Channel Matters: Estimating Channel Influence for Multivariate Time Series
Authors: Muyao Wang
Zeke Xie
Bo Chen
Hongwei Liu
James Kwok
Venue: NeurIPS 2025 poster ()
Keywords: multivariate time series; channel-wise influence function; Data-centric AI
Primary area: deep_learning

Abstract: The influence function serves as an efficient post-hoc interpretability tool that quantifies the impact of training data modifications on model parameters, enabling enhanced model performance, improved generalization, and interpretability insights without the need for expensive retraining processes. Recently, Multivariate Time Series (MTS) analysis has become an important yet challenging task, attracting significant attention. While channel extremely matters to MTS tasks, channel-centric methods are still largely under-explored for MTS. Particularly, no previous work studied the effects of channel information of MTS in order to explore counterfactual effects between these channels and model performance. To fill this gap, we propose a novel Channel-wise Influence (ChInf) method that is the first to estimate the influence of different channels in MTS. Based on ChInf, we naturally derived two channel-wise algorithms by incorporating ChInf into classic MTS tasks. Extensive experiments demonstrate the effectiveness of ChInf and ChInf-based methods in critical MTS analysis tasks, such as MTS anomaly detection and MTS data pruning. Specifically, our ChInf-based methods rank top-1 among all methods for comparison, while previous influence functions do not perform well on MTS anomaly detection tasks and MTS data pruning problem. This fully supports the superiority and necessity of ChInf.

Channel Matters: Estimating Channel Influence for Multivariate Time Series
Muyao Wang muyaowang@stu.xidiaan www..:@diitt.duuu...om
Zeke Xie zekexie @ takust-gz.edus
Bo Chen
bchen @ meaail.xdiianeduue
Hongwei Liu hwliu xidian.edu.cn
James Kwok*
jamesk @sse.ust.hk
Intitute of Information Sensing School of Electronic Engincering, Xidian University 'Hong Kong University of Science and Technology (Guangzhou) *Hong Kong University of Science and Technology
Abstract
The influence function serves as an efficient post-hoc interpretability tool that quantifies the impact of training data modifications on model parameters, enabling enhanced model performance, improved generalization, and interpretability in- sights without the need for expensive retraining processes. Recently, Multivariate Time Series (MTS) analysis has become an important yet challenging task, tracting significant attention. While channel extremely matters to MTS tasks, channel-centrie methods are still largely under-explored for MTS. Particularly, no previous work studied the effects of channel information of MTS in order explore counterfactual effects between these channels and model performance. To fill this gap. propose novel Channel-wise Influence (ChInf) method that is the first to estimate the influence of different channels in MTS. Based on ChInf, naturally derived two channel-wise algorithms by incorporating ChInf into classic MTS tasks. Extensive experiments demonstrate the effectiveness of ChInf and ChInf-based methods in critical MTS analysis tasks, such MTS anomaly detection and MTS data pruning. Specifically, our ChInf-based methods rank top-1 among all methods for comparison, while previous influence functions do not perform well on MTS anomaly detection tasks and MTS data pruning problem. This fully supports the superiority and necessity of ChInf. Code is available https:: //t::://gtttttb httpub/.ccmmlaree2000///iiiiiia
Introduction
Multivariate time series (MTS) plays important role in wide variety of domains, including internet services I indusstial devices 2 health care 5) finance 6 and so on. Thus, MTS modeling is crucial across wide array of applications, including disease forecasting, traffic forecasting, anomaly detection, and action recognition. recent years, researchers have focused [1, [.S.. Due to the large number of different channels MTS, numerous studies aim to analyze the importance of these channels 122 16 [8). Some of them concentrate on using graph attention structure to capture the channel dependencies [22 19), while some them try to use Channel Independence to enhance the generalization ability on different channels of MTS model [[7 20). While these methods try to
on deep learning-based MTS analysis methods g g 10 12
13
"Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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fully utilize channel information, they fail to quantify the specific influence of each channel on model performance.
To quantitatively understand and improve MTS from a data-centric perspective, we report that influence functions may serve a useful tool for MTS tasks. The influence function [2] is proposed to study the counterfactual effect between training data and model performance, which has been widely used in computer vision and natural language processing tasks, achieving promising However, in MTS, different channels exhibit complex correlations, making it particularly important to explore counterfactual effects between these channels and model performance. Considering that, it is essential to develop an appropriate influence function for MTS. would provide extensions like increasing model performance, improving model generalization, and offering interpretability of the interactions between data channels and MTS models.
results 12 23
24
25
26 27
To the best of our knowledge, the influence of MTS in deep learning has not been well studied. It is
nontrivial to apply the original influence function in [2] to this scenario, since different channels of MTS usually include different kinds of information and have various relationships [28] 12]. which
are important to MTS analysis. While recent work Timeinf [29] try to address temporal dependencies in time series, it overlooks the importance of channel-wise information and fails to achieve empirical success in practical MTS tasks. In principle, the previous influence functions can not distinguish the influence of different channels in MTS because they are designed for whole data sample, according to their definitions. In practice, we also observe that the previous influence functions do not support anomaly detection effectively and perform not very well on MTS data pruning task, while they may perform well on computer vision and natural language process tasks [22 24). Thus, how to estimate the influence of different channels in MTS remains critical problem.
To better understand and address the data-centric issues in MTS, we made three main contributions:
First, we propose novel Channel-wise Influence (ChInf) method that is the first to be able to characterize the data influence of different channels for MTS.
Second, based on Chinf, we naturally derived two channel-win enhancement algorithms for MTS anomaly detection and MTS data pruning tasks. This shows that ChInf can be easily applied to practical MTS tasks and methods.
Third, extensive experiments on various benchmark datasets demonstrate the superiority of ChInf and Chinf-based methods on the MTS anomaly detection and pruning tasks. Specifically, our ChInf-based methods rank top-1 among all methods, while previous influence functions do not perform well on some classic MTS anomaly detection tasks and MTS data pruning problem.
Related Work
In this section, we discuss related work.
2.1 Influence Functions
Influence functions estimate the effect of a given training example, z', on test example, z, for a pre-trained model. Specifically, the influence function estimates how removing a training ex- ample z' affects the loss on a test example z. (2 derived the aforementioned influence to be 12',z) = VoL 0) H VoL((zz 0), where He is the loss Hessian for the pre-trained model: He 1/n (zL(z; 0). evaluated the pre- pre-trained model's final parameter checkpoint. The loss Hessian is typically estimated with a random mini-batch of data. The main challenge in comput- ing influence is that it is impractical to explicitly form Ho unless the model is small, if one only considers parameters in a few layers. Tracin
27 address this problem by utilizing first-order gradi- ent approximation: TracIn 2z', 2) VoL (z':0) VoL(z; 0), which has been proved effectively in various tasks (4 22
30 31 Recent work, Timeinf 22), proposed an influence function capable of handling inherent temporal dependencies to better detect anomalies, which is literally orthogonal to our method. Moreover, it also did not touch the channel-wis analysis and failed to identify the value of channels for MTS.
23

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2.2 Multivariate Time Series
There are various types of MTS analysis tasks. We mainly focus on unsupervised anomaly detection and preliminarily explore the value of our method in MTS forecasting.
MTS Anomaly detection: MTS anomaly detection has been extensively studied, including complex deep learning models [32 9
These models are trained to forecast or reconstruct presumed normal system states and then deployed to detect anomalies in unseen test datasets. The anomaly score, defined as the magnitude of prediction or reconstruction errors, serves as an indicator of abnormality each timestamp. However, [34] have demonstrated that these methods create an illusion of progress due to flaws in the datasets (35] and evaluation metrics [36], and they provide more fair and reliable benchmark. Additionally, unlike model-centric approaches, data-centric methods, which assess anomalies through influence measures that quantify the sensitivity of the model input perturbations. Representative examples include Timelnf 29 and our proposed ChInf.
19 33
MTS Forecasting: MTS forecasting, many methods try to model the temporal dynamics and channel dependencies effectively. An important issue in MTS forecasting is how to better generalize to unseen channels with a limited number of channels [12]. This places high demands on the model architecture, as the model must capture representative information across different channels and effectively utilize this information. There are two typical state-of-the-art methods to achieve this. (12), which uses attention mechanisms capture channel correlations. The other is PatchTST 177. which enhances the model's generalization ability by sharing the same model parameters across different channels through Channel haannel-Indeeedennee strategy.
One is iTransformer
State-of oftt-oof-the-at methods in MTS forecasting and MTS anomaly detection aim to fully utilize channel- wise information. Unlike previous model-centric approaches, propose data-centric method that leverages whin influence information to enhance training data analysis and address MTS downstream tasks.
3 Channel-wise Influence
In this section, we formulate channel-wise influence with theoretical analysis, a novel and promising tool for MTS.
The influence function 21 requires the inversion of a Hessian matrix, which is quadratic in the number of model parameters. Fortunately, the original influence function be accelerated and approximated by TracIn (27] effectively, which decomposes the difference between the loss of the test point at the end of training versus at the beginning of training along the path taken by the training process. The specific definition can be derived as follows:
TracIn (z',=) = L (z; 0) L(z;0') Rs (oL: VoI(z;0)
(1)
where z' is the studied training example, z is the testing example, 0 is the well-trained model parameter without z², 0' is the updated parameter after training with z', L(-) is the loss function, and y is the learning rate during the training process. This formulation approximates the influence of z'
by the inner product of the gradients of the training and test losses, scaled by 17
However, in the MTS analysis, the data sample x' are MTS, which means TracIn can only calculate the whole influence all channels. In other words, it fails to characterize the difference between different channels. To fill this gap we derive new ChInf, using a derivation method similar to TracIn. Thus, we obtain Theorem 3.1 which formulates the ChInf, and the proof can be found in Appendix B
Theorem 3.1. wChe Influence Function) Assuming the c Cj is the i-th channel and channel from the data sample z', respectively, 0 is the well-trained parameter of the model without z', L(-) the loss function and 17 is the learning rate during the training process. The channel-wise influence under the first-order approximation can be formulated as follows:
TracIn (z',z) = ====)=E] VoL (cl::) (e;(),;))
(2)
Given the result, define channel-wise influence matrix Meinf as follows:
McInf |aiJINxN
3

