TimeBase: The Power of Minimalism in Efficient Long-term Time Series Forecasting
Authors: Qihe Huang
Zhengyang Zhou
Kuo Yang
Zhongchao Yi
Xu Wang
Yang Wang
Venue: ICML 2025 spotlightposter (2025)
Keywords: Time series forecasting
Primary area: applications->time_series

Abstract: Long-term time series forecasting (LTSF) has traditionally relied on  large parameters to capture extended temporal dependencies, resulting in substantial computational costs and inefficiencies in both memory usage and processing time. 
However, time series data, unlike high-dimensional images or text, often exhibit temporal pattern similarity and  low-rank structures, especially in long-term horizons.  By leveraging this structure, models can be guided to focus on more essential, concise temporal data, improving both accuracy and computational efficiency. In this paper, we introduce TimeBase, an ultra-lightweight network  to harness the power of minimalism in LTSF.  TimeBase 1) extracts core basis temporal components and 2) transforms traditional point-level forecasting into efficient segment-level forecasting, achieving optimal utilization of both data and parameters. Extensive experiments on diverse  real-world datasets show that TimeBase achieves remarkable efficiency and secures competitive forecasting performance. Additionally, TimeBase can also serve as a very effective plug-and-play complexity reducer for any patch-based forecasting models. Code is available at \url{https://github.com/hqh0728/TimeBase}.

TimeBase: The Power of Minimalism in Efficient Long-term Time Series Forecasting
Qihe Huang' Zhengyang Zhou 123 Kuo Yang' Zhongchao Yi Xu Wang 12
Yang Wang 12
Abstract
1. Introduction
Long-term time series forecasting (LTSF) has been stud- ied with significant interest in various domains, ranging from energy management, traffic accident preservation, and extreme disaster warning. With the rapid advancement of deep learning, an increasing number of models have been proposed (Qiu et al., 2024; Zheng et al., 2023; Ang et al., 2023; Chen al., 2023; Wu al., 2021; Zhou al., 2023a; Wang et al., 2023b; Lin et al., 2023a; Ding al., 2018; Chen
Long-term time series forecasting (LTSF) has tra- ditionally relied on large parameters to capture extended temporal dependencies, resulting in sub- stantial computational costs and inefficiencies in both memory usage and processing time. How- ever, time series data, unlike high-dimensional images or text, often exhibit temporal pattern similarity and low-rank structures, especially in long-term horizons, By leveraging this structure, models can be guided to focus on more essen- tial, concise temporal data, improving both accu- racy and computational efficiency. In this paper. we introduce TimeBase, an ultra-lightweight net- work to harness the power of minimalism in LTSF. TimeBase 1) extracts core basis temporal com- ponents and 2) transforms traditional point-level forecasting into efficient segment-level forecast- ing, achieving optimal utilization both data and parameters. Extensive experiments on di- verse real-world datasets show that Tise achieves remarkable efficiency and secures com- petitive forecasting performance. Additionally, TimeBase can also serve as very effective plug- and-play complexity reducer for any patch-based forecasting models. Code is available at ht https: / //gi sigthuu.... www./ggw/////eee..oo www.hghhhh220.000 B/TimeBase.
et al., 2018; Wang al., 2024c; Chen et al., 2019; Wang
et al., 2023a; Huang al., 2024c; Zhou et al., 2023b; Wang
at., 2024b; Lin et al., 2024a; Miao et al., 2025; Yi et al., 2024), including MLP-based (Liu et al., 2022; Huang et al., 2024a; Miao et al., 2024), RNN-based (Lin et al., 2023b), and Transformer-based (L.iu et al., 202 1b; Zhang and Yan, 2023), approaches, all of which employ thousands mil- lions of parameters capture long-range dependencies and forecast future outcomes.
Generally, higher number of parameters increases the model capacity, which can lead to better predictive per- formance (Zhou et al., 2023c; Zhao et al., 2024). In the fields of computer vision (CV) and natural language pro- cessing (NLP), large models have achieved significant suc- (He et al., 2016; Liu et al., 2023). For instance, Vision Transformers (ViT) (Dosovitskiy, 2020) have demonstrated outstanding capabilities in image recognition, while large language models (LLM) have made breakthrough advances across various language tasks (Devlin, 2018; 8; Radford et al., 2019). Recently, large models are being explored for LTSF to capture complex temporal patterns and long-range de- pendencies (Cheng al., 2024; Liu al., 2025a; 2024d; 2025b). For example, some LLM- forecasting meth- ods are proposed with tens of billions of parameters (Jin et al., 2023). However, despite their impressive performance on specific forecasting tasks, these models suffer from high computational costs and resource-intensive requirements.
University of Science and Technology of China (USTC), Hefei, China Suuhou Institute for Advanced Research, USTC, Suzhou, China State Key Laboratory of Resourees and Environmental Information System, Beijing, China. Qihe Huang <hqh @ mail.ustc.eduu www..edu..c.......... Kuo Yang <yangkuo@ mail.ustc..du.cn. Yi <zhongchaoyie watiaaaoommmmmllt....eee... mail.ustc.edu.cmm
Zhongchao Xu Wang Cwx309@ustc.edd en>> Correspondence to: Zhengyang Zhou <zzy0929 wutt999eustt..oom www.uuuttuu....oo adu.cn.>, Yang Wang <angyan @ ustc.edee edu.cn>>.
In fact, images and text, as high-dimensional data, contain multiple dependencies and complex underlying physical rules (Liu al., 2021a), which necessitate the use of more parameters to model their rich semantic structures. How- ever, as shown in Figure I(a), one-dimensional time series data is typically much more regular, exhibiting obvious tem-
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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TimeBase: The Power of Minimalism in LTSF
60
BUThl
Traffic
50
ETTh2
Electricity
Singular Value
ETTml
Weather
Extremely Similar Patterns
Pearson Correlation Coefficient
no
Pi
40
ETTm2
1.00
0,99
0.32
1.9%
01:1
(19
0.79
0.98
30
0.99
0.94
0.99
0893
099
0.55
0.99
74
01.99
0.93
0.78
0.12
0.04
00:9
019
03.4
20
0.50
[00
035
0.93
0.98
1.00
01:2
(493
0.99
0193
0194
Low Singular value
0193
(193
0.89
019
1,00
0197
0.85
0193
0.59
0:53
0U94
0.85
150
038
0.99
053
0094
(095
0,54
0.99
0
0.19
056
(19)
0.55
0.54
0.0
005
(193
0193
0.51
5
10
15
20
P1
P2
P4
P6
P8
P10
P12
Index of Singular Values (b) Low-Rank Characteristics
(a) Temporal Pattern Similarity
Figure 1. Illustration of temporal pattern similarity and approximate low-rank characteristies in long-term time series.
105
TimeMixer
RLM
Infomen2021)
(71.59ms, 1376 14MB)
1132 2360.31MB)
Autoformen
10
(72:67ms, 99MB)
FEDfomer
(203.31mg) 3 2 2361,76MB
Mean Squared Ener (MSE)
0.35
Fil M(2022
iTransfomer
130 41ms, 706B)
Autaformer
(126 75ms, 4346.89MB)
(2400 03ms, 18004.33MB
PanhTST(3023)
DLinen12023)
MACS
10a
Diinear
0.30
[imeMixen2124)
(3.25ms, 150.21MB)
Momory Footpeint (Batch Size 12)
FTTS(2024)
10a
025
SpuncTSP1202)
13.95ms. 32M))
TimeBase(Ours)
SparseTSF
12.59ms, 125. 20MB)
10
4. 5x Macs Save
TimeBase
(0.98ms, 88.89MB)
1GB
4GB
18GB
Scale Smaller & Comparable Performance
50
100
150
200
250
300
0.15
10a
5x102
102
10c
10a
10
107
10a
CPU Inference Speed (ms)
Parameter Number
Figure 2. Comparison of forecasting performance and model efficiency in terms of MSE, MACs, parameters, memory usage, and CPU inference speed for 720-horizon forecasting in Electricity.
poral patterns. Moreover, in long-term time series data, this regularity can even manifest as low-rank characteristics (Liu et al., 2012), which can be reflected through singular value decomposition as shown in Figure 1(b), indicating that there a considerable amount of redundant information (Jones and Brelsford, 1967; Hochreiter and Schmidhuber, 1997). This raises an important question: Is it truly necessary to employ such a large number of parameters to learn these regular time series patterns (Tan et al., 2024; Zuo et al., 2024; Khayati et al., 2024)?
domains and scales. Additionally, TimeBase can also serve as a very effective plug-and tool for patch-based fore- casting methods, enabling extreme complexity reduction, i.e., 77.74% for PatchTST in MACs, prompting prediction accuracy. Our contributions can be summarized as follows:
Considering the temporal pattern similarity and low- rank characteristic, we demonstrate that basis compo- nent extraction with segment-level forecasting is an effective approach for LTSF to fully utilize both data and model. This method can significantly reduce the unavoidable ultra-high complexity and large model parameters associated with current LTSF models.
In this study, we design an extremely lightweight time series forecasting network, TimeBase, which is centered around basis component extraction and segment-level forecasting. As illustrated in Figure/ 2, TimeBase utilizes only 0.39k parameters, reducing MACs by 120x and parameter count by more than 2.6 1 times compared to DLincar. Com- pared to the standard linear lightweight model SSF it reduces MACs by 4.5x, parameter count by 2.5x. Despite being minimal model, TimeBase demonstrates superior predictive performance on real-world datasets of various
We propose TimeBase, which is currently the lightest time series forecaster and an effective plug-and-play complexity reducer. It requires only 0.39k param- eters, achieving 4.5x reduction in MAC and a 2.5x decrease in the number of parameters compared to the previously lightest model Sparse'TSF. Besides,
2

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TimeBase: The Power of Minimalism in LTSF
could make for PatchT'ST.
computation reduction
multi-granularity data, enriching the information contained within the dataset. In this paper, we propose TameBase, which further leverages the approximate low-rank nature of long teggtermmin time series and significantly reducing the parame- ter scale.
TimeBase not only maintains an extremely small model size but also achieves competitive forecasting perfor- mance across various real-world datasets. Specifically, TimeBase ranks Top2 on 29 out of 34 average metrics (MSE and MAE) across 17 normal scale datasets when compared to the ten state-of ofate-of-the-arr baselines.
3. Method
3.1. Problem Definition
In LTSF, the objective is to predict future values over an extended time horizon based on very long look-back win- dows. Formally, let X = [In, 20,, vr) R denote the historical time series data, where T > 1 is the length of bock-back window. The goal is to forecast the future val- ues Y = #T+2:................. ZT+L] E R with forecasting horizon L > 1. However, the exceptionally long horizon scale and substantially increases model size, leading to a rapid and considerable growth in the need of computa- tion resources, which may be unnecessary for time series data that follow simple and regular patterns. Consequently, our focus shifts to designing models that not only deliver robust and efficient performance but also remain extremely lightweight.
TimeBase offers a potential strategy for designing LTSF models with more efficiency and could provide valuable insights for the development of backbone ar- chitectures in large pre-trained LTSF models.
2. Related Work
Long-term time series forecasting (LTSF) aims to predict future sequences of considerable length using extended his- torical windows (Ang et al., 2024). The advancement of deep learning has significantly enhanced the accuracy of LTSF, with various foundational models, such as Transform- ers (Zhou et al., 2021; Zhang and Yan, 2023), Temporal Convolutional Networks (TCNs) (Luo and Wang, 2024), and Recurrent Neural Networks (RNNs) (Lin et al., 2023b), being employed to design long-term forecasting networks. These models are designed based on the different proper- ties of time series, such as series decomposition (Wu al., 2021), frequency domain (Xu et al., 2024), and periodic characteristics (Wu et al., 2023). As to series decomposi- tion, for instance, Autoformer (Wu et al., 2021) introduces a series decomposition block that utilizes moving average techniques to decompose complex temporal variations into seasonal and trend components, each undergoing separate time series modeling. Additionally, FEDformer (Zhou et al., 2022) further enhances the representation capabilities of the series decomposition block by employing multiple kernels moving average to decompose data at various granulari- ties, thereby improving forccasting performance. Consid- ering the frequency domain characteristics of time series, FITS (Xu al., 2024) operates on the principle that time series can be manipulated through interpolation in the com- plex frequency domain, achieving performance comparable to state-of-the-ut models for time series forecasting, On the other hand, periodicity is significant factor considered by many LTSF methods. TimesNet (Wu et al., 2023)pro- poses the use of Fourier Transform to capture multiple pe- riodic lengths of time series, expanding -iedimmesio time series into several two-dimensional components, which are processed through two-dimensional networks to handle high-dimensional data. SparseTSFF (Lin et al., 2024b) di- rectly utilizes the prior periodicity, thereby reducing the scale of network parameters. CrossGNN (Huang et al., 2024b) employs moving average techniques based on pe- riodicity to expand single-gran -eanllrrraullattty time series data into
3.2. TimeBase
In practical, regular time series often exhibit prominent segment-level patterns (Lin et al., 2024b), with approxi- mate low-rank characteristics (Jones and Brelsford, 1967). For example, traffic flow typically follows daily period, with similar patterns recurring each day. To effectively lever- age time series data and accomplish efficient forecasting, we propose TimeBase, implemented through Basis Extraction and Segment-level Forecasting by two extremely small- scale linear layers. This approach drastically reduces the model parameters to the hundred level while maintain- ing state-of-the-art (SOTA) forecasting performance. Most existing multivariate time series (MTS) are homogeneous, meaning that each sequence within the dataset exhibits simi- lar patterns. Based on this property, employ the Channel Independence (Nie et al., 2023) to simplify the forecast- ing of MTS into separate univariate forecasting tasks. An overview of TimeBase is shown in Figure 3.
3.2.1. BASIS COMPONENT RECONSTRUCTION
First, we need determine the basis length P from the time series and segment the series accordingly. The de- termination process can be categorized into two scenarios: (1) The time series has predefined prior period much smaller than look-back window, for instance, in domains like electricity or traffic, the cyclic patterns often follow a daily periodicity, allowing us to directly assign P 24 as the segment length from hourly sampled data. (2) When the time series lacks clear periodicity or the period exceeds
3

