Rethinking Channel Dependence for Multivariate Time Series Forecasting: Learning from Leading Indicators
Authors: Lifan Zhao
Yanyan Shen
Venue: ICLR 2024 poster (2024)
Keywords: Multivariate time series forecasting
channel dependence
lead-lag relationships
distribution shift
Primary area: general machine learning (i.e., none of the above)
TL;DR: We rethink the distribution shift in multivariate time series, where channel dependence varies over time. We propose LIFT to dynamically select and leverage leading indicators, which emprically improves SOTA forecasting methods by 5.5% in average.

Abstract: Recently, channel-independent methods have achieved state-of-the-art performance in multivariate time series (MTS) forecasting. Despite reducing overfitting risks, these methods miss potential opportunities in utilizing channel dependence for accurate predictions. We argue that there exist locally stationary lead-lag relationships between variates, i.e., some lagged variates may follow the leading indicators within a short time period. Exploiting such channel dependence is beneficial since leading indicators offer advance information that can be used to reduce the forecasting difficulty of the lagged variates. In this paper, we propose a new method named LIFT that first efficiently estimates leading indicators and their leading steps at each time step and then judiciously allows the lagged variates to utilize the advance information from leading indicators. LIFT plays as a plugin that can be seamlessly collaborated with arbitrary time series forecasting methods. Extensive experiments on six real-world datasets demonstrate that LIFT improves the state-of-the-art methods by 5.4% in average forecasting performance. Our code is available at https://github.com/SJTU-Quant/LIFT.

Published as a conference paper at ICLR 2024
RETHINKING CHANNEL DEPENDENCE FOR MULTI- VARIATE TIME SERIES FORECASTING: LEARNING FROM LEADING INDICATORS
Lifan Zhao
Yanyan Shen
Shanghai Jiao Tong University mogiciand an2338 jtu. edu. cn
Shanghai Jiao Tong University shenyy@sjtu. edu. on
ABSTRACT
Recently, channel-independen methods have achieved state-of-the -fet---rrr perfor- mance in multivariate time series (MTS) forecasting. Despite reducing overfitting risks, these methods miss potential opportunities in utilizing channel dependence for accurate predictions. We argue that there exist locally stationary lead-lag re- lationships between variates, i.e.. some lagged variates may follow the leading indicators within a short time period. Exploiting such channel dependence is ben- eficial since leading indicators offer advance information that can be used to re- duce the forecasting difficulty of the lagged variates. In this paper, we propose a new method named LIFT that first efficiently estimates leading indicators and their leading steps at each time step and then judiciously allows the lagged variates to utilize the advance information from leading indicators. LIFFT plays as a plugin that can be seamlessly collaborated with arbitrary time series forecasting methods. Extensive experiments on six real-world datasets demonstrate that LIFT improves the oate-oof-thear methods by 5.4% in average forecasting performance. Our code is available at https:: / thub. com www.TUU--uunttLttee
INTRODUCTION
Multivariate time series (MTS) forecasting, one of the most popular research topies, a fundamental task in various domains such as weather, traffic, and finance. An MTS consists of multiple channels (a.k.a,, variates'), ), where each channel is a univariate time series. Many MTS forecasting researches argue each channel has dependence on other channels. Accordingly, numerous approaches adopt channel-dependent (CD) strategies and jointly model multiple variates by advanced neural archi- tectures. including GNNs (Wu et al., 2020; Cao et al., 2020; Huang et al., 2023; Yi et al., 2023a), MLPs (Chen et al., 2023; Ekambaram et al., 2023; Wang et al., 2024a; Yi et al., 2023b), CNNs (Wu al., 2023), Transformers (Zhou et al., 2021; Ni et al., 2023; Wang al., 2024b; Liu et al., 2023a), and others (Shen et al., 2024; Jia et al., 2023; Fan et al., 2024).
Unexpectedly, CD methods have been defeated by recently proposed channel-indepenade methods (Nie et al., 2023; Lee et al., 2024; Zhou et al., 2023; Jin et al., 2024; Cao et al., 2024; Chen al., 2024; Dai et al., 2024) and even a simple linear model (Zeng et al., 2023; Li al., 2023a; Xu et al., 2023). These CI methods seperately forecast each univariate time series based on its own historical values, instead of referring to other variates. While only modeling cross-time dependence, CI Transformers (Nic et al., 2023; Zhou et al., 2023) surprisingly outperform CD Transformers that jointly model cross-ting time and cross-vard vross-variate dependence (Grigsby et al., 2021; Zhang & Yan, 2023). One reason is that existing CD methods lack prior knowledge about channel dependence and may encounter the overfitting issue (Han et al., 2023). This gives rise to an interesting question: is there any explicit channel dependence that is effective to MTS forecasting?
(CI)
In this work, we turn the spotlight on the locally stationary lead-lag relationship between variates. An intriguing yet underestimated characteristic of many MTS is that the evolution of variates may
"corresponding author.
We use the terms "variate" and "channel" interchangeably.

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Published as a conference paper at ICLR 2024
Lookback Window
Horizon Window
Lookback Window
Horizon Window
1/1
2/2
1/2
1/3
V/
(a)
(b)
Figure I: Illustration of locally stationary lead-lag relationships. (a) On training data, three variates V1. 02 and by share similar temporal patterns (see colors) across the lookback window and horizon window, while 2n and 12 run ahead of Byy by four and two steps, respectively. However, the leading indicators and leading steps can only keep static for short period. (b) On test data, U1 is no longer a leading indicator, and 1/2 also changes its leading steps to five.
lag behind some other variates, termed leading indicators. Leading indicators may directly influ- ence the wave of other variates, while the influence requires a certain time delay to propagate and take effect. For example, increasing concentration of an anti-fever drug in the blood may cause a decrease in body temperature after an hour but not immediately. On top of this, another common case is that both leading indicators and lagged variates depend on some latent factors, while the leading ones are the first to get affected. For example, typhoon first cools down coastal cities and, after a few days, cools down inland cities.
As such effects typically change little within a certain period, the lead-lag relationships are locally stationary once established. As illustrated in Figure la, the lagged variate and its leading indicators share similar temporal patterns across the lookback window and the horizon window. If a leading indicator evolves d-step ahead of the target variate, the latest d steps of its lookback window will share similar temporal patterns with the future ā steps of the lagged variate. Particularly, when the lagged variate completely follows its leading indicator, the difficulty of forecasting H steps for the lagged variate can be reduced to forecasting H 8 steps by previewing the advance information. Despite the advent of lead-lag relationships, the dynamic variation in leading indicators and leading steps poses the challenge to modeling channel dependence. As shown in Figure 1, the specific leading indicators and the corresponding leading steps can vary over time.
In light of this, we propose method named LIFT (short for Learning from Leading Indicators For MTS Forecasting), involving three key steps. First, we develop an efficient cross-correlen corselcorn com- putation algorithm to dynamically estimate the leading indicators and the leading steps at each time step. Second, as depicted in Figure 2, we align each variate and its leading indicators via a target- oriented shift trick. Third, we employ a backbone to make preliminary predictions and introduce a Lead-aware Refiner to calibrate the rough predictions. It is noteworthy that many MTS are hetero- geneous, where the variates are different dimensions of an object (e.g., wind speed, humidity, and air temperature in weather). In these cases, the lagged variates may be correlated with the leading indicators by sharing only a part of temporal patterns. To address this issue, we exploit desirable sig- nals in the frequency domain and realize the Lead-aware Refiner by an Adaptive Frequency Mixer that adaptively filters out undesirable frequency components of leading indicators and absorbs the remaining desirable ones. The main contributions of this paper are summarized as follows.