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where each element asjj is defined as:
aijj : : (c;;
VoL (cz; 0).
According to the theorem 3.1 the TracIn can be treated sum of these elements in the channel-wit influence matrix Monn falling to utilize the channel-wise information in the matrix specifically. Thus, the final ChInf can be defined as follows:
CIF (c).,)) == (c: VoL(e;;;
where ci c, is the i-th channel and j-th channel from the data sample z, z' respectively. This Chinf describes the influence between different channels among MTS.
Remark 3.2. (Characteristics of Channel-wise Influence Matrix) The channel-wise influence matrix reflects the relationships between different channels in a specific model. Specifically, each element dt,j in the matrix MCInf represents how much training with channel i helps reduce the loss for channel j. which means similar channels usually have high influence score. Each model has its unique channel influence matrix, reflecting the model's way of utilizing channel information in MTS. Therefore, we can use Mctnf f for post-hoe interpretable analysis of the model.
4 Methodology and Application
In this section, we discuss how to incorporate ChInf into classic MTS tasks, including anomaly detection and forecasting, naturally resulting in two Chinf-based methods. As channel matters to MTS tasks, we proposed novel channel pruning challenge, which aims at achieve satisfying performance with less data channels and computational costs.
4.1 Multivariate Time Series Anomaly Detection
Defining the training MTS as x = ( 22...., ZT): where T is the duration of and the observation at time æ; R.V is N dimensional vector where N denotes the number of chamnels, thus æ E R ,The training data only contains non-anomalous timestep. The test set, xe (ar1, 2222 ...y ) contains both normal and anomalous timestamps and y = [v' y -... yrl {0, 1} represents their labels, where y = 0 denotes a normal and yí 1 an anomalous timestamp Then the task of anomaly detection to select a function fe X + R such that fo( = ye estimates the anomaly score. When it is larger than the threshold, the data is predicted anomaly.
Problem Definition:
Relationship between self-influence and anomaly score: According the conclusion in Section 4.1 of (27), influence can be an effective way to detect the anomaly sample. Specifically, the idea is to measure self-influence, i.e., the influence of training point on its own loss, i.e., the training point z' and the test point z in TracIn are identical. From an intuitive perspective, self-influence reflects how much a model can reduce the loss during testing by training on sample z' itself. Therefore, anomalous samples, due to their distribution being inconsistent with normal training data, tend to reduce more loss, resulting in greater self-influence. Therefore, when we sort test examples by decreasing self-influence, an effective influence computation method would tend to rank anomaly samples at the beginning of the ranking.
Apply in MTS anomaly detection: Based on these premises, we propose to derive an anomaly score based the Chinf in Section for MTS. Consider test sample x' for which wish to assess whether it is an anomaly. We can compute the channel-wise influence matrix Mc at first and then get the diagonal elements of the Mcna Since, according to the Remark [3.2 the nature of self-influence, the diagonal elements reflect the Chinf, it is an effective method to reflect the anomaly level of each channel. Consistent with previous anomaly detection methods, we use the maximum anomaly score across different channels as the anomaly score of MTS x' at time t as:
indicate the anomaly score of each channel.
Score (w() == mar(n (c;:0) VoL (c::
where c₁ is the i-th channel of the MTS sample æ/. Ø is the trained parameter of the model, and is the learning rate during the training process. To ensure a fair comparison, we adopt the same anomaly score normalization and threshold selection strategy Details regarding this methodology can be found in Appendix C The comprehensive process for MTS anomaly detection is further claborated in Algorithm
outlined in [3 for detecting anomalies.

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Algorithm Chinf based anomaly detection
Algorithm 2 Chinf based MTTS channel pruning
Require:
test dataset Diose: a well-trained network 0; loss function L()): threshold / empty anomaly score dictionary + ADscorefl; empty prediction dictionary + ADPrediet]] for x DDeat do
Require: val dataset mati; a well-trained network loss function L(); sample interval
empty channel set D >> 0 empty channel score dictionary - [Score[]
for æ E Dent do
ADscore [a] max(1 (c::)) VoL(c; end for
for c; E de CScore += (Vollcc) Vo
end for end for Sort(C
Normalize ADScore([] Score normalization.
if A Dscore [a)] then
I/ Sort influence scores
ADP Predict [ar]
I/ Anomaly sample.
for i to N do
if mod Add c, to 2 end if
then
else
ADPredict [xc
// Normal sample.
end if
return detection result ADPredict [-].
end for
return pruned channel set D.
4.2 Multivariate Time Series Data Pruning
Dataset Pruning: : Dataset pruning remains a critical research focus, particularly as demonstrated by 337's findings showing its potential to overcome scaling law limitations through strategic dataset reduction while maintaining model performance. While the technique has been successfully applied to image [23] 22 and text (38|3) datasets, its implementation in MTS analysis remains underdeveloped, presenting both technical challenges and untapped opportunities for efficiency optimization.
iTransformer [12 demonstrates that training on part of channels enables effective full-channel predictions in MTS, indicating that there exists redundancy between MTS channels. Considering the importance and the necessity of dataset pruning (37] the excellent performance of the influence function in dataset pruning tasks [23] 22 and the discovery of iTrrasformer, we propose new algorithm suitable for MTS named channel pruning to achieve the purpose of dataset pruning. With the help channel pruning, we can accurately identify the subset of channels that are most representative for the model's ''s training without retraining the model, resulting in MTS dataset pruning.
Motivation:
Goal of Channel Pruning: Given an MTS x (cI....,cy' cv}.y = .c., dy containing N channels where c, E RTx x is the input space and is the label space. Channel pruning aims to identify a set of representative channels from æ as few as possible to reduce the training cost and find the relationship between model and channels. The identified representative subset. D (c1.... ..., em 3 and D C D, where D is the full dataset, should have a maximal impact on the learned model, i.c. the test performances of the models learned the training sets before and after pruning should be close, as described below:
(5) where P(D) is the channel distribution, L(-) is the loss function, and θ and 6, are the em- pirical risk minimizers on the training set D before and after pruning D, respectively, i... arg minace + EcGE L (c,, (C,,0) 0) and 03- = arg minace 1 m Eese Eeeb (c,
P(p) ~ (EEPP(D) (c,
Apply in channel pruning: Considering the channel-wise data pruning problem, our proposed Chinf method can effectively address this issue. According to the Re.ark(3.2] our approach can use Metaf to represent the characteristics of each channel by calculating the influence of different channels. Then, we use a concise approach to obtain representative subset of channels. Specifically, we can rank the diagonal elements of MCInf, i.c., the ChInf, and select the subset of channels at regular intervals for a certain model. Since similar channels have a similar self-influence, we can adopt regular sampling on the original channel set D based on the ChInf to acquire a representative subset of channels DD for a certain model and dataset, which is typically much smaller than the original dataset. The detailed process of channel pruning is shown in Algorithm Consequently, we can train or fine-tune the model with a limited set of data efficiently. Additionally, can serve as an explainable method to reflect the channel-modelin ability of different approaches. Specifically, the
5

