Periodicity Decoupling Framework for Long-term Series Forecasting
Authors: Tao Dai
Beiliang Wu
Peiyuan Liu
Naiqi Li
Jigang Bao
Yong Jiang
Shu-Tao Xia
Venue: ICLR 2024 poster (2024)
Keywords: Long-term time series forecasting
Transformer
CNN
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose a novel Periodicity Decoupling Framework (PDF) for long-term series forecasting by capturing 2D temporal variation modeling.

Abstract: Convolutional neural network (CNN)-based and Transformer-based methods have recently made significant strides in time series forecasting, which excel at modeling local temporal variations or capturing long-term dependencies. However, real-world time series usually contain intricate temporal patterns, thus making it challenging for existing methods that mainly focus on temporal variations modeling from the 1D time series directly. Based on the intrinsic periodicity of time series, we propose a novel Periodicity Decoupling Framework (PDF) to capture 2D temporal variations of decoupled series for long-term series forecasting. Our PDF mainly consists of three components: multi-periodic decoupling block (MDB), dual variations modeling block (DVMB), and variations aggregation block (VAB). Unlike the previous methods that model 1D temporal variations, our PDF mainly models 2D temporal variations, decoupled from 1D time series by MDB. After that, DVMB attempts to further capture short-term and long-term variations, followed by VAB to make final predictions. Extensive experimental results across seven real-world long-term time series datasets demonstrate the superiority of our method over other state-of-the-art methods, in terms of both forecasting performance and computational efficiency. Code is available at https://github.com/Hank0626/PDF.

Published as a conference paper at ICLR 2024
PERIODICITY DECOUPLING FRAMEWORK FOR LONG- TERM SERIES FORECASTING
Tao Dai Beiliang Wu', Peiyuan Liu Naiqi Jigang Bao", Yong Jiang', Shanng"SShuTTee Shu-Tao Xia
1 College of Computer Science and Software Engineering. Shenzhen University, China
National Engineering Laboratory for Big Data System Computing Technology. Shenzhen University, China TTsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, China
"WeBank Institute of Financial Technology, Shenzhen University, China {daitao.edu peiyuanliu.edu, linaiqi.thu/ @ gmmail.com; {jiangy, xiast) @sz. www.ttiinnga.uueee edu.en
ABSTRACT
Convolutional neural network (CNNN-based and Transformer-based methods have recently made significant strides in time series forecasting, which excel at mod- eling local temporal variations or capturing long-term dependencies. However, real-world time series usually contain intricate temporal patterns, thus making it challenging for existing methods that mainly focus temporal variations model- ing from the ID time series directly. Based on the intrinsic periodicity of time series, propose novel Periodicity Decoupling Framework (PDF) cap- ture 2D temporal variations of decoupled series for long-term series forecast- ing. Our PDF mainly consists of three components: multi-periodic decoupling block (MDB), dual variations modeling block (DVMB), and variations aggrega- tion block (VAB). Unlike the previous methods that model ID temporal varia- tions, our PDF mainly models 2D temporal variations, decoupled from ID time series by MDB. After that, DVMB attempts to further capture short-term and long-term variations, followed by VAB to make final predictions. Extensive ex- perimental results across seven real- world long-term time series datasets demon- strate the superiority of our method over other state-of-the-art methods, in terms both forecasting performance and computational efficiency. Code is available at httttps:///ithhi com/ 626/PDD
INTRODUCTION
Time series forecasting plays an essential role in multiple applications, including weather prediction (Angryk et al., 2020), energy management (Zhou et al. 2021). financial investment (Patton 2013),
and traffic flow estimation (Chen et al.) 2001). Recently, with the rapid development of deep learn- ing, plenty of deep learning bDse methods have been developed for time series forecasting
(Lim & Zohren 2021), which can be roughly divided into CNN-based (Wang al. 2022 Liu al.
2022a) and Transformer-based methods (Li et al. 2019 Zhou et al., 2021).
Existing DL-based methods mainly focus on ID temporal variation modeling directly, which plays a crucial role in time series forecasting. Among them, CNN-based methods (Bai et al.) 2018 Wang al. 2022 Wu al., (023) have shown the powerful ability capture short-term variations. For example, TCN (Bai et al.) 2018) incorporates the local information of time series along the tempo- ral dimensions by utilizing convolution operations, and exhibits superior performance in short-term and medium-tere term predictions. However, this type of method usually fails to work well for long-term time series, due to the limited representation of long-term dependencies. By contrast, Transformer- based methods (Li et al. 2019 Zhou al. 2021 Wu et al., 2021) excel at capturing long-tem nergterm dependencies due to the use of self-attention mechanism. For example, Autoformer (Wu et al.) 2021] attempts to exploit the series-wise temporal dependencies with auto-correlation mechanism. PStchTST (Nie et al. (023)) proposes a novel patching strategy to retain local semantic information within each patch. Although the Transformer-based methods have shown competitive perfor-
*Correspondence Peiyuan Liu and Naiqi Li

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Published as a conference paper at ICLR 2024
slast--amm varistiun
long-izmm variation
645
100
055
ConoFence
0.39
1645
1.49
033
20024
0:31
31
3ia
(a)
(b)
Figure 1: (a) Illustration of periodicity- decoupling into short- and long-term series. (b) Com- parison of MSE and patch number of our PDF over other Transformer-bas future 96 time steps on Traffic dataset. Tranformer-based methods obtain worse MSE results with more patch numbers. By contrast, our PDF achieves the lowest MSE with only 24 patches on the look-back window of 960 length. The radius of the circle is the number of patches.
methods to predict
mance than CNN-based methods, they often suffer from heavy computational costs, especially for long-term time series input, which thus limits their real applications.
worth considering that the modeling of ID temporal variations can be a complex task due to the intricate patterns involved. These variations can come in various types, including short-term fluctuations, falling, and rising, which can often overlap with each other (see Figure lap. Despite the challenges, researchers have made significant progress in this area, and the most effective way to model temporal variation remains an open question. However, it is important to note that real-world time series often exhibit multi-perie celltiipriooiiity, such as daily and weekly variations for traffic forecasting, which has been confirmed in recent work (Wu et al. (203).. Furthermore, long-term time series can be simplified decoupled based on a predetermined period. For example, as shown in Figure Ia the original time series can be decoupled into short-term series and long-temn term series, which contain short-term changes and long-tery long-term dependencies, respectively. By taking these observations into account, we can utilize period information to decouple long-term time series.