We propose a novel method called LIFT that exploits the locally stationary lead-lag relation- ship between variates for MTS forecasting. LIFT works as a plug-and-play module and can seamlessly incorporate arbitrary time series forecasting backbones.
We introduce an efficient algorithm to estimate the leading indicators and the corresponding leading steps at any time step. We further devise a Lead-aware Refiner that adaptively leverages the informative signals of leading indicators in the frequency domain to refine the predictions of lagged variates.
Extensive experimental results six real-world datasets demonstrate that LIFT significantly im- proves the tfate-of-the-al methods in both short-term and long-term MTS forecasting. Specifi- cally, LIFT makes an average improvement of 7.9% over CI models and 3.0% over CD models. We also introduce a lightweight yet strong method LightMTS, which enjoys high parameter efficiency and achieves the best performance on popular Weather and Electricity datasets.
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V11
D4 shift
1/2
D2 shift
1/
1/
Misplaced!
Aligned!
Overfitting training patterns in Figure
(b) Proposed target-oriented shift
Figure 2: Illustration of our key idea. In one case of test data, longer leads U3. Instead, the leading indicators of vy are v2 and u4. which lead by five and three steps, respectively. An intuitive idea is to shift 02 and 04 by the corresponding leading steps to keep them always aligned with tz.-
PRELIMINARIES
multivariate time series (MTS)² is denoted by X (x(1)).. .xO)), where is the number of variates (a.k.a. channels) and (z) is the time series of the j-th variate. Given an L-Iength lookback window X t- {x,
x(() Jj=1 E CxL, the MTS forecasting task at time & aims to predict H consecutive future time steps in the horizon window, i.e., R++1: RC
-+1 1++11+11"
- variate i leads variate j by steps at time recent observations to estimate the leading indicators and the leading steps. Specifically, the lead-lag relationship can which is be quantified by the cross-correlation coefficient between p(f) 21-41-6:t-6 and x() 14111
We assume x,(1)
*++1++H similar to xx(1
Through the lens of locally stationary lead-lag relationships,
defined as follows.
Definition 1 (Cross-conrelation coefficient), Assuming variate i is 8 steps ahead of variate j over the L-length lookback window, the cross-correlation coefficient between the two variates at time t is defined as:
R2)( ( L+1a)
x00 -
x60 -p())
Σ
L
a
P=t-L+1
where 2 E R and o() E IR represent the mean and standard variation of the univariate time series within the lookback window, respectively.
THE LIFT APPROACH
In this section, we propose our LIFT method that dynamically identifies leading indicators and adaptively leverages them MTS forecasting.
3.1 OVERVIEW
Figure 3 depicts the overview of LIFT, which involves 6 major steps as follows.
(1) Preliminary forecasting, Given a lookback window I+ we first obtain rough predic-
tions X X4 from a black-box backbone, which can be implemented by any existing time series forecasting model.
(2) Instance normalization. Given X,-++1 and X K(+1+H we apply instance normaliza-
tion (Kim et al., 2022) without affine parameters as to unify the value range across the variates. Specifically, based on the mean and standard deviation each variate in X.-L+1I. we obtain normalized lookback window Xt-L+1:: and normalized predictions X
(3) Lead estimation. Given X 1:ts the Lead Estimator calculates the cross-correlation coeffi cients for pair-wise variates. For each variate we select the K most possible leading indicators I!-) and cross-
I!-i E T K (K&C) along with the corresponding leading steps {8(5)
correlation coefficients R(1) E K
We use hald symhols denote matrices of multiple variates.
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Xi+1c+#
denormalize
F-1
cross-correlation
Lead-aware Refiner
LIFT
Adaptive Frequency Mixer
Lead Estimator
leading steps
State Estimator
normalize
X1-tt1
Backbone
X Retret
cross
aligned
lookbbck
correlation
sequences
window
Figure 3: Overview of LIFT. All layers in the grey background are non-parametric. We depict the input of the lookback window by solid curves and the predictions of the horizon window by dashed curves. As illustration, choose the two most possible leading indicators for each target variate, e.g. the orange and the yellow ones are leading indicators of the red at time
(4) Target-oriented shifts. After obtaining I( and (5 Jizz5) for variate j. we shift x!O 1-L+1 and X 141+H by 6(0) steps where E I(d) We thereby obtain j-oriented MTS segment s ERKH where the leading indicators get aligned with variate in the horizon window.
(5) Lead-aware refinement. The Lead-aware Refiner extracts signals from s( and refines the normalized preliminary predictions X(1) ++1:+ as
x()
(+1++H
(6) Instance denormalization. Finally, we denormalize x()
++++ with the original mean and ++1:+H
standard deviation, yielding the final predictions X (i)
Training scheme. We can jointly train the backbone and Lead-aware Refiner by the MSE between X Ri-1:0+H and the ground truth R++14+H. Alternatively, given a pretrained and frozen backbone, we can precompute the preliminary predictions only once on training data, reducing the time of hyperparameter tuning and GPU memory occupation during training.
non-trivial to leverage the lead-lag relationships due issues of efficiency and noise. As Eq. (1) requires OLL), brute-force estimation method that searches all possible 8 in {1..... I.} requires ( computations. Also, S(3) can contain some irrelevant patterns from leaders which are noise to the lagged variate.
Technical challenges.
Notably,
To tackle these issues, we implement the Lead Estimator by an efficient algorithm of O(L log L) complexity. And we develop the Lead-aware Refiner by an Adaptive Frequency Mixer that adap- tively generates frequency-domain filters and mixes desirable frequency components according to the cross-correlations and variate states.
3.2 LEAD ESTIMATOR
Given the normalized lookback window Xt-++1:+ the Lead Estimator first computes the cross- correlation coefficients between each pair of variate and variate based extension of Khinchhn theorem (Wiener, 1930) (see details in Appendix A). Formally, we estimate the coefficients for all possible leading steps in {0, ........ L L-1) 1} at once by the following equation:
R R - (
1 F(X F(X
where F is the Fast Fourier Transform, F-1 is its inverse, C is the element-wise product, and the bar denotes the conjugate operation. The complexity is reduced (l T.).
Note that variates can exhibit either positive or negative correlations. The leading step 8/00 be- tween the target variate j and its leading indicator is meant reach the maximum absolute cross-

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correlation coefficient, i.e.,
1 (T)1- ISTSL-1
arg
For simplicity, we denote the maximum absolute coefficient |/2 (5! as |2,7" Then, we choose K variates that show the most significant lead-lag relationships as leading indicators of variate j. which are defined as:
I arg (op Isisc
Specifically, the K leading indicators Z are sorted by cross-correlations in descending order, i.e., the k-th indicator in [ has the k-th highest |R[ w.r.f variate Furthermore, we use R2!5)) E IR 11ezz
to denote array of (12222 "1e
Notably, our Lead Estimator is non-parametric and we can precompute the estimations only training data, instead of repeating the computations at every epoch.
3.3 LEAD-AWARE REFINER
For each variate j. the Lead-aware Refiner is to refine {(0 1+++ by its leading indicators. We will describe the refinement process for variate j. and the other C - variates are refined in parallel.
