.243
0.200
0.230
0.228
0.353
00,234
0.286
0.310
0.331
|0.250
0.292
00,290
0.378
0.242
0.342
192
0.226
0.249
0:228
0.233
0.233
0.261
0,231
0.263
0.258
0.281
0.229
0.253
0.316
0.374
0.267
0.310
0.734
0.725
0.296
0.318
0.320
0.398
0.285
0,380
336
0.235
0.261
0.262
0.244
0.248
0.273
0,241
0.268
0.293
0.311
0.243
0.269
0.347
0.393
0,290
0.315
0.750
0.735
0.319
0.350
0.353
0.415
0.282
0.376
720
0.239
0.268
0.258
0.249
0.249
0.275
0.250
0.281
0.290
0.315
0.245
0.272
0.348
0.389
0.288
0.317
0.769
0.765
0.338
0.337
0.357
0.413
0.357
0.427
Aug.
00.23
0.250
0.237
0.233
0.233
0.262
0.230
0. 26264
0.262
0.288
0.229
0,256
0.324
0.377
00,270
0.307 0.641 0.639
0.301
0.319
0.330
0.401
0.292 0.381
Table 8. Full results of long-term forecasting. The input sequence length L is set to 96 for all baselines. All results are averaged across four different forecasting horizon: T' E {96, 192, 336, 720]. The best and second-bes results are highlighted in bold and underlined. respectively.
15

---

TimeFilter
Models
TimeFilter (Ours)
DUET
CCM
PDF
ModemTTNN
i'ransformer (2024b)
CATS
PatehTST
Crossformer (2023)
TimesNet (2023)
DLinear
(20248))
(2024)
(2024)
(2024)
(2024)
(2023)
(2023)
Metric
MSE
MAE|
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
00.14
0.193
0.146
0.191
0.144
0.197
0.147
0.194
0.1149
0.200
0.157
0.207
0.161
0.207
0.149
0.198
0.143
0.210
0.168
0.214
0,170
0.230
Weather
0.184
0.234
0.188
0.231
0.191
0.238
0.192
0.239
0.196
0.200
0.248
0.208
0.250
0.194
0.241
0.198
0.260
0:219
0.262
0.216
0.273
336
0.234
0.276
0.234
0.268
0.245
0.285
0.244
0.279
0.238
0.277
0.252
0.287
0.264
0.290
0.245
0.282
0,258
0.314
0.278
0.302
0,258
0.307
720
0.305
0.327
0.305
0.319
0.316
0.333
0,318
0.330
0.314
0.334
0.320
0.336
0,342
0.341
0.314
0,334
0.335
0.385
0.353
0.351
0,323
0.362
|Avg
00.216
0.258
0:218
0.252
[0.225
0.263
0.225
0.261
0.224
0.264
|0.232
0.270
0.244
0.272
0.226
0.264
0.234
0.292
0.225
0.282
0.242 0.293
10.126
0.220
0.128
0.219
(0.136
0.231
0.126
0.220
0.129
0.226
00.134
0.230
0.149 0.237
0.129
0.222
0.135
0.237
00.168
0.272
0,140 0.237
Elletticity
192
0.143
0.237
0.146
0.236
0.153
0.248
0.145
0.237
0.143
0,239
0.154
0.250
0.163 0.250
0.147
0.240
0.160
0.262
0.184
0.289
0,152 0.249
336
0.153
0.252
0.163
0.254
0.168
0.262
0.159
0.255
0.161
0.259
0.169
0. 0.265
0,180 0.268
0.163
0,259
0.182 0.282
0.198
0.300
0.169
0.267
0.177
0.275
0.195
0.282
0.210
0.301
0.194
0.287
0..191
0.286
0.194
0.288
0.219
0.302
0.197
0.290
0.246
0.337
0.220
0.320
0.203
0.301
AAv
10.150
0.246
0.158
0.248
(0.167
0.261
0.156
0.250
[0.156
0,253
[0.163
0.258
0.178
0.264
00.159
0.253
0. 181
0.279
[0.192
0.295
161 0.264
96
0.332
0.238
0.360
0.238
00.357
0.246
0.350
0.239
0..66
0,253
0.363
0.265
0.421
0.270
0.360
0.249
0,512
0.282
0.599
0.321
0.410
0.282
Traffic
192
0.348
0.246
0.384
0.250
0.379
0.254
0,363
0.247
0.379
0.261
0.384
0.273
0,436
0.275
0.379
0,256
0.501
0.273
0.617
0.336
0.423
0.287
336
0.365
0.256
0.395
0.260
0.389
0.255
0.376
0.258
0.397
0.270
0.396
0.277
0.453
0.284
0.392
0.264
0.507
0.279
0.629
0.336
0.436
0.296
720
0.396
0.275
0.435
0.278
00.430
0.281
0.419
0.279
0.444
0.296
0.445
0.308
0.484
0.303
0.432
0.286
0.571
0.301
0.640
0.350
0.466
0.315
Avg.
|0.360
0.254
0.394
0.257
00.389
0.259
0.377
0.256
00.396
0.270
0.397
0.281
00.449
0.283
00.391
0.264
0.523
0.284
[0.620
0.336
00.44 0.295
96
1.030
0.492
1.090
0.510
1,103
0.511
1.116
0,530
1,100
0.514
1.044
0.494
1.157
0.535
1.354
0,519
1.083
0.508
1.274
0.524
1.108
0,536
Climate
192
L039
0.489
1.098
0.511
1.118
0.510
1,362
0.570
1.125
0.523
1.055
0.496
1.073
0.507
1,388
0.572
1.115
0.532
1.274
0.541
1.122 0.536
336
1.056
0.489
1.120
0.515
1,130
0.513
1.396
0.568
1.153 0.530
1,069
0.495
1.072
0.495
1.410
0.581
1,246
0.575
1.285
0,584
1.146 0.545
720
1.61
0.490
1.103
0.498
1.143
0.515
1,386
0.552
1,243
0.599
1.073
0.497
1.078
0.498
1.395
0,575
1,236
0.584
1.347
0.626
1.158 0.545
|Ar..