Motivated by the above observations, we propose a novel Periodicity Decoupling Framework (PDF) for long-term series forecasting by capturing the intricate periodic information inside the time se- ries. Based on the periodicity of the time series, the original ID time series can be further decoupled into simpler short and long-term series, which respectively represent the local changes and global correlations of the ID time series. Due to the diversity of short-term variations (e.g.. fluctuation, rising, and falling), employ "frequency slicing". corresponding to different periods, to divide the look-back window into several uub-sequences. For long-term variations, we utilize "period patch- ing" to extract changes within corresponding time segments across all periods (see Figure Tap. The "period patching" ensures each patch contains rich long-term semantic information.
Technically, propose novel Periodicity Decoupling Framework (PDF) for long-term time series forecasting, As illustrated in Fig. 2 our PDF contains three main components: multi-periodic de- coupling block (MDB), dual variations modeling block (DVMB), and variations aggregation block (VAB). Unlike the previous methods that focus on ID temporal variations modeling, our PDF mod- els 2D temporal variations. Specifically, the multi-periode time series into different short- and long-term ID series based on the period of input series in the frequency domain, followed by further reshaping into 2D tensors with rich short- and long-term variations. After that, the dual variations modeling block attempts to capture short-term and long- term variations from the decoupled 2D tensors, followed by a variations aggregation block to make final predictions. Extension experiments various long-ter term time series datasets, in terms both forecasting performance and computational efficiency. Notably, seen in Figure lb. our PDF handles the long-term series (with look-back window length of 960) better while not sacrificing computational cost (with only 24 patches) than other Tranformer-basee methods.
decoupling block first decouples the ID
our PDF confirm its state-of-the-art performance across
2

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Our main contributions are summarized as follows:
We propose novel Periodicity Decoupling Framework (PDF) for long-term series fore- casting, which fully captures 2D temporal short-term and long-term variations from the decoupled series in parallel architecture.
We propose multi-periodic decoupling block to capture various periods of the input series in the frequency domain. Based on the periodicity of the time series, the ID time series can be decoupled into simpler short- and long-term series formulated with 2D tensors. To fully capture the short- and long-term variations, we propose dual variations modeling block (DVMB) with short- and long-term variations extractor, which is able to preserve the high- frequency information of short-term changes while exploiting long-term dependencies.
Extensive experiments demonstrate the effectiveness of our PDF over other state-of-the- art methods across various long-term time series datasets, in terms of both forecasting performance and computational efficiency.
RELATED WORK
Traditional time series forecasting methods such as ARIMA (Anderson & Kendall 1976) and Holt- Winter (Hyndman & Athanasopoulos) 2018) offer robust theoretical frameworks but suffer from limitations handling data with intricate temporal dynamics. Recent years have witnessed mile- stone achievements of deep learning-basce approaches in time series forecasting. which mainly include CNN-based [Wu et al. 2023), and Transformer- based methods (Lim & Zohren 202I).
Convolutional neural network (CNN) has gained widespread popularity due to its ability to capture localized features (Xia et al., 2017 Zhang et al. 2021; Woo et al. 2023). Many CNN-based time se- ries forecasting methods employ Temporal Convolutional Networks (TCN) extract local temporal
dynamics (Bai et al., 2018: Liu et al. 2022a; Wang et al., (022). where MICN (Wang et al.] 2022)
and TetesNet (Wu al.) 2023) are related to method. Typically, MICN attempts to combine local features and global correlations to capture the overall view of time series with convolution kernels. TimesNet focuses on modeling 2D temporal variations in 2D spaces from the extraction of "intra-period" and "inter-period" variations. However, these methods rely heavily on convolution kernels to model series variations, resulting in limited representations of long-term dependencies. Instead, our method can capture both short- and long-term variations simultaneously with dual vari- ations modeling block.
Another type of Transformer-basee method has shown more competitive performance in long-term time series forecasting. With the self-attention mechanism, Transformer and its variant are capable of capturing long-term dependencies and extracting global information (Dosovitskiy et al.] 2021]
Fan et al. 2021 Ryoo et al. 2021 Liu et al. 2022b). However, their scalability and efficiency are constrained by the quadratic complexity of the attention mechanism. To mitigate this, various tech- niques are proposed to reduce the complexity of the Transformer. For example, LogTrans (Li et al. 2019) utilizes convolution self-attention to reduce the space complexity. Informer (Zhou al. 2021 applies distilling strategies to exploit the most crucial keys. Pyraformer (Liu et al. (021)) proposes a pyramid attention design with inter-scale and intra-scale connections. More recent work PatchTST
(Ni et al. (2023) employs patch-based strategies to enhance the locality while improving long-term term forecasting accuracy. However, existing Transformer-basee methods still focus on ID temporal vari- ation modeling and suffer from heavy computational burden for long-term time series. Instead, we propose a more efficient Periodicity Decoupling Framework (PDF) for long-term series forecasting by fully capturing 2D temporal short-term and long-tern term variations in a parallel architecture.
