Channel Normalization for Time Series Channel Identification
Authors: Seunghan Lee
Taeyoung Park
Kibok Lee
Venue: ICML 2025 poster (2025)
Keywords: Time Series
Channel Normalization
Time Series Forecasting
Primary area: deep_learning->sequential_models_time_series
TL;DR: We propose Channel Normalization (CN), a simple yet effective normalization strategy that enhances channel-identifiability of a time series model by employing distinct affine transformation parameters for each channel.

Abstract: Channel identifiability (CID) refers to the ability to distinguish among individual channels in time series (TS) modeling. The absence of CID often results in producing identical outputs for identical inputs, disregarding channel-specific characteristics. In this paper, we highlight the importance of CID and propose Channel Normalization (CN), a simple yet effective normalization strategy that enhances CID by assigning distinct affine transformation parameters to each channel. We further extend CN in two ways: 1) Adaptive CN (ACN) dynamically adjusts parameters based on the input TS, improving adaptability in TS models, and 2) Prototypical CN (PCN) introduces a set of learnable prototypes instead of per-channel parameters, enabling applicability to datasets with unknown or varying number of channels and facilitating use in TS foundation models. We demonstrate the effectiveness of CN and its variants by applying them to various TS models, achieving significant performance gains for both non-CID and CID models. In addition, we analyze the success of our approach from an information theory perspective. Code is available at [https://github.com/seunghan96/CN](https://github.com/seunghan96/CN).

Channel Normalization for Time Series Channel Identification
Seunghan Lee 12 Taeyoung Park' *1 Kibok Lee"
Channel- uniienifinid (Non-CID")
Whannel--dentifiabll (CID)
Abstract
Channel Ta
Channel TP
Channel identifiability (CID) refers to the ability to distinguish among individual channels in time series (TS) modeling. The absence of CID often results in producing identical outputs for identi- cal inputs, disregarding channel-specific charac- teristics. In this paper, we highlight the impor- tance of CID and propose Channel Normaliza- tion (CN), a simple yet effective normalization strategy that enhances CID by assigning distinct affine transformation parameters to each chan- nel. We further extend CN in two ways: 1) Adap- tive CN (ACN) dynamically adjusts parameters based on the input TS, improving adaptability in TS models, and 2) Prototypical CN (PCN) in- troduces a set of learnable prototypes instead of per-channel parameters, enabling applicability to datasets with unknown varying number of chan- nels and facilitating use in TS foundation models. We demonstrate the effectiveness of CN and its variants by applying them to various TS mod- els, achieving significant performance gains for both non-CID and CID models. In addition, we analyze the success of our approach from an in- formation theory perspective. Code is available https: //tp:/gggtttb/t//// com/ seunghan96
"Tranafommrr Linear 2023)
Channel Nommalization
Figure I: Motivating example for channel identifiability. When two different channels receive the locally identical inputs (green), non-CID model yields the same outputs (yellow) for both, failing to distinguish between them, as shown in the left panel. In contrast, applying CN enables CID and produces distinct outputs even with the same inputs, as shown in the right panel.
Average MSE (4Hs)
ETTml
Weather
PEMS03
Imp.
iTransformer
.408
.260
.142
+ Constant vector
.397
.246
.114
6.5%
Table I: Necessity of CID. Simply adding different constant vec- tors to each channel token improves the performance. Full results and comparison with our methods are shown Appendix G
CID), while others are not (i.e., channel-unidentiffaal or non-CID), producing identical outputs for the identical input
regardless of the channel (Liu et al. 2024 Zeng et al. 2023)
Figure Dillustrates the TS forecasting results using iTrans- former (Liu et al. 2024), a widely adopted non-CID model, on a toy dataset with two channels displaying distinct pat- terns. The figure shows that the model fails on this simple task, as non-CID models lack information about channel identities, producing identical outputs (yellow) for both channels whenever given identical inputs (green). Further- more, Table [ shows that adding distinct constant vectors to each channel token, enabling the model to distinguish among channels, improves the forecasting performance. These results highlight the importance of CID in TS models.
1. Introduction
Time series (TS) forecasting is widely used in various fields,
including traffic (Cirstea et al. 2022). electricity (Dudek)
jct al.) 2021). and sales forecasting Li et al., 2022). A range of TS forecasting methods have been developed based
A naive approach to solving this issue is to use different pa- rameters for each chamnel in the tokenization layer, although this increases computational burden (Nie et al. 2024), or to add learnable vectors to each channel token (i.e., channel identifiers) (Chi al. 2024). These methods yield limited performance gains, as discussed in Section 5.3 motivating us to design a simple yet effective method to enhance CID.
on different architectures, such as Transformers (Vaswani 2017), multi-layer perceptrons (MLPs) (Rumelhart
let al.
jet al. 1986,, and state-space models (SSMs) (Gu & Dao 2023). b. Among them, some models are inherently able to distinguish among channels (i.e., channel- coammel-identffubble or
'Equal advising 'Dpparmeent of Statistics and Data Science, Yonsei University KRAFTON; work done while Yonsei Univer- sity. Correspondence to: Taeyoung Park <tpark sipark@yonsci..om yonsei.ao kr.>> Kibok Lee <kibok @ @oonsei.ac.nr.
To this end, we propose Channel Normalization (CN), simple yet effective normalization strategy designed to en- hance CID of 'S models. Unlike Layer Normalization (LN) [Ba et al. 2016) which applies shared affine transformation parameters across all channels, CN employs distinct param- eters for each channel, allowing the model to differentiate
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Channel Normalization for Time Series Channel Identification
(1) 121 (1): Channelidentifiall (2): Data-dependent
LN (base): 0.102 vs. CN: 0.087 vs. ACN: 0.085
LN (base): 0.179 vs. CN: 0.161 vs, ACN: 0.158
X
X RMLP
(ug. MSEI
Input Output
(vvg. MSE)
Input Output
X
O ITrans.
(ETTm1. Weather. PEM507) 4H
X
TSMixer
+ CN
O S-Mamba
ACN
PEMS07
ECL
(a) Model properties vs, MSE improvement (%)
(b) TS forecasting visualization (Model: iTransformer)
Figure 2: Effectiveness of CN/ACN. (a) shows that our method is effective across various backbones, where 1) non-CID models (c.g..
RMLP, iTransformer) exhibit greater improvements from CN and 2) data-indepenn -nteeedddennnet models (c.g. RMLP, TSMixer), whose parameters do not depend on the input, benefit more from transitioning from CN to ACN. (b) shows forecasting results with and without our methods.
among channels effectively. Furthermore, we introduce two variants of CN: 1) Adaptive CN (ACN), which dynamically adjusts parameters based on the input TS to improve adapt- ability, and 2) Prototypical CN (PCN), which introduces set of learnable prototypes as affine transformation param- eters after normalization to handle multiple datasets with unknown/varying number of channels using a single model, particularly useful for TS foundation models (TSFMs).
ct al. 2023) uses a linear model to capture TD, RLinear and RMLP (Li et al.) 2023) integrate reversible normalization
(RevIN) (Kim al. (02T)) to MI.ss and TSMixer (Chen)
ct al. 2023) adopts MLPs to capture both TD and CD. Re-
cently, various methods (Ahamed & Cheng 2024
Ma et al.
2024 (Zeg et al. 2024 Cai et al.
2024 have been proposed 20233. which introduces
that utilize Mamba (Gu & Dao
a selective scan mechanism to SSM to capture long-range context with linear complexity. S-Mamba (Wang et al.) 2025) and Bi-Mamba+ (Liang et al., 2024) capture CD with bidirectional Mamba, and SOR-Mamba (Lee et al.) 2024 employs a regularization strategy to effectively capture CD.
The main contributions are summarized as follows:
We propose CN to enhance CID of TS models by employ- ing channel-specific parameters, unlike Layer Normaliza- tion which uses shared parameters, offering simple and effective strategy.
Recently, several methods have emerged that enhance CID of TS models. InjectTST (Chi et al. 2024) proposes chan- nel identifier that helps Transformer differentiate among channels and C-LoRA (Nie et al., 2024) conditions a CD model on channel-specific components using a channel- aware low-rank adaptation method. Similarly, CCM (Chen) et al., 2024a) integrates channel-cluster identity model by grouping channels based on their similarities. However, these methods, aside from the channel identifier, were not primarily developed to enhance CID, and their im- pact on CID is merely a byproduct. Furthermore, they either require modifications to the architecture (Chen et al. 2024a Nie et al., 2024) or provide only limited performance gains CChi al. 2024 Nie et al.) 2024). as shown Table 6
We propose two variants of CN: 1) ACN to better capture time-varying characteristics of each channel by adapting its parameters to input TS and 2) PCN to handle multiple datasets with unknown/varying number of channels by introducing learnable prototypes where parameters are assigned to prototypes instead of channels.
TS
We provide extensive experiments on various backbones including TSFMs, achieving significant improvements for both CID and non-CID models as shown in Figure aa.)
We analyze the effect of our method from an information theory perspective, showing that it 1) enriches feature rep- resentations, 2) improves the uniqueness of each channel representation, and 3) diversifies the correlation between channel representations, supporting the performance gain.
Normalization. Various normalization methods for deep neural networks have been introduced (Ioffe 2015) Wu & He 2018 to improve convergence and training stability, differing in the dimension they normalize. Layer Normaliza- tion (LN) (Ba et al.) 2016). which uses shared affine transfor- mation parameters across channels, is commonly employed in TS backbones (Nie et al. 2023 Wang et al., 2025) to reduce inter-channel discrepancies (Liu et al. 2024). In
2. Related Works
TS forecasting models. TS forecasting in deep learning has been approached with two strategies: channel- dependent (CD) strategy, which captures dependencies among chan- nels, and each channel individually and focuses only on the temporal dependency (TD). Methods using these strategies include Transformer-based. MI.P-based, and SSM-based models.
(CI) strategy, which treats
contrast to LN, we propose assigning channel-specific pa- rameters to distinguish among channels.
For Transformer- based models, PatchTST (Nie et al. 2023) divides TS into patches and feeds them into Transformer in CI manner. iTTrnsformer (Liu et al. 2024) treats each channel as token to capture CD using the attention mecha- nism, resulting in significant performance gains. However, these models suffer from the quadratic complexity of the attention mechanism. To overcome this issue, various MLP- based models have been proposed, where DLinear (Zeng
3. Preliminaries
TS forecasting (TSF). In TSF tasks, a model predicts the future values y = (XL+1I .... XL+H) with a lookback win- dow (i.e., input TS) (x1, (X1,...; Xr.). In this setup, each x; E RC represents values at individual time steps, with L. H. and C' indicating the size of the lookback window, the forecast horizon, and the number of channels, respectively.
2