1.047
0.490
1.103
0.509
1.123
0.512
1,315
0.555
1.155
0.542
1.060
0.496
1.095
0.509
1.387
0.562
1.170 0.550
1 1.295 0.569
1.134 0.541
Table 9. Full results of long-term forecasting. The input length L is scarched from (192, 3366 law of TSF (Shi et al., 2024). All results are averaged across four different forecasting horizon: T E {96, 192, 336, 720). The best and sccond- best results are highlighted in bold and underlined, respectively.
512, 720} for optimal horizon in the scaling
Models
TimeFilter (Ours)
DUET
iTaasforeer (2004b)
Leddam (2024)
SOFTS
PatcTTTT
Crossformer (2003)
TimesNot (2013)
TIDE (2023)
DLincar (2025)
SCINet (2022)
FEDformer (2022)
(2004b)
(2004))
(2225)
Metric
MSE
MAF
MSB
MAR
MSE
MAR
MSF
MAF
MSE
MAR
MSE
MAE
MSF
MAF
MSE
MAR
MSF
MAR
MSF
MAR
MSB
MAR
MSF
MAE
0.063
0.165
0.064
0.166
0.071
0.174
0.068
0.174
0.064
0.165
0.099
0.216
0. 0.203
0.085
0.192
0.178
0.243
0.066
0.172
0.126
0.251
PEMS03
0.079
0.185
0.081
0.186
0.093
0.201
0.094
0.202
0.083
188
0.142
0,259
0.121
0.240
0.118
0.223
0.257
0.371
0.201
0.317
0.085
0.198
0.149
0.275
0.222
0.125
0.236
0.140
0.254
0.114
0.223
0.211
0.319
0.200
0.317
0.155
0.260
0.379
0.163
0.333
0.425
0.127
0.238
0.227
0.348
Aug.
|0.084
0.191
0.086
0.192
0.096
0.204
00.101
0.210
0.087
0.192
0.151
0.265
00.138
0.253
0.119
0.271
0.271
0.380
0.2119
0.295
000093
0.203
0.167
0.291
10.068
0.167
0.079
0.181
0.078
08183
00.076
0. 82
0.074
0.176
0.105
0.224
10.098
0.218
0.087
0.195
0.219
0,340
0.144
0.272
0.077
0.127
0.138
0.262
PEMS04
24
0.080
0.183
0.096
0,203
0.095
0.205
0.097
0.209
0.088
0.194
0.153
0.257
0.131
0.256
0.103
0.215
0.292
0.398
0.224
0,340
0.084
0.193
0.177
0.295
0.101
0.209
0.114
0.226
0.120
0:233
0..13
0.249
0.
0.219
0.229
0.339
0.205
0.326
0.136
0.250
0.409
0.478
0.335
0.437
0.099
0.270
0.368
|0.083
0.186
0.096
0.203
0.098
0.207
|0.102
0.213
[0.091
0.196
0.162
0.273
00.145
0.267
00.109
0.220
0.307
0.405
|0,236
0.350
00.085
0.194
0.195 0.308
10.055
0.150
0.060
0.156
0.067
0.165
0.066
0. 0.164
000057
0.152
0.095
0.207
0.004
0.200
0.082
0.181
0.173
0.304
0..115
0.242
0.068
0.171
0.109
0.225
PEMSU7
0.068
0.166
0.073
0.122
0.088
190
0.079
0.585
0.023
0.173
0.150
0.262
0.139
0.247
0.101
0.204
0.271
0.383
0.210
0.329
0.119
0.225
0.125
0.244
0.089
0.193
0.096
0.201
0.110
0.215
0.115
0.228
0.096
0.195
0.253
0.340
0.311
0.369
0.133
0.238
0.446
0.495
0.398
0.458
0.114
0.237
0.165
0,288
Aug.
|0.071
0.170
(0.076
0.176
0.088
0.190
0.087
0.192
00.075
0.173
0.166
0,270
00.18
0.272
00.106
0.208
0.297
0.394
00.241
0.343
00.112
0.211
0.133
0,282
12
0.064
0.162
0.072
0.168
0.079
08182
0.070
0.173
0.077
0.171
0.168
0,232
0.165
0.214
0.112
0.212
0.227
0.343
0.154
0.276
0.087
0.184
0.173
0,273
PEMSON
0.079
0.182
0.093
0.191
0.115
0.219
0.091
0.2000
0.104
0.201
0.224
0.281
0.215
0.260
0.144
0.238
0.318
0.409
0.248
0.353
0.122
0.221
0.210
0.310
0.105
0.214
0.123
0.217
0.186
0.235
0,145
0.261
0.164
0.253
0.321
0.354
0.315
0.355
0.1198
0.283
0.497
0.510
0.440
0.470
0.189
0.270
0.520
0,394
Avg.
[0.083
0.186
0.096
0.192
0.127
0.212
0.102
0.211 00.114
0.208
0.238
0,289 0.232
0.270
001550 0.244 0.347
0.421
[0.281 0.366
00.133
0.225
0.234 0,326
Table 10. Full results of short-term forecasting. The input sequence length is set to 96 for baselines. All results are averaged across four different forecasting horizon: T' E {12, 24, 48). The best and second- best results are highlighted in bold and underlined, respectively.
16

---

TimeFilter
Catagories
Filter Replace
MoR
Cases
TimeFilter
Top-K
Random K
RegionTop-- K
RegionThre
C C-Fitte
w/o Filter
Static Routing
w/o Cimp & drn
Metric
MSE
MAE
MSB
MAE
MSB
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0,153
0.199
0.158
0.203
0.158
0.204
0.156
0.202
0.156
0.202
0.155
0.202
0.159
0.204
0.156
0,203
0.156
0.203
Weather
192
0,202
0,246
0,207
0.250
0,207
0.251
0.205
0.249
0.202
0.247
0.206
0.249
0,208
0.251
0,205
0.248
0.206
0.248
336
0,260
0.289
0.264
0.292
0.263
0.292
0.262
0.292
0.262
0.291
0.263
0.291
0.265
0.292
0.263
0.290
0.264
0.291
720
0.342
0.341
0.344
0.343
0.342
0.341
0.342
0.342
0.341
0.341
0.343
0.342
0.345
0.345
0.343
0.342
0.344
0.344
Aug.