3 PERIODICITY DECOUPLING FRAMEWORK
3.1 THE OVERALL ARCHITECTURE
In time series forecasting, given a historical input series X, = [x], xg......x. x,1" E Rixd, it aims to predict future output series Xo [x++1. Xe42 "x4 T E R T'dd where t, T is the number
of time steps in the past and future, respectively, where d 1 is the number of dimensions. The overall architecture of our method is shown in Figure 2 In our PDF, due to the complex temporal
3

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Published as a conference paper at ICLR 2024
patterns, it is the first step decouple the ID time series for better variation modeling. To this end, we design a Multi-perioddie Decoupling Block to learn the periods of imput series in the frequency domain and convert the ID time series into short- and long-term series, followed by reshaping into 2D) tensors. Then, the obtained short-term and long-term 2D tensors are fed into serveral Dual Variations Modeling Blocks (DVMB) model short- and long-term variations in parallel way. Finally, we use a Variations Aggregation Block to merge the outputs from all DVMBs to yield the final prediction Xo. More details about our PDF are shown in the following sections.
Multi-periodic Decoupling Block
Variations
Pariodicity Extractor
Dual Variations Modeling Black
Aggregation Block
Variations
Coreataas
Lineal
Xo
X
Period based
Tamporal Varitioo Decoupler
Reshaper
Stottsonn Varitionns
Extratior
Figure 2: The architecture of Periodicity Decoupling Framework, which mainly consists of multi-periodic decoupling block, dual variations modeling block, and variations aggregation block.
3.2 MULTI-PERIODIC DECOUPLING BLOCK
The Multi-periodic Decoupling block uses Periodicity Extractor and Period-based Reshaper to trans- form ID time series into 2D) spaces. Then it utilizes Temporal Variations Decoupler to decouple the long-term and short-term information through "period patching" and "frequency slicing".
Periodicity Extractor. Previous work (Wu et al., 2023) emphasizes that the original ID structure of time series inadequately represents only adjacent time point variations, and a 2D structure can effectively capture variations both within and between periods. Therefore, for given ID input X E 4 of dimension d, we employ the Fast Fourier Transform (FFT) (Chatfield [981] to analyze the time series in the frequency domain as follows:
Avg(Amp(FFT((X))
Here, FFT and Amp denote the FFT and amplitude extraction, respectively. The channel-wise av- erage operation Avg over d channels yields A R', representing the amplitudes of frequencies. Specifically, the j-th value A, represents the intensity of the periodic basis function for frequency f. We the univariate X, E Rt instead of X, denote the input time series in the following caleu- lation, because the subsequent transformations and predictions are made in a channel-indepenee manner (Zheng et al. 2014 Nie al. 2023).
Different from Wu et al. (2023), we select frequencies not only focus on high amplitude but also incorporate those with significant values and amplitude. We assert that frequencies with high am- plitude better represent the primary components, while those with larger values facilitate a more discernible distinction between long-term and short-term relationships. We summarize the k fre- quencies selection by:
F, = arg top-m (A), FA, = arg top-ki (A), (Ji,... Sk} = Fx, Utop-ka(F, (F1)) (2) f. 1533 f. 1513
Pesiodictt
Terpooral Varitiom Doomspie
fal
X;
Porobbent Raduper
of
Ale
Figure 3: Multi-periodic Decoupling Block.
4

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where F., and FR, represents the and 1 frequencies with highest amplitudes from A, respectively. We ensure that u is greater than or equal to 11- Due to the conjugate symmetry in the frequency domain, f. only focuses the former [5] frequencies. The final set of ls frequencies is composed of Fk, and the top-k2 frequencies with the greatest values from F. Fxz-
geriod-based Reshaper: Based on the selected frequencies (f1,-... ffx and corresponding period lengths {PI,'. .Pk (p; = (1), we reshape the ID input series X, E If into 2D tensors by:
X Reshape (Padding(X1)), i {1. (1.-..,
(3)
Here, Padding(() is employed to extend the length of Xr to p: J; by filling zeros for Reshape (-), where fü and p. denote the number of rows and columns of the 2D tensor, respec- tively. For the obtained 2D tensor Xin E RRxx,, each row represents the short-term variations and each column represents long-term variations. We then employ Temporal Variations Decoupler to decouple the long-term and short-term information through "period patching" and "frequency slicing".
Period Patching: Denote the patch length p and the stride length as 8, we divide X%D E TRAxp, along dimension p; and aggregate along dimension f. to form patch. Specifically, Xin is patched into multiple patches 2a5d E TRP,,, where N = [ + 1 is the number of patches and each patch contains P = fi X time steps. xτ, denotes the j-th patch. This patching strategy condenses complete long-term variations between all periods.
"g"
Compared with former patching strategies (Nie et al. 2023 Zhang & Yan 2023). our patches capture a broader scope and richer semantic information, enhancing the capacity of the Transformer for modeling long-term variations. Meanwhile, because the number of patches decreases from t/s to max(p,)/ s, the computational cost is significantly reduced.
Frequency Slicing: Along with St dimensions, we split X% into several ID short-term slices af" E RR: where r E [1, fi] denotes the r-th row of X Each local slice represents the short-term variations within every period.
3.3 DUAL VARIATIONS MODELING BLOCK
As illustrated in Figure the Dual Variations Modeling Block is composed of two parts: long-term variations extractor and short-term variations extractor. It adopts a dual-branch parallel architecture to model the long-term and short-term variations in the time series. This parallel structure not only better preserves the high-frequency information of short-term changes but also enhances computa-
tional efficiency (Wang et al. 2022 Si et al. 2022). The details of each component will be given as follows.