---

Channel Normalization for Time Series Channel Identification
Normalization
2. Parameters (a)
(E: Instance, L: Layer, C: Channel)
Channel sim.
Channel-proto pronnoo--roooooooo
3. Comparison
IN
e
(1)
(2)
(3)
Cr-Norm
|
ACN Global Local
LN
X
X
X
CN
LN
PCN
(1) Channel identifiable
PCN
A: Channel cluster identifiable
CN
(2) Data dependent
K prototypes
Uninownivarying Cs
Figure 3: Overall framework of CN/ACN/PNN, (1) CN employs channel-specifee parameters, enabling the model to distinguish among channels. (2) ACN extends CN by adapting its parameters to the input TS by utilizing local parameters, which are attended to with different weights based on the similarity between input channels (i.e., channel similarity). (3) PCN makes CN applicable to multiple datasets with unknown/varying number of channels by assigning parameters to each prototype instead of each channel, which are attended with different weights depending on the similarity between input channels and prototypes (i.c., channel channel-rrotorpp similarity).
Framework of TSF models. General TS forecasting mod- els follow the framework below:
4.1. Channel Normalization (CN)
Layer Normalization (LN), LN applies affine transforma- tions with parameters {ar, 33 to the normalized data as:
(Optional): x < NORMALIZE(X)
1. Token Embedding: z (x)),
Norm (2b,c,a) = Zb,,,d Pb.c
2. Encoder: z +- f(z),
(L)
Tb.C
(2)
3. Projection Layer: ý < 92(z);
20,e,d = a Norm (2b.c.a) + B.
(Optional): y <- DENORMALIZE(())
Various TS methods apply LN by using shared parameters {a, 3} across channels to reduce discrepancies among the
where z is D-dimensional vector and various normaliza- tion methods apply within f for training stability (e.g.. LN). Similarly, our method applies within f. remaining orthog- onal to techniques that normalize the input (x) and denor- malize the output (ý) to address distribution shifts (Passalis 2019 Kim et al. 202T).
channels (Liu et al., 2024 Wang et al. 2025).
Channel Normalization (CN). Unlike LN, which uses shared affine transformation parameters across channels, CN employs channel-specife parameters as follows:
ct al.
bb,ed a. Norm (Zb,e,a) + Bes
(3)
Model property 1: Channel identifiability. Let X = (f1....,xe.) E LxC be an input TS with C' channels of length L. A TSF model exhibits CID if it can distinguish among channels with identical input. That is, A CID model can produce distinct outputs x(X); + @(X), even with the same inputs 2; = Zj, whereas a non- non-CID model produces the same outputs (XX;: (x);
where a and B. denotes the parameters of the e-th channel. This simple modification enables the model to distinguish among channels, with the additional computational burden of using channel-specine parameters being minor compared to the shared parameters of LN, as shown in Table g
Variants of CN. To further enhance the flexibility and ap- plicability of CN, we propose two variants, ACN and PCN, addressing the following questions, respectively:
Model property 2: Data dependency. A TSF model é is data dependent (Chen et al. 2023) if its parameters adapt to the input TS (e.g., attention in Transformers or selective scanning in Mamba). contrast, & data independent if its parameters are fixed across the input TS (e.g., linear mod-
Q1) As the parameters of CN are independent on input and unable capture the dynamic characteristics of each imput channel, how can make them adapt to the input?
els). Data-dependent models exhibit high representational capacity, whereas data-independent models are simpler and
Q2) As CN requires a predefined number of channels, how can we handle multiple datasets with unknown or varying number of channels (e.g., training TSFMs)?
less prone to overfitting (Chen et al. 2023
4. Methodology
4.2. Adaptive Channel Normalization (ACN)
The parameters of CN are fixed across time steps and in- dependent of the input TS. However, the characteristics of each channel may vary over time due distribution shifts (Han et al., 2023). To this end, we propose ACN by intro- ducing local parameters (at) to CN, which are attended to with different weights depending on the input channels. To distinguish local parameters from the original parameters of CN, we refer to the original parameters as global parameters (a (, as they are shared globally across the time steps.
In this section, we introduce CN which employs channel- specific parameters, unlike LN which employs shared pa- rameters, to enhance CID of TS models. Furthermore, we propose two variants: ACN, which adjusts the parameters based on the input TS, and PCN, which handles multiple datasets with unknown or varying number of channels. The overall framework is shown in Figure CN can serve as both a strategy (framework) and method.
3

---

Channel Normalization for Time Series Channel Identification
Algorithm 1 Channel Normalization (CN)
Algorithm 2 Adaptive Channel Normalization (ACN)
Require:
Require:
1: Input 2 BXCDD
1: Input 2 RBXCXD
2: Parameters / E RRCD
2: Channel similarity marrix & E RBBxxxC
Ensure: Output 2 E BBXCxD
3: Global and local parameters a , St / RCDD
for b= l...., do
Ensure: Ouipui
RBxCxD
4: for C,....Coo do
4: for b = 1, B do
5: for d = I...,, do
for c 1. ...... do
6: Sb.c.d ac.d" d) Pe,d
6: for d 1, D do
7:
end for
7: ab,c.d = 0 o.d EE So,ar
8:
end for
9:
end for
8:
A -2i S (nd
9: Ab.c.d ab,o, Norm( 3b,add
Channel similarity. To attend to local parameters adap- tively based on the input, we construct channel similarity matrix $ E BBC representing the similarity between the input channels, where B denotes batch size. Specifi- cally, we use cosine similarity, which is then normalized by
10: end for
11: end for
12:
end for
Algorithm Prototypical Channel Normalization (PCN)
Require:
softmax with temperature T as:
1: Input 2 R
2: Channel-proto similarity matrix Sa 38 c CBRXX
S0,e1.02 ||2 |1
2b.0, 26,02
(4)
3: Prototype parameters a. RRXD
Ensure: Output E BxCXD 4: for b= 1, B do
exp (Sb.01,0//T)
Sb,01.02
(5)
5: for c == I.....C. do
Ei= exp x/T)
6:
for 1,..,, do
7:
Ob,e,d = " S af
where E (1...., B) and c2 E (1....,C) C}. This matrix serves as dynamic weights to obtain (dynamic) parameter tb E R2D from the (static) parameter of RDD below
8:
Ab. = ΣK b.c.s Ld
is,ed abboadd (orm((z,,,e))) Bo,e,
10: end for
11: end for
12:
end for
C Σ
b,c
Sp.ci a
(6)
Channel-prototype similarity. To enable channels with arbitrary number utilize the prototype parameters, we construct channel-prototyp similarity matrix Saa e R BxCxK representing the similarity between input chan- nels and prototypes. Note that rather than employing a latent space (z) to represent channels, we apply an addi- tional projection layer (h) in the data space (æ:) to align with the prototype space. Specifically, we use cosine similarity, which is then normalized by softmax with temperature as:
where Sh.ct is the similarity between the e-th and the i-th channel of the b-th data, a. is the (static) local parameter of the i-th channel, and at__ is the resulting (dynamic) local parameters of the e-th channel of the b-th data, representing the weighted average of af using St,c,s dynamic weights.
The parameters of ACN are constructed by element-wise multiplication of the global and dynamic local parameters (ag all.)) which complement each other, shown in Table 7 Further analyses regarding the robustness to the similarity metric, т. and the space where the similarity is calculated are shown in Appendix and K respectively.
h(F0,c) a. [1( |a
S6,c.
exp (S (Stn.a(T)
Sb,ck
(8)
4.3. Prototypical Channel Normalization (PCN)
D exp S (Sh..a/T)
Since CN assign parameters to each channel, it is infeasible to handle datasets with an unknown C (c.g.. inference on unseen datasets) or to train on multiple datasets with varying Css (e.g.. require parameters for all channels in all datasets). To address this issue, we propose PCN by introducing learn able prototypes, where learnable parameters are assigned to each prototype instead of each channel, enabling it to han- dle an arbitrary number of channels. Similar to ACN, these prototype parameters (ae) are attended to with different weights depending on the input TS.
where k E (1....,K)) K)). K is the number of prototypes, and h is a linear projection layer. Similar to ACN, this matrix used as dynamic weights to obtain af from ag as below:
Sa S.
of
Further analyses of the robustness to K and the employment of are demonstrated in Appendix and K.2 respectively.
"The same procedure is applied to 8 as to a.

---

Channel Normalization for Time Series Channel Identification
Non-CID models
Trannformer
+
CN
ACN
Imp. (MSE)
RMLP
+ CN
+ ACN
Imp. (MSE)
Danasets
MSE
MAE
MSE
MAE
MSE
MAE
CN
ACN
MSE
MAB
MSE
MAE
MSE
MAE
CNN
AACN
ETTh1
A57
.499
.41
.339
.438
.338
3.5%
4.2%
A71
.453
.455
.457
.48
.435
5.5%
4.9%
ETTh2
.844
.407
.376
.004
.374
2.1%
2.6%
.380
.405
.376
.402
ETTml
408
.42
196
403
.395
.002
19%
3.2%
401
.406
_844
.997
.383
.396
4.2%
4.5%
ETTm2
.23
.337
.289
331
.288
1.4%
1.7%
.280
.326
.277
324
.277
.323
1.1%
PEMS03
142
.248
101
2044
.99
.203
31 0%
38.0%
.205
.294
.192
284
.159
.266
6.3%
22.4%
PEMS04
.121
.232
.988
196
088
.195
27.3%
27.3%
.236
.321
212
.04
.156
.265
10.2%
33,9%
PEMS07
.102
.205
.967
178
.085
.174
14.7%
16.7%
.200
.284
184
270
.131
.233
8.0%L
34.5 9
PEMS08
.254
.306
1159
225
.153
.211
37.4%
39.8%
.277
.333
247
308
.187
.279
10.85
32.5
Exchange
368
.409
.352
401
.449
.398
4.4%
5.2%
.356
.355
.400
.353
.299
0.3%
0.8%
Weather
260
.281
.247
273
2245
.271
5.0%
5.8%
.272
.292
2.19
.274
.246
.273
8.5%
9.6%
Solar
224
.261
.228
258
.200
2.6%
6.0%
.248
.276
.242
.272
5.0%
ECL
.179
.270
.161
2556
.158
.256
10.1%
11.7%
.228
.313
.100
227
.189
.276
16.7%
17.1%
Average
275
.318
244
299
.241
.295
11.3%
12.4%
.297
.346
280
.330
.262
.319
5.7%
11.8%
Best count (148)
0
9
9
46
Imp::
1.1%p
7
44
46
A Imp:
6.1 6.1%
CID models
S-Mamba
+ CNN
ACN
Imp. (MSE)
TSMixer
+ CN
+ ACN
Imp. (MSE)
ETTb1
A57
.452
.15
.150
0.11%
2.0%
.438
.435
.453
.111
1.9%
ETTH2
383
.408
375
.011
.374
.400
1.1%
2.3%
.003
.418
.387
.410
.386
.407
4.0%
4.2%
ETTml
.98
.407
3997
.006
.394
0.35
1.0%
.406
.386
399
.385
.297
3.7%
ETTm2
290
.333
.86
329
.284
.328
1.4%
2.1%
.287
.330
.266
329
.280
.325
0.3%
2.4%
PEMS0:
.133
.240
108
214
.107
.213
18 8%%
19.5%
.129
.236
.24
.288
.120
2.30
3.9%
7.0%
PEMS04
096
.205
085
.189
11.5%
.288
.114
.222
.109
.222
5.2%
PEMS07
.900
.191
078
168
.073
.167
13.31%
18.9%
11.5
.210
115
2009
.103
.203
0.0%
10.4%
PEMS08
.157
.242
133
216
15.3%
22.9%
.250
.167
.258
10.2%
10.2%
Exchange
.64
.40
362
4055
.357
.002
0.55
1.9%
.365
.406
.358
.402
.356
.400
1,9%
2.5%
Weather
.252
.277
.246
.273
.247
.274
2.4%
2.0%
.260
.285
.246
.744
.242
.272
5.4%
6.9%
Solar
244
.255
230
262
.228
.261
5.7%
6.6%
.255
.294
2266
.267
.245
224
3,5%
3,9%
ECL
.174
.269
163
261
.162
.259
6.3%
6.9%
.211
.310
181
2280
.174
.273
14.2%
17.8%
Average
253
.309
.24
298
.240
.297
4.0%
5.1%
.266
.321
2244
.309
.243
.308
4,5%
8.6%
Best count (148)
0
15
25
38
31
A Imp.:
1.1%p
10
16
40
36
à Imp.:
4.1%p
Table 2: Results of TS forecasting, We apply CN/ACN non-CID and CID models, achieving performance gains across all models.
5. Experiments
2024 and replicated using the official codes.
Experimental setups. We demonstrate the effectiveness of our method on TSF tasks with 12 datasets. For evalua- tion metrics, we use mean squared error (MSE) and mean absolute error (MAE). We follow the experimental setups from C-LoRA (Nie et al. 2024), with size of the lookback window (L) set to 96, and divide all datasets into training, validation, and test sets in chronological order. Further details of the setups are provided in Appendix A]
5.1. Application of CN/ACN
TS forecasting. Table 2 presents the average performance across four horizons (Hl E {96, 192, 336, 720)), demon- strating that both CN and ACN consistently improve across all datasets and backbones, with ACN yielding additional gains compared to CN. Below, we analyze the performance gain from CN and the additional gain from ACN in relation to the two properties of the backbones.
Datasets. For the experiments, we use 12 datasets: four ETT datasets (ETTh1, ETTh2, ETTml, ETTm2) (Zhou et al. 2021), four PEMS datasets (PEMS03, PEMS04, PEMS07, PEMS08) (Chen et al. 200T), Exchange, Weather, ECL (Wu ctt al. [221)) , and Solar-Eneen Energy (Solar) (Lai et al. 2018). De- tails of the dataset statistics are provided in Appendix A.
a) CID ys. non-CID, As CN enhances the CID of models, it provides substantial improvements for non-CI1) models (e.g., iTransformer, RMLP), as shown in Figure SggureBla a). How- ever, also benefits CID models (e.g., S-Mamba, TSMixer) that already have the ability distinguish among channels, although the improvements are relatively smaller. This is further validated by the results in Table 2 where non-CID models exhibit greater performance gains than CID models.
Backbones. For the experiments, we select four backbones: iTransformer (Liu et al. 2024), RMLP (Li et al. 2023). S-
Mamba WWag et al. 2025), ). and TSMixer (Li ct al. 2023) For backbones that utilize L.N, we replace it with our method,
b) Data dependent vs. Data independent. As ACN im- proves upon CN by adapting to the input TS, transitioning from CN to ACN provides substantial improvements for data ietr-iidepeedeett models (c.g., RMLP, TSMixer), as shown in Figure a). However, also benefits data-dependent models g.. i'rrasffrmer, S-Mamba) whose parameters already adapt to the input TS, although the improvements are relatively smaller. This is further validated by the results in Table where data-ina independent models exhibit greater additional performance gains from CN to ACN.
and for those without, we add our method. As illustrated in Figure []), a), these methods can be categorized based on their a) inherent CID ability and b) data dependency of model parameters. Furthermore, for PCN, we employ UniTS (Gao et al. 2024). a TSFM that addresses diverse tasks using prompt-tunm ttupttgttnnnn.ttm to demonstrate its capability to handle mul- tiple datasets with varying C's and perform inference on unseen datasets with unknown Cs.. The baseline results are obtained from previous works (Nie et al., 2024 Lee et al.
5