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TimeBase: The Power of Minimalism in LTSF
Complexity Reducer by TimeBase
Params
Speed
Performance
M Plug-and-play g
Patch-based
+
TB
Patch
Feature
Forecasting
Forecasting Model
Encode
X basis E CR RxP
Basis Extraction
Segment Forecasting
X e RT
Segment
Flatten
Y YERL L L
Optional
Orthonagal Restriction
his E R NxP
X xped E R N'PP
W1 +WN
-C=
G
X basis X basis
W
w' Basks
E
v
Segmon!
Segment
Noiv
Conganent
Corth = |G diag
Back 7
Notse
Fusuri Segneet
Figure 3. Overview of TimeBase. The core of TimeBase lies in extracting temporal basis components and segment-level forecasting. aims demonstrate that even the most minimalistic models can exhibit strong predictive power, providing design foundation for more effective time series models. In addition, TimeBase can also serve as plug-ane and-play reducer to decrease the complexity of any patch-based models.
1/6 the input horizon, frequency analysis (e.g., FFT) (Wu et al., 2023) on the training set Xarin can help identify domi- nant components, whose wavelengths may serve as the basis length. In this case, a relatively smaller P is recommended to enhance the expressiveness of basis components.
we can approximate the full-rank basis components using:
X busis CinsisExtract(Xaa
(2)
where BasisExtract(,)) is implemented by a simple lin- ear layer. Formally, Xhis can be expressed as a linear combination of basis components, represented as X = X busis WE+ B, where WE E RR the combined weight and the bias term B R denote the temporal noise By rearranging, we derive X
Based on the predefined basis length P. we divide the one-dimensional time series X E R² into N = P non-overlapping segments, denoted as X = (2,, Xv] RNxP, each of length P, analogous to non-overlapping patches (Nie al., 2023). When the length of X N is insufficient to meet P. the corresponding values from XN-1 will be used to fill in the gaps. The segment operation can be represented as:
TT
his Wt BWt There- fore the objective of Eq. (2) is to learn a linear layer of Whis Wf and Buis - BWt to approximate the ba- sis components. Next, we leverage the well-learned basis components realize segment-level forecasting:
basis X
X Segment(y.pe (X)
(1)
Xpred Segment (Xbssis)
(3)
where N and in Segment JN (+) represent the num- ber of rows and columns the transformed 2D ma- trix. The maximized rank of the matrix X is Rmas min(N, P). Given that typical time series exhibit similar temporal patterns and low-rank characteristics, we have R < P). In this context, directly designing model for forecasting leads to unnecessary resource con- sumption. Fortunately, inspired from Basis Representa- tion (Benson, 1998) and Linear Decomposition (Dantzig and Wolfe. 1960). series fundamental temporal pattern the basis components
Here, Xprod I. XP represents N' future segmented
time series, where N' N 1. The operation DigmentForccastt implemented also through a linear layer, aggregates the basis components for forecasting. Fi- nally, Xpre unfolded obtain prediction result Y RR
Flatten (Xpred)
(4)
3.2.2. BASIS ORTHOGONAL RESTRICTION
can be identified, referred X E R to capture compact information and min- imize the model size. Just any vector in coordinate system can be represented as a linear combination of its ba- sis vectors, the combination of basis components in specific time series can represent its any segment-level temporal pattern (Hochreiter and Schmidhuber, 1997). Conversely,
To make that the learned X effectively captures the es- sential and diverse temporal patterns, orthogonal con- straint can be applied. From the perspective of the data space, the orthogonality of the basis vectors enhances its rep- resentation power, providing them ability to express as any vector the data space through linear combination (Dantzig and Wolfe, 1960). Therefore, the temporal basis component

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TimeBase: The Power of Minimalism in LTSF
should also be diverse and distinet, preventing the extraction of very single time-series patterns. Based on this, apply the Basis Orthogonal Restriction.
4.1. Experiment Setup
Datasets We conduct experiments on 21 widely-used
and publicly available real-world datasets, including 17 normal-scale benchmarks: ETThI, ETTh2, ETTml, ETTm2', Weather?, Electricity?, Traffic* SSolar Energy (Lai et al., 2018), Wind (Li et al., 2022), METR-LA (Li et al., 2017), Exchange Rate (Lai et al., 2018), ZafNoo (Poy- atos et al., 2020) and CzeLan (Poyatos et al., 2020), AQShunyi (Zhang et al., 2017), AQWan (Zhang et al., 2017), and 4 very large datasets: CA (4.52B), GLA (2.02B), GBA ( (1.24B),SD (0.38B) (Liu et al., 2024c). Adhering to the established protocol in (Wu et al., 2021; Qiu et al., 2024; Liu et al., 2024c), we partition the datasets into training, validation, and test sets with a ratio of 6:2:2 for four ETT datasets, CA, GLA, GBA, SD, and 7:1:2 for the remaining datasets. The statics of dataset is summarized in Table 1.
Specifically, we penalize the deviation of Xoasis from an orthogonal set by adding a regularization loss Lenh:
G= = X basis X basis
(5)
Lort
G - diagg diaggG))l
(6)
where G is the gram matrix of X and ||. Frobenius norm. This term encourages Xhasis to approach an orthogonal configuration, ensuring that each basis com- ponent captures unique and uncorrelated temporal patterns. The overall training objective is then updated to:
|II denotes the
Table 1. Dataset Statistics. Var the number of variables, Length is the dataset length, T is the length of look-back window, L is the forecasting horizon, Freq is the sampling frequency, Scale
C = Ppeediction + Aarth Corth
Here, Lprestiction represents the original prediction loss, i.e., mean squared error (MSE) for regression, and Aorh is a hyperparameter that controls the weight of the orthogonal regularization term.
represents the number of data points.
Dataset
Var
Length
T
L
Freq
Scale
ETTh1
14,400
720
96~720
Thour
0.IM
ETTh2
14,400
720
96~720
Thour
0.IM
ETTml
57,600
720
96-720
15mins
0.4M
ETTm2
57,600
720
96--220
15mins
0.4M
3.3. Parameter Scale of TimeBase
Normal Scale Benchmark
Weather
21
52,696
720
96-7220
10mins
1.IM
Electricity
321
26,304
720
96~7220
Thour
8.IM
Theorem 3.1 (Parameter Scale of TimeBase). Let T denote the length of look-back window, L is the length of the fore- cast, P represents the length of the segment, and R gives the number of basis components. The parameter scale of Base can be expressed as:
Traffic
862
17,544
720
96-7200
Thour
15.0M
Solar
137
52,560
720
96~720
10mins
7.2M
Wind
7
48,673
720
96--720
15mins
0.4M
METR-LA
207
34,272
720
96-7220
5mins
7.1M
Exchange
7.588
720
96-7200
Iday
60.7K
AQshunyi
11
35,064
720
96~720
Ihour
0.4M
AQWan
35,064
720
96-720
Thour
0.4M
ZafNoo
19,225
720
96-720
30mins
0.2M
R P
R
R+ P
CzeLan
11
19,934
720
96--720
30mins
0.2M
Number
XT
XL+R
(8)
PM2.5
184
11,688
720
96~7220
3hours
2.2M
Temp
184
11,688
720
96~7220
3hours
2.2M
CA
8600
525,888
720
96~720
Smins
4820G
Large
GLA
3834
525,888
720
96-720
Smins
2020G
Theorem 3.1 shows that the parameter scale of TimeBase grows linearly with both the look-back window length T and the forecast horizon L.. In typical long-term forecasting setup where T L 720, the number of parameters in TimeBase, with complexity of baT++LL), is significantly smaller than that of DLinear (Zeng et al., 2023), which requires L parameters, and SparseTSF (Lin et al., 2024b), TL which uses pr + P, both with (TT) complexity.
GBA
2352
525,888
720
96~720
Smins
1240G
SD
716
525,888
720
96~7200
Smins
380G
Baselines We compare TimeBase with 10 baselines, which comprise the SOTA long-term forecasting models: TimeMixer (Wang et al., 2024d), iTransformer (Liu et al., 2024a), PatchTST (Nie al., 2023), DLinear (Zeng et al.,
2023), TimesNet (Wu al., 2023), FEDformer (Zhou et al., 2022), Autoformer (Wu et al., 2021), and Informer (Zhou
4. Experiment
et al., 2021). relatively efficient models: FITS (Xu et al., 2024) and Sparse'TSF (Lin et al., 2024b).
In this section, we demonstrate the advantages of Time- Base in competitive forecasting performance, extremely light efficiency and very effective plug-and-play function. More experiment details and additional experiment results are available at Appendix C.
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5

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TimeBase: The Power of Minimalism in LTSF
Table Long-term time series forecasting results in 17 normal scale datascts, comparing TimeBase with other baselines. Results are averaged across different forecasting horizons L E {96, 192, 336, 720}. The input length H is set to 720 across all models. Best results
are marked in red second- bestnd-bestton results are underlined in blue
TimeBase
SparseTSF
TimeMixer
FITS
iTransformer
DLineur
PatchTST
TimesNet
FEDformer
Autoformer
Informer
Methods
(ours)
(2024b)
(2024d)
(2024)
(2024a)
(2023)
(2023)
(2023)
(2022)
(2021)
(2021)
Metric
MSE MAE
MSE MAE
MSE MAE
MSE MAE
MSE
MAE
MSE MAE
MSE
MAE
MSE
MAE
MSE MAE
MSE MAE
MSE MAE
ETTh1
0.396 0.414
0.407
0.419
0.454
0.474
0.417
0.430
0.454 0.467
0.437
0.448
0.420
0.440
0.505
0.499
0.523
0.524
0.726 0.641
1.445 0.930
ETTh2
0,347
0.397
0.344
0.386
0.379
0.426
0.334
0,382
0.392
0.422
0.479
0.471
0.344
0.391
0.433
0.455
0..429
0.470
1.086
0.802
5.486 1.892
ETTml
0.356
0.380
0.362
0.384
0.381
0.414
0.359
0.382
0.370
0.401
0.361
0.383
0.355
0.385
0.408
0.425
0.448
0,466
0.564
0.522
1.138 0.818
ETTm2
0,250 0.314
0.252 0,316
0.283
0.347
0.252
0.314
0.278
0.338
0.271
0.337
0.251
0.319
0.300
0.354
00.409
0.462
0.431
0.463
3.594 1.473
Weather
0.219 0.263
0.244
0.286
0.2.38
0.281 1
0.244
0.286
(0.233
0.273
0.242 0.299
0.223
0.266
0.254
0.293
00.355
0.391
0.446
0.457
0.567 0.513
Electricity
0.167
0.258
0.168
0,264
0.171
0.274
0.172
0.266
0.166
0.264
0.169
0.272
0.169
0.266
0.238
0.334
0.233
0.348
0.236
0.343
0.408
0.464
Traffic
0.418
0.278
0.414
0,280
0.419
0.300
0.426
0.291
0.406
0.290
0.418
0.285
0.394
0.266
0.641
0.346
0.638
0,400
0.684
0,421
1.028 0.588
AQShunyi
0.674
0.506
0.760
0.546
0.736
0.536
0.763
0.548
0..72
0.520
0.757
0.572
0.690
0.500
0.788
0.559
1.130 0.757
1.392
0.878
1.688 0.966
AQWan
0.779
0.499
0.826
0.526
0.822
0.521
0.813
0.519
0.843
0.538
0.883
0.560
0.810
0.513
0.543 0.857
0.932 0.565
1.001
0.654
1.123 0.744
CzeLan
0,225
0.262
0,240 0.292
0.234
0,282
0,250
0.304
0.245
0.296
0.242
0,290
0.229
0.275
0.258
0.312
0,266 0.314
0.289
0,368
0.347 0.426
ZafNoo
0.495
0.445
0.531
0.490
0.505
0.464
0.520
0.480
0.540
0.500
0.540
0.495
0.510
0.469
0.545
0.502
0.601
0.539
0.667
0.635
0.726 0.696
Exchange
0.309
0.392
0,316 0.410
0.330
0.426
0.302
0.400
0.301
0.401
0,312
0.410
0.308
0.400
0.326
0.425
0.380 0.472
0.447
0.589
0.432 0.581
METR-LA
1.253
0.754
1.302
0.764
1.270
0.740
1,296
0.760
1,253
0.734
1.300
0.759
1.210
0.706
1,390
0.812
1,550
0.876
1.639
0.974
1.894 1.146
PM2.5
0.428
0.437
0.467 0.477
0.465
0.474
0.460
0.469
0,480
0.492
0.457 0.449
0.458
0.466
0.466 0.475
0,560 0,547
0.621
0.650
0.636 0.663
Solar
0.216
0.254
0,2166
0.264
0.244
0.296
0.229
0.279
0.233
0.285
0.227
0,276
0.226
0.275
0.243
0.295
0.252 0.291
0.306
0.380
0.327 0.411
Temp
0,187
0.338
0.295
0,412
0.302
0.423
0.315
0.442
0,302
0.423
0,292
0.418
0.291
0.406
0,304 0.427
0,343 0.456
0.404
0.587
0.471 0.682
Wind
0,940
0.709
1.014
0.736
1.044 0.758
1.023 0.745
L.025
0.748
1.018
0.739
0.968
0.703
1.101
0.799
1,230 0.841
1.237 0.947 1.381 1.033
4.2. Forecsating Performance of TimeBase
provide a more comprehensive comparison, examining both static and runtime metrics, which include (1) Parameters: The total number of trainable parameters, reflecting the model's size. (2) MACs Multiply-Accumulaat Opera- tions): A standard measure of computational complexity in neural networks, representing the number of multiply- accumulate operations required by the model.
As shown in Table 2, across 17 normal-scale benchmarks, TimeBase achieves Top-2 performance on 16 datasets, with the only exception being ETTh2, where its accuracy still remains highly competitive. These results further demon- strate the robustness and effectiveness of TimeBase across wide range of forecasting tasks. Compared to the standard linear model DL TimeBase reduces MSE and MAE by 8.82% and 7.64%, respectively. When compared with the currently most lightweight models, FITS and SparseTSFF TimeBase achieves an average reduction of 6. 6..34% in MSE and 4.85%%..44% in MAE. Furthermore, against some of the most powerful models to date, such as TimeMixer, iTransformer, and PatchTST, TimeBase achieves the best results on 23 out of 34 average forecasting metrics and ranks second on 6, when carefully tuned. Despite its extremely compact size, TimeBase delivers competitive forecasting ac- curacy, thanks to its compact yet effective modeling design.
(3) Max Memory: The peak memory usage during training. (4) Epoch Time:The time required to train the model for one epoch, averaged over three runs. (5) Infer Time: Infer Time indicates the average inference time per sample on CPU.
The look-back window for each model are set as 720, and the max memory is recorded with a constant batch size of 12. The efficiency analysis presented in Table 3 high- lights the remarkable advantage of TimeBase over other state-of-the-art models in terms of both static and runtime metrics. TimeBase achieves a substantial reduction in the number of parameters and computational complexity com- pared to many parameter-heavy models. Specifically, com- pared to lightest model SparseTSE Time reduces the parameter count by up to 61% % and the MACs by over 78%, while also using significantly less memory (29% reduction) and training faster (34% reduction in epoch time). These
4.3. Efficiency Analysis
Main Efficiency Comparision In addition to its impres- sive predictive performance, TimeBase offers another major advantage: its exceptionally lightweight design. Here, we
6