For each leading indicator E I(d), shift its sequence by the leading
Target-oriented shifts step as follows:
x
if
H
X 14144
(i-))
(11-6 44
(5)
x (0)
X
otherwise
141
where || is the concatenation.
For leading indicator i that is negatively correlated with the variate j. we flip its values at each time step to reflect R(()= Formally, for each i I we have:
turn turn(X X (1-5)
141+4 H) sign( R(!r)) x(i-)) (4184
We then collect (turn X (i-1)) (41:+H) | I(1) target-oriented MTS segment s E RK
State estimation For a comprehensive understanding of leading indicators, it is noteworthy that the lead-lag patterns also depend variate states. Different variates lie in their specific states with some intrinsic periodicities (or trends), egg., solar illumination is affected by rains in the short term but keeps its daily periodicity. The state of variate may also change correlation strengths with other variates, e.g., correlations between the traflic speeds of two adjacent roads strong within peak hours but much weaker within off-peak hours. Therefore, the variate states are informative signals that can guide us to filter out uncorrelated patterns.
time, exhibiting different
Assuming there are N states in total, we estimate the state probabilities variate time by:
p()) softmax (P!") faaee (x
(2 x
where p()) E N represents the intrinsic state distribution of variate j and is a learnable parameter, (3) fstate RL + N is implemented by linear layer, and p(3) (Pin) E R&N includes the
probabilities of all potential states at time t. Our adaptive frequency mixer will take p()) to generate filters to filter out noisy channel dependence according to the variate state.
Adaptive frequency mixer To extract valuable information from leading indicators, propose to model cross-variate dependence in the frequency domain. Given the normalized predictions of variate j and its target-oriented MTS segment s(1), we derive their Fourier transforms by:
(()() (1) (14+H) and T F(S!"))
(8)
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Filter Generation
Frequency Filtering
Frequency Mixing
(,,
a'kk R=
sum
R
Filter Factory
(-0)
K
Linear
(LU k==1
sum
p()
u
Figure 4: Architecture of the adaptive frequency mixer.
where FF is the Fast Fourier Transform, VO) E CLH/2]+1, and UG) E CKX({//2++)), Each element of UG), denoted as U/"), is the frequency components of the k-th leading indicator. Let A Ub v denote the difference between variate j and the k-th leading indicator.
Intuitively, the preliminary predictions deserve more refinement from the leading indicators when the estimated correlation R. is large. To filter signals in V and UG), we employ a filter factory to generate 2K + frequency-domal filters defined below:
U ()) U. - - U.K A.1
1 m=
(0)
=
( fz (R.))
TTA
where Sn R + R(2 is a linear layer with parameters specific to the n-th state. On the hand, j is directly influenced by the k-th leader, and the ground-truth v-() contains a degree of U.))
the first 2K filters to model two kinds of lead-lag relationships: (1) variate
e.g., vili VG) + +r(h) U(); (2) variate is similar to the k-th leader when they both influenced by latent factor, and the ground-truth true is the interpolation between V(J) and U e.g., v-l) true = (1 A @ v(i + AA UU v + r @ A On the other hand, we (i) use TV E R to dismiss unreliable frequency components of vo), Formally, we scale the frequency components by:
(1)
.P) = ( V(),
Jh -
o v()
A -A
(10)
Then, gather information from K leading indicators and mix the frequency components by:
(D)
K
A
(11)
where g: : (3(1/221+1) + CLH/21+1 is complex-valued linear layer.
Finally, we apply inverse Fast Fourier Transform and denormalization in order to derive the final refined predictions, which are formulated as:
500
4+1+ denorm ( (V-6)))
(12)
where we use the mean and standard deviation of x()
1+ for denormalization.
3.4 DISCUSSION
Reasoning why CD models show inferior performance. Many variates are unaligned with each other, while traditional models (e.g.. Informer (Zhou et al., 2021)) simply mix multivariate infor- mation at the same time step. Consequently, they introduce outdated information from lagged variates which noise and disturb predicting leaders. Though other models (e.g., Vector Auto- Regression (Giannone et al., 2010)) memorize CD from different time steps by static weights, they suffer from overfitting issues since the leading indicators and leading steps vary over time.
LIFT can cooperate with arbitrary time series forecasting backbones. When combining LIFT with CI backbone, we decompose MTS forecasting into stages which focus modeling time

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Published as a conference paper at ICLR 2024
dependence and channel dependence, respectively. This scheme avoids introducing noisy channel dependence during the first stage and may reduce optimization difficulty compared with traditional CD methods. When combining LIFT with CD backbone, we expect LIFT to refine the rough predictions with the actual observations of leading indicators in S.
LIFT alleviates distribution shifts by dynamically selecting and shifting indicators. Existing normalization based methods (Kim et al., 2022; Fan et al., 2023; Liu et al.. 2023b) handle distri- bution shifts of the statistical properties (e.g.. mean and variance) in the lookback window and the horizon window. Our work is orthogonal to them as we take a novel investigation into a different kind of distribution shifts in channel dependence (see visualization in Appendix D.2).
4 LIGHTWEIGHT MTS FORECASTING WITH LIFT
Thanks to the flexibility of LIFT, we introduce a lightweight MTS forecasting method named LighIMTS, where simple linear layer serves as a CI backbone. Following Li et al. (2023a), we conduct instance normalization before preliminary forecasting to alleviate distribution shifts.
As we do not learn representations in the high-dimensional latent space, LightMTS is more lightweight than popular CD models, including Transformers (Zhang & Yan, 2023; Liu et al., 2023a) and CNNs (Wu et al., 2023). Empirical evidence is provided in Appendix D.1, where the parameter efficiency of LightMTS keeps similar to DLinear Zeng et al. (2023).
EXPERIMENTS
5.1 EXPERIMENTAL SETTINGS
Datasets. We conduct extensive experiments on six widely-used MTS datasets, including Weather (Zeng al., 2023), Electricity (Wu et 2020), Traffic (Lai al., 2018), Solar (Liu al., 2023a), Wind (Liu et al., 2022), and PeMSD8 (Song et al., 2020). We provide the dataset details in Appendix C.I and conduct experiments on more datasets in Appendix D.3.
Comparison Methods. As LIFT can incorporate arbitrary time series forecasting backbones, we verify the effectiveness of LIFT with (i) two state-of-the-art CI models: PatchTST (Nie et al., 2023) and DLinear (Zeng et al., 2023); (ii) the state-of-the-ar CD model: Crossformer (Zhang & Yan, 2023): (iii) a classic CD model: MTGNN (Wu al., 2020). We use them instantiate the backbone of LIFT, while we keep the same model hyperparameters for fair comparison. We also include the baselines of PatchTST, such as FEDformer (Zhou al., 2022) and Autoformer (Wu et al., 2021).
Setups. All of the methods follow the same experimental setup with the forecast horizon H E (24, 48, 96, 192, 336, 720] for both short-term and long-term forecasting. We collect some baseline results reported by PatchT'S to compare performance with LightMTSS, where PStchTST has tuned the lookback length L of FEDformer and Autoformer. For other methods, we set L to 336. We use Mean Squared Error (MSE) and Mean Absolute Error (MAE) as evaluation metrics.
5.2 PERFORMANCE EVALUATION
Table compares the forecasting performance between the four state-of-the-arr methods and LIFT on the six MTS datasets, showing that LIFT can outperform the SOTA methods in most cases. Specifically, LIFT improves the corresponding backbone by 5.4% on average.