---

Channel Normalization for Time Series Channel Identification
(N: # datasets, C₁ f channels of i-th dataset)
Metric (Best #)
UniTS
+ PCNN
Imp.
12 Datasets
MSE
Imp.
Parameters
Zero-shot
20 FCST
Sup.
.699 (4)
.433 (16)
7.7%
iTransformer
.275
CN
> D
(MSE)
Pmt.
.48 (3)
.453 (20)
5.2%
+ CN
.244
11.3%
X
ACN
4 -i=1 C.
18 CT. (Acc.)
Sup.
80.6 (2)
83.0 (16)
3.0%
+ ACN
.241
12.4%
PCN
2K
Pmt.
75.1 (3)
79.5 (16)
5,5%
PCN
.252
8.4%
Table 3: PCN for TSFMs.
Table
4: PCN to TSFMs.
Table 5: PCN single-task models.
Averaee MSE across horizons
ETTh1
ETTh
ETTml
ETTm2
PEMS03
PEMS04
PEMSU7
PEMSOS
Exchange
Weather
Solar
ECL
Avg.
Imp.
A57
.384
.408
.293
.142
.121
.102
.254
.368
.260
.234
.179
.275
TTranfformer
C-token
450
.389
.400
.290
.123
.109
.106
.177
.376
2:16
.255
.169
.256
6.9%
C-project
452
.381
399
.286
.119
-109
097
.63
.366
.244
-230
.163
.251
8.7%
Channel idcutifier
4415
.382
397
.293
100
093
.082
.168
.365
2488
.231
.165
2:18
9.81%
+ C-LoRA
450
.392
398
.289
.114
.13
.066
.169
.364
.488
.241
.167
254
7.6%
Non-
+ACN
A38
.374
.355
.258
.998
.088
.085
.153
.349
.215
.220
.158
.241
12.4%
471
.381
.401
.280
.205
236
.200
.277
.356
.272
.261
.228
.297
C-token
455
.391
338
.277
.220
211
.196
.286
.368
2446
.277
.205
.293
1.4%
RMLP
C-project
455
.389
384
.277
186
190
.72
.233
.366
.245
.249
.195
.288
5.3%%
Channel ideenifier
452
.191
2019
.356
-250
.254
.199
4.4%
+ C-LoRA
451
379
.383
2799
.192
.98
.182
.264
.359
.245
.256
.19
.282
5.1%
+ACN
448
.376
.383
.159
.156
.131
.187
.353
.246
.242
.189
.262
11.8%
457
383
398
.290
.133
096
.090
.157
.364
.252
.244
.174
.253
S. S.Mambu
463
.117
.245
.244
1.2%
C-project
466
.400
.005
.291
.122
.089
.099
.151
.391
.219
.266
.176
.259
-2.4%
Channel identifier
457
.287
.112
086
.78
.248
.239
.167
2.0%
+ C-LoRRA
457
.105
399
.289
112
J084
.092
.144
359
2217
.238
.169
250
1.2%
CID
+ACN
.107
.095
244
5.1%
.62
.403
.401
.287
.129
.115
.115
.186
.365
.660
.255
.211
.266
TSMixer
C-token
456
.230
.221
.413
.271
.279
.291
-16.5%
C-project
457
.412
.001
.344
.232
.222
.154
268
.477
.211
.275
.281
.309
-16.2%
Channel identifier
454
390
.124
.144
106
.355
.245
.251
186
.257
2.3%
+ C-LoRR
460
.007
399
.283
.122
.110
.103
.181
.366
.245
.251
.187
.260
3.4%
+ACN
.453
.386
.385
.280
.120
.109
.103
.167
.356
.242
.245
.174
.243
8.6%
Table 6: Comparison with other methods. We compare ACN with 1) baseline methods, which employ channel-specific parameters
for token embedding (C-token) or projection layers (C-project), and 2) previous methods, including channel identifier and C-LoRA
5.2. Application of PCN
resulting in a weaker enforcement of CID.
Application to TSFMs. As shown in Table 3 applying CN and ACN to TSFM is infeasible due to 1) the substantial increase in parameters, as it requires parameters for all chan- nels across all datasets and 2) their inability to handle unseen datasets during training, as the number of channels may dif- fer between training and inference datasets. In contrast, PCN addresses these limitations by employing prototypes. 4 presents the application of PCN to UniTS, showing the average results for 20 forecasting and 18 classification tasks under supervised (Sup.) and prompt- prompt-tuning (Pmt.) settings, with consistent improvements observed across all tasks. Full results of Table improvements on zero-shot forecasting tasks are provided in Appendix N and L
5.3. Comparison with Other Methods
demonstrate the effectiveness of ACN, we compare with two categories of methods: 1) baseline methods, which use a naive strategy of employing channel-specific param- eters for token embedding (C-token) or projection layers (C-project), and 2) previous methods, including channel identifier (Chi et al. 2024), a learnable vector added to channel tokens and C-LoRA (Nie et al. 2024), which ap- plies a channel-awe -wware LoRA to TS models. Table 6 demon- strates that our method outperforms these approaches across all backbones, while two baseline methods (C-token and C-project) even degrade the performance of CID models.
Table
Application to single-task signed for TSFMs, it also improves the performance of single-task models trained a single dataset, even when the number of channels is unknown. As shown in Table applying PCN with K = 5 to iTransformer improves perfor- mance by 8.4% on average across 12 datasets and 4 horizons, though the improvement is smaller than that achieved by CN and ACN. We attribute this to the fact that, unlike CN and ACN which assign each channel a distinct parameter, PCN assigns each prototype (channel cluster) a distinct parameter, 'A single-task model trained on single dataset.
Although PCN is de-
6. Analysis
In this section, we conduct (1) ablation studies on ACN, (2) entropy analyses explain the proposed method from information theory perspective, and (3) other analyses including both qualitative and quantitative evaluations.
6.1. Ablation Study
To demonstrate the effectivenness of ACN, we conduct an ablation study of using the global and local parameters with iTransformer. Table 7 presents the results, indicating that using all components yields the best performance.
6

---

Channel Normalization for Time Series Channel Identification
ACN
Average MSE across 4 horizons
Avg.
Imp.
Adaptive CN
ETThl
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
BCL
457
.384
.408
.293
.142
.121
.102
.254
.368
.260
.234
.179
.255
439
.375
.395
.289
.108
.10
.099
.174
.347
.247
.26
.162
.247
10.2%
441
.396
.289
.101
.087
.159
.352
.28
_161
11.3%
438
.374
.395
.288
.098
.088
.085
.153
349
.245
.220
.188
.241
12.4%
Table Ablation study. None: (aa) vs. Adaptive (local parameters): ,, vs. CN (global parameters): (a vs. ACN:
Entropy gain
10-
47.
800
O
Imp.
Dataset (C+ 100)
Datasct (C'>
30
19.1%
22.0%
25.49 45.4%
600
channel
20-
9.5%
6.4%
10
0.2%
0.1%
0.5%
0.65
0.1%
1.0%
200
0.724
ETTh1
ETTh2
ETTml
ETTmZ Exchange Weather
PEMS08
PEMS01
ECL
PEMS03
PEMS07
Entropy gain (8)
(a) Entropy gain vs. Number of channels (C).
(b) Entropy gain vs. MSE.
Figure 4: Channel entropy gain by CN. (a) Datasets with higher C. show a higher entropy gain. (b) Datasets with higher entropy gain show a higher performance gain (average MSE across four horizons).
Feature Ent. Channel Ent.
iTransformer
RMLP
S-Mamba
TSMixer
LN
CN
ACN
-20
+ Ours
58
-244
7m
KL Ki-divereence btw 8 attention heads (8x8)
iTranafomer
RMLP
S-Mamba
TSMixer
Figure 6: Diversity of altention heads.
VS.
(a) Correlation matrix
Correlation matrix (CN)
[(b) Elements of correlation mairix
50-
133
500
CN
153
350
Figure 5: Entropy gain by CN. Non-CID models show greater
203
253
200
740
64
channel/feature entropy gains from CN than CID models.
422
Figure 7: Diversity of correlations btw channel representations.
6.2. Entropy Analysis
CN for both non-CID and CID models. The figure shows that non-CID models exhibit higher gains compared to CID models, indicating richer feature representations and greater uniqueness in channel representations. This supports our ar- gument that the proposed method benefits non-CID models more than CID models, which aligns with the greater per- formance gain of non-CID models, shown in Figure 2 a).
We demonstrate the effect of our method through entropy analyses with four different backbones, showing that it 1) enriches feature representations (feature entropy 1), 2) in- creases the uniqueness of each channel representation (chan- nel entropy 1), and 3) diversifies the attention heads and correlation between channel representations.
Gaussian entropy. Let Z E RR be a random vector follow- ing multivariate Gaussian distribution with covariance matrix Σ E DxD Then, the Gaussian entropy of Z is de- fined as (Z) = 2 log (22Te)) det(2)) which can be es- timated by N samples z [21, E2, 2,] E NxD as H (z) = 1 log ((2Te) det ( z =f)), with sI added avoid non-trivial solutions, following the previous works (Yu et al., 2020: Chen al. 2024b 2025).
Entropy gain by datasets. To evaluate the effectiveness of CN across datasets, we analyze the channel entropy gain achieved by CN using iTransformer with respect to (1) the number of channels and (2) the performance gain for each dataset. Figure 4(a) illustrates the relationship between the entropy gain and C. showing that datasets with higher C' presents the rela-
achieve greater entropy gain. Figure [ tionship between the entropy gain and MSE improvement, with a correlation (p) of 0.724, indicating that datasets with higher entropy gain show greater performance improvement.
For the analysis, we compute the average over test dataset with ž CxD and use the normalized entropy averaged over the last dimension for comparison across different di- mensions. Then, we define the entropy of z and z as the feature entropy and channel entropy respectively, as they measure 1) the richness of the feature dimension and 2) uniqueness of each channel representation.
Diverse attention heads & correlations btw channels. Figuurrojillustaates the KL divergence (KLD) between the distributions of eight attention heads of iTrrnsformer on PEMS03 (Chen et al. 2001), showing that our method en- ables the model to maintain greater diversity across the heads. Specifically, the average KLD between the heads of the first and last encoder layers is 0.289 and 0.077 for
Entropy gain of non-CII CID vs. CID models. Figure 5 illus- trates the gains in channel and feature entropies achieved by
7