Long-term Variations Extractor: Given the patches xyy E NxP with long leng-term information, we initially project them into the latent space via linear projection: 2وS = Linear(x)")) E NND, where D is the dimension of latent space. Subsequently, xr/1 will go through several Transformer encoder layers. The specific process of each layer can be described as follows:
5. g BatchNorm (xr)3 + MSA MSA(( , ,=gg,""")) arr))))
(4)
L.j BatchNorm ( + M))(
Here, Norm(() denotes batch normalization Toffe & Szegedy 2015). MLP(.) is a multi-
layered linear feedforward neural network. Multi-head self-attention MSA(.) mechanism enhances the representation capacity by employing multiple independent self-attention heads. Each head captures different types of long-term dependencies among different patches. All these heads are combined to obtain more comprehensive dependencies by:
X Linear (Flatteen(?:")) R²
(5)
This module contains a sequence of convolution blocks, each consisting of a Convld layer and non-linear activation function. These blocks are sequentially structured to gradually expand the receptive field, accommodating periods of various lengths. For each local slice f.r the process of each block is:
Short-term Variations Extractor:
SELU (CovvlConl))
(6)
5

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Long-ferm Variations Extractor
Transfermer Encaddrr
Lincar
8 R²
Variations
Conv SFI.U
Concat & Trincate
Figure 4: Dual Variations Modeling Block.
where SELU denotes scaled exponential linear units (Klambauer et al. 2017). To get the final prediction of the convolution part, we use the concatenate and truncate operations:
X = Truncate ((nncate(Concat((""))
The final output of the Dual Variations Modeling Block is the summation of X and X):
X
(8)
3.4 VARIATIONS AGGREGATION BLOCK
The Variations Aggregation Block consolidates the results from DVMBs. Specifically, we concate- nate these & results and then map them through parameter-shared linear layer to produce univariate prediction Xo E RT:
Xo = Linear(Concat(X
(9)
The final multivariate prediction Xo E 7xx is obtained by stacking d univariate prediction Xo-
4 EXPERIMENTS
Datasets We conduct extensive experiments on seven popular real-world datasets (Zhou et al. 2021 including Electricity Transformer Temperature (ETT) with its four sub-datasets (ETThl, ETTh2, ETTml. ETTm2), Weather, Electricity, and Traffic. We adopt the same train/val/test splits ratio 0.6:0.2:0.2 Zhou et al. (202I) for the ETT datasets and split the remaining three by the ratio of 0.7:0.1:0.2 following Wu et al. (202T).
Baselines We select select representative state-of-the-arr methods from the recent LTSF landscape base- lines, including the following categories: 1) Transformer- based models: PatchTST NNe et al.) 2023) and FEDformer (Zhou al.) 2022); 2) CNN-based models: TimesNet (Wu al. 2023) and MICN (Wagg et al. 2022); 3) Linear-based models: TiDE (Das al. 2023) and DLinear (Zeng et al.) 2023). Considering varying look-back window size leads to different performances, we pick up their best performance baselines, and corresponding results reported from the original papers.
Setups Following Zhou et al. (00II), we normalize the train/val/test sets to zero-mean using the mean and standard deviation from the training set. The Mean Square Error (MSE) and Mean Ab- solute Error (MAE) are selected as evaluation metrics, consistent with previous methods. All of the models adopt the same prediction length T (96, 192, 336, 720}. For the look-back window t, we conduct experiments on PDF using = 336 and 720 while TiDE, PatchTST, and DLinear employ 720, 512, 336, and all other models / 96.
4.1 MAIN RESULTS
We present multivariate long-term forecasting results in Table Regarding the Count value, PDF(720) and PDF(336) achieve the best and second-best results, outperforming all other methods across different categories. Quantitatively, compared with Transformer-bast based models, PDF(720) yields an overall 14.59% reduction in MSE and 10.77% reduction MAE, Compared with CNN- based models, PDF(720) yields overall 24.61 % reduction in MSE and 19.91 % reduction in MAE

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Published as a conference paper at ICLR 2024
Table 1: Multivariate long-term forecasting results with different prediction lengths T E {96, 192, 336, 720). The numbers in parentheses next to the method represent dow t. The best and the second best results are in bold and underlined. The last row Count indicates the number of times each method achieves the best or the second best results.
the look-back win-
Catagorics Models
Dars
Transformers
CNN:
Linears
PDF(720)
PDFO36)
PachTST(5125
FEDlomer
TimesNett56)
MICN(S6)
TIDE2220)
313)
Metric
MSF
MAF
MSF
MAF
MSF
MAR
MSF
MAF
MSF
MAF
MSF
MAR
MSF
MAF
MSF
MAR
0.356
0.39
0.388
0.370
0,000
0.11
0.402
0.375
B1'h2 ET
192
0.390
0.515
0.397
0.412
0.413
0.429
0.420
0.448
9.436
0.429
0.474
0.487
0.412
0.422
0.605
0.416
336
0.402
0.421
0.009
0.422
0.422
0.440
0.459