Improvement over CI Backbones. LIFT makes an average improvement 7.9% over PatchTST and DLinear on the six datasets. Notably, PatchTST and DLinear surpass Crossformer and MTGNN by a large margin on Weather, Electricity, and Traffic datasets, indicating the challenge of modeling channel dependence. Intriguingly. LIFT significantly improves CI backbones by an average margin of 4.7% on these challenging datasets, achieving the best performance in most cases. This confirms that LIFT can reduce overfitting risks by introducing prior knowledge about channel dependence.
Improvement over CD Backbones. LIFT makes an average improvement of 3,0% over Cross- former and MTGNN the six datasets. As CD backbones outperform CI ones Solar, Wind, and
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Table 1: Performance comparison in terms of forecasting errors. We highlight the better results between each pair of backbones and LIFT in bold and the best results among all methods on each dataset with underlines. We show the relative improvement of LIFT over the corresponding back- bone in the rightmost column.
Method
MSE
PatchTST
MAE
MSE
+ LIFT
MAE
MSE
DLincur
MAE
MSF
+ -LIFT
MAE
MSE
Crassfomme
MAE
MSB
+ LIFT
MAE
MSE
MITGNN
MAE
MSE
+ LIFT
MAE
Impe
24
0.091
0.122
0.089
0.119
0.104
0.152
0.090
0.125
0.086
0.126
0.086
0.126
0.090
0.128
0.090
0.126
4.7%
Weathe
0.158
0.137
0.194
0.114
0.163
0.112
0.112
0.165
0.170
0.115
0.167
5.3%
95
0.152
0.199
0.146
0.196
0.176
0.237
0.145
0,203
0.185
0.209
0.136
0.210
0.157
0.215
0.154
0.212
5.0%
192
0.197
0.245
0.190
0.238
0.220
0.282
0.189
0.249
0.197
0.264
0.196
0.262
0.206
0.269
0.203
0.266
4.3%
336
0.249
0.283
0.243
0.281
0.265
0.319
0.243
0.292
0.246
0.309
0.245
0.385
0.258
0.312
0.256
0.308
3.0%
720
0.320
0.335
0.315
0.333
0.323
0.362
0.317
0,349
0.323
0.361
0.321
0.360
0.335
0.369
0.333
0.365
1.4%
0,190
0.110
0.209
0.099
0.097
0.195
0.094
0.193
3.6%
Elctrriityy
0.210
0.110
0.205
0.125
0.223
0.113
0.209
0.116
0.216
0.113
0.211
0.116
0.215
0.112
0.211
4.0%
96
0.130
0.222
0.128
0:222
0.140
0.237
0.130
0.225
0.142
0.243
0.138
0.238
0.138
0.258
0.133
0.233
2.9%
192
0.148
0.240
0.147
0.239
0.153
0.249
0.145
0.242
0.159
0.259
0.154
0.251
0.160
0.261
0.153
0.252
2.7%
356
0.167
0.261
0.163
0.257
0.169
0.267
0.163
0.261
0,192
0 0.293
0.176
0.276
0.193
0.284
0.187
0.275
3,8%
0.2609
0,203
0.301
0.198
0,242
0.327
0.216
0.305
6.6%
0.214
0.371
0.267
0.347
0,402
0.260
0.392
0.259
7.3%
48
0.342
0.240
0.529
0:236
0.393
0.276
0.367
0,260
0.513
0.290
0.428
0.289
0.450
0.274
0.436
0.281
4.4%
Trafiic
95
0.367
0.251
0.357
0:242
0,410
0.282
0,394
0.273
0,519
0 0.93
0.462
0.284
0.479
0.289
0.464
0.286
4.2%
192
0.385
0,254
0.373
0.251
0.423
0.287
0,413
0,281
0,522
0 19296
0.490
0.283
0.507
0.307
0.491
0.301
3.3%
0.266
0.25%
0.426
0.530
0.314
0.519
0.309
0.286
0.166
0.315
0.454
0.307
0.616
0.352
0.532
0.340
5.4%
24
0.095
0.160
0.087
0.147
0.133
0.219
0.093
0.149
0.082
0.134
0.079
0.129
0.070
0.125
0.069
0.122
11.0%
48
0.153
0.227
0.143
0.200
0.190
0.267
0.145
0.197
0,146
0.203
0.140
0.178
0.131
0.180
0.130
0.177
11.0%
Solar
96
0.176
0.227
0.174
0.224
0.222
0.291
0.185
0.238
0.17u
0.345
0.174
0.224
0.167
0.234
1.166
0.22
6.2%
0.205
0,260
0.190
0.245
0.269
0.309
0.194
0.253
0,204
0 0254
0.197
0.250
0,180
0.243
0.179
0.239
7.6%
0.249
0.269
0.324
0.198
0.260
0,216
0.257
0.254
0.191
0:251
0.190
0.245
7.5%
720
0.229
0.282
0.293
0.261
0.271
0.327
0.207
0.260
0.211
0.250
0.202
0.255
0.197
0.256
0.195
0.251
8.5%
24
0.137
0.179
0.131
0.175
0.151
0.198
0.136
0.182
0.122
0.173
0.121
0.168
0.124
0.172
0.124
0.170
3.8%
48
0.163
0,200
0.155
0.196
0.175
0.214
0,159
0,200
0.147
0.194
0.147
0.189
0.149
0.192
0.148
0.191
3.4%
0.186
0,216
0.175
0.213
0.197
0.230
0.177
0.214
0,172
0218
0.169
0.208
0,170
0.211
0.169
0.208
4.1%
0.224
0.218
0.245
0.193
0.226
0.223
0.184
0.220
4.4%
336
0.316
0.239
0.202
0:234
0.233
0.258
0.205
0.238
0,281
0.340
0.199
0.252
0.195
0.253
0.192
0.227
4.6%
720
0.231
0.253
0.215
0.247
0.254
0.278
0.225
0.256
0.237
0.256
0.224
0.254
0.200
0.236
0.200
0.232
6.0%%
24
0.289
0.247
0.185
0.246
0
0
0,306
0.265
0.303
0.253
0.299
0.252
0.314
0.257
0.306
0.256
5.2%
PeMSDD
48
0.367
0,281
0.356
0.277
0.475
0.378
0
0,303
0,312
0.271
0,340
0.270
0.357
0.281
0.356
0.279
5.8%
0.562
0.304
0.386
0.297
7.6%
0.519
0.354
0,471
0.337
0.611
0.443
0.502
0.364
0.409
0.312
0.399
0.4L3
0.440
0.333
0.429
0.324
7.6%%
336
0.562
0.366
0.511
0.353
0.648
0.162
0.532
0.379
0.139
0.318
0,430
0.310
0.168
0.350
0.441
0.333
9.1%
720
0.653
0.403
0.563
0,378
0.748
0.519
0.597
0.414
0,488
0.356
0,468
0.338
0.511
0.379
0.484
0.342
12.3%
PeMSD8, we conjecture that these datasets have fewer distribution shifts in channel dependence, leading to fewer overfitting risks. Even though the CD backbones have benefited from channel de- pendence, LIFT can still refine their predictions, e.g., improving Crossformer by 4.1 % on Solar. This indicates that existing CD approaches cannot fully exploit the lead-lag relationships without prior knowledge about the dynamic variation of leading indicators and leading steps. Moreover, Crossformer mixes information from the variates that show similarity at the same time step but pays insufficient attention to the different yet informative signals of leading indicators. MTGNN learns a static graph structure among variates on the training data and aggregates information within fixed subset of variates. MTGNN may well suffer from distribution shifts in channel dependence, while LIFT dynamically selects leading indicators and reduces overfitting risks.