0.239
0.269
0,243
0.272
0.243
0.272
0.241
0.271
0.241
0.270
0.242
0.271
0.244
0.273
0.242
0.271
0.243
0.272
96
0.133
0.230
0.138
0.235
0.137
0.235
0.139
0.235
0.138
0.234
0.137
0.234
0.139
0.237
0.137
0.234
0.136
0.233
Electricity
192
0.154
0.248
0.163
0.255
0.161
0.254
0.163
0.255
0.164
0.257
0.159
0.252
0.159
0.252
0,156
0,250
0.155
0.250
336
0,162
0.261
0,168
0.266
0,174
0.270
0,169
0.265
0.166
0.264
0.169
0.267
0.168
0.265
0.167
0.264
0,166
0.263
720
0,184
0,284
0,200
0,297
0,200
0.297
0,199
0.295
0.220
0.314
0.214
0.308
0,200
0.298
0.195
0,290
0,193
0.288
Arg.
0.158
0.256
0,167
0,263
0,168
0.264
0,168
0.263
0.172
0.267
0.170
0.265
0.167
0.263
0,164
0,260
0,163
0.259
96
0,375
0,251
0,379
0,253
0,378
0.253
0.377
0.251
0.378
0.251
0.379
0.252
0,379
0.254
0,377
0,253
0.378
0.254
Traffic
192
0.395
0,262
0.404
0.265
0.402
0.264
0.403
0.263
0.406
0.264
0.405
0.264
0.404
0.266
0.398
0.264
0.397
0.264
336
0,414
0.271
0.420
0.274
0.418
0.273
0.416
0,272
0.417
0.275
0.418
0.274
0.422
0.275
0.417
0.273
0.417
0.272
720
0.445
0.289
0.449
0.293
0.448
0.292
0.449
0.292
0.447
0.291
0.448
0.291
0.450
0.295
0.447
0.292
0.293
Arg.
0.407
0.268
0,413
0.271
0.412
0.271
0.411
0.270
0.412
0.271
0.413
0.271
0.414
0.273
0.410
0.271
0.410
0.271
Solar- Energy
96
0.193
0.223
0,203
0.229
0.198
0.225
0.202
0.224
0.204
0.225
0.203
0.225
0,201
0.234
0.196
0.224
0.194
0.224
192
0,226
0.249
0,225
0.255
0.228
0.257
0.225
0.254
0.224
0.254
0.233
0.257
0,230
0.262
0,230
0,262
0.227
0.262
336
0,235
0.261
0.249
0.270
0.244
0.269
0.253
0.271
0.254
0.274
0.251
0.273
0,249
0.272
0,242
0,269
0.242
0.270
720
0,239
0,268
0,248
0.276
0,252
0,280
0.250
0.277
0.256
0.281
0,250
0.278
0,256
0,280
0,245
0.276
0.244
0.275
Aug.
0.223
0.250
0.231
0.258
0.231
0.258
0.233
0.257
0.235
0.259
0.234
0.258
0,234
0.262
0.228
0.258
0.227
0.258
96
0.370
0.394
0.379
0.399
0.377
0.396
0.379
0.398
0.379
0.398
0.377
0.399
0.379
0.398
0.374
0.397
0.374
0.394
E11h1
192
0.413
0.420
0.432
0.429
0.430
0.427
0.432
0.429
0.434
0.428
0.424
0.428
0.431
0.429
0.421
0.424
0.421
0.423
336
0.450
0.440
0.454
0.453
0.454
0.469
0.449
0.467
0.452
0.475
0.453
0.460
0.446
0.460
0.445
720
0.448
0.457
0.474
0.468
0.482
0.473
0.475
0.469
0.466
0.467
0.457
0.463
0.481
0.471
0.465
0.467
0.465
0.467
Arg.
0.420
0.428
0.441
0.438
0.441
0.437
0.441
0.438
0.437
0.436
0.431
0.436
0.442
0.438
0.440
0.434
0.430
0.432
96
0.169
0.255
0.172
0.257
0.172
0.257
0.170
0.257
0.172
0.257
0.172
0.258
0.173
0,257
0.172
0,256
0.171
0.256
E/1m2
192
0,235
0.299
0,237
0.300
0,238
0.301
0,233
0.299
0.237
0.299
0.235
0.300
0,237
0,300
0,236
0.298
0.235
0.298
336
0,293
0,336
0,298
0,340
0,296
0.338
0.298
0.340
0.297
0.340
0.299
0.342
0,298
0,340
0,295
0.337
0.294
0.336
720
0.390
0.393
0.395
0.398
0.396
0.397
0.392
0.396
0.399
0.398
0.400
0.400
0.400
0.399
0.392
0.396
0.393
0.397
Aug.
0.272
0.321
0,276
0.324
0,276
0.323
0.273
0.323
0.276
0.324
0.277
0.325
0,277
0.324
0,274
0.322
0.273
0.322
12
0.064
0.162
0.065
0.165
0.066
0.165
0.064
0.164
0.065
0.163
0.064
0.164
0.065
0.164
0.064
0.163
0.064
0.162
PEMS08
0.079
0.182
0.081
0.185
0.081
0.184
0.081
0.186
0.081
0.185
0.082
0.185
0.082
0.186
0,080
0.184
0,080
0.183
48
0.105
0.214
0.112
0.224
0.108
0.217
0.110
0.219
0.108
0.219
0.111
0.222
0.110
0.223
0,108
0.216
0.108
0.217
Aug.
0.083
0.186
0.086
0.191
0.085
0.189
0.085
0.190
0.085
0.189
0.086
0.190
0.086
0.191
0.084
0.188
0.084
0.187
Table 11. Component ablation of TimeFilter.
17

---

TimeFilter
2-
2
1
2
Ground
Prediction
Prediction
-1
GroundTruth Prediction
0
50
100
150
200
0
100
150
200
50
100
150
200
TimeFilter
SOFTS
Leddam
3-
2-
1-
GroundTruch
Groundlruth Prodiction
Prediction
Prediction
Mala
-1
Mw
100
150
100
150
200
MSGNet
i'Transformer
PStchTST
Figure 7. Visualization of predictions from different models on the Electricity dataset.