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smaller the size of the representative subset D for method, the fewer channels' information it uses for predictions, and vice versa. Thus, good MTS modeling method should have large size of D.
5 Empirical Analysis
In this section, we mainly discuss the performance of our method in MTS anomaly detection and explore the value and feasibility of our method in MTS data pruning tasks. All the datasets used in our experiments are real-world and open-source MTS datasets.
5.1
Mutivariate Time Series Anomaly Detection
5.1.1 Baselines and Experimental Settings
We conduct model comparisons across five widely-used anomaly detection datasets: SMD(40, MSL 41] SMAP [1] SWaT [42,, and WADI [9]. Given the point-adjustment evaluation metric is proved unreasonable 36 we use the standard precision, recall and Fl score to measure the performance,
aligning with (34 Moreover, due to the flaws in the previous methods, (34) provides more fair benchmark, including many simple but effective methods, such as GCN-LSTM, PCA ERROR and on, labeled Simple baseline in the Table Thus, for fair comparison, we use the same data preprocessing as described in [34] and use the results cited from their paper or reproduced with their code strong baselines. Considering iTransformer [12] can capture the channel dependencies with attention block adaptively and the good performance of Timeinf in anomaly detection, we also add them as new baselines.
Table 1: Experimental results for SWaT, SMD, MSL, SMAP, and WADI datasets. The bold and
underlined marks are the best and second best value. F1: the standard Fl score; P: Precision; R: Recall. (For SMD, MSL, and SMAP, the Precision, Recall, and Fl scores are computed for each trace and then averaged to obtain the final results.) Higher values indicate better performance.
Dasaeets
Method
DAGMM 13
77.0
99.1
63.0
43.5
$6.4
49.7
33.3
39.5
56.0
38.4
40.1
59.6
27.9
99.3
16.2
OmniiAnomuis
10
77.3
99.0
63.4
41.5
56.6
46.4
35.1
37.2
62.5
38.7
40.7
61.5
28.1
100
16.3
USAD 24
TranAD
80.0
99.0
67.1
45.7
57.9
48.1
35.8
37.8
52.5
38.1
40.1
59.7
34.0
29:3
40.4
Anumally Transormerr
33
76.5
94.3
64.3
42.6
41.9
$2.8
31.1
42.3
60.4
53.8
31.3
59.8
20.9
12.3
74.3
PCA ERROR (Simple hoseline)
83.3
96.5
73.3
57.2
58.4
39.2
43.4
65.5
39.6
63.5
88.4
35.0
I-Layer MLP (Simple haseline)
77.1
51.4
Single block MLPMixer (Simple baseline)
78.0
85.4
71.8
51.2
60.8
55.4
36.3
45.1
61.2
39.7
34.1
62.8
27.5
86.1
16.3
Single Transfommrr block (Simple baseline)
78.7
86.8
72.0
48.9
58.9
53.6
36.6
42.4
62.9
40.2
42.7
56.9
28.9
90.8
17.2
L-Layer CCN-LSTMM (Simple buseline)
82.9
98.2
71.8
55.0
62.7
59.9
12.6
46.9
61.6
46.3
45.6
58.2
13.9
74.1
31.1
Using Chial (Ouns)
58.8
Inverted Tranformer
$3.7
96.3
74.1
55.9
65.0
57.0
39.6
49.7
60.8
45.5
44.8
66.6
18.8
61.3
39.1
Using Chlaf (Ouns)
84.0
96.4
74.4
59.1
63.6
63.8
46.3
52.9
61.3
46.1
41.9
68.4
50.5
58.7
44.2
Timeinf 20
79.0
91.7
69.4
54.1
64.4
61.2
35.1
45.6
65.9
39.7
41.7
64.5
27.1
89.4
15.6
5.1.2 Main Results
In this experiment, we compare our ChInf method with other model-centric methods. Apparently, Table [ showcases the superior performance of our method, achieving the highest F1 score among the previous state-of-the-art (SOTA) methods. The above results demonstrate the effectiveness of our Chinf and Chinf-based anomaly detection method. Specifically, the use of model gradient information in self-influence highlights that the gradient information across different layers of the model enables the identification of anomalous information, contributing to good performance in anomaly detection.
5.1.3 Additional Analysis
In this section, we conduct several experiments to validate the effectiveness of the Chinf and explore its characteristics.
Ablation Study: In our method, the most important part the design of ChInf and replacing the reconstructed or predicted error with our ChInf to detect the anomalies. We conduct ablation studies

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6.8
0.8
0.7
Prees Influence
Mrea rours
0.7
CCN com
chem Influence
0.50
0.5
0.6
0.48
0.5
0.5
0.4
0
0.46
0.3
0.3
0.44
0.2
0.2
WADI
0.42
0.0
SMAP
SMD
0.0
SHAP
SMD
0.40
Dataset
Dataset
Layer
(a)
(b)
Figure 1: (a)-(b): The ablation study of Chinf for iTTrasffommer and GCN-LSTM SMAP and SMD dataset. Our Chinf can enhance MTS performance, while the conventional influence fails. (c): The relationship between the number of parameters used to calculate influence and the anomaly detection performance on different datasets.
on different datasets and models. Fig la and Fig [ show that the ChInf is better than the original TracIn [27] and the original TracIn is worse than the reconstructed error. is because that the original TracIn fails to distinguish which channel is abnormal more specifically. Additionally, both figures demonstrate that our method achieves strong performance across different models, underscoring the effectiveness and generalization capability of our data-centric approach. Given the superiority of Chinf over the TracIn, the design of dedicated ChInf becomes essential.
Trade-off Analysis: According to the formula Eq. 3 we need to compute the model's gradient. Considering computational efficiency, we use the gradients of subset of the model's parameters to calculate influence. Therefore, we tested the relationship between the number of parameters used and the anomaly detection performance, with the results shown in Fig. Ic Specifically, we use the GCN-LSTM model as an example, which has an MLP decoder contaiming two linear layers, each with weight and bias parameters. Therefore, we can use these four layers to test the effect of the number of parameters used. The results in Fig. Ic indicate that our method is not sensitive to the choice of parameters. Hence, using only the gradients of the last layer of the network is sufficient to achieve excellent performance in approximating the influence, which aligns with the conclusions 22 24 277. In addition, we also test the inference time to show the complexity of our method in Appendix
D.5
Model architecture generalization: To further demonstrate the generalizability of our method, we applied our ChInf to vari- ous model architectures and presented the results in Table 2, bold marks indicate the best results. As clearly shown in the table, our method consistently exhibited superior performance across different model archi- tectures. Therefore, we can conclude that our method is suitable for different types of models, proving that it is a qualified data- centric approach. Full results of the analy- sis are Table (Appendix).
Table 2: Generalization ability of our method with dif- ferent model architectures. Bold indicates best perfor- mance.
Method
MI MixeMiixr
Transformer
Dataset
F1
R
F1
P
R
SMD
Recon Chinf
51.2
60.8
55.4
48.9
58.9
53.6
55.5
58.3
52.1
62.9
58.2
SMAP
Recon Chinf
36.3
45.1
61.2
36.6
42.4
62.9
Recon
39.7
34.1
62.8
40.2
42.7
56.9
MSL
Chinf
46.2
44.6
57.1
47.7
42.8
64.9
Visualization of Anomaly Score: To highlight the differences between our ChInf method and traditional reconstruction-batt based methods, We visualized the anomaly scores obtained from the SMAP dataset. Apparently, as indicated by the red box in Fig. iin the Appendix, the reconstruction error fails to fully capture the anomalies, making it difficult to distinguish some normal samples from anomalies, as their anomaly scores are similar to the threshold. The results show that our method can detect true anomalies more accurately compared to reconstruction-ban methods, demonstrating the advantage of ChInf.
7

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Table 3: Channel pruning experimental results for Electricity, Solar Energy, and Traffic datasets. We use the MSE metric to reflect the performance of different methods. The bold marks are the best. The predicted length is 96. The red markers indicate the proportion of channels that can achieve satisfying performance with Chinf-based selection.
Datase
Solar
Tradtic
Proportion vaniables retained
10%
20%
30%
15%
21/6
30%
5%%
1.%
Comtiaauu selection
0.188
0.176
0.228
012115
IIM selection
0.185
0.173
0.151
0.244
0.231
0.214
0.311
0.452
0.436
0.414
0.408
TTrnnfommer
SILAP selection
0.193
0.173
0.171
0.156
0.151
0.248
0.228
0.220
0.217
0.212
0.449
0.423
0.416
0.410
0.405
NIS selection
0.201
0.185
0.180
0.177
0.167
0.250
0.248
0.227
0.222
0.214
0.428
0.408
0.402
0.399
0.397
Iufluence solection
0.187
0.174
0.170
0.165
0.150
0.229
0.224
0.220
0.219
0.210
1.419
0.405
0.398
0.397
0.395
Full variates
0.148
0.206
0.395
Proportion variahles assined
592
10%
15%
2066
4.%
5%
10%
15%
20%
20%
56
1098
1.5%
20%
209
Cominnous selectoon
0.304
0.222
0.205
0.200
0.203
0.250
0.244
0.340
0.230
0.330
0.501
0.478
0.47
0.476
0.476
LIME selettion
0.210
0.196
0.199
0.190
0.150
0.245
00.230
0.228
0.226
0.226
[1477
0.478
0.467
0.461
0.461
PachTST
STIAP selection
0.215
0.211
0.200
0.195
0.186
0.259
0.216
0:229
0.227
0.227
0.505
0.487
0.482
0.480
0.480
NFS selection
0.222
0.199
0.195
02.192
111166
0.240
00.211
012228
11.236
0.226
114994
0.474
0.465
0.460
0.450
Influence selection
0.205
0.191
0.190
0.186
0.176
0.228
0.226
0,226
0.223
0.323
0.483
0.470
0.456
0.452
0.452
Full variates
0.176
01223
0.452
5.2
Multivariate Time Series Data Pruning
5.2.1 Channel Pruning Experiment
To demonstrate the effectiveness of our method, we designed a channel pruning experiment similar to dataset pruning [22] in MTS forecasting task. In this experiment, we used three benchmark datasets with a large number of channels for testing: Electricity with 321 channels, EnleryE with 137 channels, and Traffic with 821 channels. According to 5 the specific aim of the experiment was to determine how to retain only M% of the channels while maximizing the model's generalization ability across all channels. In addition to our proposed method, with some naive baseline methods, including training with the first M% of the channels, Lime selection 45,, SHAP selection [46], and NES channel selection method [7]. M is changed to demonstrate the channel-pruning ability of these methods.
Set Up:
compared it
The bold mark results in the 3 indicate that, when retaining the same proportion of channels, our method outperforms other methods in most cases. Besides, the red mark results in the table also show that our method can maintain the original prediction performance while using no more than half of the channels, outperforming other baseline methods. Although LIME and SHAP can achieve comparable performance to our method under certain settings, approach offers several distinct advantages. These include higher computational efficiency, the ability to perform not only post-hoc analysis but also anomaly detection, and the capability to analyze inter-channel correlations. For a detailed discussion, please refer to Appendix [P.) D.3] To further prove the effectiveness of method, we test our method more datasets and models in the Appendix [D.3] Additionally, considering our selection strategy is different from conventional wisdom, such as selecting the most influential samples, add comparison experiments in Appendix D.2 to prove the effectiveness of our method.
Results Analysis:
Interpretable analysis: In addition to the superior performance shown in the Table 3 the size of the representative subset D, mentioned in Table can be served as a metric to measure the perfor- mance of an MTS model. Specifically, it is evident that iTransformer [1] has a larger representative subset than PST [17,, which means iTrans- former can utilize different kinds of channel infor- mation more effectively. This explains why iTTans- former has a better predictive performance.
Table 4: The size of representative sets of dif- ferent models on different datasets, summarized from Table
Model
ECL
Solar
Traffic
Trannformmr
50%
50%
30%
PaichTST
45%
20%
20%
Based on the results in Table 4 the iTransformer has larger representative subset, we believe that in addition to using the Chinf for channel pruning to improve the efficiency of model training and fine-tuning, another important application is its use as a post-hoe interpretable method to evaluate model's s quality. As our experimental results demonstrate, a good model should be able to fully utilize the information between different channels. Therefore, to achieve the original performance, such method would require retaining higher proportion of channels.
Outlook:
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5.2.2 Case Study and Interpretability Analysis
Chomal
Choraal
Figure 2: Visual illustration of different channels on ETTh dataset.
To further investigate whether the channels ranked by Chinf correspond to Comain-criticnn variables, we conduct qualitative case study on the ETTh/ dataset, which contains seven channels with explicit physical meanings: HUFL(O) (High Utilization Factor Load), HULL(1) (High Utilization Low Load), MUFL (Medium Utilization Factor Load), (U) (Medium Utilization Low Load), LUFL (4) (Low Utilization Factor Load), (ULL(5) (Low Utilization Low Load), and DT (6) (Oil Temperature).
According to our experiment, ChInf-based pruning yields forecasting performance comparable to using all seven channels. Specifically, the top-ranked channels selected by ChInf are HULL (1), MUFL (2), LUFL (4), and OT (6).
We further analyze the cross-channel dependencies using visualization baasea analysis as shown in Fig
The temporal patterns indicate that:
HUFL (0) and MUFL (2) exhibit similar temporal dynamics.
HULL (1) and MULL (3) exhibit similar temporal dynamics.
LUFL (4) and LULL (5) show relatively weak similarity.
OT (6) is largely independent the other load-related channels.
These results demonstrate that ChInf successfully identifies representative and diverse channels that capture distinct temporal behaviors while preserving cross-channel coverage. The selected subset (1, 2, 4, 6) aligns well with known domain structure, indicating that the ChInf rankings reflect meaningful channel importance consistent with real-world physical interpretation.
5.2.3 Sample Pruning versus Channel Pruning
Set up: To further demonstrate the superiority and the necessity of channel pruning. we conducted comparative experiment between sample-wise data pruning and channel pruning. Specifically, we reduced the training data size using two pruning strategies: for sample pruning, we applied MoSo [ effective sample pruning approach, alongside random sample pruning, which involved randomly selecting data samples. For channel pruning, we utilized our ChInf. We compared each pruning method the same remaining ratio. For example, when the horizontal axis in Fig. indicates that 50% is retained, it means the size of the entire dataset is reduced to half of its original size. the case of sample pruning. half of the training samples will be discarded; whereas in channel pruning, half of the channels will be discarded.
As shown in Fig. our channel pruning method achieved better performance while retaining the same proportion of data on all settings. This suggests that channel pruning is a more suitable method for reducing MTS data size. Additionally, we previously highlighted the value of channel pruning as a post-hoc method for analyzing MTS models. Therefore, we believe that channel pruning holds greater exploratory value in MTS.
Result Analysis:
Furthermore, we found that the performance of the MoSo-based pruning method was not as effective as that of the random pruning method. We believe this may be due to the traditional influence method underlying MoSo, which assumes that each data sample is calculated in isolation. However, the samples in time series forecasting usually have strong dependencies, thus resulting in the failure of the MoSo method. Therefore, we consider designing an effective dataset pruning method specifically for time series forecasting to be noteworthy problem.