LightMTS Strong Baseline. Moreover, we compare the performance of LightMTS and all baselines on Weather, Electricity, and Traffic datasets. We borrow the haseline results from the paper of PatchTST with H E (96, 192, 336, 720). As shown in Figure 5a, LightMTS with simple linear layer as its backbone still shows considerable performance among the state-of-the-art models. In particular, LightMTS surpasses PatchTST, the complex Transformer model, by 3.2% on Weather and 0.7% on Electricity. However, PatchTST significantly outperforms LightMTS on the Traffic dataset. As Traffic contains the greatest number of variates with complex temporal patterns, requires strong backbone to model the intricate cross-time dependence. Nevertheless, LightMTS still the most competitive baseline on Traffic.
5.3 ABLATION STUDY
To verify the effectiveness of our designs, we introduce three variants of LightMTS by removing the influence term KK (+0) in Eq. (11), removing the difference term > A in Eq. (II), and k
K directly using V Ex=1 U1()) and ΣK= Ag) in Eq. (11), respectively.
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LighMMT PatenTBT
Cresslamer MTGNN
FEDformer
LighMIS
Autoffomer
Wtb diference filter
0.64
0.00
0.42
0.45
140%
LighMTS
0.21
100%
0.19
MSE
120%
0.18
110%
0:17
100%
100%
0.18
0.14
90%
90%
0.14
0.13
80%
Weather
Electriaiyy
Traffe
80%
Weather
Eledridty
Traffc
8
12
N
12
(a)
(b)
(c)
Figure 5: (a) Performance comparison between LightMTS and all baselines; (b) Performance com- parison between variants of LightMTS; (c) Performance of DLinear+LIFT under different numbers of the selected leading indicators (i.e., K) and the states (i.e., N).
As shown in Figure 5b, we conduct experiments these variants with H relative MSE w.r.t. LightMTS on Weather, Electricity, and Traffic datasets. With both the influ- ence and the difference involved, LightMTS considers two kinds of lead-lag relationships and keeps the best performance across the datasets. In contrast, LightMTS w/o influence and LightMTS w/o difference only consider one-sided information of leading indicators, thus showing inferior perfor- mance, especially on the Electricity dataset. Furthermore, LightM'TS w/o filter achieves the worst results in all the cases, which fails adaptively filter out the noise in leading indicators.
reporting the
5.4 HYPERPARAMETER STUDY
Our method introduces merely two additional hyperparameters, i.e., the number of selected leading indicators K and the number of states N. Thus it requires little labor for hyperparameter selection.
With DLinear the backbone and H set to 96, study the hyperparameter sensitivity of LIFT. As shown in Figure 5c, LIFT achieves lower MSE with an increasing K on most datasets. Nevertheless, LIFT may well include more noise with a too large K (e.g., on the Wind dataset), resulting in per- formance degradation. Besides, LIFT cannot enjoy significant improvement with a larger K on the Electricity dataset, where the lead-lag relationships are perhaps more sparse. for variate states, LIFT achieves lower MSE with an increasing N in most cases. We observe the most significant performance drop on Weather when ignoring the variate states. It is noteworthy that the variates of Weather (e.g., wind speed, humidity, and air temperature) are recorded by various kinds of sensors, and the lead-lag patterns naturally vary with the variate states.
CONCLUSION
In this work, we rethink the channel dependence in MTS and highlight the locally stationary lead- lag relationship between variates. We propose novel method called LIFT that efficiently estimates the relationships and dynamically incorporates leading indicators in the frequency domain for MTS forecasting. LIFT can work as a plug-and-play module and generally applicable to arbitrary forecasting models. We further introduce LightMTS as lightweight yet strong baseline for MTS forecasting, which keeps similar parameter efficiency to linear models and shows considerable per- formance. We anticipate that the lead-lag lag relationship channel dependence MTS, which a promising direction for the future development of channel- dependent Transformers other complex neural networks.
offer novel cross time perspective on the

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Published as a conference paper at ICLR 2024
ACKNOWLEDGEMENTS
This work is supported by the National Key Research and Development Program of China (2022YF00200500), Shanghai Municipal Science and Technology Major Project (2021SHZDZX01022), and SJTU Global Strategic Partnership Fund (2021 SITU-HKUST).
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A MATHEMATICAL PROOFS
A.1 DISCRETE-TIME FOURIER TRANSFORM
Discrete Fourier Transform (DFT) provides a frequency domain view of discrete time series. Given particular frequency f E {0, T..... [L/2]] ). the corresponding frequency component Uy of uni- variate time series -+11 is derived by
L-1 Ut L+1+E 22
U = F(u
22.e
(13)
where i is the imaginary unit and Uy E C is complex number. The inverse DFT is defined by
F F- -
Un. 221l e
(14)
We calculate the amplitude [Uy] and the phase <UUr) of the corresponding cosine signal by
J [ 9i {U,
|U,l = 1U,
and &UU)) = tan -1
(15)
where } and }{U denotes its real and imaginary components of Uy. With R(U,) and H{ scaled in the same rate, proposed real-valued filters only scale the amplitude but keep the phase unchanged.
A.2 EFFICIENT CROSS-CORRELATION ESTIMATION
Given another time series that lags behind by d steps, denote its frequency components V and define the cross-correlation between their lookback windows by
R(0)=
Ut L+148-8----- U-L---- I. Σ ull 1 8)v(().
(16)
With ull 8) denoted as ū(6 E) and (8 - E) denoted as 8', we derive the Fourier Transform of L-1
Σ o())(e - 5)
as follows.
6=0
F
Σ ul))ū( -0)
(f) (())(((- -0)
e -i2x16
1-1
Σ ū(6 -l)e -)c--2207 12
5=0
(17)
Σ
v(()e- -127ff
Σ ū(6 -)e-- (20=25((600)
6=D
4-1
=V,
Σ
ū(8')e
12x
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Assuming is L-periodie, we have
L-1-t
L-1-t
e 12x18'
Σ
Ut - -
6'=-t
8'=0
L-1-8
U M--L+1+8+L e
1 -227((8++
i 11---1-6 e
-2m588 112m
8'=0
L-1-E
(18)
i 1+-1+1+8 22
12
i4 L+1+8'-0
8'
L+1++ e 12588'
F(
Due to the conjugate symmetry of the DFT on real-valued signals, we have
F(ū), = F()) =
(19)
where the bar is the conjugate operation. Thereby, we can obtain
Σ u(l)u(e =(())(e-6)) -6) -,J
F
(20)
Finally, we can estimate Eq. (16) as
R(8) R 1 F (F(C @ Flum-141als
I
(21)
Note that - < R(6) < 1 when ut 1+1 and 1t- have been normalized.
To obtain more accurate results, Eq. (3), and then compute Eq. (1) with (r E c S T where c < L. We would like leave this improvement as future work.
first obtain approximate leading step by Eq. (21) and
B DETAILS OF LEAD ESTIMATOR
Given the cross-correlation coefficients {R( (T) - -1} between variate i and variate j, we identify the leading step by
( arg 0,2
max 1STST-2
(T)(()))
(7)1
(22)
s.t. [R 1)1 < R (7) < | (T
(23)
which targeted at the globally maximal absolute cross-correlation. Note that Eq. (21) only esti- mates cors-coorssstions with in (0, L - 1}. If the real leading step is greater than L - -1 (e.g., [[(( > R2 (L- - 1)), we could mistakenly estimate as 1. Therefore, we only consider the peak values constrained by Eq. (23).