GroundTruth Prediction
6-
GroundTruth Prediction
6-
Groundiruth
5-
5-
4-
3-
3-
2-
1-
1-
0
0-
-1-
-11
100
150
0
100
150
200
TimeFilter
SOFTS
Leddam
6
GroundTruth
GroundTruth Prediction
GruundTruth Prediction
5
Predictoon
5-
4-
4-
3-
3-
3-
2
2
1-
1
0-
-1
0
50
100
150
200
50
100
150
200
200
MSGNet
iTransformer
PatchTST
Figure 8, Visualization of predictions from different models on the Traffic dataset.
18

---

TimeFilter
2.25
Groundiruth
Pridictionn
Prediction
2.25
2.00
1.75
1.75
1.50
1.50
1.50
1.25
1.25
1.25
1.00
1.00
TimeFilter
SOFTS
Leddam
2.25
Predition
Prediction
1.75
1.75
1.50
1.50
1.25
1.25
1.00
1.00
Cecundfruth
Peddcctoon
100
200
50
100
150
200
MSGNet
iTransformer
PatchTST
Figure 9. Visualization of predictions from different models on the Weather dataset.
Commnefh.ath Pediittion
Ground thhthh
GmandThhhh
Predittion
0.5
0.5
0.5
0.0
0.0
-0.5
-1.0
-1.0
-1.0
-1.5
1.5
-1.5
25
50
75
100
125
150
25
75
100
125
150
25
75
100
125
150
TimeFilter
SOFTS
Leddam
GrouncTruth
1.0
GroundThuth
0.5
0.5
0.0
0.0
0.5
-0.5
0.5
11.0
-1.0
-1.5
-1.5
-1.5
75
100
125
150
25
75
100
125
150
25
100
125
150
Crossformer
iTransformer
PatchTST
Figure 10. Visualization of predictions from different models the PEMS08 dataset.
19


Reviews:
[{"review_id": "MAyRUcsW9W", "reviewer": "Reviewer_1FL4", "summary": "In this paper, the authors propose to imporve multivariate time series (MTS) forecasting by proposing the TimeFilter framework, which introduces patch-specific spatial-temporal graph filtration to model dynamic dependencies. Traditional MTS forecasting approaches either follow a channel-independent (CI) approach, ignoring inter-channel dependencies, or a channel-dependent (CD) approach, which captures all dependencies indiscriminately, often leading to noise and reduced robustness. The authors argue that both methods have limitations, particularly in capturing time-varying dependencies. To address this, TimeFilter employs a fine-grained dependency modeling technique that filters out irrelevant correlations and retains only the most significant ones in a patch-specific manner. The framework is based on a graph neural network (GNN) architecture and dynamically adjusts its approach to each dataset's characteristics, using a mixture of experts mechanism. Extensive experiments across several real-world datasets demonstrate that TimeFilter outperforms state-of-the-art methods in both long- and short-term forecasting tasks.", "questions": "", "limitations": "", "rating": 2, "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": "Thanks for your valuable comments. Here are responses to your insightful concerns and questions:\n\n---\n**`Q1`: Novelty and Contribution. (The proposed idea is conceptually similar to existing Granger causality-based methods [1, 2] and instantaneous time series [3], yet the authors fail to discuss this connection.)**\n\n**`R1`:** We would like to clarify that our core innovation lies in proposing a novel fine-grained dependency modeling paradigm. Existing methods predominantly focus on channel-wise correlations at a coarse granularity, whereas our approach explicitly models time-varying fine-grained dependencies that cannot be effectively captured by channel-wise interactions alone.\n\n  The foundation motivation of TimeFilter stems from the following observation:\n  Real-world time series often exhibit hybrid characteristics containing both causal signals (spa.-tem.&spa.) and inertial signals (tem.). However, the global time series signals do not conform to the causal, auto-generative nature assumed in modeling, but are influenced by unobserved factors [5]. Such forecasting are similar to the natural language understanding (NLU), where integrating all latent signals is more crucial. As the reviewer pointed out, Granger causality[1,2] only accounts for temporal inertia and spatio-temporal causality, while the instantaneous effects[3] only capture spatial causality. In contrast, TimeFilter integrates all potential signals during graph construction and then adaptively filters out invalid signals under specific segments, achieving superior forecasting capability.\n  \n  To better demonstrate our novelty, we provide a comprehensive comparison across critical dimensions in the following table. (C channels, L lookback horizon, D hidden dimensions and K lead-lag steps).\n||TimeFilter|Granger causality|Instantaneous effects|\n|-|-|-|-|\n|Signals|Tem.&Spa.&Spa.-Tem.|Spa.-Tem.&Tem.|Spa.|\n|Granularity|Patch-wise|Channel-wise|Channel-wise|\n|Assumption|None.|Static|Synchronization|\n|Robustness|Yes|No|No|\n|Complexity|$O(C^2(\\frac{L}{P})^2)$|$O(C^2K+LCD+LD^2)$|$O(LC^3)$|\n\n[5] Time Series Prediction: Forecasting The Future And Understanding The Past\n\n[6] MTEB: Massive Text Embedding Benchmark\n\n---\n**`Q2`: Analysis of why TimeFilter improve the performance. (The paper lacks theoretical grounding or in-depth analysis to explain why TimeFilter leads to performance improvements.)**\n\n**`R2`:**  We appreciate this insightful question regarding the validation of the effectiveness of our method. The core mechanism can be explained from two complementary perspectives: \n\n- **Theoretical foundation**: As noted in Q1, our filtering mechanism addresses the spurious regression artefacts prevalent in dependency modeling. Specifically, while some signals provide genuine predictive patterns, others exhibit localized spurious correlations arising from short-term segment analysis - a phenomenon particularly pronounced in non-stationary time series [7]. TimeFilter's dynamic pruning acts as a variance reduction operator, suppressing these transient noise signals while preserving true causal relationships.\n  \n- **Empirical verification**: As shown in https://i.postimg.cc/Jn42zB0b/rebuttal-fig.png, we quantitatively demonstrate that in the Weather dataset. The filtered adjacency matrix exhibits clearer cluster structures aligned with physical sensor relationships. This fine-grained opeartion allows it to focus on high-spatial and high-temporal-spatial correlations (e.g., Fea0-Fea1 and Fea1-Fea3) while eliminating irrelevant dependencies (e.g., Fea0-Fea3).\n  \nThis dual validation by both causal theory and data-driven evidence confirms that our adaptive filtering successfully disentangles persistent patterns from transient noise, directly contributing to the observed performance gains.\n\n[7] Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization.\n\n---\n**`Q3`: Compare with LIFT[4]. (The proposed method is very similar to [4], but does not consider it as a baseline.)**\n\n**`R3`:** LIFT[4] is added into the baseline. Our method still outperforms it (PatchTST+LIFT) and we will include LIFT in the revised version. While LIFT models lead-lag dependency modeling through lead estimation and refinement, it is sensitive to data distribution and less effective for synchronous data without significant lead-lag relationships. TimeFilter enhances such generalization ability and customizes dependency types for datasets from different domains.