0.465
0.491
0.469
0.569
0.551
0.435
0.433
0,439
0.443
0.432
0.455
0.447
0.468
0.506
0.507
0.500
0.720
0,672
0.454
0,472
0.490
0.270
0.332
0.272
0.274
0.337
0.358
0.397
0.374
0.799
0.364
0.270
0.336
0.289
0.353
192
0.334
0.375
0.335
0.375
0.341
0.382
0.429
0439
0.402
0.414
0.441
0.454
0.332
0.380
0.383
0.418
336
0.324
0.379
0.325
0.377
0.329
0.384
0.496
0.487
0.452
0.452
0.654
0.567
0.360
0.407
0.448
0.465
0,423
0.74
0.956
0.716
0.551
96
0.277
0.357
0.280
0.335
0.293
0.346
0.339
0.19
0.338
0.315
0.316
0.362
0.306
0.340
0.299
0.343
Traffic Eleeticity Westber ETTa2 ET1ml
192
0.316
0.364
0.317
0,359
0.333
0.570
0.436
0.441
0.374
0.387
0.365
0.290
0.325
0.366
0.335
0.365
336
0.346
0.381
0.354
0.382
0.369
0.392
0.445
0.459
0.410
0.411
0.408
0.426
0.364
0.384
0.369
0.386
7201
0,402
0,409
0,405
0.413
0.41
0.420
0.90
0.450
0.481
0.776
0.413
0.413
0.421
96
0.159
0.251
0.162
0 055
0.166
0.256
0.201
0.287
0.187
0.267
0,179
0.275
0.161
0.251
0.167
0.260
0.217
0.292
0,319
0.21
0.223
0.296
0.269
0.328
0.219
0.309
0.307
0.376
0.215
0.289
0.224
0.303
0,325
0.200
0.376
0.274
0.329
0.325
0.388
0.342
720
0.345
0,775
0.358
0.362
0.385
0.411
0415
0,408
0.403
0.502
0.400
0.357
0.383
0.997
0.421
96
0.143
0.199
0.147
0.149
19
0.217
0.296
0.172
0.220
0.161
(1.29
0.166
0.222
0.176
0.237
0.188
0.236
0.19
0.194
0.241
0.276
0.336
0.261
0.220
0.281
0.209
0.220
0.282
0.240
0.279
0,244
0.279
0.345
0.283
0.380
0.306
0.278
0.331
0.254
0.301
0.265
0.319
720
0.308
0.728
0.318
0 130
0.314
0.334
0.403
0.228
0.365
0.359
0.311
0.356
0.313
0.340
0.523
0.362
0.126
0.200
0.127
0.319
0.129
0.222
0.193
0.308
0.168
0.272
0.164
0.389
0.132
0.229
0,140
0.237
0.235
0.147
0.240
0.315
0.177
0.885
0.147
0.243
0.249
336
0.159
0.255
0.162
0.255
0.163
0.259
0.214
0.329
0.198
0.300
0.193
0.301
0.161
0.261
0.169
0.267
720
0.194
0.287
0,200
0.290
0.197
0.20
0.266
0.355
0.220
0.320
0,212
0.321
0.196
0.294
0,200
0.301
0.366
0.519
0.309
0,410
0.282
0,363
0.247
0.374
0.348
0.379
0.256
0.733
9.617
0.336
0.537
0.315
0.346
0,423
0.287
336
0.376
0.258
0.386
0.253
0 0.392
0.266
0.621
0.383
0.620
0.336
0.534
0.313
0.355
0.260
0.436
0.706
720
0.419
0.279
0,421
0 0.278
0.432
0.28%
0.636
0.382
0.640
0,350
0.577
0.325
0.386
0,273
0.466
0.315
Count
Compared with Linear-based models, PDF(720) yields an overall 7.05% reduction in MSE and 5.51% reduction in MAE. These results affirm that PDF can effectively utilize a long historical look-back window. Furthermore, PDF(720) consistently outperforms all baselines, except for TiDE which exhibits a lower MSE on the traffic dataset. However, this superior performance of TiDE on the traffic dataset largely attributed to the prior knowledge of static covariates (Das et al. 2023
4.2 EFFECTIVENESS OF PERIOD PATCHING
Analysis of patch information. Recent works (Nie et al. 2023 Lin et al. 2023; Zhang et al.) 2023)
point out that enhancing the semantic information within patches can lead to improved predictions. To assess the performance of patches emphasizing more semantics information versus long-term information, we conduct the following comparative experiments: 1) PatchTST(336): Following the original PatchTST experimental setup, we set each patch length p = 16 and stride = 8, yielding a total of 42 patches; 2) PatchTST(336)": We set p = 64, = 14 and obtain 24 patches. Compared with PatchTST(336), each patch is longer and encompasses more semantics information. 3) PDF(33)): We employ single-period patching with a period length p1 = 24 and choose p = s = 1. Given that fi = 336/p pı = 14. each patch has a length of p x fi = 14. This resulted in 24 patches, each rich in long-term information.
The experimental results in Table 2 show that compared with PatchTST(336) and PatchTST((36)', PDF(336) demonstrates noticeable performance improvements on most datasets. These findings emphasize the importance of long-term information contained within the patches. is notewor- thy that both P and PDF(336) have the same number of patches. Even though each patch in PatchTST(336)* longer, theoretically suggesting potential for better prediction results, its performance does not improve and is even worse than PatchTST(336) in some cases. This fur- ther indicates that merely extending the semantics information within patch is not sufficient for enhancing prediction. The key is to ensure each patch captures more long-term information and our period patching method can effectively address this concern.
Analysis of efficiency. To further validate the computational efficiency of our period patching ap- proach, we conduct experiments comparing the Multiply-Accumuu Operations (MACs) (Cao et al. 2022 of our PDF with two other patch-based methods across various look-back windows
7

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Published as a conference paper at ICLR 2024
Table 2: Results of each patch with various semantic information. PatchTST(336))
denotes the variant of PatchTST with longer patches (e.g. more semantics informa- tion). The best results are bold.