Besides, we further normalize the cross-correlation coefficients R kez())- As the evolution of the target variate is affected by both itself and the K leading indicators, it is desirable to evaluate the relative leading effects. Specifically, we derive normalized coefficient for each leading indicator
i E Z((7 by:
exp
R
(24)
(0) + Everoo exp R R exp 1,t
(
where R(5) = 1. Though I( may also involve the variate itself in periodic data, we can j. only include variate in its last period due to Eq. (23). Note that time series contains not only
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the seasonality (i.e., periodicity) but also its trend. Thus we use R (0) consider the current evolution effect from variate itself beyond its periodicity.
In terms of the proposed filter factory, generate filters based on R. ( i I R which represents the proportion leading effects.
C EXPERIMENTAL DETAILS
C.1 DATASET DESCRIPTIONS
Table 2: The statisties of nine popular MTS datasets.
Datasets
Weather
Electricity
Traffic
Solar
Wind
PeMSD8
ETTml
ETTh1
ILI
# of variates
21
321
862
137
28
510
7
7
7
# of timestamps
52,696
26,304
17,544
52,560
50,000
17,856
69,680
17,420
966
Sampling Rate
10 min
hour
1 hour
10 min
hour
5 min
15 min
hour
1 week
We provide the statistics of the nine popular MTS datasets in Table 2. The detailed descriptions listed as follows.
Electricity includes the hourly electricity consumption (Kwh) of 321 clients from 2012 to 2014.
Weather includes 21 features of weather, egg,, air temperature and humidity, which recorded every 10 min for 2020 in Germany.
Traffic includes the hourly road occupancy rates recorded by the sensors of San Francisco freeways from 2015 to 2016.
Solar includes the solar power output hourly collected from 137 PV plants in Alabama State 2007.
Wind? includes hourly wind energy potential in 28 European countries. We collect the latest 50,000 records (about six years) before 2015.
P includes the traffic flow, occupation, and speed in San Bernardino from July to August in 2016, which are recorded every 5 min by 170 detectors. We take the dataset an MTS of 510 channels in most experiments, while only MTGNN models the 170 detectors with three features for each detector.
ETT (Electricity Transformer Temperature) includes seven oil and load features of elec- tricity transformers from July 2016 to July 2018. ETTml is 15-minutely collected and ETTh1 is hourly collected.
ILI 10 includes the ratio of patients seen with influenzalike illness and the number of pa- tients. It includes weekly data from the Centers for Disease Control and Prevention of the United States from 2002 to 2021.
To evaluate the forecasting performance of the baselines, divide each dataset into the training set, validation set, and test set by the ratio 7:1:2.
hnttss::/ww.bbbceejnaaaarn
"nttp://pems.ddt.cc..g///
htm1
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Table 3:
Comparison of practical efficiency of PatchTST, Crossformer, and LIFT with И 720.
MACs are the number of multiply-accumulan operations per sample. The batch size is set to 1.
Relative additional cost
Relative
PatchTST
+LIFT
Crossformer
+LIFT
additional cost
Parameter (M)
4.3
4.7
9.3%
11.7
12.5
7.1%
MACs (G)
0.5
0.5
1.8%
4.4
4.4
0.2%
Weather
Memory (MB)
43
46
7.7%
206
209
1.6%
Time (ms)
4.7
6.0
29.6%
37.7
40.6
7.6%
Parameter (M)
4.3
9.3%
2.4
2.8
16.7%
MACs Memory (MB)
7.1
1.8%
8.6
8.7
1.4%
Electricity
0.4
1.2%
873
878
0.6%
Time (ms)
31.5
33.1
5.1%
Parameter (M)
4.3
4.7
9.3%
3.2
3.6
12.4%
MACs (G)
19.0
19.4
1.8%
10.6
10.9
3.2%
Traffic
Memory (MB)
1061
1092
2.9%
1558
1578
1.3%
Time (ms)
5.1
5.8
14.5%
32.7
36.4
11.3%
Parameter (M)
2.0
41.2%
1.9
23.4%
MACs (G)
0.9
5.3%
3.7
3.7
1.6%
Solar
Memory (MB)
92
4.3%
377
1.0%
Time (ms)
4.9
6.0
23.4%
31.9
33.4
5.0%
Parameter (M)
2.0
2.5
20.6%
11.8
12.2
3,4%
MACs (G)
0.2
0.2
5.6%
5.7
5.8
0.2%
Wind
Memory (MB)
25
29
13.1%
256
259
1.3%
Time (ms)
4.7
6.1
28.9%
45.2
47.2
4.4%
Parameter (M)
2.0
2.8
35.8%
2.9
4.0
36.5%
MACs (G)
3,5
3.7
5.6%
13.6
13.8
1.5%
PeMSD8
Memory (MB)
319
325
1.9%
1381
1387
0.4%
Time (ms)
5.6
6.0
7.0%
32,4
33.7
4.3%
C.2 IMPLEMENTATION DETAILS
All experiments are conducted on single Nvidia A100 40GB GPU. We use the official implemen- tations of all baselines and follow their recommended hyperparameters. Typically, the batch size is set to 32 for most baselines, while PatchTST recommends 128 for Weather. We adopt the Adam optimizer and search the optimal learning rate in (0.5, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001, 0,00005, 0,00001). }. As for LIFT, we continue the grid search with K in {1. 2. 4, 8, 12, 16) and N in (1, 4. 8. 12, 16). Note that we stop the hyperparameter tuning for consistent performance drop along one dimension of the hyperparameters, i.e., we only conduct the search in a subset of the grid.
As the Lead-aware Refiner has few dependencies on the preliminary predictions from the back- bone, it is sometimes hard to train LIFT in the early epochs, especially when using complex CD backbones. speed up the convergence, one alternative way is to pretrain the backbone for epochs and then jointly train the framework. In our experiments, we report the best result in Table 2, while DLinear+LIFT and LightMTS are always trained in an end-to-end manner.
D ADDITIONAL EXPERIMENTS
D.1 EFFICIENCY STUDY
Following Zeng et al. (2023), we compare the backbones and LIFT by the number of parameters, the number of operations, the GPU memory consumption, and the inference time. As shown in Table 3, LIFT additionally requires an average of 10.7% parameters, 1.7% MACs, 2.5% GPU memory, and 14.2% inference time more than its backbone. It is also noteworthy that our Lead Estimator parametric and can pre-compute estimation only once on the training data, reducing the practical training time.
Moreover, we compare the parameter efficiency of LightMTS and all baselines on the six datasets. As shown in Figure 6, LightMTS keeps similar parameter efficiency with innear, simple linear
16

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Published as a conference paper at ICLR 2024
UgMOMTS
PatdiTST
Dineer
Crossfomer
MTONN
Autoformr
Inforee
167
1+7
1.0
modee paramakars
0.5-
0.8
2.6
0.6
1.5
0.6
0.4
0.4
0.4
0.2
0.0
400 Eloctricity
Weather
Traffic
Wind
PcNiSDe
Solar
Figure 6: The number of model parameters on six datasets with horizon H in {24, 48, 96, 192, 336. 720}. For Crossformer, follow its recommended lookback length L. For Informer and Autoformer, is 96. For other methods, is 336.
model. On average, the parameter size of LightMTS is 1/5 of PatchTST, 1/25 of Crossformer, 1/50 of MTGNN, and 1/70 Informer and Autoformer. It is noteworthy that a larger enlarges the gap between PatchTST and LightMTS because PatchTST employs a fully connected layer decode the H-length sequence of high-dimensional hidden states, Although the parameter sizes of Informer and Autoformer are irrelevant to they are still the most warameter-heay due their high- nigh-dimensional learning throughout encoding and decoding.