\n||TimeFilter|LIFT|\n|-|-|-|\n|Metric|MSE/MAE|MSE/MAE|\n|Weather|0.216/0.258|0.229/0.262|\n|ECL|0.150/0.246|0.158/0.252|\n|Traffic|0.360/0.254|0.386/0.260|\n\n---\nOverall, thanks again for your valuable comments. We hope the detailed experiments and clarifications provided above address your concerns. If you have any further questions, please feel free to reach out, and we would be happy to provide additional clarifications.", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "One of the main concern is the limited contribution of this paper.  The proposed idea is conceptually similar to existing Granger causality-based methods [1, 2] and instantaneous time series [3], yet the authors fail to discuss this connection. Specificially, the temporal dependencies and spatial-temporal dependencies can be described by Granger Causality, and the inter-channel (spatial) dependencies can be described as instantaneous dependencies.\n\n\n[1] Marcinkevičs, Ričards, and Julia E. Vogt. \"Interpretable models for granger causality using self-explaining neural networks.\" arXiv preprint arXiv:2101.07600 (2021).   \n[2] Tank, Alex, et al. \"Neural granger causality.\" IEEE Transactions on Pattern Analysis and Machine Intelligence 44.8 (2021): 4267-4279.   \n[3] Lippe, Phillip, et al. \"Causal representation learning for instantaneous and temporal effects in interactive systems.\" arXiv preprint arXiv:2206.06169 (2022).", "theoretical_claims": "The paper lacks theoretical grounding or in-depth analysis to explain why the proposed method leads to performance improvements. A deeper understanding of the underlying mechanisms would clarify and strengthen the contribution.", "experimental_designs_or_analyses": "The proposed method is very similar to [4], but the authors do not consider it as baseline.\n\n[4] Zhao, L. and Shen, Y. Rethinking channel dependence for multivariate time series forecasting: Learning from leading indicators. In The Twelfth International Conference on Learning Representations, 2024.", "relation_to_broader_scientific_literature": "N.A", "essential_references_not_discussed": "Please refer to \"Methods And Evaluation Criteria\" and \"Experimental Designs Or Analyses\"", "other_strengths_and_weaknesses": "N.A.", "other_comments_or_suggestions": "N.A.", "questions_for_authors": "N.A.", "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": "t8lCtzSbhD", "reviewer": "Reviewer_us7m", "summary": "This paper introduces a novel approach that addresses the limitations of channel-independent (CI) and channel-dependent (CD) and channel-claustering (CC) strategies. The proposed TimeFilter transition from previous coarse-grained, channel-wise clustering approaches\nto a finer-grained, patch-wise partitioning strategy. Specifically, TimeFilter constructs a spatial-temporal graph based on patch-level distances by k-Nearest Neighbor methods. Moreover, a mixture-of-experts mechanism is applied to dynamically route and filter dependencies for each time patch. Adaptive Graph Learning (AGL) Module updates the time series embedding through neighborhood aggregation and performs forecasting. Extensive experiments show that TimeFilter generally outperforms other baselines, validating the effectiveness of the proposed TimeFilter framework.", "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": "Thanks for your valuable comments. Here are detailed responses to your questions:\n\n## Claims And Evidence\n**E1:** Claims of appropriate dependency modeling strategies.\n\n**R1:** We conducted ablation studies in the table below with variants: tem.-only (T), spa.-only (S), spa.-tem.-only (ST), and their combinations without filtering. The results show that no single dependency combination outperforms others across datasets. The context-aware filtering mechanism reduces prediction errors by adaptively suppressing noise. This confirms that TimeFilter’s performance results from the principled fusion of complementary dependencies, not isolated components. We will expand this analysis in revision.\n||TimeFilter|T|S|ST|T&S|T&ST|S&ST|T&S&ST|\n|-|-|-|-|-|-|-|-|-|\n|Weather|**0.239**/**0.269**|0.242/0.272|0.241/0.272|0.241/0.271|0.241/0.271|0.241/0.271|0.241/0.272|0.243/0.274|\n|ECL|**0.158**/**0.256**|0.168/0.265|0.170/0.269|0.162/0.260|0.171/0.271|0.162/0.260|0.163/0.261|0.166/0.264|\n\n**E2:** Irrelevant dependency.\n\n**R2:** \"Irrelevant dependencies\" refer to transient interactions caused by non-stationary patterns (extreme values, missing records, or noise) that lead to unstable statistical associations. As shown in https://i.postimg.cc/Jn42zB0b/rebuttal-fig.png, TimeFilter captures finer dependencies, distinguishing irrelevant ones. In contrast, PatchTST focuses on mutation points, and iTransformer spreads attention too broadly. This evidence shows how unfiltered dependencies propagate irrelevant ones, while our filtering mechanism preserves crucial context-aware relationships.\n\n## Methods And Evaluation Criteria\n**M1:** Discussions exploring real-world scenarios.\n\n**R1:** We provide real-world examples based on intuition: user behavior with low inter-channel correlation benefits from tem. dependencies, while spa. dependencies are key for real-time monitoring. In traffic flow prediction, spa.-tem. dependencies capture dynamic interactions across locations. For more details, refer to Section 4.2 (Routing Network, lines 205-233).\n\n## Weaknesses\n**W1:** Parameter Sensitivity.\n\n**R1:** The number of nodes $n$ is set as $n = C \\times \\lceil L/P \\rceil$, where $P$ is the patch length. The result in table below shows that performance varies with $P$, as different patch lengths capture distinct temporal semantics based on the dataset's characteristics. The threshold $p$ influences the number of favored dependencies. Except for $p = 1.0$ (full retention), the model already removes irrelevant dependencies, making it less sensitive to $p$.\n\n||TimeFilter($p=0.5$)|$P=96$|$P=48$|$P=32$|$P=16$|$p=0.3$|$p=0.9$|\n|-|-|-|-|-|-|-|-|\n|Weather|**0.239**/**0.269**|0.242/0.271|**0.239**/**0.269**|0.243/0.272|0.245/0.275|0.241/0.272|0.240/0.270|\n|ECL|**0.158**/**0.256**|0.163/0.259|0.166/0.263|**0.158**/**0.256**|0.160/0.258|0.162/0.258|0.159/0.257|\n|Traffic|**0.407**/**0.268**|**0.407**/**0.268**|0.427/0.278|0.433/0.283|0.430/0.281|0.415/0.276|0.413/0.272|\n\n**W2:** Paradigm Transformation.\n\n**R2:** Thank you for recognizing the novelty of our work. We’ll change \"Paradigm Transformation\" to \"Novel Paradigm\" and acknowledge prior channel strategies and GNNs.