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TimeBase: The Power of Minimalism in LTSF
Table Efficiency comparison of TimeBase and other state-of-the-ard models the Electricity dataset with a forecasting length of 720.
To ensure fair comparison, the look-back window is set as 720 for all models. MACs here indicate the computational cost for processing a single sample, and memory usage measured with batch size of 12 all models.
Model
Parameters
MACs
Max (Mm.(MB)
Epoch Time(s)
Infer Time (CPU)
Informer (2021)
22.45M
7,85G
1424.99
143.05
72.67ms
Autoformer (2021)
22.14M
8,97G
4348.89
225.78
126. 75ms
FEDformer (2022)
22.14M
10.48G
2361.76
558.03
203.31ms
PatchTST (2023)
8.69M
14.17G
18034.33
827.34
249.02ms
DLincar (2023)
1.04M
333.04M
158.21
41.08
3.25ms
FITS (2024)
133.60K
42.73M
496.70
43.13
3,95ms
iTransformer (2024a)
5.47M
1.79G
828.32
65.62
30.41ms
TimeMixer (2024d)
5.58M
30.30G
1376.14
537.26
71.59ms
SparseTTSFF (2024b)
1,0K
12.71M
125.20
31.30
2.59ms
TimcBasc(ours)
0.39K
2.77M
88.89
20.60
0.98ms
Reduction
2.56x4
4.58x4
1.40x4
1.57x4
2.64x
results further demonstrate that TameBase not only main- tains strong predictive performance but also offers superior efficiency, making it well-suited for resource-come conttrc-cnnttraieed environments.
with limited computational resources or extremely large input scales.
4.5. Plug-and-Play Complexity Reducer for PatchTST
Settings TimeBase can be seamlessly integrated into any natch-based forecasting model to extract basis com- ponents prior to patching the input time series. For patch- based methods, the general prediction pipeline can be ab- stracted as Y = Head (Enoode (Patch(X)) With the incorporation of TimeBase, the pipeline becomes Y = Head Head (Encode [Encode(BasisEttraac)) aaasisEEttacctPaatch/a
Efficiency in Ultra-long Look-back Window Addition- ally, we evaluate the efficiency of TimeBase under ultra- long look-back windows, comparing it with DLinear. The comparison focuses on three key metries: running time per iteration, GPU memory usage, and parameter count, as shown in Figure 4. As the look-back window increases from 720 6480, with fixed batch size of 12 and predic- tion length of 720, TimeBase consistently demonstrates its lightweight nature. Even with a ninefold increase in input sequence length, TimeBase's s running time only increases by 0.05 seconds, GPU memory usage expands by a factor of 3.8, and the number of parameters grows by only 3.1 times. These results highlight the model's 's extreme efficiency and scalability in handling ultra-long sequences.
where the ex- traction of basis components significantly reduces the num- ber of required patches, thus greatly simplifying model complexity. We validate the plug-and-plas capability of TimeBase by applying it to the widely recognized PatchTST model, resulting in the variant PATCHTST (w/ TAMEBASE). To ensure a fair comparison in terms of forecasting accu- racy, computational complexity, and parameter count, we set the input sequence length to 720 for both PatchTST and PATCHTST (w/ TIMEBASE).
4.4. Performance on Large-Scale Datasets
To further validate the robustness of TimeBase under ex- treme conditions, conduct experiments four large- scale datasets. As shown in Table 5, TimeBase consis- tently outperforms or matches strong baselines across all forecasting horizons, underscoring its effectiveness in han- dling large-scale forecasting tasks. Notably, TimeBase ranks within the top two across all horizons, highlighting its advan- tages in both prediction quality and computational efficiency. Compared to state-of-the- -htte-rrt- models such iTransformer, DLinear, and PStchTST, TimeBase achieves lower forecast- ing enrors while significantly reducing computational cost, as reflected by its much lower Multiply-Acomuular Op- erations (MACs). Even on the largest dataset, CA with 8600 variables, TimeBase requires only 42. 12.13MM77.30M MACs, showcasing its exceptional capability in scenarios
Quantitative Analysis of TimeBase Integration As shown in Table 4, PATCHTST (w/ TIMEBASE) achieves comparable or even slightly improved forecasting accuracy, measured by MSE and MAE, while significantly reducing both parameter count and computational cost. Specifically, across 56 forecasting metrics on 7 datasets, PATCHTST (w/ TIMEBASE) outperforms the original PatchTST on 43 met- rics, while the remaining 13 show only a marginal average degradation of 1.54%, which can be considered negligible. In terms of overall performance, PSTCHTST (w/ TIME- BaSE) reduces MSE and MAE by 1.27% % and 1.13%, re- spectively, indicating that the basis extraction does not harm predictive power, and may even offer slight improvements. On the efficiency front, Time enables dramatic reduc- tions: MACs are reduced by 77.74% % to 93.03 %, and the

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TimeBase: The Power of Minimalism in LTSF
0.175
TimeBese DLinear
TimeBase
Iec
0.150
600
DLinear
TimeBase DLinear
Memory (M8)
500
0.125
Time
400
0.100
300
Perameter
0.075
0.050
200
132
2
0.025
100
100
0.000
720
2160
3600
5040
6480
720
2160
3600
5040
6480
720
2160
3600
5040
6480
(a) Running Time Comparison
(b) GPU Memory Comparison
(c) Parameter Number Comparison
Figure 4. Comparison of efficiency metrics between TimeBase and other lightweight model with varying look- back windows. (a) Running time per iteration (s/iter), (b) GPU memory consumption, and (c) Parameter count the look-back window increases from 720 to 6480.
Table 4, Performance of TimeBase a Plug-ane Play Component for Patch Based Methods. The input length is set as 720.
Model
PatchTST
PATCHTST (W/ TIMEBASE)
Metric
MSE
MAE
MACs
Params
MSE
MAE
MACs
Params
96 0.377 0.408
11.05 M 0.15 M
0.364 13.45%
0.398|22.450
0.77M
193.03%
0.03M
880 00.00%
ETTh1
192 0.413 0.431
12.02 M 0.29 M
g2.6292.66%
0.424.1.6229
0.84M 193.01%
0.04M
186.21%
336 0.436 0.446
13.47
0.50 M
0.423
J2.98%
0.437
p2.02%
1.24M
190.79%
0.06M
888 88.000%
720 0.455 0.475
17.34 M
1.05 M
0.475
0.49
1..%
1.58M
190.80%
0.11M
88.652%
96 0.276 0.339
11.05
0.15 M
0.275
10.36%
0.339
10.00%
1.28M
488.42%
0.03M
80 00%
ETTh2
192 0.342 0.385
12.02 M
0.29 M
0.334
12.34%
0.381
11.04%
1.39M
188.44%
0.05M
J52 76%
336 0.364 0.405 13.47
0.50 M
0.36
p0.40%
1.25M
190.72%
0.06M
188.00%
720 0.395 0.434 17.34 M. 1.05 M
0.397
0,436
1.19M
193.14%
0.09M
439%
96 0.298 0.352 258.69 M. 1,51 M 0.29
12.68%
0.345
11.99%
28.87M
188.849
0.52M
665 56%
ETTm1
192 0.335 0.373 266.43 M 2.61 M 0.331
11.19%
0.368
0.368
(1.34%
29.73M
188.849%
0.65M
10%
336 0,366 0.394 278.05 4.27 M 0.364
50.55%
0.386
02.03%
31.02M
188.849
0.83M
180 56%
720 0.420 0.421 309.01 M 8.69 M 0.419
10.24%
0.41611.19%
34.46M
188.88%
1.32M
16%
96 0.165 0.260 258.69 M 1.51 M 0.1
10.00%
10.00%
0.256,1.54%
11.54%
57.59M
J77.74%
0.65M
166.93%
ETTm2
192 0.219 0.298 266.43 M 2.61 M 0.222
0.293311.889
29.73M
188.8496
0.65M
17%.10%
336 0.268 0.333 278.05 4.27 M 0.273
0.332
10.30%
61.89M,
177.74%
1.26M
470.49%
720 0.352 0.386 309.01 M 8.69 M 0.353
po.23%
0.385
10.26%
68.77M
177.76%
2.24M
174.22%
96 149 0.199 776.08 1.51 M 0.145 12.68%
12.05%
86.60M
188.84%
0.52M
.65.66%
Weather
192 0.199 0.243 799.30 M 2.61 M 0.189 12.07%
0.238
j2.08%2.00%%
89.18M
188.847%
0.65M
17%.10%
336 0.240 0.281 834.14
4.27 M
0.243
0.284
92.62M
188.90%
0.83M
50.56%
720 0.312 0.334 927.04 M 8.69 M 0.3 0.314
10.64%
0,334
10.00%
103.38M
188.88%
1.32M
184.8.%%
96 0.141 0.240 11,86
1.51 M 0.128 19.22%
0.223
17.08%
1.32G
188.87%
0.52M
.65.66%
Electricity
192 0.156 0.256
12.22 G 2.61 M 0.145 17.05%
17.08%
1.36G
188.87%
0.65M
10%%
336 0.172 0.267
12.75 G 4.27 M 0.160
0.160]6.98%
16.98%
0.255
0.25514.499%
14.49%%
1.42G
188.86%
0.83M
,60 58%
720 0.207 0.299
14.17 G 8.69 M 0.197 14.83%
0.288
13.68%
1.58G
188.85%
1.32M
164.61%
0.363 0.250
31.86
1,51 M 0.360 j0.83%
0.252
4.27G
186-60%
0.55M
163.68%
Traffic
192 0.382 0.258
32.81 G
2.61 M 0.37 12.88%
0.256,0.7899
0.256
10.78%
4.39G
186.62%
0.70M
173.18%
336 0.399 0.268
34.24
4.27 M
0.396
11996o..75
75%
p3.78%
4.58G
686
0.92ML
178 43%%
720 0.432 0.289 38.05
G
8.69 M 0.422
(2.31%
0.284
11.7%
5.09G 686.62%
1.51M
J62 62%
number of parameters is decreased by 56.95% to 91.43% %. For example, on the Traffic dataset, the parameter count of PatchTST ranges from 1.51M to 8.69M, whereas the Taieeeaae-enhanceed variant achieves the same task with substantially fewer parameters and computations. These results demonstrate that the low-rank nature of long-term time series often leads to substantial computational redun- dancy, which can be effectively mitigated by TimeBase via basis component extraction. Its plug-and-plas design al-
lows seamless integration with existing patch-based models, enabling users to boost efficiency significantly without the need for architectural re-design or retraining from scratch.
4.6. Hyperparameter Analysis
This section explores the impact of two key hyperparame- ters on the performance of TimeBase: the number of basis functions R and the orthogonal loss weight Aorth- Fig-
8