D.2 DISTRIBUTION SHIFTS
To investigate the dynamic variation of lead-lag relationships, we adopt the proposed lead estimator
(K = 2) to count the lead-lag relationships in the training data and test data. As shown Figure 7, some leading indicators of specific variate in the training data cannot keep the relationships in the test data, while the test data also encounters new patterns. GNN-based (Wu et al., 2020) and MLP- based (Li al., 2023b) methods susceptible to such distribution shifts of leading indicators due to their static parameter weights in modeling channel dependence.
Furthermore, visualize the distribution of leading steps between the pair of variates. We choose a lagged variate and its leading indicator that is the most commonly observed across training and test data. As shown in Figure 8, some of the leading steps (e.g., 250) observed in training data rarely reoccur in the test data. By contrast, the leading indicator show new leading steps (e.g., 40 and 125)
0.15
005
271918877 122111122111
0.30
0.25
-0.06
0.20
6.18
201918777
-0
9 5 101 20F19109110/0111110000
0 2.2 4 3 1 1111111111111711100
0 2 4 9-101 1200411111417
Training
Changes
Figure 7: Distributions of leading indicators (K 2) the training data (left) and test data (mid) on the Weather dataset, where each cell represents the occurrence frequency of the lead-lag relationship between each pair of variates. The right shows the changes in occurrence frequency from training data to test data.
17

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Published as a conference paper at ICLR 2024
0.0175
train
test
0.0150
0.0125
Density
0.0100
0.0075
0.0000
leading step
Figure 8: The histogram of the leading step between selected pair of variates in the training data and test data on the Weather dataset. We also estimate the distributions with kernel density estimator.
Table 4: Performance comparison in terms of forecasting errors. We highlight the better results
between each pair of backbones and LIFT in bold.
PatchTST MSE
+ LIFT
DLincar
+ LIFT
Method
MAE
MSE
MAE
MSE
MAE
MSE
MAE
0.307
0.358
0.303
0.356
0.367
0.316
0.360
48
0.338
0.374
0.337
0.374
0.345
0.378
0.342
0.376
96
0.375
0.399
0,370
0.395
0.375
0.399
0.372
0,395
ETTh1
192
0.414
0.421
0.410
0.419
0.405
0.416
0.413
0.423
336
0.431
0.436
0.433
0.435
0.439
0.443
0.453
0.453
720
0.449
0.466
0.447
0.464
0.472
0.490
0.509
0.512
0.193
0.270
0.211
0.285
0.196
48
0.254
0.319
0.252
0.316
0.272
0.326
0.258
0.321
96
0.290
0.342
0,287
0,338
0.299
0.343
0.293
0,342
ElTml
192
0,332
0.369
0.329
0.367
0.335
0.365
0.334
0.365
336
0,366
0.392
0.365
0.390
0,369
0.386
0.369
0.387
720
0.420
0.424
0.412
0.425
0.421
0.426
0.424
0,909
0.562
1.115
0.713
0.974
1.523
0.764
1,378
0.726
1.844
0,941
1.705
0.904
24
1.767
0,830
1.677
0.841
2.215
1.081
2,083
1.002
36
1.651
0.857
1.629
0.830
2.301
1.076
2.172
1.033
48
1.711
0.859
1.668
0.844
2.316
1.091
2.187
1.052
1.816
0.884
1.123
2.342
the test data. Furthermore, the leading step is not fixed but dynamically varies across phases, increasing the difficulty of modeling channel dependence.
D.3 PERFORMANCE ON OTHER DATASETS
We conduct more experiments on Illness, ETTml, and ETThl datasets. As PatchTST and DLinear perform the best on these benchmarks, we employ them as backbones. As shown in Table 4, LIFT fails in cases. We reason that these datasets composed of only 7 variates and perhaps have insufficient leading indicators for each variate. Nevertheless, is worth mentioning that we can collect abundant variates in practical applications, paving the way for adopting LIFTT
18

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Published as a conference paper at ICLR 2024
Table Performance comparison in terms of forecasting errors. The best results are in bold and the second best are underlined
Moddl
LightMTS
PatchTST
Dlincar
Crossforter
MTGNN
PEDiormer
Autoformer
Informer
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Wealber
96
0.147
0.198
0.152
0.199
0.176
0.237
0.145
0.209
0.157
0.216
0.238
0.314
0.249
0.329
0.354
0.405
192
0.190
0.238
0.197
0.243
0.220
0.282
0.197
0.264
0.205
0.269
0.275
0.329
0.325
0.370
0.419
0.434
336
0.240
0.278
0.249
0.283
0.265
0.319
0.246
0.309
0.258
0.312
0.339
0.377
0.351
0.391
0.583
0.545
720
0.318
0.332
0.320
0.335
0.323
0.362
0.323
0.364
0.335
0.369
0.389
0.409
0.415
0.426
0.916
0.705
Traflte Electiicly
96
0.131
0.224
0.130
0.222
0.140
0.237
0.142
0.243
0.139
0.242
0.186
0.302
0.196
0.313
0.304
0.393
192
0.148
0.240
0.148
0.240
0.153
0.249
0.159
0.259
0,160
0.261
0.197
0.311
0.211
0.324
0.327
0.417
336
0.167
0.169
0.267
0.293
0.193
0.328
0.333
0.422
720
0.200
0.288
0.202
0.291
0.205
0.301
0.225
0.516
0.242
0.327
0.253
0.344
0,236
0.342
0,351
0.427
96
0.386
2628
0.367
0.251
0.410
0.282
0.519
0.293
0.479
0.289
0.576
0.359
0.597
0.371
0.733
0.410
0.423
0.287
0.296
0.507
0.610
0.380
0.777
0.438
336
0.417
0.282
0.398
0.265
0.436
0.296
0.530
0.300
0,539
0.314
0.608
0.375
0.623
0.387
0.776
0.434
720
0.444
0.298
0.434
0.287
0.466
0.315
0.584
0.369
0.616
0.352
0.621
0.375
0.639
0.395
0.827
0.466
0.359
0.307
0.375
0.399
0.448
0.440
0.390
0,510
0,492
0.626
0.560
ETTml ETTh
192
0
0.374
0.338
0.374
0.405
0.416
0.471
0.174
0.449
0.133
0.365
0.415
0.514
0.495
0.725
0.619
336
0.369
0.391
0.375
0.399
0.439
0.443
0.570
0.546
0.598
0.554
0.392
0.425
0,510
0.492
1.005
0.741
730
0.416
0.472
0.490
0.621
0.685
0.446
0.458
0.493
1,133
0.845
96
0.285
0.340
0.290
0.342
0.299
0.343
0.315
0.370
0.330
0.388
0.326
0.390
0.510
01.492
0.626
0.560
192
0.325
0.364
0.332
0.369
0.335
0.365
0.348
0.390
0.376
0.419
0.365
0.415
0,514
0.495
0.725
0.619
0.384
0.425
720
0.422
0.423
0.420
0.424
0.425
0.421
0.511
0.552
0.485
0.488
0.446
0.458
0.527
0.493
1,133
0.845
D.4 PERFORMANCE OF LIGHTMTS
In Table 5. we compare the forecasting errors of LightMTS with all baselines. The lookback length L is set to 336 for LightMTS, PatchTST, DLinear, Crossformer, and MTGNN. We borrow the results of other basclines from the paper of PatchTST. As shown in Table LightMTS achieves comparable performance in all the cases.