\n\n## Questions\n**Q1&Q3:** Complexity.\n\n**R1:** We compare the theoretical complexity of TimeFilter with other Transformer-based models in the table below. $C$ is the number of channels, $L$ is the input length, and $P$ is the patch length.\n|TimeFilter|iTransformer|PatchTST|Crossformer|\n|-|-|-|-|\n|$O(C^2·(\\frac{L}{P})^2)$|$O(C^2)$|$O(C·(\\frac{L}{P})^2)$|$O(\\frac{C}{P^2}·L^2)$|\n\nThe graph construction and filtering modules have complexity $O((\\frac{CL}{P})^2)$, and the GNN module is $O(CL)$. However, theoretical complexity alone doesn't fully capture real-world performance, as shown by testing on a NVIDIA A100 GPU with training (1 epoch) and inference times averaged over 5 runs:\n|||TimeFilter|iTransformer|PatchTST|Crossformer|\n|-|-|-|-|-|-|\n|Weather ($C=21$)|Training|19s|23s|41s|74s|\n||Inference|4s|6s|9s|11s|\n|Electricity ($C=321$)|Training|91s|74s|280s|248s|\n||Inference|14s|11s|36s|34s|\n|Traffic ($C=862$)|Training|77s|101s|352s|360s|\n||Inference|14s|18s|66s|81s|\n\nDespite higher theoretical complexity, TimeFilter outperforms PatchTST in training and inference speed. This efficiency gain likely comes from structured sparsity in filtering, where einsum-based operations produce sparse matrices, reducing FLOPs and memory overhead.\n\n**Q2:** Is $m=2$ sufficient for modeling?\n\n**R2:** The value of $m$ is not a fixed hyperparameter. It represents the number of filter types, determined dynamically by the confidence threshold ($p$). Typically, one or two filters exceed $p$, so the dynamic routing mechanism selects one or two dependencies. When all three filters are active, it indicates a need for full channel dependency structures, though only a few patches require all dependencies.\n\nThank you again for your careful review and constructive suggestions, which have inspired us to improve our paper further.", "claims_and_evidence": "-  The paper claims appropriate dependency modeling strategies, However, there is no direct quantitative breakdown of how much each component of the constructed dependency graphs contributes to the performance improvement. There is a noticeable lack of ablation studies on the importance of temporal,  spatial, and spatial-temporal subgraphs. \n- The paper shows the impact of the filtering component. However, the definition of \"irrelevant\" is implicit in the model's learning process. Visualizations or statistical analysis of the filtered connections could be beneficial.", "methods_and_evaluation_criteria": "The proposed TimeFilter and the decomposition of temporal, spatial, and spatial-temporal subgraphs are reasonable for cross-channel time series modeling. Discussions exploring real-world scenarios and specific use cases where these three graph types prove necessary would be beneficial.", "theoretical_claims": "There is no proof for theoretical claims in the paper.", "experimental_designs_or_analyses": "The paper uses standard benchmark datasets for time series forecasting and includes a good range of baseline methods.", "relation_to_broader_scientific_literature": "The paper clearly articulates the limitations of existing CI, CD, and CC methods and unifies these channel strategies in Figure 2. TimeFilter achieves the best trade-off between complete separated and complete collaborative channel dependencies.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Pros:\n- The motivation for dependency filtering and graph modeling for cross-channel time series forecasting is clear and reasonable.\n- The paper is well-written, clear and easy to follow.\n- Codes are available, which promotes the reproducibility of the work.\n\nCons:\n- The sensitivity of hyper-parameters (e.g., number of nodes $n$, threshold $p$) is unclear.  There is a lack of detailed ablation study on these necessary hyper-parameters.\n- The claim of \"Paradigm Transformation\" might be seen as exaggerated. While the method is somewhat novel, it builds upon existing research in time series channel strategies and graph neural networks.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- What is the computational cost of graph construction, compared with other modules in TimeFilter? \n- In Figure 4, the scores for the 3 filter for each dependency subgraph are close to 0. Does that mean $m=2$ is sufficient for TimeFilter modeling?\n-  What is the complexity of each component in the proposed TimeFilter? There is no formal analysis of the model's complexity.", "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": "VHnZsENo2X", "reviewer": "Reviewer_1eke", "summary": "This paper proposes a Patch-wise filtering modeling approach to select important dependencies and remove irrelevant noisy relationships. It integrates the benefits of CI and CD strategies and offers a more fine-grained and adaptive consideration of dynamically evolving dependencies over time compared to the CC strategy. The paper conducts extensive experiments, demonstrating significant improvements across various lookback lengths. Additionally, detailed ablation studies are provided. Overall, the paper is of good quality.", "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 insightful advice. Here are responses to your questions:\n\n## Other Strengths And Weaknesses\n\n**`W1`: Some modules exhibit limited novelty. For instance, the idea of Dynamic Expert Allocation is not first proposed in this paper.**\n\n**`R1`:** We sincerely appreciate your deep expertise in identifying this important aspect. Dynamic MOE's adaptive token-wise expert allocation to address predetermined expert selection constraints, our TimeFilter does indeed build on this previous work[1,2], but we would like to clarify that our paper does not claim this as our primary contribution. Our key innovation lies in fine-grained dependency modeling beyond CD/CI, which can be realized through various mechanisms. Dynamic MOE serves as our implementation choice, but alternative approaches like attention-gated dependency routers or hierarchical graph convolutions with adaptive receptive fields could also instantiate our core paradigm. The novelty lies in the proposed dependency disentanglement framework rather than the specific routing mechanism.\n\n[1] Harder Tasks Need More Experts: Dynamic Routing in MoE Models.\n\n[2] Dynamic Mixture of Experts: An Auto-Tuning Approach for Efficient Transformer Models.\n\n---\n**`W2`: It would be better if the authors also visualize the dependency graphs of the other baselines.**\n\n**`R2`:** We've visualized PatchTST's and iTransformer's attention on Weather dataset in https://i.postimg.cc/Jn42zB0b/rebuttal-fig.png. TimeFilter achieves more precise dependency modeling by selectively focusing on specific types of relationships, unlike PatchTST's narrow focus on mutation points and iTransformer's overly generalized attention across periods. This selective approach allows TimeFilter to better identify meaningful interactions while filtering out noise.