---

Channel Normalization for Time Series Channel Identification
(a) (-SNE CN parameters
(b) Visualization of 3 channels for each cluster (Cluster 2, 3,
(c) Correlation matrix of 21 channels
Cluster
Shide legg
4.
Cluster
Cluster
Cluster 3
Figure 8: Visualization of parameters and channels. (a) shows the I-SNE of the parameters CN, with four clusters formed. (b) visualizes three channels from each cluster, demonstrating that channels in the same cluster share similar patterns except for those in the 4th cluster. (c) visualizes the correlation matrix of the channels, where channels in the 4th cluster lack close relationships with others.
F1Tm?
L 96, H
iTrans.
Ch. identifier
C-LoRA
Ours
0.20
CN
+ ACN
ACV
ACH
8.000
Train (seclepoch)
7.7
9.4
11.1
7.8
10.8
Inference (ms)
2.0
2.3
2.8
2.1
2.5
3.33
155
300
100
726
56
324
75
Parameters
3.2M
+ +..IM
+2.8
+.7M
+14M
313
ECL
PEMS04
PEM507
CN
65
Avg. MSE
.254
.168
.169
.559
.153
4GH
Table 8: Efficiency analysis.
Six: Ingutwindow
Size of
LN, compared to 0.369 and 0.395 for CN. Additionally, Figure 7 a) presents the correlation matrices of channel representations of PEMS03 using iTransformer with and without CN, along with the distribution of matrix elements in F demonstrating that CN enhances the diversity of the correlations. This increase in diversity in both aspects supports the performance improvements achieved by our method, with the average MSE across four horizons being 0.142, 0. 101, and 0.098 for LN, CN, and ACN.
Figure 9: Effectiveness of CN/ACN under various L.
K=10 10
K = 20
K = 50
K
100
1-SNE
*Distortion
Redundant prototypes.
Chaelere
Chintere
Cluziere
"MAE
0,460
0.453
0.454
0.454
MAE 20 FCST taoks (Bassline: 0.478)
"Sum sqquared distemces to cluater cantera
6.3. Other Analyses
Figure 10: t-SNE & distortion plot of PCN parameters.
Visualization of CN params. To demonstrate that the pa-
numbers of prototypes (K), using UniTS as the backbone. The distortion plots, shown below, are obtained by perform- ing K-means clustering on these parameters to assess the redundancy of the prototypes. The figures indicate that in- creasing K leads to performance stabilization after certain point (K 20), as redundant prototypes begin to emerge.
rameters of CN effectively capture the CID, we visualize the
parameters (a) of 21 channels in Weather (Wu et al. 202T]
using t-SNE (Van der Maaten & Hinton 2008). Figure X
shows the result, displaying (a) four distinct clusters and (b)
the visualization of channels corresponding each cluster.
The figure indicates that channels with similar patterns be-
For further analyses, please refer to the below sections:
long to the same cluster, except for the fourth cluster (blue), whose channels show no close relationship with other chan- nels, as also shown by the (c) correlation matrix.
Theoretical entropy analysis: Appendix C
Comparison with Instance Normalization: Appendix
Efficiency analysis. Table g shows the 1) number of pa- rameters, 2) training time (per epoch), and 3) inference time
Robustness to K, T, similarity space: Appendix ||
K
PCN for zero-shot forecasting with TSFM: Appendix
(per data instance) of iTransformer on PEMS08 (Chen et al. 200T) across various methods for CID. The results indicate that applying our methods has minimal impact on the num- ber of parameters and computational time, while providing greater performance gain compared to other methods.
Application of multiple methods for CID: Appendix
E
Visualization of TS forecasting results: Appendix
0
7. Conclusion
In this work, we introduce CN, a normalization strategy to enhance CID of TS models with channel-specific parame- ters. Furthermore, we propose ACN to adapt to input TS single-task models, and PCN to handle multiple datasets with unknown/varying number of channels on TSFMs. A potential direction for future work involves developing a method to automatically determine the number of proto- types for PCN based on the dataset. We hope that our work highlights the importance of CID in TS analysis.
Performance under varying Ls. To validate the effective- ness of our method under various sizes of lookback windows (L), we evaluate our method on iTransformer with forecast horizon of H- = 12 for the PEMS datasets and H = 96 for the other datasets. Figure 2
indicates that the performance gain remains robust across all datasets regardless of L.
Various Kss for PCN. Figure 10 shows the tES visualiza- tions of prototype parameters (a?) of PCN across varying
8

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Channel Normalization for Time Series Channel Identification
Impact Statement
Chen, X., Qiu, P., Zhu, W., Li, H., Wang, H., Sotiras, A., Wang, Y., and Razi, A. Sequence complementor: Com- plementing transformers for time series forecasting with learnable sequences, In AAAI, 2025.
This paper aims to advance time series modeling by empha- sizing the importance of channel identification and propos- ing Channel Normalization to enhance it. We do not identify any specific societal consequences that require emphasis.
Chi, C., Wang. X., Yang. K., Song, Z., Jin, D., Zhu, L., Deng, C., and Feng. Injecttst: transformer method of injecting global information into independent chan- nels for long time series forecasting. arXiv preprint arXiv:2403 203.:24003.2814,, 2024.
Acknowledgements
This work was supported by the National Research Foundation Korea (NRF) grant funded by the Ko- rea government (MSIT) (2020R1 A01055959, 0022RIA4A0033386 RS-2023-00217703 00341749), the MSIT(Ministry of Science and ICT), Korea, under the ICAN(ICT Challenge and Advanced Network of HRD) support program ASS-2023-00259934 supervised by the ITTP(Ins for Information & Communications Technology Planning & Evaluation), Yonsei University Re- search Fund (2024-22-0148), and a grant from KRAFTON AI.
Cirstea, R.-G., Yang, B., Guo, C.,, Kieu, T., and Pan, Towards spatio-temporal aware traffic time series fore- casting. In 2022 IEEE 38th International Conference on Data Engineering (ICDE), pp. 2900-2913. IEEE, 2022.
RS-2024-
Dudek, G., Petka, P., and Smyl, S. A hybrid residual di- lated Istm and exponential smoothing model for midterm electric load forecasting. IEEE Transactions on Neural Networks and Learning Systems, 33(7):2879-25) 2891, 2021.
Gao, S., Koker, T., Queen, O., Hartvigsen, T., Tsiligkaridis, T., and Zitnik, M. Units: Building a unified time series model. arXiv preprint arXiv:2403.0013/ 2024.
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Channel Normalization for Time Series Channel Identification
Appendix
Experimental Settings
12
A.1 Dataset Statistics
12
A.2 Experimental Setups
12
Properties of TS Backbones
13
B.1 Channel Identifiability
13
B.2 Data Dependency
13
Theoretical Entropy Analysis
14
D Various Backbones
15
Application of Multiple Methods for CID
16
Comparison with Instance Normalization
16
Comparison with Constant Vectors
16
H Robustness to Similarity Metric
17
Robustness to Number of Prototypes K
17
J Robustness to Temperature
18
Robustness to Similarity Space for ACN & PCN
19
K. Similarity Space for ACN
19
K.2 Similarity Space for PCN
20
Application of PCN to Zero-shot Forecasting with TSFMs
21
M Full Results: Application of CN & ACN
22
Full Results: Application PCN
24
N.I Application of PCN to non-TSFMs
24
N.2 Application of PCN to TSFMs
24
Visualization of Forecasting Results
26
0,
Visualization of TSF with Trransformer
26
Visualization of TSF with RMLP
28
Visualization TSF with TSMixer
30
0.4
Visualization of TSF with S-Mamba
32

---

Channel Normalization for Time Series Channel Identification
A. Experimental Settings
A.1. Dataset Statistics
Dataset statistics. For the experiments, 12 datasets from various domains are used, with their statistics detailed in Table A.1 where C' and T represent the number of channels and timesteps, respectively.
Dataset split. We follow the same data processing steps and train-validation-tesl split protocol used in S-Mamba (Wang al. 2025), maintaining chronological order in the separation of training, validation, and test sets, using 6:2:2 ratio for the Golar-Energy, ETT, and PEMS datasets, and 7:1:2 ratio for the other datasets. Hyperparameters are tuned based on the validation loss.
Size of lookback window (L). Following the previous works (Nie al. 2024 Liu et al. 2024). L is uniformly set to 96 for all datasets and models. Further analysis regarding the performance under different I. is discussed in Figure
Statistics
Dataset Split
Size of Input & Output
Dataset
T
(Nuain Nyal, MNist)
L
ETThl
Zhou et al.
2021
17420
(8545, 2881, 2881 1)
ETTh2
Zhou et al.
2021
17420
(8545, 2881, 2881)
7
ETTml
Zhou et al.
2021
69680
(34465, 11521, 11521)
ETTm2
Zhou et al.
2021
69680
(34465, 11521, 11521)
{96, 192, 336, 720)
Exchange (Wu et al.
2021)
8
7588
(5120, 665, 1422)
Weather (Wu et al.
2021)
21
52696
(36792, 5271, 10540)
96
ECL Wu et al.
202
321
26304
(18317, 2633, 5261)
Solar- Eneryy (Lai et al. 2018
137
52560
366011,5611, 10417)
PEMS03
(Luu al.
2022
358
26209
(15617, 5135, 5135)
PEMS04
(Liu et al. 2022
307
15992
(10172, 3375, 3375)
{12, 24, 48, 96}
PEMS07
(Luu et al. 2022)
883
28224
(16911, 5622, 5622)
PEMS08
(Liu et al. 2022
170
17856
(10690, 3548, 3548)
Table A.1: Datasets for TS forecasting.
A.2. Experimental Setups
Application of CN/ACN. For all experiments regarding TS forecasting with four different backbones, we use the official code from C-LoRA (Nie et al. 2024), except for S-Mamba (Wang et al. 2025), as C-LoRA does not use S-Mamba as a backbone.
Application of PCN. For all experiments involving TSFM, UniTS (Gao et al. 2024 is trained across multiple tasks using a unified protocol. To accommodate the largest dataset, samples from each dataset are repeated within each epoch. The training protocol, as outlined in the original paper, is as follows:
Supervised training: Models are trained for 5 epochs with gradient accumulation, yielding an effective batch size of 1024. The initial learning rate is set to 3.2e-2 and adjusted using multi-step decay schedule. Self-supervieed pretraining: Models are trained for 10 epochs with an effective batch size of 4096, starting with a learning rate of 6.4e-3 and utilizing cosine decay schedule.
The embedding dimension is set to 64 for the supervised version and 32 for the prompt-tuning version. Note that we encountered a convergence issue in the prompt-tuning setting, which was also reported by others in a GitHub issue. To resolve this, we set the hidden dimension to 32, which led a performance decrease compared to the results in the original paper. For fair comparison, this setting applied uniformly both UniTS and its application to PCN.
12

---

Channel Normalization for Time Series Channel Identification
Parameter initialization. The initialization of the parameters for Channel Normalization (CN), Adaptive Channel Normalization (ACN), and Prototypical Channel Normalization (PCN) is designed ensure that no normalization occurs when learning has not yet taken place:
The scale parameter (ax) is initialized to 1.
The shift parameter (3) initialized
This choice is consistent with the default initialization used in PyTorch (Paszke et al. 2019) normalization layers, including Layer Normalization and Batch Normalization Therefore, the parameters for CN, ACN, and PCN are initialized follows:
CN: a = 1,3 = 0
ACN: og = 1, = 0. = 1, 0
PCN: ap = 1, =0 0
B. Properties of TS Backbones
B.1. Channel Identifiability
Definition. A MTS forecasting model f R -> RxC exhibits channel identifiability (CID) if, for any input TS x E _xC the output fx) depends on the channel index e of x, such that the forecasted value f(x).. is unique to even when all channels x have identical values.
Using the above property, MTS forecasting models can be classified into two categories: models without CID and models with CID.
1) Model without channel identifiability. If model f lacks CID, then for any x E RLXC with all channels having identical input values, the forecasted outputs for all channels will also be identical:
x F: ei] = x |. 02) + ((x) & ai] =
Vei, 02.
(B.1)
2) Model with channel identifiability. If a model / possesses CID, then for any E RLxC with channels having identical input values, the forecasted outputs for different channels will be distinct due to the model's ability to incorporate channel positional information:
c. = x [3) A f(x) [c, fxx (2) Vei, C2,
(B.2)
where the inequality arises from the model's recognition of channel positions.
B.2. Data Dependency
Definition. A MTS forecasting model : LXC T'C exhibits data dependency if the model parameters 0 depend on the input TS x. Specifically, for a given input TS x E RA the model parameters 0 may vary based on the content structure of x, affecting the model's output.
Using the above property, MTS forecasting models can be classified into two categories: models without data dependency and models with data dependency.
1) Model without data dependency. If model f does not exhibit data dependency, then the model parameters 0 are fixed and independent of the input TS x:
y ff (x
(B.3)
2) Model with data dependency. If model f exhibits data dependency, then the model parameters depend on the input
TS
y = f (x, ((x)).
(B.4)
13