19


Reviews:
[{"review_id": "TyRcvi7Kcz", "reviewer": "Reviewer_JmQw", "summary": "This paper proposes a new method, called LIFT, which first efficiently estimates leading indicators and their leading steps at each time step, and then judiciously allows the lagged variates to utilize the advance information from leading indicators. This method can be used as a plugin, which is seamlessly collaborated with arbitrary time series forecasting methods. Extensive experiments on six real-world datasets demonstrate the effectiveness of the proposed method with respect to the average forecasting performance.", "questions": "(See above)", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This proposed method is somewhat novel, making benefits of the channel dependence. It can efficiently estimate leading indicators and the leading steps, and allow the lagged variates to utilize the advanced information from leading indicators.\n\n2. This method can be regarded as a plugin to collaborate with any other time series forecasting methods.", "weaknesses": "1. It remains unclear to me how instance normalization and denormalization could effect the results and performances.\n\n2. In Figure 3, “all layers in the grey background” refer to which part in the figure?  \n\n3. Inappropriate expression: “one of the hottest” -> “one the most popular”.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DpsIFsdxgw", "reviewer": "Reviewer_MEf1", "summary": "This paper considers the problem of multivariate time series forecasting. The authors propose a forecasting algorithm called LIFT that exploits dependencies among the time series by first finding a set of leading time series and their corresponding leading steps for a given time series and then uses a Lead-aware Refiner that adaptively leverages the informative signals of leading indicators in the frequency domain to refine the predictions of lagged variates. The first step, i.e., finding the leading time series and their corresponding steps is designed based on Wiener–Khinchin theorem that uses fast Fourier transformation. The second step contains two main parts: state estimation and a frequency mixer. They evaluate the performance of LIFT through several experiments and their result shows that LIFT makes an average improvement over both CI and CD methods.", "questions": "Is it possible to discover causal relationships (e.g., Granger causality) among time series via modifications of LIFT?\n\n\nWhy there is subscript $\\Delta$ and $U$ in equation (9)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "They study an important problem. The paper is well-written. \n\nUsing interdependencies among the time series to better forecast is more intuitive than CI methods and this is showing by this work empirically. \n\nBased on the presented evidence in empirical study, the proposed method shows improvement compared to the state of the art.", "weaknesses": "The main concern is the generality of the proposed method to capture general forms of interactions among time series. The leading relationship in this work is detected based on pairwise cross-correlation. What if there are more complex interactions in which two or more time series jointly influence another time series? A reason that in some applications CI is outperforming CD methods could be the result of such misspecified interactions in the CD methods.   \n\nConsider a setting in which the influence structure among  the time series is a chain, i.e., $X^{(1)} -> X^{(2)} -> \\cdots -> X^{(C)}$. In this case, there are scenarios in which the pairwise cross-correlations between $X^{(C)}$ and {$X^{(i)}: 1\\leq i\\leq C-2$} are all higher than the correlation between $X^{(C)}$ and $X^{(C-1)}$. This means that LIFT will not pick  $X^{(C-1)}$ for forecasting while the only relevant time series for forecasting $X^{(C)}$ is $X^{(C-1)}$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8rfgwXrx9P", "reviewer": "Reviewer_cNYF", "summary": "This paper presented LIFT, a post-hoc plugin that enhance performance for time series forecasting tasks.\n\nLIFT works on top of black-box backbone forecasting models. It first estimates leading indicators and then shift those indicators to sync with target variate. Finally, filter based Adaptive frequency mixer was used to extract valuable information from leading indicators.\n\nExtensive experiments on six real-world datasets demonstrate the effectiveness of LIFT.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The design to sync the series with leading indicators is intuitive.\n2. Clear design of LIFT starting from preliminary forecasting to lead estimation and target-oriented shifts to reconstruction for the prediction.\n3. Very smart post-hoc process which in orthogonal to existing time series models.\n4. Proposed LightMTS a simple yet effective lightweight baseline method for MTS forecasting.", "weaknesses": "1. The proposed correlation based leading indicators discovery method may over-simplify the dynamics in the time series data. If the time-delayed relation among those variates is highly complex which cannot easily be captured by cross-correlation coefficient, the estimation of the leading indicators may not be accurate. Some discussion on this case is expected.\n2. As introduced in locally stationary lead-lag relationship, such relationship should be static only in a short period of time, which may indicate that it may not work well in long-term forecasting. However, the author also claims good performance for that case. Some elaboration on this will be very helpful.\n3. As a plug-and-play module that works on top of the baseline predictions, some fair comparison about the computation cost such as the total computation load should be considered. Otherwise, it is not surprised to see that after injecting some inductive bias, the performance is better than baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DPGUNo8RRk", "reviewer": "Reviewer_4A7m", "summary": "The paper introduces LIFT, a novel method aimed at enhancing multivariate time series (MTS) forecasting by leveraging locally stationary lead-lag relationships between variates. The authors suggest that recognizing and utilizing the dependence between channels can significantly improve forecasting accuracy. LIFT is designed to be a flexible plug-in that can integrate with any existing time series forecasting methods. The method was tested across several real-world datasets, showing remarkable improvement. Additionally, the paper presents LightMTS, a new baseline model for MTS forecasting that maintains parameter efficiency while offering competitive performance.", "questions": "Please allow me to preface my questions by mentioning that I am relatively new to this field, and as such, my inquiries may not reflect a deep understanding of the intricate concepts presented.\n\nThe following question is my major concern:\n\nQ1. While the motivation lies in the exploitation of dependence between variables, the experiment shows better performance and improvement for CI methods compared with CD methods, which is kind of contradicted. Could you provide more insight into the possible reasons?\n\n&nbsp;\n\nThe following questions stem from what I believe are confusions arising from differences between different fields (which won't influence my rating): \n\nQ2.  Is it possible to provide a formal definition of  **locally stationary lead-lag relation** as \"influence requires a certain time delay to propagate and take effect\" is not clear to interpret mathematically? Furthermore, in the context of cross-correlation used to quantify such relationships, does this indicate that a lead-lag relationship is inferred whenever we observe relatively larger cross-correlation in differing time slots between two variables? (In the causality field, the correlation is not equivalent to influence and effect).\n \nQ3. As a continuation of my previous question, the use of cross-correlation for identifying lead-lag relationships suggests a focus on linear associations. May I inquire if this suggests that the algorithm's applicability is confined to variables that share a linear relationship (e.g., $X_1 = X_2^2$, the cross-correlation will return zero)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "**Originality** The paper proposes a novel approach to leverage the lead-lag relation to contribute to MTS forecasting.\n\n**Quality** The experiment is solid and improvement is fruitful.\n\n**Significance** The proposed framework is flexible and can be incorporated with time series forecasting backbones, indicating its potential to improve predictive performance in a myriad of applications that depend on time series analysis.\n\n**Clarity** The paper is well-written and easy to follow.", "weaknesses": "To me, the paper is well written, logically fluent and the experiment result is fruitful. My only concern is about potential contradiction in results and the slightly limited applicability. Please see the Question part for more details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)