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Channel Prusing
Charnel-Pnniag
DuEa Frusieg
Made Data-Praning
0.8
Pharme
0.7
0.200
0.00
Prrmnng
0.255
0.20
0.225
6.05
0.1
0.35 0.2
0.5
0.45
0.1
8.15
0.2 0.5
0.05
0.2
0.35
0.2
0.3
(a)
(b)
(C)
Figure 3: (a)-(c): The comparison experiment between sample pruning and channel pruning on three datasets. From left to right are the Electricity dataset, the Solar Energy dataset, and the Traffic dataset. The evaluation metric used is mean squared error (MSE), with lower values indicating better performance. The horizontal axis represents the remaining ratio of the dataset.
Conclusion and Discussion
Channel matters to MTS tasks. However, channel-centric methods are still largely under-explored for MTS. In this paper, we introduce novel ChInf method, which is the first capable of estimating the influence of individual channels MTS. Unlike traditional model-centric approaches, our data- centric method demonstrates superior performance in anomaly detection and channel pruning and address the limitations of existing influence functions for MTS. Extensive experiments on real-world datasets validate the effectiveness and versatility of our approach for various MTS analysis tasks. We report that channel pruning, a novel dataset pruning method, can prune more redundant MTS data than existing sample- wise data pruning methods without notable loss of performance. We also revealed two usually overlooked issues in MTS forecasting: the issue of insufficient data utilization by current MTS models and the potential redundancy in existing datasets. In summary, method not only provides a powerful data-centric tool for MTS analysis but also offers valuable insights into the characteristics of channel information in MTS. Our work paves the way for future development of MTS methods from a data-centric perspective.
While we have successfully applied our method to two fundamental MTS tasks and demonstrated effectiveness over previous influence functions and model-centric methods, our empirical analysis mainly focused MTS anomaly detection and MTS data pruning tasks. There remains a vast landscape MTS-related tasks that are yet to be explored and understood.
Limitation:
Targets Anomalies of ChInf: Inspired by previous research [48
49 50), we discuss the types of anomalies that our method is designed target. ChInf demonstrates particular effectiveness in detecting localized, channel-wis. anomalies, where individual channels deviate from their normal interactions with others. such cases, the model parameters exhibit abnormal sensitivity specific inputs-an effect naturally captured by self-influence estimation. This property enables Chinf to identify "broken channels" or malfunctioning sensors, aligning with prior findings emphasizing the importance of modeling inter-channel dependencies. Conversely, ChInf may be less effective for globally synchronized anomalics 52] that occur uniformly across all channels, as these do not induce distinct channel-wise influence patterns. Regarding the datasets used in our experiments, many contain subset-sensor failures or transient spikes, which make them particularly suitable for evaluating ChInf's strengths.
Looking ahead, primary focus of our research will the further application of the ChInf. We believe that delving deeper into this area will yield valuable insights and contribute significantly to advancing the field. Our findings also highlight two usually overlooked issues in MTS forecasting: the issue of insufficient data utilization by current MTS models and the potential redundancy in existing datasets. These insights also suggest two wide-scope future research directions: improving model efficiency in data utilization and enhancing dataset quality.
Future Directions:
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Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 62576266 and U21B2006: in part by the Fundamental Research Funds for the Central Universities QTZX24003 and QTZX23018; in part by the 111 Project under Grant B B10039, This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region (Grant 16202523 and HKU C7004-22G). This work was supported by Guangdong Provincial Key Lab of Integrated Communication, Sensing and Computation for Ubiquitous Internet of Things (No.2023B 1212010007)
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A
Broader Impact
Our model is well-suited for multivariate time series analysis tasks, offering practical and positive impacts across various domains, including disease forecasting, traffic prediction, internet services, content delivery networks, wearable devices, and action recognition. However, we emphatically discourage its application in activities related to financial crimes or any other endeavors that could lead to negative societal consequences.
B
Proof
Proof. The proof of Theorem 3.1
TracIn (z', ,)) (z; L(z;0'
N
Σ L (cn 0) Σ L(c;00)
1=1
Σ
VL (0' 00 +0 -010)
(6)
VL(c; (V
VL (z
where the first equation is the original definition of TracIn; rectify the equation and derive the second equation, indicating the sum of the loss of each channel. The third equation calculated by the first approximation of the loss function and then we replace (6' 0) with 1) VL ()) Therefore, we can derive the final equation which demonstrates the original TracIn the channel-wiss level. The proof is complete.
C Details of Experiments
C.1 Dataset details
Anomaly detection:
Since SMD, SMAP, and MSL datasets contain traces with various lengths in both the training and test sets, we report the average length of traces and the average number of anomalies among all traces per dataset. The detailed information of the datasets can be found in Table.
Table The detailed dataset information.
Dataset
(ensors(traces)
Train
Test
Anomalies
SWaT
51
47520
44991
4589(12.2%)
WADI
127
118750
17280
1633(9.45%)
SMD
38(28)
25300
25300
1050(4.21%)
SMAP
25(54)
2555
8070
1034(12.42%)
MSL
55(27)
2159
2730
286(1
Time series forecasting:
The detailed information of these datasets can be found in the Table 6
22