Datasets
ETTh1
Electricity
Traftic
Method
Metric
96
192
336
720
96
192
336
720
96
192
336
720
PatehTST((366)
MSE
0.370
0.413
0.422
0.447
0.129
0.147
0.163
0.197
0.360
0.379
0.392
0.432
MAE
0.400
0.429
0.440
0.468
0.222
0.240
0.259
0.290
0.249
0.256
0.264
0 0.286
PathTTTT(33))
MSE
0.389
0.425
0.435
0.451
0.130
0,148
0,164
0,202
0.367
0.376
0.393
0.428
MAE
0.411
0,429
0.440
0.468
0.223
0.241
0.257
0.291
0.253
0.260
0.263
0.283
PDF(336)
MSE
0.357
0,397
0,409
0,432
0.128
0.147
0.166
0.204
0.383
0.374
0.388
0,423
MAE
0.388
0.412
0.422
0.455
0.223
0.243
0.263
0.294
0.239
0.250
0.257
0.277
t (336, 512, 720, 960} and prediction length T E {96, 192, 336, 7203. The results are sum- marized in Table 3 Overall, the MACs for PDF reduced by 34.64% compared to PatchTST and 74.38% compared to Crossformer. For a fixed look-back window t. the increase in MACs for PDF corresponding to the growth in prediction length T typically resides in the magnitude of millions, whereas for PatchTST and Crossformer, is in the magnitude of gillions. The same observation is noted when keeping the prediction length constant and increasing the size of the look-back window. In extreme cases, specifically for ETThl with t = 960 and T 720. PDF demonstrated superior lightweight performance, with reductions in MACs of 54. 12% and 99.71% % compared to PatchTST and Crossformer, respectively.
Table 3: Comparison of moultply-AAcummin Operations (MACs) among PDF and two other patch-based Transformer methods (PatchTST (Nie al. 2023) and Crossformer
(Zhang & Yan 2023)) for different look-back window / E (336, 512, 720, 960) and prediction lengths T E {96, 192, 336, 7203. "M" and "G" stand for million and gillion operations. The lowest computational costs are in bold.
Look-back
336
Models
PDF
Patch.
Cross.
PDF
Patch.
Cross.
PDF
Patch.
Cross.
PDF
Patch.
Cross.
96
3,97M
5,21M
0.81G
4.67M
7.94M
1,08G
5.49M
11.17M
1.37G
6.49M
14.89M
1.72G
192
4.19M
5.66M
1.13G
5.01M
8.63M
1,40G
5.98M
12.14M
1.69G
7.13M
16.18M
2,04G
336
4.53M
6,34M
161G
5.53M
9.66M
1889
6.70M
13.59M
2.17G
8.10M
18. 18.12M
2.52G
720
5.44M
8.15M
2.89G
6.11
12.42M
3.16G
8.64M
17.46M
3.45G
10.68M
28M
3,80G
Avg.
4.53M
6.34M
1.61G
5.53M
9.66M
1,88
6.70M
13.59M
2.17G
8.10M
18.12M
2.52G
Electricity
96
3.410
5,53G
2.17G
3.600
8,420
2.89G
3.820
11.843
3.66G
4,08G
15.79G
4,39G
192
3,42G
3.61G
8.67G
3.74G
3.84G
12,20G
4.52G
16.26G
5.45G
336
3.43G
5.94G
4.31G
3.64G
9.05G
5.03G
3.88G
12.73G
5.81G
4.16G
16.97G
6.74G
720
3.48G
6.60G
7.74G
3.70G
10.06G
8.46G
3.97G
14.15G
9.24G
4.27G
18.86G
10.17G
Avg.
4.31G
3.64G
9.050
5,03G
3,88G
12.733
3.816
4.16G
16.97G
6.74G
96
9.15G
14.84G
5.81G
9.67G
22.61G
7.74G
10.27G
31.80G
9.82G
10.97G
42.40G
12.32G
Traflic
192
9.18G
15,28G
8.11G
9,71G
23.29G
10.04G
10.33G
32.75G
12.12G
11.05G
43.67G
14.62G
336
9.22G
15.95G
11.566
9.77G
24.31G
13,50G
10,42G
34.18G
15.57G
11.17G
45.57G
18.08G
720
9.33G
17.73G
20.77G
9.94G
27.02G
22.70G
10.66G
37,99G
24.78G
11,49G
50.66G
27.28G
Avg.
9.22G
15.95G
11.56G
9.77G
24.31G
13.50G
10.42G
38G
15.57G
11,17G
45.58G
18.08G
4.3 ABLATION STUDIES
Convolution Module. To investigate the impact of convolution in short-term variations modeling, we conduct a study comparing the following three cases: 1) Parallel Convolution; 2) Sequential Convolution; 3) Without Convolution. We perform these comparisons in four datasets. The results in Table 4 show that parallel convolution consistently outperforms its sequential counterpart, an advantage possibly stemming from the training challenges posed by deeper networks in serial archi- tectures. Interestingly, models without convolution yield better results than those using sequential convolution, highlighting the drawbacks of overly deep serial networks. Furthermore, when com- pared to the model without convolution, the parallel approach achieves notable performance im- provements on datasets with weaker periodicity, demonstrating its effectiveness in preserving short- term information without increasing network depth. The observed degradation in performance for datasets with strong periodicity, such as Traffic, underscores the necessity of placing emphasis on the long-term variations across periods.
Variations A Aggregation Method. We explore two methods for aggregating the outputs of multiple DVMBs within the variations aggregation block: 1) Concat: Concatenate the outputs of all DVMBs
8

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Published as a conference paper at ICLR 2024
Table 4: Ablation study of convolution module in PDF. "Par Conv", "Seq Conv", and "w/o Conv" denote parallel convolution, sequential convolution, and without convolution. The best results are in bold.
Darasets
ETTh2
Weather
Electricity
Traftic
Method
Metric
96
192
336
720
96
192 336
720
96
192 336
720
96
192 336
720
Par Conv
MSE
0.270
0.334
0.324
0.378
0.143
0,188
0.240
0.308
0.126
0.145
0.159
0.194
0.360
0.363
0.376
0.419
MAE
0.332
0.375
0.379
0.422
0.193
0.236
0.279
0.328
0.220
0.238
0.255
0.287
0.239
0.247
0.258
0.279
Soq Conv
MSB
0,279
0.342 0.356
0.418
0.146
0.192
0,244
0.316
0.128
0,145
0.161
0.19
0.306
0,376 0.386
0.426
MAE
0.339
0.381 0,391 0.452
0.197
0.241
0.283 0.336
0.225
0,241
0.257
0.289
0,255
0.260 0.267
0,287
MSE
0.273
0.340
0.334
0.399
0.145
0.190
0,243
0.307
0.127
0.144
0.100
0.196
0.348
0.363
0.375
0.420
wlo Conv
MAE
0.338
0.382 0,390
0.441
0.196
0.239
0,281
0.331
0.221
0.239
0.255
0.288
0.237
0.247
0.256
0.282
and map them through linear projection: 2) Mean: Compute the average outputs of all DVMBs. The experimental results of these two aggregation strategies are presented in Table [5] which shows that the Concat operation generally has better performance than the Mean operation.