---

TimeBase: The Power of Minimalism in LTSF
ETTh1
ETTh2
Traffic
Electricity
0.46
0.46
0.21
0.44
0.45
0.20
0.38
0.44
0.19
MSE
0.42
0.36
0.43
0.18
0.40
0.34
0.42
0.17
0.38
0.32
0.41
0.16
0.15
0.36
0.40
0.14
0.16 Orthogonal Woight
0.08
0.00
0.08
0.16
0.00
0.04
0.08
0.12
0.16 0.20
0.04
0.12
0.16
Orthogonal Weight
Orthogonal Weight
Orthogonal Weight
Figure 5. Effect of orthogonal loss weight Aorth across Traffic, Electricity, ETThl, and ETTh2.
R 2) leads to a noticeable performance drop. However, once R exceeds a modest threshold (typically R 2 4), the model performance stabilizes. These findings indicate that TimeBase can achieve strong forecasting performance with small number of basis components. In practice, we typi- cally set R for most datasets. For datasets with more
Table 5. Long- teng-termmne time series forecasting results in large scale
datasets. Input lengths all models are set 720.
Methods
TimeBase
iTransformer
DLinear
PatcTTST
Metric
MAE
MACs
MAB
MACs
MAE
MACs
MAE
MACs
0.259
42.11M
0,309
7,24G
00.279
1.78G
0.266
7,61G
0.299 47.06M
0.340
7,45G
0.312
3.57G
0.358
15.22G
CA
336
0.317
54.49M
0.337
7.76G
0.355
6.24G
0.321
26.63G
complex temporal patterns, larger R is recommended to
720
0.386
74.30M
00.441
8.61G
0,386
13,38G
0.439
57.07G
enhance the expressive capacity of the basis components.
96 00,302 18.77M
00.661
3.23G
0,358
795.39M
0.314
3.39G
192 0.338
20.98M
00,345
3.32G
0,385
1.59G
0.397
6.78G
GL.A
336
0.397 24.29M
00.413
3.46G
0..44
2.78G
0.407
11.87G
5. Conclusion
720 0.452
333113M
(0.489
3.840
0,488
5.97G
0.473
25.44G
96 [0.183
11,52M
(0,206
1.98G
0.189
487.94M
94M
0.198
2,08G
192 0.194 12.87M
00.198
2.04G
0.212
975.87M
0.224
4.16G
Considering the temporal pattern similarity and low-rank characteristics, we design TimeBase, the lightest known model for long-term forecasting, with only 0.39K param- eters, 2.77M MACS computation, 88.89M memory usage, and a CPU inference speed of 0.98ms. This demonstrates that even the most minimalist models can achieve strong predictive performance, providing a design foundation for more efficient time series models. Furthermore, TimeBase can function as plug-and-play tool to reduce the complex- ity of any patch-based models. This approach (1) enables long-term time series forecasting models to be deployed eesource-constaing
GBA
336
0.199 14.90M
00.231
2.13G
0.2337
1.71G
0.208
7.28G
720 0.206
20.32M
0.229
2.36G
0.237
3.66G
0.227
15.61G
96
0.163
3.51M
0.199
606.99M[|.
0.7
148.54M
0.175
633.47M
192
0.173
3.92M
00.199
624.73M
0.184
297.08M
0.174
1.27G
SD
336
0.179
4.54M
0.185
651.35M
00.208
519.88M
0.195
2.22G
720
0.183
6.19M
0.216
722.32M
0.202
1.1GG
0,218
4.75G
Traffic
Electricity
0.50
0.24
0.48
0.22
MSE
0.46
MSE
edge devices, and (2) offers new insights for lightweight model designs, encouraging time series researchers to fully leverage sequential data and inspir- ing the development of backbone networks for pre-trained large LTSF models.
0.44
0.18
0.42
0.16
0.40
0.14
12
2
24
Basis Number R
Basis Number R
Figure 6. Effect of basis number R.
A Acknowledgement
This work was supported by the National Natural Science Foundation of China (No. 122279119 Natural Science Foun- dation of Jiangsu Province 20240066, 80224440,,, the grant from State Key Laboratory of Resources and Envi- ronmental Information System. The Al-driven experiments, simulations and model training performed on the robotic Al-Scientist platform of Chinese Academy of Science.
ure 5 shows the MSE results as the orthogonal loss weight North is varied across [0.00, 0.04, 0.08, 0.12, 0.16, 0.20]. For datasets such as ETThI, and ETTh2, prediction perfor- mance fluctuates with different values of Aorth. However, for Traffic and Electricity, the performance remains rela- tively stable. Overall, appropriate value of Aorth can enhance forecasting performance, which are varied across different datasets. In Figure 6, the number of basis number R is varied from [2, 4, 6, 12, 18, 24, 30], and the correspond- ing MSE results for Traffic, Electricity datasets are reported. The results show that using too few basis components (e.g.,
9

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TimeBase: The Power of Minimalism in LTSF
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TimeBase: The Power of Minimalism in LTSF
A. Lightweight Forecasting Survey
As for lightweight forecasting, many Linear-based models have emerged, aiming to achieve lightweight forecasting solutions (Wang et al., 2024a). The main differences of them from TimeBase are summarized in Table 6. DLinear (Zeng et al., 2023) introduces a linear model based on trend and seasonal decomposition, whose competitive forecasting performance empirically demonstrated the feasibility of using Linear for LTSF. Following this, TiDE (Das et al., 2024) provides theoretical proof that the simplest linear analogue could achieve near-optimal error rates for linear dynamical systems. Later, numerous Mixer-based works emerge, such as MTS-Mixe (Li et al., 2023), TSMixer (Ekambaram et al., 2023) and HDMixer (Huang et al., 2024a), which stack standard MLP layers to efficiently capture correlations across different dimensions of multivariate time series. Furthermore, Koopa (Liu et al., 2024b) addresses the challenge of dynamic and unstable time series systems by disentangling time-variant and time-invariant components using Fourier filters and designing a Koopman Predictor advance the respective dynamics. TimeMixer (Wang et al., 2024d) tackles the issue of different granularity levels in miero and macro series by proposing mixing blocks, fully leveraging disentangled multi-scale series in both past extraction and future prediction phases. These works represent efficient time series forecasting models based on Linear structures (1.03 M ~ 31.07 M). More recently, FITS (Xu et al., 2024) further reduces model complexity to 10K parameters by employing low-pass filtering to extract a compact set of frequency-domain features via linear layers, while Sparse (Lin et al., 2024b) leverages periodic structures in the data to achieve an even more compact design with only 1.0K parameters. However, still remains challenging when faced with stricter deployment constraints on edge devices and higher efficiency demands (Chatfield, 2013). To address this, we propose TimeBase to significantly reduce data complexity by extracting temporal basis components using the low-rank characteristies in long-term time series. Our approach could require only 0.39K parameters to achieve competitive predictive performance.
Table 6. Differences between Tame and other lightweight Linar-basee models. Model size refers to the parameter count, and
performance is evaluated using the MSF metric, both assessed the 720-horizon electricity forecasting task.
Linear-hased Model
Tiuuelase(Ours)
SparseTFF
FITS
TimeMixer
Koopa
DLiiner
MTS-Mixer
TSMixer
HDMixer
TIDE
Extremely Light
Extremely Light
Extremely Light (10
Light
Normal
Light
Light (2,2M)
Light (LO5M)
Light (481M)
Normal
Model Size
(1.0 K)
(5,57M)
(30.04
(L.03M
(31,07M)
Perfect (0.208)
Perfect (0.208)
Perfect (0.209)
Perfect (0.200)
Perfect
Perfect (0.200)
Good (0:213)
Good
Good (0.243)
Good
Performance
(0.215)
(0.236)
(0.241)
Forecusting Type
Segment-level
Frequency-Jeve Seyment-level
Point-
Point-level
Point-level Point-level
Point-level
Point level
Point-level
Plug-ani alayay
/
X
x
x
X
x
X
X
X
B. Effectiveness Analysis of TimeBase
Low Rank Characteristics Assume we have historical time series matrix X E RNX,, whereN represents the number of temporal segments and P denotes the length of each segment. According to the definition of singular value decomposition (SVD), any matrix E Nx can be decomposed as:
his- = USVT
where U is an N x N orthogonal matrix, E is an N x P diagonal matrix containing the singular values, and V is an P x P orthogonal matrix. We can establish a threshold to compute the approximate rank:
Rank(Xnis) Re #la; E E: o)
(10)
where is the singular values of X for small threshold > 0, and # (.) is the number. Figure (b) illustrates the singular value distribution from real-world long-term time series data. The rapid decay of the singular values indicates that a large portion of the matrix's ' information is captured by only a few dominant components, thereby confirming its low-rank nature. Thus, Due to the extreme similarity temporal patterns across different segments, long-term time series Xpis E Nx often exhibits approximate low-rank characteristics:
Rank(Xn < min(N,P)
(11)
Generalization of TimeBase Specifically, in the context of TimeBase, the right singular vectors in V of Eq(9) can be interpreted as candidate basis components E RXP, where & min(N, P) specifies the number of significant
14

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TimeBase: The Power of Minimalism in LTSF
components to be retained. Next, reconstruct the time series data X using these basis vectors, it can be achieved linear layer in deep learning framework, which applies the following transformation: X E B. Based on this, we can get ET [X his B) x W and the prediction is X
E pred + Bpeed The prediction error is defined as:
peed
E W pred Bpred
(12)
pred
By ignoring the bias term Bpeed the norm of the enror is:
| 12
X
pred
W pred preaBBl22
(13)
To derive the optimal coefficient matrix W predd, we need to solve the following optimization problem:
min Xrred
W pred
Wpst
min Tr ((Xpree pred W
W preed ( pred
W Wreed E)7)
(14)
min Tr (rr(reedde Xr(ppreXXpres) 2 ( Expea)) + Tr prod EE"w""""
pred)
Went
Next, we take the derivative:
Woss X - pred E
Wiss Th X pedd X preaa
Tr(W pred EX EX + Tr(W pred EE" BEETes W prea)
(15)
2X 2X E
+ 2W pe EE
Setting the derivative equal to zero, we get:
W prod
Xped E" (EET)-1
(16)
Substituting the optimal coefficient W pred into the error expression, obtain the error:
pred X prea E (EE") "'E 'E Xpre (I E (EE")+ E)
(17)
Here, = E (EE")
-'E
-'E is a projection matrix. The norm property of the projection matrix can bounded:
P I E EE
El|2
1
minn
(18)
where amin represents the smallest singular value of EET. Since singular values are equivalent to eigenvalues in this context:
Omin
Amin
EET")
(19)
Next, based on the fact that the spectral norm of the projection matrix is bounded above by Amin (Golub and Van Loan, 2013), and utilizing the inequality property of matrix norms,
|| ||| S ||A B/2 (Horn and Johnson, 2012), we obtain:
| Amin EET)
X Xpeadl2
(20)
where Amin (EE*) denotes the smallest eigenvalue of the Gram matrix EET. This result highlights that the generalization capability of TimeBase to arbitrary time series relies on learning a well-represented basis matrix E. If E exhibits favorable eigenvalue distribution (i.e., Amin (EET) is large), the upper bound on prediction error is lower, highlighting the importance of learned high-quality basis components.
C. Experiment Details
C.1. Experiment Setups
We briefly describe the selected 10 state-of-the-e baselines as follows: 1) iTransformer (Liu et al., 2024a) simply inverts the duties of the attention mechanism and the feed-forward network encode each individual series into variate tokens and
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TimeBase: The Power of Minimalism in LTSF
for forecasting. 2) PatchTST (Nie et al., 2023) is a strong versatile transformer baseline using channel-independence and patching. 3) TimesNet (Wu et al., 2023) is task-general foundational model for time series, reshaping 1-dim temporal data to 2-dim space and using 2-dim backbone to deal with the data. 4) FEDformer (Zhou et al., 2022) introduces enceeneee decomposer to model seasonal- -tenn time series in an efficient manner. 5) Autoformer (Wu et al., 2021) employs an auto-correlation mechanism and series decomposition block to improve long-sequence forecasting. 6) Informer (Zhou et al., 2021) utilizes a sparse self-attention mechanism and a distilling operation to handle long time series more efficiently. 7) DLinear (Zeng et al., 2023) is a simple linear-based model combined with a decomposition module. 8) FITS (Xu et al., 2024) introduces an innovative method for time series forecasting using a complex -vauud neural network, effectively capturing both the magnitude and phase of the data. This dual representation enables more thorough and efficient analysis of time series signals. 9) Sparse (Lin et al., 2024b) simplifies the time series forecasting process by down sampling the original sequences at fixed intervals. 10) TimeMixer (Wang et al., 2024d) tackles the issue of different granularity levels in micro and macro series by proposing mixing blocks.
Implementation Details We build TimeBase using Py Torch 1.13.0 (Paszke et al., 2019). The model is trained with the Adam optimizer (Kingma, 2014) with L.2 loss over 30 epochs. After the first three epochs, a learning rate decay of 0.8 applied, and early stopping is employed with a patience threshold of five epochs. In TimeBase, Channel Independence (CI) (Nie et al., 2023) is involved to simplify the multivariate forecasting process univariate time series forecasting. Due to its highly simplistic design, TimeBase requires minimal hyperparameter tuning. The segment length P is set to the natural period of the dataset (e.g., P = 24 for ETTh1), or respectively shorter when dealing with datasets that exhibit extremely long periods (e.g., P P=4 for Weather). We perform a grid search for TimeBase to find the optimal hyperparameters, specifically for the regularization parameter Aorth = [0.00, 0.04, 0.08, 0. 1.12, 0.16, 0.20] to accommodate variances between datasets, as well as the learning rate between 0.01 and 0.5, The loss function is MSE. To enhance the reliability of our results, we re-run baselines in an uniform and fair setting. For SparseTSF, PatchTST, DLinear, FITS, TimeMixer, Fedformer, and TimesNet, utilized their official code repositories. For Autoformer and Informer, DLinear repository to run these models. To ensure fair comparison with standard LTSF baselines such as (Xu et al., 2024; Lin et al., 2024b), which utilize uniform input length of 720, we also adopt an input length of 720 for all models. It is important to note that we have corrected the bug involving _est_loader where drop_last= True during testing on the test set, ensuring that drop 1 as sttt=aaleee is used instead, which ensures a fair comparison in LTSF(Qiu et al., 2024).
leverage the code provided in the official
C.2. Additional Results
[ossT)]sin(1007)
1.0
Test 300%)
0.9
0.5
2-
0.4
0.2
0.0
1.5
3.0
4.5
X
Figure 7. Visualization of the synthetic data generated using the equation Y coo((X) sin(100X) The dataset comprises 5000 samples, which are split into training (60%), validation (20%), and testing (20%) sets.
Effectiveness on Synthetic Data The synthetic dataset derived from the function Y = cos(X)) (i000) is shown in Figure 7, and provides a controlled and challenging benchmark to evaluate the forecasting performance of TimeBase. The data was sampled with high frequency to capture intricate oscillations, divided into training, validation, and testing sets in a 6:2:2 ratio. The basis number is set as 6 for TimeBase to achieve efficient basis extraction. TimeBase consistently demonstrated superior accuracy and efficiency across varying prediction lengths. For a forecasting length of 100, it achieved an MSE of 0.007 and an MAE of 0.070, utilizing only 0.19K parameters and 0.077M MACs, substantially less than its competitors. At a forecasting length of 200, TimeBase maintained robust accuracy (MSE = 0.010, MAE = 0.082) while still requiring minimal resources (0.23K parameters, 0.093M MACs). For a length of 300, it continued to excel in efficiency and delivered competitive accuracy (MSE- = 0.013, MAE 0.094), outperforming ininn and approaching PatchTST's
16