---

Channel Normalization for Time Series Channel Identification
C. Theoretical Entropy Analysis
Following the previous work (Chnn et al. 2025). we analyze our approach using theoretical entropy analysis.
Justification 1. Applying CN achieves more informative representation (ZeN) compared to LN (ZEN) without any normalization Non,,), as it increases in the entropy:
H (Zane) S ((IN)) S ((CN)).
(C.1)
Proof. The joint entropy can be decomposed as follows:
(Z)
- H(Z) + H(o1: 31)
(C.2)
H Nose
H(Z.a..31)
(C.3)
=Hix
H(Z, ,1, (( 211
(C.4)
HZ, (a,, 11)
(C.5)
=Hes
This follows from the non-negativity of conditional entropy TTomas & Joy 2006).
Justification 2. more informative representation (i.e., higher H (Z)) can potentially lower forecasting error, as under the Gaussian assumption, the minimum mean-squared error (MMSE) is bounded by:
exp (2H (Y 2))
MMSE
(C.6)
2Te
Proof. Following Equation ! we construct the chain with modification where the last layer of g2 separated:
X
2 pre
I Z
82
Zpona
92 Y
(C.7)
This allows the propagation in the final layer 92 to be expressed as:
Y 9/2 (Zpoa)
WZ
(C.8)
Assuming a Gaussian distribution for Zpose Y, and Y, we can derive the following bound, as shown in previous works
(Carson al. 2012 Prasad 2010):
MMSE exp 2H (Y) Zpost)
(C.9)
2xe
Since Zpost = J/2 (Z), the chain property (Thomas Joy 2006) ensures that Z contains least as much information about Y as Zpwats i.e., knowing Z reduces the uncertainty about Y:
H (Y Z post 2 H (Y Z)
(C.10)
By substituting Equation C.10 into Equation C.9 we obtain:
MMSE > exp (2H (Y) Zposs)
exp (2Hl (Y) (Z))
(C.11)
2xe
2Te
14

---

Channel Normalization for Time Series Channel Identification
D. Various Backbones
Backhones for CN/ACN. The four backhones used in the experiments are categorized based on their 1) channel identifiability
(CID) and 2) data dependency, as shown in Figure D.I
RMLP (Li et al. 2023) captures temporal dependencies within each channel in a Gelannel-indeeendee manner, applying identical weights across all channels.
Transformer (Liu et al. 2024) captures channel dependencies using self-attention mechanism that is order- inder-invariint,,
rendering channels unidentifiable (non-CID)
TSMixer (Chen et al. 2023) employs MLPs to capture both temporal and channel dependencies, with distinct weights assigned to each channel.
S-Mamba (Wang al. 2025) utilizes the Mamba architecture to capture channel dependencies, leveraging the inherent properties of Mamba (e.g., state-space modeling) to differentiate between channels.
For architectures that utilize Layer Normalization (LN), such iTransformer and S-Mamba, replace LN with our proposed method. For architectures without any normalization, we incorporate our method directly.
Data-dependent
Non-CID
CID
X
0
Ch11
Ch2
Ch3
Ch4
Ch1
Ch2
Ch3
Ch4
Ch1
Ch2
Ch3
Ch4
Ch1
Ch2
Ch3
CM
X
Ch1
Ch2
Ch3
Ch4
Ch1
Ch2
Ch3
Ch4
CID
RMLP (Linear models)
iTransformer
Ch1
ch2
Chs
Ch4
Ch3
Ch3
Ch2
Ch2
Ch1
Ch2
Ch3
Ch
S-Mamba (SSM-based)
TSMixer
Parameters depend on input (data)
Parameters do not depend on input (data)
Figure D.1: Four different backbones and their properties.
Backbones for PCN. UniTS (Goo et al. 2024) is designed with three distinct UniTS blocks, as well as GEN tower and a CLS tower. Each data source is assigned unique prompt and task tokens, while tasks within the same source that require different forecast lengths use a shared prompt and GEN token. To facilitate zero-shot learning for new datasets, universal prompt and GEN token utilized across all data sources.
15

---

Channel Normalization for Time Series Channel Identification
E. Application of Multiple Methods for CID
As plug-in method, ACN is applicable to TS models along with other CID methods. To demonstrate that our method complements these techniques, we evaluate its performance when combined with channel identifier (Chi et al. 2024 and C-LoRA (Nie et al. 2024), using iTransformer (Liu et al. 2024) as the backbone, as shown in Table E.1
Average MSE berizons
Thansformer
Avg.
ETThl
ETT52
ETTml
ET7m2
PEMS03
PEMS04
PEMS07
PEMSOS
Exchange
Weather
Solar
ECL
.293
.142
ACANN
.A38
.374
.395
.288
.098
.088
.885
.153
.349
.245
.220
.158
.241
+ACN++ C-LoRA
.438
.380
-397
.285
.109
.096
162
.360
.245
.227
.162
.246
ACN Channel identifier
.422
.377
_394
-289
.099
.089
.054
.158
.344
.245
.222
.158
242
ACN C-LoRA
Channel identilier
440
.381
.400
286
.105
.090
.089
163
346
.245
.266
162
.444
Table E.1: Application of multiple methods CID.
F. Comparison with Instance Normalization
Table F.1 shows the comparison of our methods (CN, ACN) with Instance Normalization (IN)
(Ulyanon 2016) on
I'Transfommer (Liu et al. 2024) in terms of average MSE across four horizons for various datasets, demonstrating that our methods outperforms IN.
Average MSF across horizons
iTransforner
Avg.
ETTh1
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
ECL
+IN
.42
.377
.397
.291
.011
.02
.088
.165
.356
.249
226
.162
.246
CN
441
376
.396
.289
.101
.088
.087
.159
.352
.24
.228
.161
.244
ACN
.438
.374
.395
.288
.098
.088
.085
.153
.349
.245
.220
.158
.241
Table F.1: Comparison with Instance Normalization (IN).
G. Comparison with Constant Vectors
Table G.1 presents the performance of adding different constant vectors each channel token, allowing the model to distinguish channels iTransformer (Liu al., 2024). The results indicate that this simple addition improves performance, while our methods (CN, ACN) achieves better performance in terms of average MSE across four horizons for various datasets.
Average MSE across 4 harizons
iTranformer
Avg.
Imp.
ETTh1
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
ECL
457
384
.088
.293
.142
.121
.102
.254
.368
.200
.234
.179
.275
Constant vector
443
.378
.290
246
.235
.252
8.4%
CN
376
.289
.159
227
.228
2244
ACN
.488
.374
.395
.288
.998
.088
.085
.153
.349
.245
.220
.158
.241
12.4%
Table G.I: Comparison with contant vectors.
16

---

Channel Normalization for Time Series Channel Identification
H. Robustness to Similarity Metric
To construct a channel similarity matrix S E BRCXXC for ACN, various similiarity metric can be employed. To evaluate whether the proposed method sensitive to the choice of similarity metric, we compare several options, including (negative) cosine similarity, & distance, and &g distance. Table presents the average MSE across four horizons for various datasets, demonstrating that the performance remains robust to the choice of similarity metric.
Average MSE across horizons
iTransformrr
Avg.
ETThl
ETTh2
ETTml
ETTm2
Exchange
Weather
Solar
ECL
LN (Base)
.457
.384
.408
.293
.368
.260
.234
.179
.329
.440
.374
.395
.288
.350
.245
.220
.179
.309
CN
l2
.439
.375
.395
.288
.350
.245
.221
.158
.309
Cosine
.438
.374
.395
.288
.349
.245
.220
.158
.308
Table H.l: Robustness similarity metric for ACN.
I. Robustness to Number of Prototypes K
Table [.] shows the results of applying PCN with various values of K. The results indicate that the performance remains robust to the choice of K.
Average MSE across horizons
Avg.
ETThl
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
457
384
.408
.293
.142
.121
.102
.254
368
.260
.244
.179
.275
.440
376
.004
.290
.115
.121
.102
.180
.340
.257
.235
.169
.252
.439
.375
.403
.290
.16
.121
.101
.345
.257
.235
.169
.252
.437
.376
.404
289
.117
.120
.101
.349
.257
.232
.438
.366
.403
.289
.166
.101
.182
.339
.257
.233
.169
.252
20
.434
376
404
.288
.17
120
102
183
.336
.257
233
.169
.252
Table L.: Robustness to K for PCN.
17

---

Channel Normalization for Time Series Channel Identification
J. Robustness to Temperature
Tables and 1.4 display the average MSE across four different horizons for the 12 datasets, with four different backbones, using various values of the temperature (r) in ACN. The results show that the effectiveness of ACN is consistent
across different values
Average MSE across 4 horizons
Avg.
ETTh1
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMSO8
Exchange
Weather
Solar
ECL
0.05
.499
.766
.366
.288
009
.088
.087
.16
.351
.245
.221
.159
.242
0.1
.439
376
396
.288
099
.088
.085
.156
.351
.266
.222
.158
.242
.439
.375
.395
.289
.099
.088
.086
.154
.266
.158
0.5
.439
.375
.395
.289
.098
.088
.086
.159
.350
.247
.224
.158
.242
1.0
439
376
.395
289
.098
.088
.087
.159
.350
.248
.224
.158
.242
Table J.1: Robustness for ACN with iTransformer.
Backbone: S-Mamba
Average MSE across horizons
Avg.
ETTh1
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
ECL
.499
.375
.285
.088
.093
.075
.122
.248
.64
.241
0.1
.49
.375
.394
.285
.107
.995
.074
.127
.358
.248
.229
168
.241
0.2
.449
.375
.35
.285
108
.095
.076
.129
.358
.249
.230
.163
.241
0.5
.499
.375
395
.285
.109
.095
.076
.124
.358
.250
.231
.165
.241
449
.35
.395
.285
.108
.095
.076
.124
.250
.231
.165
.241
Table J.2: Robustness to r for ACN with S-Mamba.
Backbone: TSMixer
Average MSE across horizons
Avg.
E1Th1
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
ECL
0.05
.453
.387
.311
.280
.130
.112
.105
.178
.356
.242
.248
.174
.255
.453
.387
.391
.280
.124
.109
.103
.177
.356
.242
.247
.174
.254
.453
.388
.391
.280
.120
.109
.103
.168
.356
.242
.246
.175
.253
0.5
455
388
.390
.800
.121
.110
.103
.171
.356
.242
.246
.174
.253
1.0
455
.88
.390
.280
.122
.110
.104
.173
.356
.242
.246
.174
.254
Table J.3: Robustness to for ACN with TSMixer.
Backbone: RMLP
Average MSE across horizons
Avg.
E
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
ECL
0.05
.449
.378
.384
.277
.162
.164
.136
.200
.354
.246
.244
.189
.265
0.1
.450
.377
384
.277
.159
.157
.131
.187
.354
.246
.243
.189
.263
0.2
450
.377
.83
.277
160
.157
.131
.188
.353
.247
.251
.189
.264
0.5
.488
.377
384
.277
.178
.168
.136
.199
.353
.247
.257
.190
.268
.448
.377
384
.277
.180
.168
.138
.202
.353
.247
.257
.191
.268
Table J.4: Robustness to 7 for ACN with RMLP.
18

---

Channel Normalization for Time Series Channel Identification
K. Robustness to Similarity Space for ACN & PCN
K.1. Similariiy Space for ACN
The similarity between the channels in TS for ACN be calculated either in the data space (X) the latent space (Z). Table [olbee/K.]iniicattte that performance is robust to the choice of space across various datasets with four different backbones, further validating the effectiveness of ACN.
Average MSE across 4 horizons
Sim. space
Avg.
ETTh
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PRMS07
PEMS08
Exchange
Weather
Solar
ECT.
iTrans.
457
.844
.142
.121
102
.254
368
.260
.234
.275
X
.438
.375
.395
.288
100
.099
.085
.16
.349
.247
.221
.158
.242
Z
.438
.374
.395
.288
.098
.088
.085
.153
.349
.245
.220
.158
.241
S-Mam.
.457
.383
.398
.290
.133
.096
.090
.157
.364
.252
.244
.144
.253
X
.488
.355
.394
.285
.107
.092
.073
.121
.357
.247
.228
.162
.240
Z
.488
.374
.394
.284
.107
.092
.073
.121
.357
.247
.228
.162
.240
TSMixer
462
.401
2887
.129
.15
.186
.365
260
.255
.211
.266
ACN
.433
.387
.387
.280
.111
.109
.103
.168
.356
.242
.245
.178
.244
Z
.433
.366
.385
.280
.120
.109
.103
.167
.356
.242
.245
.174
.243
RMLP
.71
381
401
.280
.255
.236
200
.277
.366
.222
.261
.288
.297
ACN
2 x
.448
.376
383
.277
161
.17
131
.186
.353
.246
.243
.189
.262
.448
.376
.383
.277
.159
.156
.131
.187
.353
.246
.242
.189
.262
Table K.1: Robustness to similarity space for ACN.
19