Table 5: Ablation study of the variations aggregation method. The best results are in bold.
Darasets
ETTh2
ETTm2
Weathr
Electricity
Method Metric
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336 720
MSE
0.277
0.334
0.324
0.378
0.159
0.217
0.266
0.345
0.143
0.188
0.240
0.508
0.126
0.145
0.159
0.194
Concal
MAE
0.332
0.375
0.379
0.422
0.251
0.292
0,325
0.375
0.193
0.236
0.279
0,328
0.220
0.238
0,255
0.287
MSE
0.274
0,340
0,328
0.396
0.163
0.219
0.270
0,344
0.144
0.191
0.244
0,308
0.127
0.144
0.161
0.194
Mean
0.337
0.381
0.384
0.437
0.254
0.295
0.329
0.377
0.240
0.282
0.330
0.221
0.240
0.257
0.287
4.4 COMPUTATIONAL COMPLEXITY ANALYSIS
Table 6 compares the theoretical complexity per layer across different Transformer-based models. The complexity of the encoder layer in the origi- nal Transformer is O(t²). Subsequent works man- age to reduce the complexity of the encoder layer to (1 log t) or even (t). While the patch-based approaches retain quadratic complexity, the intro- duction of the patch length p makes (())))) fa- vorable over O(t) when is not excessively large. Notably, expect for PDF, all existing Transformer- based methods have the complexities of en- coder layer tied to the length of the look-back window f. The computational complexity of PDF is only related to the maximum decoupled peri- odic length p;. This ensures that even when the t is extremely large, computational costs remain low. For example, if we select the Electricity dataset with / = 10s and choose its most repre- sentative periodic p; = 24 with the patch length p 24, our computational complexity will be sig- nificantly lower than all other methods.
Table 6: Theoretical computational complexity per layer in Transformer-based models. t and T' denote the length of the look-back window and prediction window, respectively. d denotes the number of variates. denotes the length of each patch in the patch-based methods.
Mahhod
Eneader layer
Deceder layer
Trans.
Varwani
o(e2)
O(T(( + T))
In
Thaaet al
2021
O(t log f)
O(T(T + log C)
Auto.
Wu
2021
O(t loge)
o(
T) log( T))
Pym.
Lin
2021
O(t)
O(t(t T))
FED.
Zhou al
2022
O(1)
o TT)
ETS.
Woo ct
2022
O(t logg)
Tog
Cross.
Zhong & Yan
2023
01 (1)
0 T))
MIP Zhung
2023
o(g))))
=(
PET.
Lin ct al.
2003
o(())))
Putch.
Nie el
2023
o(()))
PDF (Ours)
O(( (2()))))))))
CONCLUSIONS
This paper introduces an efficient Periodicity Decoupling Framework (PDF) for long-term series forecasting, The PDF captures both short- and long-term temporal variations in 2D spaces, The approach involves breaking down complex 1D time series using a multi-periodic decoupling block (MDB) based on periodicity. Additionally, dual variations modeling block (DVMB) is proposed to learn short- and long-term variations from the decoupled 2D series in parallel. Compared to previous methods that only model ID temporal variations, our PDF performs better by effectively extracting both short- and long-term variations. Experiments on real-world datasets demonstrate the superior forecasting performance and computational efficiency over other state-of-te oate-of-the- art methods.

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Published as a conference paper at ICLR 2024
ACKNOWLEDGMENTS
This work is supported in part by the National Key Research and Development Program of China, under Grant 2022YFF1202104, National Natural Science Foundation of China, un- der Grant (62302309, 62171248), Shenzhen Science and Technology Program (Grant No. (
WDZC20231 12 Open Fund of National Engineering Laboratory for Big Data System Computing Technology (Grant No. SZU- BDSC-OFF024-23), and Swift Fund Fintech Funding 2023.