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TimeBase: The Power of Minimalism in LTSF
Table 7. Performance of TimeBase on synthetic data for forecasting length of 100, 200, 300.
Model
MSE
MAE
MACs
Params
TimeBBse
0.007
0.070
0.077M
0.19K
DLinear
0.015
0.081
0.1M
100K
PatchTST
0.011
0.085
1.28M
219K
Tase
0.010
0.082
0.093M
0.23K
200
DLinear
0,021
0.124
0.2M
200K
PatchTST
0.017
0.083
1,48M
420K
TimeBase
0.013
0.094
0.099M
0.26K
300
DLinear
0.039
0.180
0.3M
300K
PatchTST
0.023
0.136
1.69M
622K
accuracy but fraction of the computational cost.
C.3. Ablation Study
ETTh1
ETTh2
Traffic
Electricity
0::10
Hittees Conrtat
With Orthges Contrait
2:10-
Widde:: Jekagera Conuraza
With Ontigga Conories
12003
048
Wrtbaa Orthe Conandi
Uneaeaaee
10000
0.22
11in Orthegana Conatraee
Couarate:
00.0
-9.53
5:15
0:155
10,00
6.2
0379
HUX21
10:00
10/06
8.78
6:2
1-0K2
Let3
1-133
1-52)
Figure 8. MSE comparison with and without orthogonal constraint across different prediction lengths for Traffic, Electricity, ETThl, and ETTh2 datasets.
Figure 8 illustrates the MSE results for various prediction lengths, both with and without the orthogonal constraint. The inclusion of the orthogonal constraint consistently leads to improvements across all datasets, with gains up to 0.036 in MSE. This indicates that the orthogonal constraint helps the model learn more representative basis components, enhancing both the model's 's representational capacity and predictive performance. The positive impact across multiple datasets and forecasting horizons demonstrates the value of incorporating the orthogonal constraint into the training process.
C.4. Segment Length Analysis
Traffic
Electricity
ETTh1
ETTh2
0.65
0.26
0.48
0.60
0.24
0.46
0.42
MSE
0.50
0.55
0.20
0.18
0.22
0.44
0.40
0.42
0.36
0.40
0.38
0.16
0.32
0.40
0.14
0.36
0.30
12
6
18
30
6
18
30
18
30
Segment Length
Segment Length
Segment Length
Segment Length
Figure MSE results for different segment lengths (P [6, 12, 18, 21, 30)) various prediction lengths Traffic, Electricity, ETTh1, and ETTh2 datasets.
This section explores the impact of segment length on the forecasting performance of Time across the Traffic, Electricity, ETTh1, and ETTh2 datasets. The analysis evaluates how varying the length of sub-sequence segments, denoted as P, affects prediction accuracy for different forecasting horizons. Figure shows that across all datasets and forecasting horizons, the best performance is consistently achieved when the segment length is set to P 24. In contrast, shorter or longer segment lengths (P = 6, 18, 30) result noticeably higher MSE values, indicating suboptimal performance. This suggests that
17

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TimeBase: The Power of Minimalism in LTSF
the choice of segment length significantly affects the model's ability capture temporal patterns effectively. The superior performance at P 24 highlights the importance of aligning the segment length with the inherent periodicity of the data. Deviation from this optimal segment length reduces the model's capacity to accurately represent the underlying time series dynamics, thus leading to a degradation in forecasting accuracy. This analysis underscores the necessity of selecting an appropriate segment length that corresponds to the temporal nature of the data. The findings suggest that segmenting the time series into periods of P = 24 yields the most representative and predictive sub-sequences, enhancing overall forecasting performance.
C.5. Performance on Further Prediction
To better illustrate its strengths in long-term time series forecasting, we extended the maximum prediction horizon beyond 720 to include 1080, 1440, and 1800 time steps. We compared its performance against well-established LTSF models, such as iTransformer and DLinear, across multiple datasets (ETTh1, ETTh2, and Electricity). As shown in Table 8, the results underscore that TimeBase consistently outperforms these models in ultra-long-tern forecasting tasks. It achieves this while maintaining linear growth in model complexity (measured by parameters, MACs), demonstrating scalability as the prediction length increases. Specifically, TimeBase not only yields lower Mean Squared Error (MSE) values but also achieves these results with significantly fewer parameters and MACs compared to its counterparts. For example, in the ETTh1 dataset at a prediction length of 1800, Bame achieves an MSE of 0.714 with only 0.7K parameters and 0.1M MACs. In contrast, iTransformer and DLinear exhibit higher MSEs of 0.812 and 0. 0.796, respectively, while using 523.9K and 2.59M parameters, and 6. 6.78M and 18.15M MACs. Similar trends are observed across the ETTh2 and Electricity datasets, where TimeBase demonstrates robust accuracy and efficiency advantages. These findings validate TimeBase's effectiveness in ultra-long-te forecasting tasks, particularly when resouree efficiency is critical. Moreover, the linear growth in computational cost ensures its feasibility for deployment on edge devices.
Table 8. Further prediction length on Electricity, ETTh2 and ETTh1. The input length is set as 720 for all models.
1080
1440
1800
Metric
MSE
Param
MAC
MSE
Param
MAC
MSE
Param
MAC
TimeBase
0.234
0.5 K
3.47 M
0.264
0.6 K
4.16 M
0.245
0.7 K
4.85 M
iTrans.
0.253
5.65 M
1.85 G
0,272
5.84 M
1.91 G
0.325
6.03 M
1.97 G
DL
0.255
1.6 M
499.45 M
0,290
2.1 M
665.86 M
0.321
2.59
832.26 M
ETTh2
TimeBase
0.478
0.5 K
0.07 M
0.543
0.6
0.09 M
0.552
0.7 K
0.1
iTrans.
0.501
431.1
5.58
477.4
0.597
$23.9
6.78
DLinear
0,583
1.6 M
10.89
0.672
2.1
14.52 M
0.652
2.59
18.15
TimeBase
0.551
0.5 K
0.07 M
0.636
0.6
0.09
0.714
0.7 K
0.1
iTrans.
0.602
431.1 K
5,58 M
0.708
K
6.18 M
0.812
523.9 K
6.78 M
DLinear
0,582
160 M
10.89 M
0.693
2.1 M
14.52 M
0.796
2.59 M
18.15 M
C.6. Extension to Multi-Seasonality
In time series forecasting, many real-world datasets exhibit multiple seasonalities, which are crucial for accurate modeling and prediction. For example, traffic data often contains both daily and weekly seasonal patterns, which can significantly influence forecasting accuracy. TimeBase, initially designed for univariate time series forecasting, can extended to handle such multi-seasonal data by learning distinct period bases for each individual seasonality and combining their outputs to generate more accurate predictions. This extension allows TameBase to model complex periodic patterns while retaining its minimalistic architecture. Mathematically, the multi-seasonal extension of TimeBase can be formulated as follows:
MSTimeBase = Σ TimeBase P PP):) p:)
(21)
where X represents the input data and P p; denotes the different period bases corresponding to each seasonality. By learning multiple period bases (p E [24, 168] hours, for example), the model can capture both short-term and long-term seasonal patterns and combine them to enhance the accuracy of the forecast. To evaluate this extension, we applied the multi-seasonality approach to the Traffic dataset, considering both daily and weekly seasonalities. The results, summarized in Table 9, demonstrate that incorporating multiple seasonalities into TimeBase improves the prediction performance with only a modest increase in computational cost and model complexity. Specifically, the model's ''s Mean Squared Error (MSH)
18

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TimeBase: The Power of Minimalism in LTSF
Pearson Conclation Coefficiont
6.5
2- 4,43
1A00
4:03
4.05
0.04
0.03
0.37
4/15
4.02
100
0100
0.11
630
0.25
4.10
4012
4.03
4.00
0.11
-00
(a) Learned Basis Compoments
(b) Similurity Between Basis Compoments
Figure Visualization of learned basis components on Electricity dataset and the corresponding Pearson correlation coefficients.
and Mean Absolute Error (MAE) improve when compared to the original TimeBase model, despite only a slight increase in the number of parameters and computational cost. This extension underscores the versatility and power of TimeBas in dealing with complex, multi-seasona, patterns, making suitable for wide range of LTSF tasks. The ability extend TimeBase while maintaining its lightweight nature reflects its potential for scalable deployment in real-world applications, where scasonalities often vary and need to be captured for accurate forecasting.
Table 9. Performance of Tase extended to multi seattiseaasonaliyy The prediction length is 720 for Traffic dataset.
Model
MSE
MAE
MACs
Params
iTransformer
0.450
0.313
1.01 G
11.61 M
TimeBase
0.456
0.301
9.93 M
0.51 K
MasemeBase
0.450
0.295
16.76 M
0.49 K
C.7. Basis Visualization
Figure 10 displays the basis components extracted by TimeBase from the Electricity dataset, along with the Pearson correlation matrix between them. The results show that the Pearson correlation coefficients between most basis components are close to zero, indicating low correlation among them. This suggests that TimeBase is capable of extracting distinct and representative basis components from the approximate low-rank structure of long-term time series data. By identifying these representative basis components, TimeBase can perform segment-level forecasting using compact set of basis components. This approach significantly reduces the number of parameters required by the model while maintaining competitive forecasting performance. The ability to leverage such compact representations is key to TimeBase's 's parameter efficiency and contributes to its strong performance on long-term time series forecasting tasks.
C.8. Full Results
19