---

Channel Normalization for Time Series Channel Identification
K.2. Similarity Space for PCN
The similarity between the channels TS and the prototypes for PCN can be calculated either in the data space (X), latent space (Z), or the latent space with an additional linear layer (h), which is used to align the space between the input TS and the prototypes. Table iinniaeee that performance is robust the choice of space across various datasets with different numbers of prototypes (K), further validating the effectiveness of PCN.
Average MSE across 4 horizons
PCN (K 5)
Avg.
ETThl
ETTh2
ETTml
ETTm2
Exchange
Weather
.457
.384
.088
.293
.368
.260
.362
Space
X
440
.378
.404
.289
.342
.288
.352
Z
.443
.381
.404
.290
.340
.259
.353
(Z)
.477
.376
.404
.289
.349
.257
.352
Average MSE across horizons
PCN (K~ 10)
Avg.
ETTh1
ETTh2
ETTm1
ETTm2
Exchange
Weather
.457
.384
.408
.293
.368
.260
.362
Space
X
440
.377
.406
.289
.342
.259
.352
7
443
.378
.405
.290
.341
.260
.355
((2)
.438
.377
.403
.289
.339
.257
.351
Average MSE across horizons
PCN (K 20)
Avg.
ETTh1
ETTh2
ETTml
ETTm2
Exchange
Weather
.457
.384
.408
.293
.368
.260
.362
Space
.439
.377
.404
.289
.333
.258
.350
.441
.379
.404
.290
.341
.259
.352
h(Z)
.434
.375
.404
.288
.336
.257
.349
Table K.2: Robustness to similarity space for PCN with K 5, 10, 20
20

---

Channel Normalization for Time Series Channel Identification
L. Application of PCN to Zero-shot Forecasting with TSFMs
We conduct TS forecasting tasks under two types of zero-shot settings with UniTS (Gao et al. 2024); the 1) Zero-shot dataset, which involves evaluation on dataset not seen during training, and the 2) Zero-shot task, where we evaluate a new forecasting horizon not included in the training process by appending mask tokens at the end the TS to predict future time steps.
Zero-shot dataset. In the TS forecasting task on unseen datasets, we evaluate our method with three datasets (NREL 2006 McLeod Gweon 2013 Hyndman et al. 2008). The results, shown Table L.I highlight consistent improvements by incorporating PCN.
Zero-shot horizon. For the TS forecasting task with new horizons, we predict an additional 384 time steps beyond the base forecasting horizon of 96 by appending masked tokens of length 16 the end of the TS. Table [..2 presents the results on four datasets (Zhou al. 2021 Wu al. 2021), showing performance improvements across all datasets.
UniTS
+ PCN
Imp.
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
Solar
.597
.607
.592
.514
0.8%
15.3 %
River
1.374
.698
1,272
.580
7.4%
16.9 %
Hospital
1.067
.797
1.046
.787
2.0%
1.3%
Avg.
1.013
.701
.970
.627
4.2%
10.6% %
Table L.:: Results of forecasting with zero-shot dataset.
UniTS
PCN
Imp.
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
ECL
.237
.329
.229
.322
3.4%
2.2%
ETTh1
.495
.463
.486
.459
1.8%
0.9%
Traffic
.632
.372
.616
.362
2.5%
2.7%
Weather
.335
.336
.334
.335
0.3%
0.3%
Avg.
425
.375
.416
.369
2.1 %
1.6%
Table L.2: Results of TS forecasting with zero-shot horizon.
21

---

Channel Normalization for Time Series Channel Identification
M. Full Results:
Application of CN & ACN
Table M.1 and Table M.2 present the results of TS forecasting for non-CIID and CID models, respectively. The proposed method shows greater improvement in non-CID models, highlighting its role in enabling channel identifiability.
Modele
iTransfrmmc
CNN
AACN
RMLP
CCN
AACN
Mani:
MSF:
MAF
MSF
MAB
MISE
MAF
MSF
MAF
MSSE
MAF
MSR
MAF
.425
.382
.001
.405
192
.316
.429
460
.433
.225
316
.458
.422
.531
505
455
.410
.822
720
.009
.194
J.26
.724
.470
.409
.114
.440
2223
.576
.492
.433
202.
.449
.441
439
.77
.337
192
.375
2015
336
428
.322
720
.446
.441
.113
.436
.43
A49
.31
.445
Ang.
.581
96
.364
336
4750
477
4113
.001
291
96
186
272
.181
764
.262
179
.259
1122
246
.175
.577
192
31.
200
336
355
150
.298
720
Arg.
.203
.357
209
331
.288
.330
200
.326
.277
30.4
.277
.313
.071
.174
10:9
1770
.667
1165
180
.183
1112
1887
.871
.179
1017
200
1860
154
.178
1181
1125
.256
1120
222
.102
316
.161
.72
.11
315
.108
2144
231
.324
216
.312
.176
258
7.80
1118
.3.6
-01%
.285
Avg.
.142
2348
103
2014
J198
.013
2015
-794
1192
.284
.159
.266
[61
.188
.071
.775
.071
.174
197
-215
1123
.202
.833
.191
-4
211
.079
.166
010
181
140
.200
1188
250
.113
.266
133
246
1855
2011
J995
2011
266
.355
7712
.172
.255
172
.109
1.51
.109
-219
443
.463
5111
.258
.35W
Avg.
.121
.222
.065
.166
.058
.195
.236
-321
-212
.304
.156
.265
.007
.105
0150
J.51
.056
.150
1074
.177
1132
J75
/665
.165
1990
1776
173
1175
169
121
.228
116
225
.913
.199
.113
2218
.007
.185
.096
1883
.26
.316
204
2055
.144
251
.172
.25
119
200
.114
.155
.379
.416
344
385
.221
318
Arg.
1013
.205
1801
1728
J185
-174
.210
-764
1.60
.211
.151
.253
PEMS36
243
214
155
.216
2.00
209
331
.166
2211
752
.504
2005
5.31
.334
2vg.
330
1159
.23
.21
.277
247
.305
1611
Ebictange
192
336
.12
.122
225
.13
720
/91
667
Ang.
96
2115
205
2723
96
2354
1801
201
7966
241
.217
192
261
771
151
216
260
316
774
.199
.212
273
700
720
.217
Ang.
.134
.651
221
1558
.230
.249
.261
.313
248
.276
.242
96
.148
.240
.133
.229
.132
.226
201
.257
.164
.253
.162
.22
192
.167
-258
.12
342
.150
.244
209
.277
17%
.362
.173
.262
336
.170
.272
165
762
.164
.260
226
316
191
.20
.198
.218
720
.220
.310
191
256
.157
.250
.273
3.50
212
.312
.230
.312
A02.
.179
.270
1611
225
.158
.256
.258
.313
1100
.222
.189
.276
Average
.25
318
244
700
-241
.205
207
346
2401
.262
.19
Count
46
7
Count
39
39
41
Table M.I: TS backbones w/o CID ability. Full results of TS forecasting tasks.
22

---

Channel Normalization for Time Series Channel Identification
Modks
S-Mambii
CN
+ ACN
TSM
CC
ACN
Mersi
MSE
MAE
MSE
MAE
MIE
MAE
MSB
MAE
MSE
MAE
MIE
MAE
.442
.338
.430
.665
.473
720
.497
.501
.022
.492
.501
.422
.470
.466
.036
Aug.
.455
.418
.449
.435
.358
.500
.11.
A44
.428
.435
720
A32
.648
.423
.841
4225
.441
A54
.462
.031
.449
.426
.445
Aag
3375
.401
.374
.418
.587
.110
.326
.110
192
3375
5022
.374
.301
-364
336
410
/414
4419
.115
.906
.412
A17
.415
.994
.015
.794
720
.744
.551
.414
.511
.472
.448
4544
.450
.466
.446
.466
.444
Axg
.148
.4117
.147
.406
.794
.404
.011
.400
-186
.146
.855
.797
96
.182
.126
.260
.261
.122
197
120
.313
2.46
.907
245
.305
2488
.M08
-901
336
.213
.349
311
.308
.345
313
.345
3111
.345
.303
.342
720
A16
.409
4000
.015
.009
.015
.166
.006
410
4013
.401
.400
Asg.
.290
.353
286
2544
.318
.257
.330
.266
.329
.250
.325
12
1066
.171
.063
1668
1962
1164
1066
.371
0065
.169
.169
J088
1977
080
185
1779
115
1090
.202
.69
197
.085
.195
.165
.277
121
211
.120
.230
.142
1.57
.244
.135
245
.313
.313
1720
2276
.168
.275
2218
.319
2014
.500
.1%6
.112
Axg
.189
.240
105
21.6
.107
-218
.128
-2966
1124
.228
.120
220
12
.073
.177
.069
.170
0072
1775
274
.181
0014
.179
.072
.176
24
1064
190
.077
.192
1855
191
2100
[M11
2003
.187
.197
1011
.213
(091
1166
102
212
121
.299
-121
.234
.117
.24
.125
.236
.103
2200
124
.311
.173
.294
.168
.274
.159
250
Avg
.215
.085
1189
1855
.002
.115
.224
.114
.223
.109
.212
.060
.157
(196
.145
.154
144
1366
.167
1161
.85k
.158
-4
.082
1151
1160
1965
.164
1088
1140
0811
115
.079
.179
.100
.204
264
.175
.150
177
1125
2.20
.127
.224
.113
.155
150
1194
(371
1164
Aug:
.290
.191
.078
1668
.073
.167
.115
.210
.115
.009
.103
.203
.006
.175
.071
.169
.171
12.1
.081
.190
080
152
.079
.181
-4
.110
.216
2003
.992
.192
195
.115
.222
.113
.117
.110
.144
227
.135
2112
-789
181
.274
2045
Avg
215
.216
.225
.167
.250
96
.086
.005
.066
206
1156
.286
.066
2005
.085
2005
.084
.213
Eschange
191
.012
.179
175
206
Axg
764
405
.357
.402
.405
1558
4022
.356
.400
95
.165
2115
162
2002
.224
110
201
.156
.244
268
245
350
Avg.
247
.285
2.00
214
-281
2210
.301
.265
209
Aag.
2335
.135
-275
.147
250
255
.155
-203
720
.214
.10
2289
.352
-223
.316
Avg
.211
2211
-110
181
Averigt:
254
25
Table M.2: TS backbones w/ CID ability. Full results of TS forecasting tasks.
23

---

Channel Normalization for Time Series Channel Identification
N. Full Results: Application of PCN
N.1. Application of PCN to non-TSFMs
Although PCN is developed for scenarios with multiple datasets and varying (e.g. TSFM), can also be applied single-task models trained results of applying PCN with K = 5 to i'Transformer. The results, averaged over 12 datasets and 4 horizons, show an 8.4% performance improvement, which is smaller than the improvement achieved by CN and ACN.
single dataset, assuming the number of channels remains unknown. [N. the
Average MSE
horizons
MTaas.
Avg.
Imp.
ETTh1
ETTh2
ETTml
ETTm2
PEMS03
PEMS04
PEMS07
PEMS08
Exchange
Weather
Solar
ECL
.57
.384
408
.293
.142
.111
.102
.254
.368
.260
.234
.179
.275
CN
.441
.376
.396
2889
.01
.088
.087
.159
.352
.247
.228
.61
.244
11.3%
ACN
.438
.374
.395
.098
.088
.085
.153
.349
.245
.220
.158
12.4%
PCN
.437
.376
4044
289
.117
.120
.101
.176
.349
.257
.232
.169
.252
8.4%
Table N.I: Application of PCN to iTTrasformer.
N.2. Application of PCN to TSFMs
Table 4 summarizes the results of forecasting and classification tasks under supervised and prompt-tuning settings, with full results for both tasks shown in Table N.2 and Table N.3 respectively.
Supervised
Prompt-Tuning
UniTS (LN)
PCN
PCN
Dataset
H
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
NN5
112
.635
.556
.610
.545
.611
.552
.602
.543
96
.172
.233
.168
.272
.174
.277
.173
.278
192
.185
2884
.182
.283
.189
.289
.189
.292
ECL
336
.196
.297
.197
.296
.205
.304
.205
.306
720
238
.321
.227
.321
.251
340
.241
.334
96
.390
.408
.388
.406
.390
.411
.384
.405
192
.428
.432
.438
.434
.432
.439
.433
.432
ETTh1
336
.422
.451
.477
.454
.480
.460
.472
.450
720
489
.476
.484
.475
.532
500
.492
.475
192
.239
.422
.202
.323
.221
.337
.207
.329
Exchange
336
.479
.486
.383
.446
.387
.453
.366
.441
ILI
60
2.48
.94
1.93
.895
2.45
.994
2.14
.940
96
.466
.325
.483
.320
.502
.330
.481
.318
192
.47
.327
.495
.324
.523
.331
.505
.322
Traffic
336
.509
.388
.506
.326
.552
.388
.535
.330
720
.25
350
.536
.341
.626
369
.591
.352
96
.161
.111
.157
.207
.175
.214
.166
.217
192
212
.255
.205
.251
.226
266
.219
.261
Weather
336
266
.255
.262
.293
.880
.033
.275
.299
720
.343
.34
.338
.342
.352
.350
.350
.348
Best Count (/20)
4
16
18
3
16
Average
.469
386
.433
.378
.478
.333
.453
.384
Table N.2: Results of multi-tad task forecasting with UniTS.
24