\n\n## Other Comments Or Suggestions\n\n---\n**`C1`: Explanation of the **COMB** operation in TimeFilter.**\n\n**`R1`:** Sorry for the confusion. COMB refers to the combination of ego and neighbour representations. We implement COMB using a simple FFN, similar to GCN [1]. We'll clarify this in the revised manuscript for better understanding.\n\n[1] Semi-Supervised Classification with Graph Convolutional Networks.\n\n---\n**`C2`:** Hyper-parameters explanation.\n\n**`R2`:** In Equation 18, $\\lambda_1$ and $\\lambda_2$ are scaling factors for the loss functions. In our experiments, we set $\\lambda_1$ to 0.05 and $\\lambda_2$ to 0.005. These values ensure the three loss functions are balanced, facilitating gradient optimization.\n\n## Questions For Authors\n\n---\n**`Q1`: Why is the dependency graph of TimeFilter in Figure 2(c) not symmetric along the diagonal?**\n\n**`R1`:** The dependency modeling graph in Figure 2(c) is not symmetric because each patch has its own customized dependency relationships. Specifically, the filtering process for patch i's ego graph is independent of patch j's, and the selected filters may differ between them. This asymmetry is intuitive in real-world scenarios. For example, in financial systems, macroeconomic policy factors can directly influence price-volume factors (e.g., a positive policy change boosts market activity), but the reverse influence (e.g., price-volume changes trigger policy adjustments) is often weaker or absent. Thus, asymmetric dependencies naturally arise in such contexts.\n\n---\nOnce again, we are deeply grateful for your recognition of our work and your constructive feedback. If you have any further questions or suggestions, we would be more than happy to address them.", "claims_and_evidence": "The paper substantiates its claim of dynamic, fine-grained relationships between patches by visualizing the dependency graphs and the distribution of gating mechanisms during the modeling process.", "methods_and_evaluation_criteria": "The proposed method, TimeFilter, offers a new and better alternative to the conventional modeling approaches of CI and CD. Like many other papers, the paper also employs commonly used time series forecasting datasets such as ETT, Traffic, and so on, as well as the widely adopted error metrics of MSE and MAE.", "theoretical_claims": "There are no theoretical claims.", "experimental_designs_or_analyses": "The experimental design in this paper is more interesting compared to other studies. In addition to using fixed lengths, the authors introduced extended experiments from the perspective of the scaling law of TSF. The ablation studies compared TimeFilter with other intuitive filtering methods to demonstrate the rationality of their design. They also provide visualisations to illustrate the performance improvements brought by the proposed modules. The code is provided, too.", "relation_to_broader_scientific_literature": "In addition to TSF, other domains such as spatio-temporal prediction, speech and acoustics also involve similar fine-grained, dynamically evolving dependencies. This paper offers new insights into leveraging such dependencies.", "essential_references_not_discussed": "I think there are no essential references that have not been discussed.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The paper is well-motivated, as the challenge of effectively modeling both temporal and inter-channel relationships.\n\n2. This paper is well written. The notations are clear.\n\n3. The experiments are comprehensive, and the compared baselines are up-to-date.\n\nWeaknesses:\n\n1. Some modules exhibit limited novelty. For instance, the idea of Dynamic Expert Allocation is not first proposed in this paper.\n\n2. It would be better if the authors also visualize the dependency graphs of the other baselines.", "other_comments_or_suggestions": "1. In Equation 14, the specific operation of **COMB** in TimeFilter is not clearly explained.\n\n2. In Equation 18, the value of λ1 and λ2 are not specified.", "questions_for_authors": "Why is the dependency modeling graph of TimeFilter in Figure 2(c) not symmetric along the diagonal? Shouldn’t the relationship between patch i and patch j be filtered simultaneously in both directions, removing the influence of patch i on patch j and vice versa?", "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": {"ethical_review_concerns": "None"}}, {"review_id": "RCtfXufRBl", "reviewer": "Reviewer_Utah", "summary": "This paper introduces a novel dependency modeling strategy for time series forecasting, distinct from the commonly used channel independent (CI) and channel dependent approaches (CD). The proposed Patch-wise Filtration method balances CI and CD, allowing for fine-grained and dynamic modeling of spatiotemporal relationships between patches while considering computational complexity. The experimental results look convincing, and the ablation studies enumerate several alternative solutions, demonstrating the benefits of TimeFilter's filtering mechanism based on the MoE architecture.", "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 insightful advice for polishing our manuscript. We have conducted sufficient experiments and analysis to dispel your concerns. The details can be found below.\n\n## Other Strengths And Weaknesses\n\n---\n**`W1`: Alalysis of theoretical complexity.**\n\n**`R1`:** We include a detailed comparison of the theoretical complexity of our method and other SOTA models. The Tab. below summarizes the theoretical computational complexity, where C is the number of channels, L is the input lengths, and P is the patch length.\n|TimeFilter|iTransformer|PatchTST|Crossformer|\n|-|-|-|-|\n|$O(C^2·(\\frac{L}{P})^2)$|$O(C^2)$|$O(C·(\\frac{L}{P})^2)$|$O(\\frac{C}{P^2}·L^2)$|\n\nMoreover, theoretical complexity alone cannot fully capture real-world performance due to implementation differences. We tested on one NVIDIA A100 GPUs, measuring training (1 epoch) and inference times for three datasets of increasing size, with results averaged over 5 runs in tha table below.\n|||TimeFilter|iTransformer|PatchTST|Crossformer|\n|-|-|-|-|-|-|\n|Weather ($C=21$)|Training|19s|23s|41s|74s|\n||Inference|4s|6s|9s|11s|\n|Electricity ($C=321$)|Training|91s|74s|280s|248s|\n||Inference|14s|11s|36s|34s|\n|Traffic ($C=862$)|Training|77s|101s|352s|360s|\n||Inference|14s|18s|66s|81s|\n\n---\n**`W2`: Description of the hyperparameter tuning process, especially the patch length parameter.