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TimeBase: The Power of Minimalism in LTSF
Table 10. Full Long-term time series forecasting results normal scale datasets, comparing TimeBase with other baselines. Input lengths of all models are set to 720. Top 2 results are highlighted in bold.
Methods
TimeBase
SpareTSF
TimeMixer
FITS
irransoommr
DLincar
PaccTTTT (2023)
TimesNet
FEDformer
Autoformer
Informer (2021)
(ours)
(2024b)
(2024d)
(2024)
(2024a)
(2023)
(2023)
(2022)
(2021)
Metric
MSE
MAE|
MSE
MAE]
MSE
MAE]
MSE MAE MSE MAB|
MSE
MAE
MSE
MAE MSE MAE MSE MAE]
MSE MAE
MSE MAB
96
00.349 0.384
0.362
0.389
0.410
0.441 00,38 0.402 0.389
0.421
0,378 0.402
0.377
0,408
00.437 0.454 0,485 0.500
0.555 0,558
1.269 0.855
192
0.387 0.410
11401
0.412
0.448
0,465 0.415 0.424 0.424 0.446
0.415 0.425
0..13 0.431
0.456 0.469 0.481 0.49%
0.599 0.575
1.487 0.943
ETTh1
336 0.408 0.418
0,435
0.428
0.482
0.490 0.433 0.439 0.456
0.469
0.449
0.449
0.436
0.446
0.494
0.494 0.522 0.521
0.853
0.702
1.544 0.945
720 0.439 0.446
0.426
0.448
0.475
0.500
0.433 0.457 0.545
0.532
0.507
0.517
00.455
0.475
0.632
0.578 0.604
0.575
0.899
0.730
1.481 0.975
96 00:292 0.345
002994
0.346
0.313 0,350
0.271 0.336 0.305
0.561
0.2994
0.360
0.276
0.339
03,349
0.403 0.401 0:451
00.54
0.359
3.789 1.312
192 0.339 0.387 0.340 0.377
0.383
0,415
0.332 0.374 0.405
0,421
0,412
0.437
00.342 0,385
0.500
0.488 0.425 0.464
1.207 0.866
6.514 2.011
ETTh2
336 0.358 0.410
013600
0.79%
0,385 0,4.3%
0.355
0.396 0.411 0.436
01.771
0.478 01.364 0.405
0..45 0.465 0.427 0.471
(1.825
0.719
5.284 1.859
720 0.400 0.448
0,383 0.425
0,432 0.471 0.377
0.423 0.448
0.470
0,740
0. 0.609
0.395 0,434
0.438 0.465 0,462 0.493
1.772 1. 1.062
4,955 1.884
96 0.311 0.351
0.314 0.359
0.332 0.384
0.313 0.357 0.315
0.369
0,314
0.350 0.298
0,352
0.359 0.391 0,406 0.441
0,455 0,464
0.632 0.574
ETTml
192 0.338 0.37
0.348 0.376
0.355
0. 3.39% 0.339 0.369 0.349 0.388
0.347
0. 3.381 0.335
0.373
O. 0.368
O. 0.398 0.450 0.477
0.562 0.514
1.131 0.802
336 0.364
0 386
0.368
0.386
0,386
0,416 0,367 0.385 0.381 0,409
0.367
0.387 0.366
0,394
0.429 0.438 0,436 0.466
0.737 0.608
1.391 0.923
720 0.413 0.414
0.419
0.413
0.452
0.457
0.417
0.417
0.437
0.439
0,415
0.415
0.420
0.421
0.477 0.474 0.462 0.479
0.503
0,502
1.397 0.973
96 0.166 0.256
0.167 0.259
0.192
0.285
0.166
0.256
0.179
0.274
0.
0.257
0.165
0.260
0,200
0.288 0.339
0.406
0.325 0.391
1.870 1.00Z
ETTm2
192 0.218 0.293
0.219 0.297
0,253
0.329
0.218
0.293 0.239
0.314
0,223
0.304
0.219
0.298
0.274
0.337 0.397 0.452
0.369 0.414
2.807
1.314
336 0.270 0.328
0.271 0.330
0.307 0.362
0.271
0.32K 0.309 0.356
0.291
0.355
0,268
0.333
0. 0.340 0.382 0.449 0.491 0.418 0.452
4.442 1.661
720 00.352 0.380
0.353 0.380
0,380
0.412
0.352
0.380
0.387 0.407
0.400
0.433
(0.352 0.386
0.384 0.407 0.451 0.499
0.612 0.594
5.258
1.914
96 0.146 0.198
0.174 0.231
0.163 0.223
0.176
0.232 0.139 0.212
0.174
0,242
0.149
0.
0.176 0.234 0.289 0 0.342 0.523 0.389
0,283 0.361
Weather
192 0.185 0.241
00.216 0.267
0,201
0.254
00.216
0.268 0.203 0.252
0,215 0.277 0.193
0.243
0.219
0.270 0,340
0. 0.394 0.389 0.423
0.445 0.461
336 0.236 0.281
0,260 0.299
0,258 0,300 0.261
0,299 0.253 0.291
0.262 0.319 0.240
0.281
0.277
0,311 0,370
0. 0,408 0.497 0.495
0.587 0,526
720 0.309 0.331
0,325 0.345
0.329 0.348 0.325 0,346 0.317 0.337
0.319 0,359
0.312
0.334
0,344
0.356 0.420
0.421 0.573 0.520
0.953 0,703
96 00.139 0.231
0.139 0.239
0. 142 0.247
0..147 0.235 0.135 0.233
0,141
0,244
0.141
0.240
0.202 0.308 0,226 0.341
0.225 0.334
0.395 0,460
Electricity
192 0.153 0.245
0.
0,5 0.250
0,164 0,275 0.9
0.256 0.155
0.253
0.155
0.258
0.156
0.256
0.218 0.322 0.220 0.336
0.223 0.332
0.405 0.460
336 0.169 0.262
0.171 0.265
0.171 0.260 669 0.270 0.169
0.267
0.170
0.275
0.172
0,267
0.232 0.332 0.224
0.337
0.233
0.341
0.404 0.460
720 0.207 0.294
0,208
0.300
0,209
0,313 0.214
0.302
0.204
0,301
0.2009
0,309
0.208
0.299
0.299 0.375 0.271 0.378
0.261 0.364
0.429 0.477
96 0..39
0.267
0.389
0.268
0,396 O, 292
0.402 0.275
0.374
0.273
0.396
0.272
0,363
0.250
0.605 0.325 0.664 0.43 0.668
0.401
0.829 0.487
Traffic
192 0..40
0.271
0.399 0.272 0.404 0.295 0.419 0.286
0.393
0.283
0.404 0,275
0,382 0,258
0.627 0,340 0.613 0.382 0,703
0.439
0.902 0.525
336 01.417 0.278
01.477 0.279
0.419 0. 0.302 0..433 0.292
0.409
0.292
01.47 0,283
0.399
0.26%
0.631
0,349 0.612 0. 0.379 0.666
0.421
0.949 0.545
720 0..456
0 298
0,449) 0.299
0.458 0.309 0.459 0,311 0.450
0.314
0.457 0.310 0.432
0.289
0,700 0.371 0.664
0.410
0.697
0.424
1,430 0.793
96 00.629 0.489
[1.711 0.553
0,703 0,534 0.759 0.555 0.686
0.508
01751 0.579 0.643
0.476
0.759 0.560 1.247 0.819
1.394
0,931
1 1,221 0.864
AQShunyi
192 0.661 0.500
0.774 0.551
0.720 0.524 0.774 0,553 0.728
0.520
0.771 0.571 0.692
0.501
0.785 0.557 1,218
0.813
1.394
0.872
1, 1.595 0.951
336 0.683 0.510
0.752 0.544
0.747
0.546 0.755
0,544 0.732
0.529
0.758
0,580 0.695
0,514
0.802
0,566 1.071 0.742
1.438
0,900
l 1.699 0.957
720 0.725 0.523
0,763
0.537
0.772
0.541
0.766
0.538 0.744
0.524
0.748
0.559 0.732
0.520
0.807
0.554 0.985
0.655
1.344
0.809
2.236 1.094
96 00.726 0.477
0.758
0,486
0.762
0. 08444
0.795
0.510 0.798
0.311
0.835
0.531 0.735
0.467
0.828
0.527 0.865 0.523
0.895
0.592
1.027 0.699
192 0.761 0.492
0,851 0.542 0.779
0.492
0.765
0.485 0.785
0.501
0.884
0.558 (0.792
0.500
0.807
0.510 1.001
0.603
0.935
0.600
1 1.293 0.834
AQWan
336 0.788 0
0,810
0.531
0.857
0.559
0.861 0.564 0.818
0.537
0.842
0.550 0.804
0.525
0.810
0.529 0.893 0.569
0.990 0.6666
1.017 0.711
720 0.847 0.524
01883
0.546
0.891
0.549
0.331
0.517 0.971
0.604
0.972
0.600
0.909
0.561
0.982
0.606 0.967
0. 566
18585 0.759
1.156 0.731
96 00.177 0.224
0,204 0.263
0.182
0,253
0.2008
0.268 0,202
0.259
0.187
0.239
0.184
0.236
0.200
0.262 0.208
0. 260 0.251 0.334
0.299 0.384
CzeLan
192 00,211 0.250
01.277 0.274
0,216
0.259
0.224
0.292 0.233 0,282
012266
0,271
0.214
0.257
0.256
0.308 0.258 0.307 0.285 0.360
0.329 0,399
336 0.243 0.276 0,239 0.305
0.246
0,312
0.247 0,316 0.239
0.305
0.238
0.301
0.229
0.290
0,274
0,347 0.299 0. 3.365 0.263 0.354
0,366 0.476
720 0.269 0.297
0.292 0.325
0.291
0.322
0.303 0.338 0.305
0.340
0,315
0.349
0.288
0.319
0.299
0.332 0 0.300 0.324 0.357 0.423
0.399 0.446
96 0..435 0.407
01,435
0.439
0,410
0.411
0.458
0,461 0.442
0.447
0.423
0,423
0.412
0.413
0.4433
0.434 0.484 0.473 0.548 0.566
0.651 0.672
192 0,470 0.435
0.548 0.509
0.473
0.437
0.506
0,468 0.560
0,522
0,534
0. 0.4
0.518 0.479
0.572 0.529 0.579 0.512 0.738 0.712
0. 0,745 0.730
ZaiNoo
336 0.511 0.453
0.525 0.474
0,576
0.520
0.528
0.477 0.536
0.486
0.596
0.537
0.528
0.477
0.579 0.523 0.566 0.499
0.624 0,564
0.662 0.601
720 0.566 0.484
0.617
0.540
0.560
0,487
0.590
0.515 0.623
0.546
0.606
0.527
0.581
0.506
0.587 0.511 0.776 0.671
0.758
0.698
0.847 0,783
96 0.087 0.207
0,083
0.090
0.229
0.094
0.241 0.096
0.244
0.096
0.243
0.0K3
0.211 0.088
0.224 0.092 0.224
0.119 0.316
0.106 0.276
Exchange
192 0815500008 0.306
0.172
0.317
0.167
0.305
0.167
0.307 0.176
0.325
0.169
0.310
0.168
0.307
0.173 0.317 0.208 0.355
0.237
0.444
0.252 0.482
336 0.355 0.436
0.308
0.395
0.342
0.437
0.319
0.409 03312 0.399
0.344
0.438
0.324
0.413
0.362
0.462 0.405 0.512
0.437
0.561
0.536 0.700
720 0.608 0.619
0.701
0.716
0,721
0.734
0.629
0,643 0.619
0.635
0.637
0.648
0.659
0.670
0.683
0.696 0,813 0.795
0.994
1.035
0.835 0.865
96 1.028 0.655
1.056
0.667
1,051
0.637
1.147
(166T 1.023
0.640
1.095
0.685
1053
11658
1.21/ 0.755 1.440
0.882
1.443
0.918
T .6666 1.088
METR-LA
192 1.112 0.711
1.232
0.730
1.163 0.689
1.295
0.772 1.196 0.713
1.330
0. 788
1.134
0.672
1.359 0.806 1.478 0.839
1.366
0.830
1 1.52 0.904
336
1.317 0.771
1.356
0.772
1.304
0.737
1.309
0.743 1.320 0.750
1.353
0. 0.764
1.250
0.706
1.376 0.778 1.555 0.860
1.525
0.883
2.036 1.209
720 1.558 0.877
1.562 0.885
1,564
0.879
1.534 0.865 1.474
0.834
1.423
0. 800
1.404
0.789
1.619 0.910 1.725 0.921
2.221
1.277
2.420 1.384
96 0.442 0.435
0.475
0.482
0,445
0.449
0.495
0.501 0.466
0.474
0.437
0.44
0.450
0.454
0.4462 0.467 0.560 0.550
0.596
0.615
0.566 0.593
192 0.429 0.432
0.458
0.465
0.479 0.483
0.455
0,460 0.505
0.511
0.454
0.458
0.461
0.465
0.452
0.456 0.480
0.474
0.567
0.614
0.670 0.678
PM2.5
336 0.426 0.433
0.478
0.482
0.473
0.475
0.450
0.453 0.461
0.467
0.449
0.451
0.475
0.477
0.490
0.492 0.586 0.555
0.671
0.684
0.749 0.793
720 0.425 0.448
0.456
0.479
0.464
0.487
0.440
0.461 0.489
0.514
0.479 0.457
0.448
0.470
0.461
0.484 0.612 0.609
0.652
0.686
0.561 0.589
96 00.192 0249
0.205
0.244
02232
0
0.218
0.217
0.255
0.200
0.205
02284 0.231 0.254
0.279
0.308 0.377
Solar
192 0.213 0.252
0.215
0.265
0.238
0.293
0.207
0.256 0.208
0.257
0,239
0.294
0.227
0,280
0.253
0.311 0.257 0.301
0.337 0.418
0.332 0.422
336 0.222 0.261
0.213
0.276
0.234
0.301
0.229
0.295 0.238
0.309
0.231 0.298
0..225 0.290
0.222 0.287 0.260 0.319
0.263 0.352
0.314 0.421
720 0.2235 0.264
0.232
0.272
0.273
0.319
0.261
0.307 0,270
0.319
0.237
0.277
0.249
0.291
0.254
0.298 0,260 0.290
0.345 0.413
0.355 0.424
96 0.142 0.286
0.149
0.300
0.158
0,316
0.168
0.338 0.155
0,310
0.165
0.329
0.145
0.291
0.155
0.309 0.173 0.338
0.201 0.411
0.227 0.459
Temp
192 00.15 0.305
0.168
0.323
0.163
0.312
0.181 0.351 0.168 0.324
0.335
16166 0.319
0.187 0.358 0.179 0.326 0.246 0.491
0.281 0.564
336 0.176 0.323
0,220 0.373
0.249
0.421
0.223 0,381 0:232 0.395
0.246
0.417
0.221 0.374
0.236 0.400 0.250 0.412 0.312 0,554
0.349 0.606
720 0.274 0.437
0.642 0.654
0.637
0.643
0.
0 0,699 0.652 0.663
0.584
0.590
0.632
0.639
0.636 0.642 0.769 0.747
0.859
0.893
1.026 1.101
96 (1.785 0.626
(1.
0.799 0.589
0.908
0.669
0.787
0.581 0.824 0.609
(1824
0.607
0.802
0.591
0
0.892 0.657 1.008 0.718
1.110 0.847
1.132 0.836
Wind
192 0.902 0.690
0.949 0.698
1.000
0.764
0.910 0.674 1.054 0.779
0.961
0.707
0.944
0.694
1.008 0.741 1.256 0.876
1.218 0.954
1.540 1.188
336 0.997 0.737
1.182 0.851
1.039
0.747
L100 0.795 1.025
0.742
1.022
0.734
1.043
0.750
L185 0.853 1.375 0.923
1.209 0.935
1.399 1.015
720 1.075 0.782
1.1 0.808
1.189 0.851
1 295
0.932
1.196
0.863
1,266
0.907
1,084 0.776
1,320 0.946
1.281
0.846
1.410
1.053
1.433 1.094
20