---

Channel Normalization for Time Series Channel Identification
Supervised
Prompt-Tunme
UniTS (LN)
+ PCN
PCN
Heartbeat
59.0
71.7
69.3
73.1
Japanese Vowels
93.5
92.7
90.8
92.7
PEMS-SP
83.2
84.9
85.0
82.7
SellRegulationSCP2
47.8
55.0
53.3
51.7
SpokenAArabicDigits
97.5
98.0
92.0
94.9
UWaveGeesturll.ibran
79.1
85.3
75.6
84.1
ECG5000
92.6
93.6
93.4
94.0
NonInvasive.
90.5
89.7
27.1
54.8
Blink
99.1
99.8
91.1
98.0
FaceDetection
64.1
66.7
57.6
60.7
DeeccessDevices
60.3
62.1
55.4
59.4
Trace
91.0
96.0
82.0
92.0
FordB
76.0
76.5
62.8
67.2
MotionSenseHAR
92.8
93.2
93.2
94.7
EMOPain
78.0
79.2
80.3
85.1
Chinatown
97.7
98.0
98.0
98.3
Pedesraann
87.3
88.2
77.0
78.5
SharePriceInerease
61.9
63.1
68.4
68.4
Best Count (/18)
2
Average Score
80.6
83.0
75.1
79.5
Table N.3: Results multi task classification with UniTS.
25

---

Channel Normalization for Time Series Channel Identification
O. Visualization of Forecasting Results
To validate the effectiveness of our method, we visualize the predicted results for various L, and H across different backbone architectures and four datasets from diverse domains: ETTml (Zhou et al. (021), Weather (Wu et al. 2021). ECL (Wu
et al. 2021), and PEMS (Liu et al. 2022), using three types of normalizations: base (LN), CN, and ACN.
0.1. Visualization of TSF with iTransformer
qula
Aa
4
mm
M
H: (96,192,336,720)
Base (LN) CN / ACN
Figure O.1: TS forecasting results of ETTml with ITTransformer
H: (96,192,336,720)
Base (LN) CN / ACN
Figure 0.2: TS forecasting results Weather with iTransformer.
26

---

Channel Normalization for Time Series Channel Identification
1
H: (96.192,3366,720)
Base (LN) CN / ACN
Figure O.3: TS forecasting results of ECL with iTransformer.
H: (12,24,48,96)
Base (LN) CN ACN
wiw
Figure 0,4: TS forecasting results of PEMS07 with iTransformer.
27

---

Channel Normalization for Time Series Channel Identification
O.2. Visualization of TSF with RMLP
H: (96, 196,192,336,720
Base (LN) / CN / ACN
Figure 0.5: TS forecasting results of ETTml with RMLP.
H: (96,192,336,720)
Base (LN) CN / ACN
Figure 0.6: TS forecasting results of Weather with RMLP.
28

---

Channel Normalization for Time Series Channel Identification
H: (96.192,3366,720)
Base (LN) CN / ACN
Figure O.7: TS forecasting results of ECL with RMLP.
MM
H: (12,24,48,96)
Base (LN) CN ACN
100
Mul wmmmmn
Figure 0,8: TS forecasting results of PEMS07 with RMLP.
29

---

Channel Normalization for Time Series Channel Identification
0.3. Visualization of TSF with TSMixer
H: (96, 196,192,336,720,
Base (LN) / CN / ACN
Figure 0.9: TS forecasting results of ETTml with TSMixer.
H: (96,192,336,720)
Base (LN) CN / ACN
Figure 0.10: TS forecasting results of Weather with TSMixer.
30

---

Channel Normalization for Time Series Channel Identification
H: (96.192,3366,72)
Base (LN) CN / ACN
Figure 0.11: TS forecasting results of ECL with TSMixer.
H: (12,24,48,96)
Base (LN) CN ACN
M/ M w/yh
Figure O.12: TS forecasting results of PEMS07 with TSMixer.
31

---

Channel Normalization for Time Series Channel Identification
0.4. Visualization of TSF with S-Mamba
Mel m
Mr
H: (96, (96,192,336,720
Base (LN) / CN / ACN
Figure 0.13: TS forecasting results of ETTml with S-Mamba.
H: (96,192,336,720)
Base (LN) CN / ACN
Figure 0.14: TS forecasting results Weather with S-Mamba.
32

---

Channel Normalization for Time Series Channel Identification
H: (96,192,3366,720
Base (LN) CN / ACN
Figure 0.15: TS forecasting results of ECL with S-Mamba.
H: (12,24,48,96)
Base (LN) CN ACN
masees mmm
Figure 16: TS forecasting results of PEMS07 with S-Mamba.
33