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Table The detailed dataset information.
Dataset
Dim
Prediction Length
Datasize
Frequency
Electricity
321
96
(18317, 2633, 5261)
Hourly
Solar-Energe
137
96
(36601, 5161, 10417)
10min Hourly
Traffic
862
96
(12185, 1757, 3509)
C.2 Training Details
All experiments were implemented using PyTorch and conducted on a single NVIDIA GeForce RTX 3090 24GB GPU.
For anomaly detection: Models were trained using the SGD optimizer with Mean Squared Error (MSE) loss. For both of them, when trained in reconstructing mode, we used time window of size 10.
For channel pruning: Models were trained using the Adam optimizer with Mean Squared Error (MSE) loss. The input length is 96 and the predicted length is 96.
Anomaly Score Normalization Anomaly detection methods for multivariate datasets often employ normalization and smoothing techniques to address abrupt changes in prediction scores that are not accurately predicted. In this paper, we mainly use two normalization methods, mean-standard deviation and median-IQR, which aligns with (34 The details are as follows:
S. -4
Oz
The jú and the median and inter-quartile range (IQR2) across time ticks of the anomaly score values respectively.
For median-IQR:
For mean-standard deviation: The jú and 5 are the mean and standard across time ticks of the anomaly score values respectively.
For fair comparison, we select the best results of the two normalization methods the final result, which aligns with 34
Threshold Selection
Typically, the threshold which yields the best Fl score on the training or validation data is selected. This selection strategy aligns with [34]. for fair comparison.
D Additional Empirical Analysis
D.1 Visualization anomaly detection
To highlight the differences between our Chinf method and traditional reconstruction- methods, visualized the anomaly scores obtained from the SMAP dataset. Apparently, as indicated by the red box in Fig. 4 the reconstruction error fails to fully capture the anomalies, making it difficult to distinguish some normal samples from the anomalies, as their anomaly scores are similar to the threshold. The results show that our method can detect true anomalies more accurately compared to reconstruction based methods, demonstrating the advantage of ChInf.
D.2 Utilization of Chinf'
We conducted new experiments comparing different selection strategies based on ChInf. The results, shown in the table indicate that our equidistant sampling approach is more effective than selecting the most influential samples. This is because it covers a broader range of channels, allowing the model learn more general time-series patterns during training.
D.3 Generalization results
To demonstrate the generalizability our method, we applied our ChInf to various model architectures and presented the results in the following table As clearly shown in the table, method
23

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Threshold Groundtruth
04
Tim
Groundtruth
205
450
600
800
1000
Reconstruct
Figure 4: Visual illustration of the anomaly score of different methods on SMAP dataset.
Table 7: Variate generalization experimental results for Electricity, Solar Energy, and Traffic datasets. We use the MSE metric to reflect the performance of different methods. The bold marks are the best. The predicted length 96. The red markers indicate the proportion of channels that need be retained to achieve the original prediction performance.
Propection vaiiables
10%
50%
20%
30%
self infuence sumple
0.181
0.220
0.359
l'rranffmmer
0.174
0.170
0.229
0.310
0.347
0.345
Full varintes
0.148
0.205
0.385
consistently exhibited superior performance across different model architectures. Therefore, we conclude that our method is suitable for different types of models, proving that it is qualified data-centric approach.
Table 8: Full results of the generalization ability experiment.
Method
L-Layer MLP
Single block MLPMixer
Single Transformer block
Dataset
F1
P
R
Fl
P
R
F1
P
R
Reconstruct Error
51.4
59.8
51.2
60.8
48.9
58.9
53.6
SMD
Chlnf
55.9
63.1
60.6
55.5
64.8
58.3
52.1
62.9
58.2
Reconstruct Error ChInf
32.3
43.2
58.7
36.3
45.1
36.6
42.4
62.9
SMAP
47.0
54.5
60.9
48.0
57.5
58.9
48.5
54.1
64.6
Reconstruct Error ChInf
37.3
34.2
64.8
39.7
34.1
62.8
40.2
42.7
56.9
MSL
45.8
42.2
65.4
46.2
44.6
47.7
42.8
64.9
Reconstruct Error ChInf
77.1
98.1
63.5
78.0
85.4
71.8
78.7
86.8
72.0
SWAT
80.1
87.7
73.7
80.6
97.6
68.6
81.9
97.7
70.6
Reconstruct Error ChInf
26.7
83.4
15.9
27.5
86.2
16.3
28.9
90.8
17.2
WADI
44.3
84.6
30.0
46.6
83.0
32.4
47.5
71.3
35.6
D.4 Additional Dataset and Baseline results
To demonstrate the effectiveness of our approach, we validated our channel channel-pruning pruning method on new datasets. Additionally, we incorporated new baseline, DLinear, time series forecasting method based on channel- -nddeeedeed strategy. The specific results are shown below:
New dataset analysis:
Since the original number of channels in ETThl and ETTml is only 7, the horizontal axis the table directly represents the number of retained channels.

---

Table 9: The additional dataset results of the channel- experiment.
Dataset
ETTh1
ETTml
number of channels retained
7
3
2
Continuous selection
0.396
0,502
0.573
0.332
0.756
0.826
iTrrnsformee
Random selection
0.396
0.428
0.434
0.332
0.362
0.372
Influence selection
0.396
0.403
0.420
0.332
0.333
0.355
Continuous selection
0.400
0.460
0.491
0.330
0.539
0.687
PatchTST
Random selection
0.400
0,415
0.424
0.330
0.352
0.364
Influence selection
0.400
0.400
0.405
0.330
0.336
0.347
The results in the table demonstrate the effectiveness channel pruning based on the ChInf, high- lighting that PatchTST and iTTransfommer exhibit comparable utilization of channel information the ETTh1 and ETTml datasets.
New forecasting length analysis:
We have added experimental results for the prediction length of 192. The results follows:
Table 10: The 192 forecasting length of the channel-pruning experiment.
Mathed
Diteut
ECT.
Sefar
Truffi
Propertion vatatler
105
35%
50%
10%
37/5
1075
555
50%
ITansforme
Contenzen adactice
0.122
0.270
0.231
I1480
Teflance stletion
0.191
0.181
0.133
0.111
0.155
0.359
0.756
0.354
0.364
0.342
0.860
0.430
0.423
0.415
0.413
Prppetiaua vasallo sctanei
40%
30%
56
2076
202
PaciTST
Coviranes seiciie
0.23
0.16
0.30
0.300
1.999
0.383
0.70
16565
0.564
0.500
0.50
0.488
0.490
0.474
0.675
sclection
0210
0.25
0.19
0.301
0.483
0434
From the results shown in the table, it can be observed that channel-pruning based on ChInf is more effective. Additionally, iTTranffommer still exhibits larger core subset, demonstrating its superior ability to model channel dependency.
New baseline analysis:
Table 11: The channel- pranningg--unigggggg experiment results of DLinear model.
Propetton of taroNis mosieed
5%
1055
15%
305
50%
100%
59
10%
195
30%
505
0005
5%
105
15%
205
30%
B.
158
0.00
N/W
0.96
0.39
0.10
0.31
0.00
6301
0699
0.65
1635
4545
0.197
0.96
0.96
0/96
0.96
030
0.30
0.00
0.0
639
0666
0545
0.65
0635
0665
The experimental results in the table show that the core channel subset of DLinear less than 5%, which highlights the limited ability of simple linear models to utilize information from different channels effectively.
D.5 Difference between Chinf and Channel Pruning Baselines
In experiment, LIME and SHAP can indeed provide similar channel-wist insights. However, Chinf offers the following unique advantages over these methods:
1. C.ooputational Efficiency: Unlike LIME and SHAP, Chinf does not require additional retraining. Once the model is trained, ChInf can computed directly without the need for perturbation- retraining (as required by LIME) an exponential number of model evaluations for different feature combinations (as required by SHAP, which has complexity of O(2") for features). This makes Chinf significantly more efficient in high-dimensional multivariate time series scenarios.
25

---

2.Beyond Post-hoc Explanations: To the best of our knowledge, LIME and SHAP serve post-hoe interpretability tools, meaning they primarily provide insights into feature importance after model training. In contrast, Chinf not only quantifies feature importance but can also be directly leveraged for practical tasks such as anomaly detection, as demonstrated our paper.
3. Chhannel correlation: While LIME and SHAP are effective interpretability tools, they fail to capture the inter-channel influence during training-a distinctive capability provided by Chinf matrix.
From the table 5 we can observe that when the number of channels is relatively small (e.g., ECL contains 321 channels), LIME and SHAP achieve results similar to ours. However, when the number of channels is large (e.g.. Traffic contains 862 channels), these methods perform worse than ours due to their limited approximation capabilitiis.
Additionally, we have provided a comparison of the time required to compute the weights for different channels in single MTS on i'Transfomme. On ECL dataset Chinf 0.071s, SHAP: 146s LIME:9s.
From the results, it evident that method is significantly faster than the other approaches.
D.6 Additional Complexity analysis results
The computational complexity of ChInf is analyzed as follows: Let n be the number of channels in multivariate time series, and let d be the number of parameters for which we compute the gradient. The complexity of computing the gradient is (d)), so the overall complexity Chinf can be expressed (nd) This indicates that the computational cost increases with both the number of channels (n) and the parameter dimensionality d.
To further illustrate the complexity of our method, added complexity analysis experiments in both time series anomaly detection and forecasting tasks. In these experiments, we measured the time required to compute the influence of all channels of single multivariate time series data sample.
Anomaly detection:
We have added an experiment measuring the time required for detection at each time point to demonstrate the complexity of approach, shown in the table below:
Table 12: The time required for our method on different time series model.
Dataset
GCN_Istm+oul
iTransformer+ours
SWAT
1.4ms
1.5ms
WADI
6.4ms
6.5ms
The results in the table indicate that our detection speed is at the millisecond level, which is acceptable for worldworld scenarios.
Channel-pruning:
By measuring the time required for calculating single-instance influence, we demonstrated how the computational time scales with the number of channels.
Table 13: The time required for channel-pruning method on different time series datasets.
ETTml
Solar-Eneryy
Electricity
traffic
T
0.0025s
0.023s
0.071s
0.18s
From the table, it can be observed that the computational complexity approximately increases linearly with the number of channels.
26