Reviews:
[{"review_id": "qZcyuwmO1w", "reviewer": "Reviewer_n697", "summary": "This manuscript focuses on improving efficiency in long-term time series forecasting. The proposed framework comprises only two linear layers, yet it achieves superior efficiency compared to recent state-of-the-art methods. Besides, it provides extensive theoretical analysis and plenty of experiments to demonstrate the effectiveness of the proposed method.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer n697,  \n\nThank you for taking the time to review our paper and for your valuable suggestions to improve its quality.\n\n---\n>**W1. Basis Orthogonal Restriction Analysis:**\n\nWe appreciate your suggestion. From the perspective of the data space, the orthogonality of the basis vectors enhances its representation power, providing them ability to express as any vector in the data space through linear combination [1]. Therefore, the ts basis components should also be diverse and distinct, preventing the extraction of very single time-series patterns [2]. Based on this, we apply the Orthogonal Constraint. In the time-series space, each time series segment can be viewed as a composition of several orthogonal basis components. The goal of TimeBase is to transform the long-term forecasting task at the time-step level into a task of basis extraction and prediction at the period level, thus enabling an efficient time-series forecasting model (0.39K, 1000 times smaller than DLinear). In the revised version, we will add a detailed analysis discussing the potential advantages of \"Basis Orthogonal Restriction.\" \n\n---\n>**W2. CPU Detail:**\n\nThank you for pointing this out.  The CPU used in our experiments is an Intel Xeon E5-2609 v4 CPU (16 cores, 2 sockets, 1.7 GHz base frequency), and we will update the manuscript accordingly.\n\n---\n>**W3. TimeBase for Multivariate Time Series:**\n\nThanks for your question. TimeBase does not consider relationships between variables; instead, it transforms the multivariate time series forecasting task into multiple univariate forecasting tasks. We have  explained this in the original manuscript on **Page 3, lines 150-156**, “ Most existing multivariate time series are homogeneous, meaning that each sequence within the dataset exhibits similar periodicity [3]. This characteristic allows them to be organized as a unified multivariate time series. Based on this property, we employ the Channel Independence [4] to simplify the forecasting of MTS data into separate univariate forecasting tasks.”\n\n---\n>**W4&Q1. Handling Multiple Time Series Patterns:**\n\nMost forecasting scenarios (e.g., traffic, electricity) exhibit a dominant period or overlapping periods (e.g., a weekly period encompassing a daily period). Therefore, we set the shortest period as the fixed P.  For datasets with multiple periods, TimeBase can extract different basis components using various P. In **Appendix C.6**, we demonstrate that  \n\n$\\text{MSTimeBase} = \\sum_{i} \\text{TimeBase}(X; P = p_i)$\n\ncould achieve a 0.006 MSE improvement on the traffic dataset. The marginal improvement suggests that the primary patterns could been effectively captured by single&primary basis length .\n\n---\n>**C1. Minor Typo**\n\nWe acknowledge the typo regarding Figure 7 not being referenced in the Appendix. We will correct this and ensure that Figure 7 is properly referenced in the revised manuscript.\n\n---\n>**Question**\n\n**Q2.** TimeBase can be applied to MTS data by independently extracting basis components from each time series in the multivariate dataset. Each time series is processed to capture its key temporal patterns, and these components are then aggregated to form a unified forecast. \n\n---\n\n[1]  Time series with periodic structure. Biometrika 1967.\n\n[2] Decomposition principle for linear programs. Operations research 1960.\n\n[3] TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis, ICLR 2023.\n\n[4] A Time Series is Worth 64 Words: Long-term Forecasting with Transformers, ICLR 2023.", "claims_and_evidence": "Yes. The claims are well-supported by theory analysis and extensive experiments.", "methods_and_evaluation_criteria": "Yes, the proposed method is validated on several real-world datasets. Its efficiency and accurate forecasting capabilities have a positive impact on long-term time series forecasting.", "theoretical_claims": "Yes. In Section 3.3, the authors provide a derivation of the parameter scale for the proposed method, and the derivation demonstrates that the model scale, as well as the input and output lengths, exhibit an extremely linear relationship, which is significantly smaller than that of the DLinear method.", "experimental_designs_or_analyses": "Yes, I have carefully reviewed the experimental design and analysis in the paper. I found that both the design and validation of the experiments are thorough, primarily in two aspects: (1) The paper uses 21 widely-used and publicly available real-world datasets, ensuring a sufficient and representative set of experiments. (2) The experimental validation is comprehensive, not only confirming the proposed method's high accuracy in predictions but also providing a detailed analysis of its efficiency, including model training time, memory usage, and computational complexity.", "relation_to_broader_scientific_literature": "This paper focuses on the design of ultra-lightweight long-term time series forecasting models and achieves remarkably impressive results. I think this research is inherited from the base decomposition, such as [1].\n\n[1] Bonizzi P, Karel J M H, Meste O, et al. Singular spectrum decomposition: A new method for time series decomposition[J]. Advances in Adaptive Data Analysis, 2014, 6(04): 1450011.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths:**\n\nS1. This paper proposes an extremely minimalistic model for long-term time series forecasting, significantly reducing the model's parameter count and computational complexity. It is an innovative and interesting contribution to the field of time series forecasting.\n   \nS2. The paper is well-structured, with a clear and concise methodology, accompanied by good figure illustrations and comprehensive tables, making it highly readable.\n\nS3. The experiments are extensive, with validation conducted on a large number of real-world datasets, including 17 medium-scale and 4 large-scale datasets, thoroughly demonstrating the effectiveness of the proposed method.\n\nS4. The plug-and-play nature of TimeBase significantly reduces the complexity for many patch-based methods, making it a promising area of research.\n\n**Weaknesses:**\n\nW1. Including an analysis that discusses the potential advantages of \"Basis Orthogonal Restriction\" could further enhance the contributions of TimeBase.\n\nW2. Table 1 provides an efficiency comparison between TimeBase and other forecasting models, including Infer Time (CPU). The specific CPU model used in the experiments should be mentioned in the experimental setup.\n\nW3. Section 3 describes the workflow for univariate time series data. Providing an explanation of how TimeBase can be applied to multivariate time series (MTS) data would help clarify and enhance the workflow description.\n\nW4. I think the authors primarily focus on reshaping time series patterns of a single length. If a dataset contains multiple time series patterns of varying lengths, would TimeBase be well-suited to handle this?", "other_comments_or_suggestions": "There is a minor typo to note, that Figure 7 is not referenced in Appendix. I recommend that the authors correct this in the revised version.", "questions_for_authors": "Q1. Does TimeBase support multiple temporal patterns? Such as Traffic could show daily and weekly periods.\n\nQ2. I wonder how TimeBase can be applied to multivariate time series (MTS) data?", "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": "NjlF396jS2", "reviewer": "Reviewer_2K2o", "summary": "This paper proposes an ultra-lightweight framework for long-term time series forecasting, TimeBase, which segments the time series, extracts basis components, and then performs forecasting. Furthermore, TimeBase can serve as a plug-and-play module to reduce the complexity of other patch-based models.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer 2K2o,\n\nWe greatly appreciate your constructive feedback and have made our best to address your concerns.\n\n---\n>**C1. Sentence Explanation**  \n\n|Term| Explain|  \n|-|-|  \n|Essential Patterns|The compact set of basis components extracted from redundant segments|\n|Interactions|Inter-correlations between extracted basis components by PatchTST|\n\n**This sentence (Page 6, Line 316-318) describes the plug-and-play integration of TimeBase into PatchTST.** TimeBase enhances the extraction of compact 'Essential Patterns' and enables a more effective 'Patch Interactions' for PatchTST, which leads to improved accuracy and reduced model complexity (as shown in Table 3 of main text) . We will refine the description to make it clearer and more understandable.\n\n---\n>**M1. Fixed Basis Length**  \n\nMost forecasting scenarios (e.g., traffic, electricity) feature a single dominant period or overlapping periods (e.g., a weekly period encompassing a daily period). Balancing data efficiency and accuracy, we set the shortest dominant period P to capture key temporal patterns.\n\nBesides, TimeBase can extract various-length basis using different P. In **Appendix C.6**, we show that \n\n$MSTimeBase = \\sum_{i} TimeBase(X; P = p_i)$\n\nyields a 0.006 MSE improvement. The minor improvement suggests that the only capturing primary pattern by TimeBase is sufficient for lightweight forecasting.\n\n---\n>**M2. Contribution of TimeBase**\n\n||FITS|SparseTSF|TimeBase\n|-|-|-|-|  \nFocused Features|Frequency Domain|Time Domain|Time Domain\nCompression Strategy |Low-Pass Filtering|Downsampling|Basis Extraction \nParas|10.5K|1K|**0.39K**\nMACs|79.9M|12.71M|**2.77M**  \nTime|35s |31.3s|**20.6s**\n\nTimeBase simplifies LTSF by extracting  basis components. In fact, it has distinct differences from FITS and SparseTSF  (Analysis is in `D.1 response to Reviewer PC2P`). Overall, **TimeBase’s Key Contributions** are three-fold, \n- **Modeling Techniques**: Propose basis extraction with orthogonal constraints to capture primary  patterns and avoid redundant computation from surge of time-series.\n- **Lightweight, Efficient and Effective Forecaster**: Achieve ultra lightweight but effective forecasting (0.39K Params, 2.77M Macs).\n- **Plug-and-Play Usability**:  Act as a complexity reducer for patch-based methods, cutting PatchTST’s MACs to 1/9 of its original one (e.g., 14.17G → 1.58G, **Tab.7, Appendix C.2**).\n\n---\n>**M3. Basis Extraction and Low-Rank Adaptation**\n\nThe key distinction between them lies in how to handle long-term dependencies in LTSF.  \n- **Basis extraction** identifies a compact set of representative time patterns from historical data. Unlike low-rank approximation, TimeBase explicitly extracts key basis components and uses them as fundamental units for prediction.  \n- **Low-rank adaptation**, in contrast, often relies on SVD to approximate data with a reduced-rank structure but does not explicitly extract  interpretable components.\n\n---\n>**T1. Theoretical Supplement.**\n\nTimeBase|SparseTSF|DLinear|\n|-|-|-| \n$aT+bL+c$|$(TL)/P^2$|$2TL$| \n$a =(R+1)/P,b=R/P,c=R$\n\nTimeBase with **O(T+L)** scale is more efficient compared to SparseTSF/DLinear with **O(T*L)**  in LTSF. This will be added in revised version.\n\n---\n>**E1. Baseline Settings**\n\n|Old Data|New Data|  \n|-|-|  \n|Default setting from official code|1. Search lr in [1e-4, 1e-3, 1e-2]|  \n||2. Fix & Unify other parameters: dim=64, enc_layers=2|\n\n---\n>**E2. Inference Time**\n\nWe provide CPU inference time as it is more practical for edge deployment. Thanks for your advice! We now include GPU (A100-80GB) inference times (ms) for 720-output Elec.\n|TimeBase|DLinear|Sparse|FITS|iTrans|\n|-|-|-|-|-|\n|**0.47**|0.58|0.65|1.09|3.3|\n\n---\n>**E3. Function of Figure 4**\n\nTheorem 3.1 analyzes model scale theoretically. To intuitively verify TimeBase's efficiency, we visualize three computational metrics (Mem, Time, Paras) under ultra-long look-back windows in Figure 4.\n\n---\n>**E4. Impact of Input Length**\n\nThanks for your advice! We provide comparisons and analysis for input lengths [96, 336, 720] in the `W2 response to Reviewer JfQc`.\n\n---\n>**D1. Discussion of FITS and SparseTSF**\n\nThanks for your valuable advice. We have discussed the differences between TimeBase, FITS, and SparseTSF in the `D.1 response to Reviewer PC2P`, and we will add this discussion in the related work.\n\n---\n>**Question**\n\n**Q1.** We apply a sliding window to segment the dataset into (num, patch) samples of 720 input length, perform SVD on each sample, sort the singular values, and then average them to derive the dataset’s typical singular value distribution.\n\n**Q2.** Figure 2 primarily highlights the comparison between TimeBase and DLinear. We have updated SparseTSF (1.0K Parameters, 125.2M Memory, 2.59ms CPU Inference Time, 12.71M MACs) in Figure 2, and you can check it in **our `original anonymous code link` in Abstract**. Thank you!\n\n**Q3-4.** Thank you very much! We will update them.\n\n**Q5.** P=4 (ETTm/Weather) | P=24 (Others)", "claims_and_evidence": "For the claim \"TimeBase effectively captures essential patterns and interactions within the time series data, allowing it to retain or even slightly improve accuracy\".  it does not clearly outline what essential patterns and interactions are being captured.", "methods_and_evaluation_criteria": "1. **Is a fixed basis length P sufficient?** Time series data often exhibit multiple periodic patterns, yet the basis length P must be predefined, which may not fully capture these variations.\n2. **Limited contribution:** The proposed method builds on SparseTSF and FITS, with its main novelty being basis extraction. This may constrain the paper’s overall contribution.\n3. **Distinction between basis extraction and low-rank adaptation:** What differentiates basis extraction from low-rank adaptation in the context of this work?", "theoretical_claims": "1. Theorem 3.1 is similar to the theorem proposed in SparseTSF. However, it only compares the parameter scale with DLinear and lacks a comparison with SparseTSF, which limits the analysis of its relative efficiency and novelty.", "experimental_designs_or_analyses": "1. **Hyperparameter Search for Baselines:** Since the experiments are conducted on new datasets, was hyperparameter tuning performed for the baseline models? If so, can you provide details on the search space used for each baseline?\n\n2. **Inference Time on CPU:** Why is inference time measured on a CPU? Would the performance gap between models be smaller if measured on a GPU? A comparison on GPU could provide a more balanced evaluation.\n\n3. **Redundancy of Figure 4:** In the experiment \"Efficiency in Ultra-long Look-back Window,\" Figure 4 appears redundant, as Theorem 3.1 already presents the parameter scale of TimeBase. Consider whether Figure 4 adds additional insights beyond what is already stated in the theorem.\n\n4. **Impact of Look-back Window Size:** The study lacks experiments analyzing the effect of different look-back window sizes. Since the look-back window size significantly impacts performance, evaluating its influence would strengthen the analysis.", "relation_to_broader_scientific_literature": "Encouraging time series researchers to fully utilize sequential data and inspiring the development of backbone networks for pre-trained large-scale LTSF models.", "essential_references_not_discussed": "FITS and SparseTSF are two fundamental prior works that underpin this study, making it important to clarify their methodologies and highlight the distinctions and contributions of TimeBase.", "other_strengths_and_weaknesses": "Mentioned above.", "other_comments_or_suggestions": "Mentioned above.", "questions_for_authors": "1. In Figure 1(b), how is the singular value of the entire dataset calculated?\n2. In Figure 2, why does the figure not include the latest baseline, SparseTSF?\n3. The resolution of Figure 3 is low. Consider improving its clarity.\n4. In line 251, TimeMixer is not an LLM-based method.\n5. Can you provide the selected segment length P for all datasets?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A"}}, {"review_id": "J42STCnuHE", "reviewer": "Reviewer_JfQc", "summary": "This submission focuses on designing a simple yet efficient model to tackle complex forecasting problems and presents TimeBase, an ultra-lightweight network for long-term time series forecasting (LTSF). The experiments show that TimeBase has a small number of parameters, low computation and memory usage, and efficient CPU inference speed, which is significantly lower than the lightest DLinear. Moreover, TimeBase can not only function as a time-series forecaster but also serve as an efficient complexity reducer for patch-based methods.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer JfQc,  \n\nThank you for your interest in our paper and the constructive feedback you provide. We will address your questions point by point.\n\n---\n>**W1. Real-world low-resource scenarios**  \n\nThank you for your suggestion! TimeBase can be applied in various resource-constrained environments, such as edge computing, mobile devices, and IoT-based forecasting. In these scenarios, the ability to handle large-scale time series data with minimal computational overhead makes TimeBase an ideal candidate.