Reviews:
[{"review_id": "RmWmFJ4LdV", "reviewer": "Reviewer_ZmeZ", "summary": "The authors propose a new method to adaptively normalize each time series channel distinctly through learned channel specific adaptive parameters. These adaptive parameters for each channel are data dependent are computed through a dynamic weighted summation of a similarity matrix computed between channel token embeddings.\n  This  normalization allows models  to incorporate channel identity information. This allows different channels to produce different outputs even when provided the same input data to forecast on\nEquipped with this adaptive normalization,  models can incorporate channel information when forecasting future values. \n\n\nThe authors also propose a variation of this normalization method that could be deployed for scenarios where the number of channels can change or are unknown (which is the case for time series foundation models). This can be done by reformulating the channel adaptive normalization parameters as the weighted sum of a learnable set of prototypes. \n\nA major contribution of the proposed method is that this normalization scheme can be applied to any existing time series forecasting method.\nThe authors test their adaptive normalization scheme with various time series models (transformers, MLPs, Mamba). These include models which are already incorporate channel identification and models do not. When tested on different time series forecasting datasets,  the results show that the adaptive normalization improves forecasting performance for all models, even models that already incorporate channel identify information .\n\nThe authors also show how their proposed prototype based adaptive channel normalization scheme helps improve performance for Time Series foundation models.", "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": "## Suggestions 1. Explanation of Legends in Figures\n> Reviewer: *\"It would be helpful for readers to provide what different legend in different figures represent. For example, LN in Figure 5? Is that layer normalization? This needs to be clarified*\n\nThank you for your feedback. Due to **space limitations**, commonly used terms such as Channel Normalization (CN) and Layer Normalization (LN) **were not explicitly labeled** in the legends. To ensure clearer understanding, **we will add these clarifications in the revised version**.\n\n&nbsp;\n\n&nbsp;\n\n## Question 1. [Figure 1] Confusing descriptions & Miscolored\n> Reviewer: *The description in Figure 1 is very confusing. The outputs are supposed to be yellow, and the inputs are supposed to be green. no?*\n\n**[Confusing descriptions]**\n\nRegarding the description in the figure, **\"Figure 1: Channel Identifiability\"** is intended to motivate the **necessity of CID**. \n\nTo illustrate this, we present two cases: **“with CID”** and **“without CID”**. Specifically, in the **left** panel, when the local inputs are identical (green), non-CID models fail to distinguish between channels, producing identical outputs (yellow). In contrast, in the **right** panel, applying our proposed CN introduces CID, such that even if the local inputs are the same (green), CID models distinguish between channels, producing different outputs (yellow). \n\nWe **will revise and clarify them** in the updated version. If there are still any confusing aspects regarding the figure, please feel free to ask us; we are happy to provide additional clarification.\n\n&nbsp;\n\n**[Miscolored]**\n\nRegarding the **coloring mistake**, we will fix the colors (green & yellow) in the revised version. Thank you for pointing that out!\n\n\n&nbsp;\n\n&nbsp;\n\n## Question 2. Negative (approximated) entropy\n> Reviewer: *Why is the entropy values negative in Figure 5?*\n\nAs the Gaussian entropy is ***approximated*** based on the data samples, the negative entropy values in **Figure 5** emerge. Previous studies ([1]--[5]) have also adopted this approximation (as mentioned in **L369--374**), and, as seen in [5] Sequence Complementor (AAAI 2025), negative values are observed in **Figure 2-(c)**, **Figure 5**, and **Figure S.6**.\n\n&nbsp;\n\n[1] Ma, et al. \"Segmentation of multivariate mixed data via lossy data coding and compression.\" TPAMI (2007)\n\n[2] Yu, et al. \"Learning diverse and discriminative representations via the principle of maximal coding rate reduction.\" NeurIPS (2020)\n\n[3] Chen, et al.  \"Learning on Bandwidth Constrained Multi-Source Data with MIMOinspired DPP MAP Inference.\" IEEE Transactions on Machine Learning in Communications and Networking (2024)\n\n[4] Chen, et al. \"Rd-dpp: Rate-distortion theory meets determinantal point process to diversify learning data samples.\" WACV (2025)\n\n[5] Chen, et al. \"Sequence Complementor: Complementing Transformers For Time Series Forecasting with Learnable Sequences.\" AAAI (2025)\n\n&nbsp;\n\n&nbsp;\n\n## Question 3. Is $\\hat{\\alpha}_{b, c}^{L}$, a vector or a scalar?\n> Reviewer: *Is $\\hat{\\alpha}_{b, c}^{L}$ a value or a vector in $d$? what dimension does it lie in (For equation 6)?*\n\n\n\n\nIn the **Equation (6)**, \n\n- $\\hat{S}_{b, c, i}$ is a scalar and \n\n- $\\alpha_i^{\\mathrm{L}}$  (or $\\alpha_{i,:}^{\\mathrm{L}}$)  is a $D$-dimensional vector, \n\nresulting in a $D$-dimensional vector. \n\nThanks to the reviewer’s feedback, we found typos in **line 6 of Algorithm 2,3**: $\\alpha_{b,c,d}$ should be the vector $\\alpha_{b,c,:}$, not a scalar, and this also applies to the bias term $\\beta_{b,c,d}$ in line 7.\n\nWe agree that this might be difficult to understand with the explanation around **Algorithm 2**, we will provide a more detailed description of the dimensions in the main text and correct the error.\n\n&nbsp;\n\n&nbsp;\n\n## **Question 4. Missing notations**\n> Reviewer: *What are $B$, $D$  in Algorithm 1, text before equation 4. This is never clarified which makes things confusing for the end user*\n\nThank you for pointing that out. We found some of these definitions were missing. $B$ and $D$ represent the **\"batch size\"** and the **\"number of channels\"**, respectively. \n\nTo improve clarity, we will explicitly define them in **Section 4** and also include them in the algorithm pseudocode.\n\n&nbsp;\n\n&nbsp;\n\n**If there are any unresolved issues, please feel free to discuss them with us!**", "claims_and_evidence": "The claims are supported by extensive experiments on different types of datasets and time series models. The improvement in performance over no adaptive normalization schemes suggest that the proposed scheme does enhance performance.\n\n\nThe authors also claim that the learned  normalization enhances feature representation and improves uniqueness of channel representation. This is backed by their experiments which show how the proposed method increases the entropy associated with representations across channels.", "methods_and_evaluation_criteria": "Yes, the proposed method and the evaluation criteria make sense for the model.\n\nThe method is compared with other normalization schemes, with various time series forecasting models on a variety of datasets", "theoretical_claims": "The main paper doesn't have e theoretical claim.", "experimental_designs_or_analyses": "Yes, the soundness of the validity and experimental design makes sense.\n\nThe datasets used, and the baselines evaluated with all make sense. Particularly for results in Table2 and Table 3.\n\n\nI checked the validity for experiments that showed how normalization learns feature representations which are more diverse/unique across channels.", "relation_to_broader_scientific_literature": "The authors appropriately relate their finding to broader scientific literature. They put their work in context with models that incorporate channel identification  and those that not. They also put their work in context with commonly used normalization schemes or incorporate channel specific parameters (such as channel specific identified or Channel specific LORA)", "essential_references_not_discussed": "None that I can think of", "other_strengths_and_weaknesses": "I think the main contribution of the paper in terms of its methodology is strong, but a major weakness is the lack of clarity in the paper which hampers reading. I will gave examples of this in my comments/suggestions/questions \n\nThis is the main reason I would incline towards giving it a weak acceptance..", "other_comments_or_suggestions": "It would be helpful for readers to provide what different legend in different figures represent.\n\nFor example, LN in Figure 5? Is that layer normalization? This needs to be clarified", "questions_for_authors": "- The description in Figure 1 is very confusing. The outputs are supposed to be yellow, and the inputs are supposed to be green. no?\n- Why is the entropy values negative in Figure 5?\n- Is $\\hat{\\alpha}_{b,c}^L$ a value or a vector in $d$? ? what dimension does it lie in (For equation 6) . \n- What are $B$, $D$ in Algorithm 1 , text before equation4 . This is never clarified which makes things  confusing for the end user", "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": "e7w2ztYvEX", "reviewer": "Reviewer_PycD", "summary": "- The Channel Normalization (CN) strategy is proposed to enhance the Channel Identifiability (CID) of Time Series (TS) models by assigning specific parameters to each channel.\n- Two variants of CN, Adaptive CN (ACN) and Prototypical CN (PCN), are introduced to dynamically adjust parameters and handle datasets with unknown or varying numbers of channels, respectively.", "questions": "", "limitations": "", "rating": 5, "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": "## Weakness 1. Miscolored Figure 1\n> Reviewer: *\"Figure 1: ~ producing different outputs (green) even with same inputs (yellow).\" Maybe the colors are mispositioned.*\n\nThank you for pointing that out. We will fix it in the revised version.\n\n&nbsp;\n\n&nbsp;\n\n## Question 1. Details about the performance of CN, ACN, PCN\n> Reviewer: *I have a concern about the performance of CN, ACN, PCN. The authors should provide some details about the performance.*\n\n**[PCN vs. CN/ACN]**  \nIf your concern is the comparison between PCN and CN/ACN, please refer to **Table 5**. Although PCN extends CN, it underperforms CN in the **”single”-task model**, as PCN is intended for scenarios where the ***number of channels is unknown***. As shown in **Table 3**, PCN is tailored for **zero-shot scenarios** and does not allocate parameters per channel but rather shares them across channels. For example, only PCN is applied in **Table 4**, as CN and ACN are not applicable to zero-shot scenarios.\n**L324–326 (Right column)** We attribute this to the fact that, unlike CN and ACN which assign each channel a distinct parameter, PCN assigns each prototype a distinct parameter,\n\n&nbsp;\n\n**[CN vs. ACN]**  \nIf your concern is the comparison between CN and ACN, ACN extends CN by incorporating local parameters that **dynamically** adapt based on the input TS, considering its dynamic nature (**L20--23 (Left column)**). As seen in **Table 8**, this enhancement improves performance while introducing only a negligible increase in computational complexity.\n\n&nbsp;\n\n**[Others]**  \nAlternatively, if your inquiry pertains to any of the following, we provide the relevant information:\n- a) **Performance results**: \n  - [CN/ACN] **Tables 2, 6, 7**\n  - [PCN] **Tables 4, 5**\n- b) **Interpretation of results**: **Section 5.1**\n- c) **Experimental details**: \n  - [Datasets] **Appendix A.1**\n  - [Settings] **Appendix A.2**\n\nIf this response does not fully address your concern, we would appreciate any further clarification or specific details you could provide.\n\n&nbsp;\n\n&nbsp;\n\n## Question 2. Initialization of parameters of CN, ACN, PCN\n> Reviewer: *How do authors initialize the parameters of CN, ACN, PCN? How do the different initialization methods affect the metrics?*\n\nThe initialization of the parameters for CN, ACN, and PCN is designed to ensure that ***no normalization occurs when learning has not yet taken place***:\n- The **scale** parameter ($\\alpha$) is initialized to **1**.\n- The **shift** parameter ($\\beta$) is initialized to **0**.\n\nThis aligns with the default initialization used in **PyTorch normalization layers**:\n- (Pytorch) **Layer** normalization initialization: https://github.com/pytorch/pytorch/blob/1eba9b3aa3c43f86f4a2c807ac8e12c4a7767340/torch/nn/modules/normalization.py#L210\n- (Pytorch) **Batch** normalization initialization: https://github.com/pytorch/pytorch/blob/1eba9b3aa3c43f86f4a2c807ac8e12c4a7767340/torch/nn/modules/batchnorm.py#L93\n\nTherefore, the parameters for CN, ACN, and PCN are initialized as follows:\n\n- CN: $\\alpha=1, \\beta=0$\n\n- ACN: $\\alpha^{\\text{G}}=1, \\alpha^{\\text{L}}=0, \\beta^{\\text{G}}=1, \\beta^{\\text{L}}=0$, so that $\\alpha=1, \\beta=0$\n\n- PCN: $\\alpha^{\\text{P}}=1, \\beta^{\\text{P}}=0$\n\nThis initialization is documented in the Python files within `/layers` in the attached anonymous GitHub link.\n\nWe acknowledge that the explanation of initialization was omitted in the paper. As the reviewer pointed out, **we will include this explanation in the revised version**. Thank you for your feedback.\n\n&nbsp;\n\nAdditionally, after testing on four ETT datasets and running experiments three times with different random initialization seeds (using PyTorch's default `nn.Parameter` initialization), **the effect on the average across four horizons was negligible**, differing only at the fourth decimal place.\n\n&nbsp;\n\n&nbsp;\n\n**If there are any unresolved issues, please feel free to discuss them with us!**", "claims_and_evidence": "The paper proves the effectiveness of CN, ACN, and PCN in improving model performance through experiments on multiple models and datasets, such as 12 datasets and 4 backbone networks, using Mean Squared Error (MSE) and Mean Absolute Error (MAE) as indicators. The evidence is relatively sufficient.", "methods_and_evaluation_criteria": "- The proposed methods of CN, ACN, and PCN are designed to address the channel identifiability problem in TS models, with clear improvement goals and reasonable method designs.\n- The selection of common TS datasets and evaluation metrics, MSE and MAE, can effectively measure the performance of models in TS forecasting tasks.", "theoretical_claims": "The paper proves that CN can obtain more informative representations and potentially reduce forecasting errors from the perspective of theoretical entropy analysis. Its universality in practical applications remains to be further verified.", "experimental_designs_or_analyses": "The experiments select multiple datasets and different backbone networks, and compare CN and its variants with other methods. The experimental design is relatively comprehensive.", "relation_to_broader_scientific_literature": "Based on the literature related to TS forecasting models and normalization methods, the paper proposes a new method to enhance CID. Compared with recent similar studies (such as InjectTST, C - LoRA, etc.), it highlights the advantages and innovativeness of its own methods.", "essential_references_not_discussed": "There are no obvious unreferenced papers.", "other_strengths_and_weaknesses": "None.", "other_comments_or_suggestions": "\"Figure 1: Channel identifiability. Applying the proposed methods to non-CID models enables to distinguish among channels, producing different outputs (green) even with same inputs (yellow).\" Maybe the colors are mispositioned.", "questions_for_authors": "- Q1: I have a concern about the performance of CN, ACN, PCN. The authors should provide some details about the performance.\n- Q2: How do authors initialize the parameters of CN, ACN, PCN? How do the different initialization methods affect the metrics?", "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": "V03yBuI6Yr", "reviewer": "Reviewer_LY88", "summary": "The paper talk about the importance of Channel Identifiability (CID) when modeling multivariate time series data. The paper talk about how existing methods failed to provide CID capability. To solve this problem, the model proposed various Channel Normalization (CN) method. CN is a type of normalization method that uses different affine transformation for different channel. In additional a simple CN method, the paper also provided two extension 1) the Adaptive CN (ACN) method which also models the dependency between input dimensions in a data dependent fashion and 2) Prototypical CN (PCN) method which uses learned prototype to model different dimensions.", "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": "## Weakness 1. PCN’s CID capability\n> Reviewer: *I am not fully convinced that PCN provides CID capability. ~ I believe PCN is more of an extension of CN (i.e., an effective channel normalization method that does not provide CID) and should be discussed in the appendix.*\n\nThank you for pointing this out. We acknowledge that PCN indeed does not provide CID capability, as PCN does not assign an identity to **“individual channels”** but rather to **“channel clusters”**.\n\nPCN is specifically designed for scenarios encountering **new** channels in **new** datasets (common settings for foundation models), where ***channel identification becomes infeasible***. Our proposed PCN address this challenge by assigning **“channel cluster”** identification via learnable prototypes as affine transformation parameters after normalization.\n\nWhile the current manuscript presents PCN as an extension of CN in that it also provides **a form of identity (to either channels or channel clusters)**, we recognize the reviewer's concern that it may require a relaxed version of CID for channel clusters. We will clarify this distinction in the revision.\n\n&nbsp;\n\n## Weakness 2. Missing notations\n> Reviewer: *The variables used in Section 4 are not introduced. What are $B$, $C$, $D$, and $K$? It is difficult to understand the methods without looking at the source code.*\n\nThank you for pointing this out. We found some of these definitions were missing. \n\n$B$ represents the batch size, $C$ denotes the number of channels, $D$ is the hidden dimension, and $K$ refers to the number of prototypes. \n\nTo improve clarity, we will explicitly define $B$, $C$, $D$, and $K$ in **Section 4** and also include them in the algorithm pseudocode.\n\n&nbsp;\n\n&nbsp;\n\n**If there are any unresolved issues, please feel free to discuss them with us!**", "claims_and_evidence": "- Claims:\n    1. Some existing methods lack CID capability, potentially leading to suboptimal performance.\n    2. CN is a simple yet effective method to provide CID capability for different models.\n    3. ACN improves upon CN by performing normalization in a data-dependent fashion.\n    4. PCN also provides CID capability.\n- Evidence:\n    1. This claim is supported by Figure 1 and Table 1.\n    2. This claim is backed by experimental results and can be easily inferred from the methodology, as the method is straightforward.\n    3. This claim is also supported by experimental results and can be easily inferred from the methodology.\n    4. This claim is not fully substantiated. Although PCN improves model performance, the source of the improvement is unclear. Moreover, I am not fully convinced that PCN provides CID, which I will elaborate on in later sections.\n- Missing: Theoretical analysis is absent, but it is not necessary as it is easy to see how CN/ACN provides CID.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria make sense for the problem at hand.", "theoretical_claims": "I am not fully convinced that PCN provides CID capability. Comparing the algorithms for CN (Algorithm 1), ACN (Algorithm 2), and PCN (Algorithm 3), we can see that PCN is the only one without a global parameter for normalization. Without a global parameter, the normalization process cannot differentiate between two channels if they have identical inputs (Figure 1). This could be the main reason why PCN performs worse than CN and ACN. The paper would be stronger if PCN were not included in the main text. I believe PCN is more of an extension of CN (i.e., an effective channel normalization method that does not provide CID) and should be discussed in the appendix.", "experimental_designs_or_analyses": "Yes, I have checked the experimental designs and analyses. No severe issues were found.", "relation_to_broader_scientific_literature": "The proposed CN methods are simple yet effective, which is a novel contribution not seen in prior work.", "essential_references_not_discussed": "To the best of my knowledge, there are no essential references missing from the paper's discussion.", "other_strengths_and_weaknesses": "- Strengths: The motivation behind the proposed methods is clear, and the proposed methods are simple yet effective. \n- Weaknesses: One of the proposed methods (PCN) seems out of place and may not fully align with the paper's main contributions.", "other_comments_or_suggestions": "The variables used in Section 4 are not introduced. What are B, C, D, and K? It is difficult to understand the methods without looking at the source code.", "questions_for_authors": "1. How does PCN provide CID?", "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 proposes Channel Normalization (CN) and its variants—Adaptive CN (ACN) and Prototypical CN (PCN)—to improve Channel Identifiability (CID) in time series models. CN assigns distinct affine parameters to each channel; ACN adapts these based on input, and PCN enables generalization to unknown or varying numbers of channels. The methods are lightweight, architecture-agnostic, and show strong empirical performance across diverse models and datasets.

The reviewers appreciated the broad applicability, consistent improvements, and simplicity of the approach. While some raised clarity concerns and questioned PCN’s CID ability, the authors addressed these through a detailed and thoughtful rebuttal, clarifying that PCN offers cluster-level identification and is tailored for foundation model settings.

Overall, this is a well-motivated and practically useful contribution to time series modeling. The revisions and response fully resolve remaining concerns. I recommend acceptance.
Decision: Accept (poster)