Reviews:
[{"review_id": "Hg8vkmkyNo", "reviewer": "Reviewer_EHTb", "summary": "This paper proposes Channel-wise Influence (ChInf), a novel influence function for Multivariate Time Series (MTS) data, which estimates the influence of each channel, allowing both interpretability and efficient channel pruning. The authors validate their method through experiments on anomaly detection and channel pruning, demonstrating its effectiveness, interpretability, and generality across multiple datasets and model architectures (e.g., iTransformer, PatchTST).", "questions": "1. For online anomaly detection, is it feasible to update ChInf incrementally?\n2. In the context of real-world datasets such as Electricity, is there any case study, qualitative analysis or interpretability experiment demonstrating whether the channels ranked by ChInf correspond to domain-critical variables or sensors? Do these rankings align with known domain knowledge from the respective domains?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. This paper introduces the Channel-wise Influence Function, a new method designed to assess and interpret the influence relationships among different channels in MTS data. In contrast, the traditional methods assess the impact of entire data samples on model performance. \n2. ChInf proves to be a versatile approach, achieving good performance in two MTS tasks—anomaly detection and channel pruning.\n\nWeakness:\n1. The method requires computing gradients individually for each channel, leading to a computational complexity that scales linearly with the number of channels, which may pose challenges when applied to large-scale multivariate time series data.\n2. The method is evaluated on two representative tasks, without investigating its applicability to other important MTS problems, which may limit the demonstration of its broader generalizability.", "was_revised": true, "final_justification": "The authors have addressed my concerns adequately.", "rebuttal": "Thank you for your valuable review and support. We sincerely appreciate your insightful feedback!\n\n\n**Q1** *The method requires computing gradients individually for each channel, leading to a computational complexity that scales linearly with the number of channels, which may pose challenges when applied to large-scale multivariate time series data.*\n\n**A1:** We appreciate the reviewer’s observation. We acknowledge that the computational complexity of ChInf scales linearly with the number of channels, since it requires computing per-channel gradients. However, in practice, we find that this cost is manageable for two reasons:\n(1) MTS models used in practice are typically lightweight, especially in industrial or real-time settings; (2) ChInf only requires partial gradient computation with respect to the input rather than full backpropagation.\n\nAs a result, the actual computation time per sample remains in the order of milliseconds for most benchmark datasets with dozens of channels, suggesting that the computational cost is not a major bottleneck in practice. We provide an empirical analysis in Figure 1(c) of the main paper and further report implementation details in Appendix D.6. While ChInf requires channel-wise gradient computation, our experiments show it remains practical for common MTS settings. Moreover, improving the efficiency of influence estimation—such as through approximation techniques or model-specific acceleration—remains a promising direction for future work.\n\n**Q2** *The method is evaluated on two representative tasks, without investigating its applicability to other important MTS problems, which may limit the demonstration of its broader generalizability.*\n\n**A2:** We thank the reviewer for highlighting this point.     We chose to evaluate ChInf on anomaly detection and time series forecasting because they are among the most widely studied and practically relevant tasks in multivariate time series research.     Moreover, these two tasks correspond to classical use cases of influence functions in machine learning: anomaly detection relates to self-influence, while forecasting and pruning relate to data selection and input attribution.     We believe this aligns with the foundational motivations behind influence-based analysis.\n\nThat said, ChInf is a general-purpose tool and is not limited to these tasks.     One promising direction we have begun to explore is using ChInf to reweight the training process.     Specifically, at the end of each training epoch, one could compute the ChInf matrix to assess inter-channel importance, and use it to adjust sample or channel weights in the next epoch.\n\nDue to time constraints, we could not include extensive experiments on this training-time integration in the current version, but we believe it is a valuable extension that further showcases the generality and practicality of ChInf.     We plan to include this direction in future work.\n\n**Q3：** *Questions:For online anomaly detection, is it feasible to update ChInf incrementally?*\n\n**A3:** Thank you for this interesting question. If we understand the reviewer’s suggestion correctly, it refers to updating ChInf in an online manner, where the influence scores are recomputed incrementally as the model updates over time. We believe this is indeed feasible. Since ChInf relies on gradients with respect to the current model parameters, as long as the model is continuously updated (e.g., via streaming or online learning), the self-influence scores can be recomputed at each time step using the latest weights, thereby enabling real-time or online influence tracking.\n\nWhile we have not explicitly implemented this setting in our current work, we consider it a promising extension. If we have misunderstood the question, we kindly invite the reviewer to clarify, and we would be happy to further elaborate.\n\n**Q4:** *In the context of real-world datasets such as Electricity, is there any case study, qualitative analysis or interpretability experiment demonstrating whether the channels ranked by ChInf correspond to domain-critical variables or sensors? Do these rankings align with known domain knowledge from the respective domains?*\n\n\n**A4:** We thank the reviewer for raising this valuable point regarding the alignment between ChInf-based rankings and known domain-critical variables. While the Electricity dataset contains a large number of channels, making qualitative interpretation difficult, we conducted a focused case study on the ETTh1 dataset, which contains 7 channels with clear physical meanings:\nHUFL(0) (High Utilization Factor Load), HULL(1) (High Utilization Low Load), MUFL(2)(Medium Utilization Factor Load), MULL(3)(Medium Utilization Low Load), LUFL(4)(Low Utilization Factor Load), LULL(5)(Low Utilization Low Load), and OT(6) (Oil Temperature).\n\nAccording to our experiment, ChInf-based pruning retains high forecasting accuracy when reducing the input to just 4 channels.\n\nBased on our ChInf rankings, when retaining 4 channels, our method selects HULL (1), MUFL (2), LUFL(4), and OT(6). These selected subsets yield performance comparable to using all channels.\n\nWe further analyzed the cross-channel relationships using dataset visualization experiment. The patterns show that:\n\n    HUFL (0) and MUFL (2) exhibit similar temporal dynamics.\n    HULL (1) and MULL (3) exhibit similar temporal dynamics.\n    LUFL (4) and LULL (5) present relatively weak similarity.\n    OT (6) shows little correlation with other channels.\n\nThese findings suggest that ChInf effectively identifies representative channels, retaining diversity. The selected channels (e.g., 1, 2, 4, 6) represent distinct temporal behaviors while covering correlated patterns, aligning with domain intuition.\n\nDue to rebuttal constraints, we cannot modify the appendix or include figures, but we plan to provide qualitative visualizations and a more detailed interpretability analysis in the final version. These findings suggest that ChInf rankings do reflect meaningful channel importance aligned with domain structure.\n\nFinally, we sincerely thank the reviewer’s feedback. It definitely inspires us to further improve our work and clarification for more readers. We respectfully hope that the reviewer could reevaluate our work given the responses addresing your main concerns."}, {"review_id": "924RrpDcCq", "reviewer": "Reviewer_MH5D", "summary": "This work proposes Channel-wise Influence (ChInf), a novel method that extends influence functions to multivariate time series (MTS) data by estimating the influence of individual channels. While influence functions have traditionally been used to analyze the effect of training data on model parameters without retraining, their application to channel-centric analysis in MTS tasks has been largely unexplored. To address this gap, the authors introduce ChInf and derive two ChInf-based algorithms for common MTS tasks. Experimental results show that these methods significantly outperform existing approaches in MTS anomaly detection and data pruning, highlighting the necessity and effectiveness of channel-wise influence analysis.", "questions": "1. Could you provide additional experiments on MTS anomaly detection using the UEA archives, as prior works on MTS anomaly detection typically include evaluations on these datasets?\n\n2. Can you provide more theoretical analyses on this work? For instance, the computational complexity?\n\n3. It's clear that the influence of MTS in deep learning has not been well studied, and how to estimate the influence of different channels in MTS is critical. However, what are the challenges in the process remains unclear. More discussions are required.\n\n4. The authors are suggested to add more discussions into the related work. For instance, the discussions on the MTS anomaly detection and forecasting are limited. Additional discussions on the categories of MTS anomaly detection or forecasting can enhance the readability of this work.", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. This study is interesting, which investigate the influence functions to multivariate time series (MTS) data by estimating the influence of individual channels.\n\n2. The writing and organization of this paper are good.\n\n3. The proposed method shows excellent performance.\n\nHowever, I have some concerns as follows.\n\nWeaknesses:\n\n1. Could you provide additional experiments on MTS anomaly detection using the UEA archives, as prior works on MTS anomaly detection typically include evaluations on these datasets?\n\n2. Can you provide more theoretical analyses on this work? For instance, the computational complexity?\n\n3. It's clear that the influence of MTS in deep learning has not been well studied, and how to estimate the influence of different channels in MTS is critical. However, what are the challenges in the process remains unclear. More discussions are required.\n\n4. The authors are suggested to add more discussions into the related work. For instance, the discussions on the MTS anomaly detection and forecasting are limited. Additional discussions on the categories of MTS anomaly detection or forecasting can enhance the readability of this work.", "was_revised": true, "final_justification": "The authors have addressed my concerns.", "rebuttal": "Thank you for your valuable review and support. We sincerely appreciate your insightful feedback!\n\n**Q1：** *Could you provide additional experiments on MTS anomaly detection using the UEA archives, as prior works on MTS anomaly detection typically include evaluations on these datasets?*\n\n**A1:** Thank you for your valuable comment. To the best of our knowledge, the UEA time series archive primarily consists of datasets designed for classification tasks and does not include benchmarks specifically curated for multivariate time series  anomaly detection. We carefully reviewed the archive and found no prior works that utilize UEA datasets for MTS anomaly detection.\n\nIf your comment refers instead to the UCR archive, we would like to clarify that UCR datasets are univariate.  