**\n\n**`R2`:** We conducted an extensive hyperparameter search covering learning rates from $5\\times 10^{-4}$ to $10^{-3}$, encoder layers from $1$ to $3$, $d_{model}$ values from $32$ to $512$, training epochs from $10$ to $30$, and patch lengths from $2$ to $96$. \n\nIn particular, the patch length $P$ has a significant impact on the results as it determines the contextual semantic information captured. We chose $P$ based on the characteristics of the dataset, experimenting with values of $L, \\frac{L}{2}, \\frac{L}{3}$, etc. This data-driven, result-oriented tuning ensures optimal performance. For transparency, we'll describe this process in detail in the appendix.\n\n---\n**`W3`: Difference with other masking (or pruning) dependencies methods.**\n\n**`R3`:** While previous work has explored dependency masking, our approach differs itself by filtering based on dependency types rather than binary causal relationships. Traditional methods often mask dependencies by assuming fixed causal relationships, which can be misleading due to spurious correlations in short-term dependencies.\nOur innovation lies in recognizing that different domains require distinct dependency types. For example: Traffic flow prediction benefits from spatial dependencies (e.g., interactions between adjacent road segments); financial markets often require filtering out short-term noise while preserving long-term trends; electricity demand forecasting relies heavily on periodic dependencies tied to daily/seasonal cycles.\nThe significant performance gains in our experiments validate the effectiveness of this type of dependency filtering and demonstrate its superiority over generic causal masking strategies.\n\n## Questions For Authors\n\n---\n**`Q1`: Details of parallel pruning.**\n\n**`R1`:** Sorry for the confusion. For each patch i, its ego graph corresponds to a row in the overall adjacency matrix M. Pruning is achieved by directly modifying M. For the three filters, we pre-generate masks. Then, by element-wise multiplication of M with the selected mask via Einstein summation, we efficiently parallelize the filtering without any additional complexity. We'll clarify this in the revised manuscript.\n\n---\n**`Q2`: Why choose a pure GNN architecture rather than attention?**\n\n**`R2`:** We appreciate this insightful technical question. The choice of a GNN-based architecture over masked self-attention is motivated by both computational efficiency and inherent advantages in modeling temporal dependencies. GNNs offer lower complexity at $O(Qm)$ where m, pruned to O(kn), is the number of edges, compared to Transformers' quadratic $O(n^2)$ complexity (Q layers, m edges, n nodes). GNNs are inherently adept at modeling instance relationships through explicit graph structures, preserving inductive bias and offering better interpretability than the often opaque attention weights. We'll ensure these advantages are clearly articulated in the revised manuscript.\n\nWe hope our response addresses your concerns.", "claims_and_evidence": "This paper asserts that the relationships between patches become effective or ineffective over time, which is difficult to capture using channel-wise modeling methods. First, this claim is intuitive. Moreover, the example illustration in the Introduction and the experimental figures in the Case Study clearly validate this assertion. Additionally, the visualization of the four modeling approaches highlights the differences between various granularity-based modeling methods.", "methods_and_evaluation_criteria": "This paper introduces a novel dependency modeling approach, offering a fresh perspective to the time series forecasting community. The usage of datasets and metrics follows the standard practices in previous works.", "theoretical_claims": "There is no theoretical claim in this paper.", "experimental_designs_or_analyses": "The experiments in this paper are comprehensive, covering both short-term and long-term forecasting tasks, ablation studies, case studies, analysis of training time and memory consumption, as well as the impact of the lookback horizon. The above results look convincing and I cannot find major flaws.", "relation_to_broader_scientific_literature": "The choice between CI and CD, and how to better model dependencies for temporal representation, is always a hot topic in time series community. The findings and implementation in this paper can be conducive to tasks such as time series forecasting, imputation, classification, and anomaly detection.", "essential_references_not_discussed": "This paper has a very good coverage of the literature.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The proposed framework is a new perspective in time series dependency modeling, which performs well in terms of both forecasting performance and efficiency.\n2. This paper conducts extensive experiments.\n3. The writing is clear and the figures are pleasing.\n4. The paper consistently presents clear intuitions and thoughtful design choices.\n5. I especially appreciate the case study in the paper, which draws a conclusion to the continuous debate between CI and CD methods in recent years.\n\nWeaknesser:\n\n1. The theoretical complexity of this method is quadratic, which does not offer the low memory footprint characteristic of linear-based methods.\n2. Though not being a deal breaker, the paper lacks a clear description of the hyperparameter tuning process, especially regarding the patch length parameter. In the appendix, the chosen patch lengths vary significantly across different datasets, raising questions about how this hyperparameter is determined.\n3. The idea of masking (or pruning) dependencies has existed before, although not as refined or effective as presented in this paper.", "other_comments_or_suggestions": "N.A.", "questions_for_authors": "1. The authors convert each patch into an ego graph and prune it, claiming that this operation can be parallelized without introducing additional complexity. However, I am curious about how exactly this parallelization is achieved, as the paper does not provide a clear explanation.\n2. Masking attention can achieve similar effects. Why did the authors opt for a pure GNN architecture instead?", "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 introduces TimeFilter, a novel patch-wise dependency modeling framework that unifies and advances beyond CI, CD, and CC strategies for time series forecasting. By constructing spatial-temporal graphs with k-NN and using a mixture-of-experts mechanism, TimeFilter dynamically filters relevant dependencies at fine granularity. Extensive experiments and ablations demonstrate good performance against multiple baselines. The rebuttal addresses most of the reviewers' concerns. The authors are encouraged to incorporate those revisions (e.g., softening novelty claims, adding complexity analysis, and adding discussion on hyperparameter tuning).
Decision: Accept (poster)