\n\n---\n>**W2. Comparison with more input length**  \n\nThank you and `Reviewer 2K2o` for suggesting an analysis of different look-back window sizes effect to improve the experimental quality of our paper.\nIn our original experiment, we choose a fixed length of 720 time steps to maintain consistency with FITS [1] and ensure a fair comparison. Additionally, we provide experimental **MAE comparisons with input lengths of [96, 336, 720]**, **all with a 720-step output length**, demonstrating that TimeBase maintains both its lightweight design and high prediction accuracy across different input lengths. Two interesting observations from this analysis are:\n- Most datasets, i.e., Electricity, ETTm1, ETTm2, show improved prediction performance as the input length increases.\n- For lightweight models such as TimeBase, SparseTSF, FITS, and DLinear, when the input length is only 96, none of them achieve good prediction performance on the Traffic dataset, but iTransformer and PatchTST could maintain prediction accuracy.\n\n| Model ||TimeBase|||SparseTSF|||FITS|||iTransformer|||DLinear|||PatchTST||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n| Input length|96|336|720|96|336|720|96|336|720|96|336|720|96|336|720|96|336|720 |\n| Electricity|0.347|0.294 |0.294 |0.357|0.299 |0.300 |0.355|0.299 |0.302 |0.354|0.305 |0.301 |0.385|0.321 |0.309 |0.346|0.298 |0.299  |\n| Weather|0.321|0.336 |0.331 |0.328|0.343 |0.345 |0.329|0.343 |0.346 |0.324|0.336 |0.337 |0.363|0.363 |0.359 |0.313|0.335 |0.334  |\n| Traffic|0.384|0.310 |0.298 |0.387|0.305 |0.299 |0.386|0.308 |0.311 |0.305|0.354 |0.314 |0.394|0.378 |0.310 |0.311 |0.311 |0.289  |\n| ETTm1|0.448|0.421 |0.414 |0.449|0.417 |0.413 |0.449|0.416 |0.417 |0.457|0.436 |0.439 |0.452|0.424 |0.415 |0.44|0.425 |0.421  |\n| ETTm2|0.397|0.383 |0.380 |0.408|0.382 |0.380 |0.408|0.383 |0.380 |0.409|0.397 |0.407 |0.526|0.465 |0.433 |0.404|0.386 |0.386  |\n| ETTh1|0.461|0.439 |0.446 |0.464|0.446 |0.448 |0.459|0.453 |0.457 |0.501|0.507 |0.532 |0.53|0.516 |0.517 |0.476|0.471 |0.475  |\n| ETTh2|0.445|0.436 |0.448 |0.457|0.434 |0.425 |0.451|0.433 |0.423 |0.446|0.447 |0.470 |0.671|0.628 |0.609 |0.442|0.429 |0.434  |\n\nWe appreciate your key suggestions for improving the quality of our paper presentation. We will include all results in the revised manuscript to provide more experimental analysis. \n\n---\n>**S1. Decimal Precision in Table 3**  \n\nThank you for pointing this out. We will standardize the decimal precision to three decimal places for consistency in the revised manuscript.\n\n---\n>**S2. TimeMixer Description Correction**  \n\nWe appreciate your attention to detail. We will correct the description of TimeMixer on page 5, lines 250-251, and clarify that it is not based on LLM. This will be updated in the revised manuscript.\n\n---\n>**Q1. Impact of the 'drop_last' correction**  \n\nThe correction to the 'drop_last' parameter ensures that incomplete batches at the end of a sequence are excluded during testing. In some previous baselines, setting 'drop_last=True' with a very high batch size led to the discarding of difficult-to-predict samples, resulting in a significant improvement in prediction accuracy. This issue was later pointed out by FITS [1] and TFB [2], helping the time series forecasting field achieve better evaluation of predictive models. TimeBase addresses this bug to provide a fairer experimental comparison.\n\n----\n\n[1] FITS: Modeling Time Series with 10k parameters. ICLR 2024.\n\n[2] TFB: towards comprehensive and fair benchmarking of time series forecasting methods. VLDB 2024.", "claims_and_evidence": "Yes. The lightweight forecasting performance is validated in Sections 4.2 and 4.3, while its ability to serve as an efficient complexity reducer for patch-based methods is demonstrated in Section 4.4.", "methods_and_evaluation_criteria": "Yes. The proposed method is valuable in LTSF and the effectiveness of the proposed solution is well-supported by extensive validation on real-world benchmark datasets.", "theoretical_claims": "Yes. In Appendix B, the author discusses the low-rank characteristics of long-term time series and confirms that the generalization lower bound of TimeBase is related to the learned time series basis, highlighting the importance of a high-quality basis vector space and the necessity of the orthogonal constraint.", "experimental_designs_or_analyses": "Yes. I think the experimental design is one of the highlights of this research. In particular, Section 4.3 demonstrates how TimeBase can reduce the complexity of PatchTST. Experimental results show that TimeBase reduces the computation of PatchTST to just 1/9 of its original MACs. This illustrates TimeBase's ability to significantly reduce the resource consumption of any patch-based model without sacrificing performance.", "relation_to_broader_scientific_literature": "I think the most significant contribution of this paper is that TimeBase can greatly reduce the spatio-temporal complexity of any model. As demonstrated in the experiments, it can reduce the computation of PatchTST by at least 63%, significantly improving the efficiency of current time series forecasting models.", "essential_references_not_discussed": "The discussion of related work is sufficient.", "other_strengths_and_weaknesses": "Strengths:\n\n1.The paper introduces and validates a novel approach for long-term time series forecasting by using temporal patterns for segment-level prediction rather than point-level forecasting.\n\n2.TimeBase significantly reduces spatio-temporal complexity for any model. As revealed in the experiments, it can reduce the computation of PatchTST by at least 63%, greatly enhancing the efficiency of current time series forecasting models.\n\n3.The paper conducts validation on 21 real-world datasets, covering data from different domains, which demonstrates the generalizability of TimeBase.\n\nWeaknesses:\n\n1.The proposed network is extremely lightweight. I suggest the authors explore real-world low-resource scenarios where TimeBase can be applied, to further deepen the motivation behind lightweight forecasting models.\n\n2.All experiments are currently conducted with a 720 input setting. Considering that different models may have optimal input lengths, could the authors provide a comparison of predictions using a 336 input length to PatchTST? This could further strengthen the experimental results.", "other_comments_or_suggestions": "1. The experimental results reported in Table 3 have inconsistent decimal precision, with some values rounded to two decimal places and others to three. The author should standardize this to three decimal places.\n\n2. On page 5, lines 250-251, the author describes TimeMixer as an LLM-based method, but in fact, TimeMixer is not related to LLM. This should be corrected.", "questions_for_authors": "I notice that the author made a correction to the 'drop_last' parameter in the code. What would happen if this correction had not been made?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "No ethical concerns."}}, {"review_id": "aiYrKKh8pu", "reviewer": "Reviewer_Pc2P", "summary": "This article presents a lightweight model requiring only 0.39k parameters, providing an extremely effective method for time-series forecasting. Extensive experiments have been conducted to verify the effectiveness of such a proposal.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer Pc2P,\n\nThank you for taking the time to review our work and for your valuable feedback.\n\n---  \n>**D1. Discussion on Other Related Works**\n\nThank you and `Reviewer 2K2o` for suggesting a comparison with related work and clarifying our distinctions and contributions. We provide an overall comparison in the table below, followed by a brief analysis and discussion.\n\n||FITS|SparseTSF|TimeBase|  \n|:-:|:-:|:-:|:-:|  \n|Focused Domain|Frequency Domain|Time Domain|Time Domain|  \n|Compression Strategy|Low-Pass Filtering|Downsampling|Basis Extraction|  \n|Parameters|10.5K|1K|**0.39K**|  \n|MACs|79.9M|12.71M|**2.77M**|  \n|Time|35s|31.3s|**20.6s**|  \n\n- **FITS** performs interpolation after low-pass filtering in frequency domain. FITS utilizes **frequency domain** feature, while TimeBase  only operates in  **time domain**, which is a fundamental difference between the two. \n\n- **SparseTSF** performs sparse prediction after local feature extraction, focusing on using TCN to **capture local features**. In contrast, TimeBase leverages the low-rank nature of long-term time series and enhances the encoding of key patterns through basis extraction. In terms of data efficiency, SparseTSF makes predictions **using all downsampled segments**, whereas TimeBase improves efficiency by **extracting only a small number of key basis components**.\n\n---\n>**W1. Integration with Patch-Based Methods**  \n\nTimeBase can serve as a lightweight complexity reducer for patch-based methods by replacing their feature extraction step with basis extraction and aggregation. Specifically for any patch-based models, after transforming the time series into **N** patches, TimeBase aggregates them into **R** core patches (**R ≪ N**). The model then performs forward computation based only on these **R** core patches, significantly reducing computational redundancy. For example, when integrated into PatchTST, TimeBase reduces the MACs from **14.17G to 1.58G** (**Table 7, Appendix C.2**).\n\n---\n>**W2. Batch Size Selection (Section 4.3)**  \n\nFor efficiency analysis, we set the batch size to 12 for all models to ensure a fair comparison of efficiency metrics. This choice considers that an overly small batch size is impractical, while an excessively large batch size may cause some models to exceed memory limits. Following the comparison method used in FITS, we also set the batch size to 12 when testing the models' peak memory usage.\n\n---\n>**W3. Why Use Linear Layers**  \n\nTimeBase is designed to avoid unnecessary complexity while maintaining strong predictive performance. CNNs and Transformers are also effective in capturing sequential dependencies, but they introduce higher computational overhead. Additionally, empirical results from methods like DLinear and TimeMixer have demonstrated the effectiveness of linear models in time series forecasting. Therefore, our method leverages **linear basis extraction**, which efficiently captures key temporal patterns while significantly reducing computational cost.\n \n---\n>**S1. Table 2 Formatting Improvement**  \n\nThank you for your suggestion. We will update Table 2 by using **different colors to highlight the best and second-best results**, improving readability.  \n\n---\n>**Q1: Replacing BasisExtract() with an Attention Mechanism**  \n\nThank you for your question. Here, we compare implementing BasisExtract() with Attention() VS the previous Linear(). The Attention-based BasisExtract() first applies a multi-head attention mechanism for global segment feature extraction, followed by a fully connected layer to map the features into a small set of basis components.  \n\nThe table below (720-input, 720-output on the Electricity dataset) shows their scale comparison, where the computational cost and parameter count of the Attention() version are 3-4 times higher than the previous implementation. The second table presents their prediction performance comparison (720-input, 720-output), indicating that on the Electricity and Traffic datasets, the Attention-based BasisExtract() achieves performance improvements. This suggests that increasing the parameter count enhances the expressiveness of the extracted basis components, leading to higher forecasting accuracy.\n\nBasisExtract($\\cdot$)|MACs|Paras\n|:-:|:---:|:---:|\n**Linear($\\cdot$)**|2.77M|0.39K\n**Attention($\\cdot$)** |8.32M|0.99K\n\nBasisExtract($\\cdot$)|Linear($\\cdot$)||Attention($\\cdot$)||\n|-|:-:|-|:-:|-|\nMetric|MSE|MAE|MSE|MAE\nElectricity|0.207|0.294|0.206|0.297\nWeather|0.309|0.331|0.324|0.351\nTraffic|0.456|0.298|0.451|0.294\nETTm1|0.413|0.414|0.411|0.412\nETTm2|0.352|0.380|0.348|0.384\nETTh1|0.429|0.446|0.437|0.452\nETTh2|0.400|0.448|0.398|0.443\n\n---\n>**Q2: Selecting an Appropriate P Value Without Prior Knowledge**  \n\nWhen no prior domain knowledge is available, **data-driven heuristics** such as Singular Spectrum Analysis (SSA) or Fourier Transform can help estimate dominant periods.", "claims_and_evidence": "The claims made in the submission are supported by sufficient experiments.", "methods_and_evaluation_criteria": "The proposed method addresses the problem of efficient time-series forecasting on resource-constrained terminals, and the author has provided a comprehensive evaluation of the model's actual resource consumption and prediction performance.", "theoretical_claims": "In Appendix B, the author proves that the convergence lower bound of TimeBase is related to the orthogonality of the extracted temporal patterns, which provides a certain degree of credibility to the model's design.", "experimental_designs_or_analyses": "I focused on the section discussing the model's efficiency. In Section 4.3, the author provides detailed efficiency metrics, showing that both runtime and actual computational load are significantly lower than existing lightweight models, demonstrating that TimeBase is indeed extremely lightweight. Additionally, in Section 4.3, the author conducts experiments on efficiency with an ultra-long look-back window. The experimental results show that, even with an extremely long input length, TimeBase remains far more resource-efficient than DLinear, consistently maintaining very low resource consumption.", "relation_to_broader_scientific_literature": "Currently, there have been some explorations into lightweight time series forecasting [1][2], but these models have only managed to keep the parameter count above 1K. However, TimeBase has made a breakthrough by reducing the parameter count to below 0.4K while maintaining extremely low computational cost, further advancing the development of lightweight time series forecasting.\n\n[1] SparseTSF: Modeling Long-term Time Series Forecasting with 1K Parameters, ICML 2024.\n\n[2] FITS: Modeling Time Series with 10K Parameters, ICLR 2024.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n1. Long-term time series forecasting is an important problem, and solving it efficiently has many real-world implications. This paper makes full use of the low-rank characteristics of long-term time series and innovatively proposes a lightweight time series forecasting model, effectively addressing this problem.\n2. The efficiency validation of the model is thorough, especially the validation with extremely long inputs, which highlights TimeBase's lightweight advantage under extreme conditions. \n3. The paper is well-structured and written, and the methodology is described clearly.\n\nWeaknesses:\n1. How TimeBase can be integrated into patch-based forecasting methods is not clearly and specifically described by the authors.\n2. The authors set the batch size to 12 in Section 4.3. Is there a specific reason for this choice?\n3. The authors only use two linear layers in the design of TimeBase, but from my understanding, both basis extraction and segment forecasting could potentially be achieved using transformers or CNNs. The authors should provide a detailed explanation of why they chose to use linear layers for the model design.", "other_comments_or_suggestions": "The authors have chosen to bold the two best results in Table 2, but this does not indicate which method performs the best in terms of prediction accuracy. To improve the readability of the table, I recommend that the authors use different colors to highlight the first and second best results.", "questions_for_authors": "Q1. I am curious—if the implementation of BasisExtract() is replaced with a more parameterized Attention mechanism instead of the linear layer, would the model’s prediction accuracy improve?\n\nQ2. In the absence of prior knowledge or specific conditions, what methodologies can be employed to determine an appropriate P value for a given dataset?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "No ethical concerns."}}]

Author rebuttal:

Meta-review: This paper introduces a new model for time series forecasting. I am pleasantly surprised by the strong performance, especially given the simplicity of the proposed model. The trade-off between efficiency and accuracy is quite impressive. In addition, the paper offers useful insights that are valuable for the design of time series forecasting models.

Overall, the reviews are very positive, and the authors have addressed all major concerns during the rebuttal phase. Notably, it appears that most of Reviewer 2K2o's concerns have been resolved as well. I also recall seeing this paper previously at ICLR 2025, and it seems that the authors have addressed the earlier feedback in this revised version.

In addition to the reviewers' comments, my main suggestion is to improve the clarity and presentation of the figures. Figures 1 and 2 are currently difficult to parse and offer limited value in their present form. Figure 3 contains useful information but is quite crowded, adding more space could help readability. The in-plot elements of Figure 4 are also hard to parse. I recommend revisiting all figures for the camera-ready version, possibly moving one or two figures from the results section to the appendix to make room for cleaner, more informative visuals.

That said, I believe this paper is a solid contribution to ICML, and I recommend accepting it.
Decision: Accept (spotlight poster)