Since our method is designed to analyze channel-wise influence in multivariate settings, univariate datasets such as those in the UCR archive fall outside the scope of our study.  We thus focus our evaluation on multivariate time series benchmarks.\n\nIf we have misunderstood your suggestion, we would greatly appreciate it if you could clarify your intent and, if possible, point us to prior work that uses UEA datasets for anomaly detection. We would be happy to include those references or consider additional experiments accordingly.\n\n**Q2:** *Can you provide more theoretical analyses on this work? For instance, the computational complexity?*\n\n**A2:** Thank you for the suggestion. We agree that theoretical insights can strengthen our contribution. We have introduced the complexity analysis in Appendix D.6. The details are as follows:\n\nChInf is based on first-order gradient approximations, and thus the per-sample complexity is linear in the number of channels N and linear in the number of parameters used for influence estimation (which can be limited to the final layer, as shown in Fig. 1c). Specifically, the complexity is: O(N⋅d)\n \nwhere d is the number of parameters chosen (e.g., final-layer parameters), and N is the number of channels. In practice, we show (Fig. 1c) that using only a small subset of parameters still yields strong performance, making ChInf highly scalable.\n \n\n**Q3:** *It's clear that the influence of MTS in deep learning has not been well studied, and how to estimate the influence of different channels in MTS is critical. However, what are the challenges in the process remains unclear. More discussions are required.*\n\n**A3:** Thank you for raising this important conceptual question. Indeed, one of the central challenges we address in this work is extending influence estimation from the conventional sample level to the channel level in multivariate time series.\n\nWhile this shift might appear straightforward at first glance, it poses a fundamental theoretical and algorithmic challenge: standard influence functions are intrinsically designed for sample-wise analysis, and cannot be directly applied to assess the impact of individual channels. This is because in most formulations, influence is computed by perturbing or removing an entire training sample — a complete MTS instance — and evaluating its effect on the test loss.\n\nIn contrast, estimating the channel-wise influence requires a rethinking of the foundational assumptions. Specifically, we must redefine what it means to \"perturb\" only a single channel, while preserving the temporal and structural integrity of the time series. This entails a new formulation that decouples the loss sensitivity with respect to each channel's contribution, which is non-trivial not only mathematically and computationally, but also conceptually.\n\nOur work provides such a formulation, along with theoretical justification and practical approximations. In this sense, the main challenge lies in conceptualizing and enabling channel-wise influence estimation as a first-class tool, something prior influence-function-based works in vision or NLP have not addressed.\n\n**Q4:** *The authors are suggested to add more discussions into the related work. For instance, the discussions on the MTS anomaly detection and forecasting are limited. Additional discussions on the categories of MTS anomaly detection or forecasting can enhance the readability of this work.*\n\n**A4:** We appreciate the reviewer’s suggestion. We will revise the related work section to include a more comprehensive discussion of MTS anomaly detection and forecasting. \n\nMTS Anomaly Detection:\nExisting MTS anomaly detection methods can be broadly categorized into two groups. The first group consists of model-centric methods, which estimate anomaly scores via prediction or reconstruction errors (e.g., TransAD,USAD, GDN, GCN-LSTM and so on). Within this group, certain models explicitly model channel dependencies. For example, GDN and GCN-LSTM use graph convolutions to represent the structural relationships among sensors and capture correlations across channels. The second group includes data-centric methods, which leverage influence measures to evaluate the impact of inputs (e.g., TimeInf, our proposed ChInf). These methods do not rely on explicit temporal or graph modeling, but instead use influence-based diagnostics to reflect how sensitive the model is to input perturbations. We emphasize that our work belongs to the latter category and is one of the first to provide channel-wise influence analysis, offering better granularity and interpretability for channel-level anomaly detection.\n\nMTS Forecasting:\nFor multivariate time series forecasting, recent studies show that the training paradigm can significantly affect model performance. There are two typical training paradiam. A number of works adopt channel-dependent models (e.g., iTransformer), which aim to capture channel dependencies by jointly modeling all channels. Others use channel-independent models (e.g., PatchTST), which make the parameter of the model shared across all channels to avoid inter-channel interference. Our proposed ChInf can be integrated into either type of forecasting model to analyze the influence of individual channels, allowing deeper insight into channel contributions regardless of the model’s training paradigm.\n\nFinally, we sincerely thank the reviewer’s feedback. It definitely inspires us to further improve our work and clarification for more readers. We respectfully hope that the reviewer could reevaluate our work given the responses addresing your main concerns."}, {"review_id": "mwcOW3sTCh", "reviewer": "Reviewer_g6Y3", "summary": "The paper introduces a method for quantifying how removing a channel from a multivariate time-series affects the loss on a another channel. This metric is applied to standard time-series tasks including anomaly detection and data pruning.", "questions": "Re-iterating my previous questions:\n- Why do simple baseline methods outperform most deep methods, and what differentiates ChInf? Even if that is mentioned in a cited work, a sentence explanation would be helpful to aid in interpretation of the results.\n- What kind of anomalies do you expect ChInf to be strong at detecting and where do you expect it to perform less favorably. Is there something about the tested datasets that indicates that they are well suited for ChInf? Are there cases where you expect the proposed method to underperform? Some additional discussion of this point would be helpful.", "limitations": "See my question regarding anomalies that may be \"missed\" by this method.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The paper is generally well written and the method seems sound albeit somewhat incremental. In the context of anomaly detection, it seems to amount to a different aggregation of per-channel influence (max instead of sum) if I understand correctly. Still the method seems useful in practice. The empirical analysis is well done and includes simple baselines that seem to outperform many competing deep models. Some discussion of this results would be helpful, along with why the proposed method is stronger. This leads me to a larger point:\n\nA common view regarding anomaly detection is that some prior knowledge is needed regarding the down-stream task in order to define what constitutes an anomaly [1,2]. This makes evaluation and comparison of anomaly detection methods challenging. I think this work could be stronger if there was at least a discussion on the *kinds* of anomalies the method targets. For instance it would seem to me that the method should be strong at detecting anomalies when individual channels are \"broken\", meaning they do not follow their usual pattern in relation to other channels (for instance, this work [3] tackles such anomalies in games by modeling correlations between channels). This might be in contrast to methods that are better at detecting when the entire time-series behaves abnormaly.\n\nI would also recommend that the authors revise some of the method explanation, for instance after eq. 1, to improve clarity.\n\n[1] Winkens, Jim, et al. \"Contrastive training for improved out-of-distribution detection.\" arXiv preprint arXiv:2007.05566 (2020).\n[2] Le Lan, Charline, and Laurent Dinh. \"Perfect density models cannot guarantee anomaly detection.\" Entropy 23.12 (2021): 1690.\n[3] Lymperopoulos, Panagiotis, Yukun Li, and Liping Liu. \"Exploiting variable correlation with masked modeling for anomaly detection in time series.\" NeurIPS 2022 Workshop on Robustness in Sequence Modeling. 2022.", "was_revised": true, "final_justification": "I have increased my score. The authors have addressed my questions and I think their revisions make their contribution more clear.", "rebuttal": "Thank you for your valuable review and support. We sincerely appreciate your insightful feedback!\n\n**Q1:** *I would also recommend that the authors revise some of the method explanation, for instance after eq. 1, to improve clarity.*\n\n**A1:** Thank you for the suggestion. We will revise the explanation after Equation (1) to better highlight the distinction between sample-level and channel-level influence computation, and clarify how the Chinf matrix term reflects the structural interactions among channels. We have revised the paragraph to enhance readability and make the derivation and intuition behind TracIn and our proposed ChnInf more accessible to readers. The revised version is as follows:\n\n> **[Modified]**  \n> \"The influence function [Koh and Liang, 2017] estimates the effect of removing a training example on the model’s performance. \nHowever, its exact computation requires inverting a Hessian matrix, which is computationally expensive. TracIn [Pruthi et al., 2020] proposes an efficient approximation by measuring the change in the loss of a test point as a model is trained with a specific training point. Formally, the influence of training sample \\( z' \\) on test sample \\( z \\) is approximated as:  \n> \\[\n> $\\text{TracIn}(z', z) = L(z; \\theta) - L(z; \\theta') \\approx \\eta \\nabla_\\theta L(z'; \\theta)^\\top \\nabla_\\theta L(z; \\theta)$\n> \\]  \n> where \\( z' \\) is a training example, \\( z \\) is a test example, \\( \\theta \\) is the model parameter before training on \\( z' \\), \\( \\theta' \\) is the updated parameter after training on \\( z' \\), and \\( \\eta \\) is the learning rate. This formulation approximates the influence of \\( z' \\) by the inner product of the gradients of the training and test losses, scaled by \\( \\eta \\).\"\n\n>However, in multivariate time series (MTS) scenarios, both $z$ and $z'$ represent MTS data. TracIn computes the overall influence of the entire input but cannot distinguish how individual channels contribute. This limitation makes it difficult to analyze channel-wise influence. To address this, we propose a new method, \\textbf{ChnInf}, that decomposes TracIn-style influence estimation into per-channel contributions. Our derivation, which follows a similar principle to TracIn, is presented in Theorem 3.1, with full proof provided in Appendix B.\n\nWe hope this revision improves the clarity of our method section. We have also ensured that similar improvements are made throughout the revised manuscript.\n\n\n\n \n\n**Q2：** *Why do simple baseline methods outperform most deep methods, and what differentiates ChInf? Even if that is mentioned in a cited work, a sentence explanation would be helpful to aid in interpretation of the results.*\n\n**A2:** We thank the reviewer for raising this important point. As noted in prior work [1], simple models often outperform deep architectures in MTS anomaly detection because normal patterns in these datasets are relatively regular and can be captured by shallow models. The added complexity of deep models provides limited marginal gain, consistent with the principle of Ockham’s Razor. Moreover, as discussed in [1], previous deep methods sometimes showed inflated performance due to flawed evaluation protocols, such as using 'point adjustment'.\n\nChInf differs fundamentally from both shallow and deep detection methods. First, ChInf is model-agnostic—it can be applied on top of any backbone model without modifying the architecture or retraining. Second, ChInf leverages gradient-based self-influence scores, which provide a model-centric perspective for anomaly detection by quantifying how much a data point (or channel) influences its own loss landscape. This diagnostic signal reflects how sensitive the model’s parameters are to perturbations in specific inputs, thus capturing subtle anomalies that may not result in large prediction errors. \n\n**Q3:** *What kind of anomalies do you expect ChInf to be strong at detecting and where do you expect it to perform less favorably. Is there something about the tested datasets that indicates that they are well suited for ChInf? Are there cases where you expect the proposed method to underperform? Some additional discussion of this point would be helpful.*\n\n**A3:** We appreciate the reviewer’s insightful question. ChInf is particularly effective at detecting localized, channel-wise anomalies, where individual channels deviate from their typical interactions with others. These cases often result in unexpected gradients, indicating that the model’s parameters are unusually sensitive to the input—a scenario naturally captured by self-influence.\n\nThis aligns with the reviewer’s suggestion regarding “broken channels”, and is corroborated by findings in the references which you list, which highlight the value of modeling channel dependencies.\n\nConversely, ChInf may be less effective in settings, where anomalies are purely temporal across all channels.\n \nAs for the datasets used in our experiments, they often contain small subset-sensor failures or spikes, making them naturally aligned with ChInf’s strengths. We will add previous discussion and cite the references you provided to the Appendix to further clarify where ChInf’s advantages lie.\n\n[1] Position Paper: Quo Vadis, Unsupervised Time Series Anomaly Detection?\n\nFinally, we sincerely thank the reviewer’s feedback. It definitely inspires us to further improve our work and clarification for more readers. We respectfully hope that the reviewer could reevaluate our work given the responses addresing your main concerns."}, {"review_id": "NkVBCpzrHX", "reviewer": "Reviewer_JiPH", "summary": "This paper proposes using influence functions, which estimate the counterfactual effect between training data and model performance, as a data-centric perspective to better understand and analyze multivariate time series (MTS). Addressing the limitation of existing influence functions, such as TimeInf, which overlook the impact of different channels in MTS, the authors introduce a new influence function called ChInf. This method is applied to two key MTS tasks: anomaly detection and data pruning. The results demonstrate that ChInf consistently performs well across various methods, validating its effectiveness.", "questions": "1. See Weakness #2 above.\n2. In Section 5.1.3, you claim that the original TracIn is ineffective because it doesn't differentiate between channels. This conclusion seems somewhat unconvincing: shouldn't a global influence function still yield meaningful results? According to your reasoning, would TracIn and ChInf perform similarly on univariate time series? Further explanation or empirical validation of this point would be helpful.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n1. Applying influence functions to analyze MTS is a novel and meaningful idea.\n2. The paper is well-structured and easy to follow, progressing logically from the introduction of influence functions, through their refinement, to the empirical validation of ChInf across multiple tasks.\n3. The experiments are thorough, demonstrating ChInf’s effectiveness from multiple perspectives and on various tasks.\n\nCons:\n1. The citation formatting is inconsistent and needs revision.\n2. There are concerns about the definition of the influence score in the two downstream tasks:\n* For example, in anomaly detection, why is the score defined as the maximum across all channels rather than the average? This choice requires further empirical justification, especially since uniformly distributed influence across channels might be overlooked.\n* In Equation (4), why is the score computed only for a channel's influence on itself? Shouldn't channel i potentially influence all channels, not just itself?", "was_revised": true, "final_justification": "I raise the score because all my concerns have been addressed, especially the definition of the final score across different channel.", "rebuttal": "Thank you for your valuable review and support. We sincerely appreciate your insightful feedback!\n\n**Q1** *The citation formatting is inconsistent and needs revision.*\n\n**A1:** Thank you for pointing this out. We apologize for the inconsistency in citation formatting and will correct all formatting issues in the final version to ensure clarity and professionalism.\n\n**Q2:** *For example, in anomaly detection, why is the score defined as the maximum across all channels rather than the average? This choice requires further empirical justification, especially since uniformly distributed influence across channels might be overlooked.*\n\n**A2:** Thank you for raising this insightful question.\n\nWe chose to use the maximum of the self-influence values across channels as the anomaly score to better capture localized, channel-specific anomalies.\n\nTo empirically justify this design choice, we highlight the following:\n\nComparison with TracIn:\nTracIn computes a global influence score by summing over all channels, effectively averaging out the individual channel effects. As a result, it cannot distinguish which specific channels contribute to the anomaly. By comparing our ChInf scores with TracIn’s aggregated results (e.g., Figure 1a/b), we demonstrate the advantage of our fine-grained formulation in capturing localized anomalies. This supports the hypothesis that averaging dilutes anomaly signals, especially when the anomaly is confined to a single or small subset of channels.\n\nAdditional datasets for validation:\nWe further validated our scoring strategy by adding two widely used MTS anomaly detection benchmarks — SWaT and MSL— in which anomalies are often confined to specific sensors. Our method continues to outperform baselines in these datasets, reinforcing that the maximum-based scoring is more effective for anomaly detection.\n\n|     Model    |  SWaT |     |  MSL  |     |\n|:------------:|:-----:|:---:|:-----:|-----|\n|              | Chinf | TracIn | Chinf | TracIn |\n| iTransformer | 84.0    |  77.1   |  46.1     |   38.3  |\n|   GCN-Lstm   |  82.9   |   76.9  |    47.1   |  40.1   |\n\nQualitative motivation:\nIn the multivariate time series datasets we evaluate on—SWaT, WADI, SMAP, and others—anomalies typically manifest in only a small subset of channels, such as a malfunctioning sensor or a corrupted reading in a single stream. In such settings, aggregating influence scores via mean or sum can dilute the anomaly signal, especially when most channels remain normal. By contrast, using the maximum self-influence across channels effectively surfaces localized anomalies that might otherwise be overshadowed in aggregate-based schemes.\n\nThis design choice is empirically supported by our results in Section 5.1.3 and Figure 1a/b, where max-based scoring outperforms global influence baselines (e.g., TracIn) that implicitly average over all channels. While mean- or sum-based scoring might be preferable in scenarios involving uniformly distributed anomalies, such patterns are less common in the evaluated benchmarks. We believe this reflects a broader trend in real-world MTS applications, where localized channel faults are more critical and frequent than global disruptions.\n\n**Q3:** *In Equation (4), why is the score computed only for a channel's influence on itself? Shouldn't channel i potentially influence all channels, not just itself?*\n\n**A3:** Thank you for this thoughtful question.\n\nOur use of self-influence in Equation (4) directly follows the idea of self-influence as introduced by Koh and Liang [1], where the influence of a data point on itself is used as a metric to assess its abnormality or correctness. This formulation has been widely adopted in anomaly detection literature[2] that leverages influence functions.\n\nIn the context of multivariate time series, this naturally extends to channel-wise self-influence: the influence of a channel's input on its own loss reflects how much that channel influence the model's behavior, hence indicating potential anomalies. Thus, channel-wise self-influence is enough to detect anomalies. We will clarify this point in the revision to avoid confusion.\n\n \n\n**Q4:** *In Section 5.1.3, you claim that the original TracIn is ineffective because it doesn't differentiate between channels. This conclusion seems somewhat unconvincing: shouldn't a global influence function still yield meaningful results? According to your reasoning, would TracIn and ChInf perform similarly on univariate time series? Further explanation or empirical validation of this point would be helpful.*\n\n**A4:** Thank you for this insightful observation.\n\nWe would like to clarify that we do not claim TracIn to be entirely ineffective, while it is less effective than the proposed Chinf. Rather, our intention is to point out that TracIn operates at the sample level and does not distinguish between individual channels. Consequently, it may overlook localized anomalies that are specific to certain channels. In comparison, per-channel reconstruction losses of the base model serve as a proxy for identifying local deviations. We acknowledge that our previous wording may have caused confusion, and we are happy to revise the text to better reflect this distinction.\n\nOn univariate time series, TracIn and ChInf reduce to the same form — we fully agree. In fact, ChInf is designed to generalize TracIn to the multivariate setting, offering better resolution in MTS scenarios.\n\nIn multivariate settings, TracIn produces one score per sample (treating all channels together), while ChInf computes a channel-wise matrix, offering significantly richer information. This is particularly helpful for tasks like channel pruning, where individual channel importance matters.\n\nEmpirically, Figure 1a and 1b show that TracIn performs worse than ChInf. This supports our claim that channel-wise resolution is essential in MTS, while TracIn is not sufficient.\n \n\n[1] Understanding Black-box Predictions via Influence Functions ICML 2017\n\n[2] Time Series Data Contribution via Influence Functions ICLR 2025\n\nFinally, we sincerely thank the reviewer’s feedback. It definitely inspires us to further improve our work and clarification for more readers. We respectfully hope that the reviewer could reevaluate our work given the responses addressing your main concerns."}]

Author rebuttal:

Meta-review: The authors introduce novel channel-wise influence functions that measure counterfactual effects. They evaluate their method for channel pruning and anomaly detection and show that the contributions clearly improve the results. 

The reviewers highlighted the novelty of the approach and that the experiments show clear advantages of introducing the channel wise formulation. Criticism included lack of clarity in certain parts and missing theoretical analysis, which the authors could partially provide during the discussion phase. All in all, the authors convinced all reviewers to converge to accept recommendations, which I intend to follow.

After checking the paper myself and discussing with the SAC, we noticed that the paper lacks detailed experimental descriptions, e.g. if and how validation splits were used, or how individual hyperparameters have been chosen. In order to enable reproduction of the work, we heavily encourage to expand the description of experimental setup for the camera ready version.
Decision: Accept (poster)