(0Y1202008181000122000
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Reviews:
[{"review_id": "DeVcZ1fgxM", "reviewer": "Reviewer_fZ8f", "summary": "This paper decouples variations of different scales in multi-variate long-term time series based on their periodicity. It then leverages the respective modeling strengths of CNNs and Transformer models to represent these scale-distinct variations. Extensive experiments on multiple long-term time series forecasting datasets demonstrate that the proposed Periodic Decoupling Framework (PDF) method outperforms the latest state-of-the-art approaches across various forecasting time intervals.", "questions": "1) What are the benefits of long-term variation and short-term variation extractors?\n2) The periodicity obtained is adaptive to the time series of input?\n3) It is reasonable to model 2D temporal short-term and long-term variations. Is it possible to model 3D variations in time series?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "Strengths:\n1) The proposed periodicity decoupling framework is a novel and efficient solution to capture 2D temporal variation modeling for long-term series forecasting. Besides, modeling long-term and short-term variations also offers a new perspective for time series forecasting. \n2) The fusion of CNNs and Transformers is prevalent in CV and NLP. This paper bridges time series studies with other advanced domains, introducing the PDF framework by leveraging the strengths of both architectures. The PDF demonstrates outstanding performance and efficiency in time series tasks, as evidenced by thorough experiments.\n3) The authors analyze the computational burden with other state-of-the-art methods and demonstrate the efficiency of the proposed method.\n4) The overall paper is well-organized and easy to follow.", "weaknesses": "1) The channel-independent strategy focuses on the dependencies of the time dimension adopted in this work and is inspired by PatchTST [1]. However, channel dependency is also helpful. In other works, such as Crossformer [2], modeling channels have also been beneficial to some extent in forecasting results. A more robust justification for this choice is needed.\n2) The innovativeness of the patching method in this work needs to be articulated more specifically. What distinguishes this patching approach from other recent patch-based works (such as PatchTST [1] and PETformer [3])?\n\nOther minor suggestions：\nIn the first sentence of Section 4.4, one of the occurrences of \"complexity\" should be removed.\n\n[1] Nie, Yuqi, et al. \"A time series is worth 64 words: Long-term forecasting with transformers.\", 2022, https://arxiv.org/pdf/2211.14730.\n[2] Zhang, Yunhao, and Junchi Yan. \"Crossformer: Transformer utilizing cross-dimension dependency for multivariate time series forecasting.\", 2022, https://openreview.net/pdf?id=vSVLM2j9eie.\n[3] Lin, Shengsheng, et al. \"PETformer: Long-term time series forecasting via placeholder-enhanced transformer.\", 2023, https://arxiv.org/pdf/2308.04791.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Yb3A1hTm4c", "reviewer": "Reviewer_gS6P", "summary": "Real-world time series forecasting is challenging, since it usually contains intricate temporal patterns. This paper focuses on exploits intricate temporal patterns by decoupling the complex series into simpler series to achieve long-term series forecasting. Thus, the authors developed  a novel Periodicity Decoupling Framework (PDF) for long-term series forecasting by capturing 2D temporal variation modeling. Extensive experimental results across  seven real-world long-term time series datasets demonstrate the superiority of the proposed method over other state-of-the-art methods, in terms of both forecasting performance and computational efficiency.", "questions": "- Is there other ways to decouple time series? Besides, how to evaluate the effectiveness of the proposed decoupled strategy?\n- It is recommended to compare the running time of other methods.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- It is reasonable and interesting to decouple the complex 1D time series into simpler series with various variations based on periodicity.\n- The proposed multi-periodic decoupling block is a novel and effective solution to capture various periods of the input series. Based on the periodicity of the time series, the 1D time series are decoupled into simpler short- and long-term series.\n- Extensive experiments demonstrate the effectiveness of the proposed over other state-of-the-art methods (e.g., TimesNet, TiDE) across various long-term time series datasets.\n- The overall paper is well-written and easy to follow.", "weaknesses": "- How to decouple the time series remains an open question. Although the authors propose a simple yet effective periodicity-based strategy to decouple the time series, how to evaluate the effectiveness of decouple strategy has been less explained.\n- Experiments show the computational efficiency of the proposed methods over other transformer-based methods. However, the authors have not compared the running time of different methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GTWeOPHmVw", "reviewer": "Reviewer_Puj1", "summary": "This paper introduces a novel approach called the Periodicity Decoupling Framework (PDF) for enhancing time series forecasting. The PDF method comprises three main components: the multi-periodic decoupling block (MDB) to extract 2D temporal variations from 1D time series, the dual variations modeling block (DVMB) for capturing short-term and long-term variations, and the variations aggregation block (VAB) for making predictions. Extensive experiments on seven real-world long-term time series datasets demonstrate the effectiveness of proposed methods.", "questions": "See weakness.", "limitations": "", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. this paper is easy to follow.\n2. this paper studies a classic problem, time series forecasting.", "weaknesses": "1. the motivation. This paper is not well-motivated. Why do we need 2D states to capture periodicity in time series forecasting? Do the authors show the necessity of the 2D modeling for periods?\n2. the novelty is limited. The biggest contribution of this paper, the formulation of 1D to 2D transformation seems to be the same as TimesNet [1], which makes the novelty largely limited.\n3. the contribution is a little weak. Compared with existing works (such as TimesNet), it only revises several blocks for time series forecasting. The methods don't contribute well to the community.\n4. missing of related work. More periodic modeling works should be discussed [2,3]. Also, differences compared with TimesNet should be discussed.\n\n[1] Timesnet: Temporal 2d-variation modeling for general time series analysis. In ICLR.\n[2] DEPTS: Deep Expansion Learning for Periodic Time Series Forecasting. In ICLR.\n[3] Bridging self-attention and time series decomposition for periodic forecasting. In CIKM.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "juyapI8u1N", "reviewer": "Reviewer_KqCU", "summary": "This paper proposes a Periodicity Decoupling Framework (PDF) to address the challenges of long-term time series forecasting, which traditionally have intricated temporal patterns. The proposed PDF decouples the time series into distinct short-term and long-term series based on its periodicity. Following this, the dual variations modeling block (DVMB) is employed to extract both short-term and long-term variations. Finally, the variations aggregation block (VAB) aggregates the extracted variations for final predictions. Experimental results show that PDF achieves state-of-the-art performance while maintaining low computational cost.", "questions": "1) What is the effects of different perodics?  Since the proposed method lies in periodic to decouple the original time series.  \n2) Can the proposed method applied to other time series applications, like time series classification?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1) Decoupling long-term and short-term variations from complex time series based on periodicity seems reasonable. Besides. the proposed method captures long-term and short-term variations, which fully utilizes the ability of Transformer to model global variations and CNNs to model local variations. \n2) The paper proposes a simple yet effective way to extract the dual variations of short-term and long-term variations. Besides, the foundational design is well-motivated and robustly substantiated.\n3) The significance of both short-term and long-term variations in time series forecasting is adeptly highlighted. The manuscript is well-written and understandable, and the figures and formulas are well-presented.", "weaknesses": "1)Unlike the frequency selection strategy in TimesNet [1], the reason for using a different strategy in the Multi-periodic Decoupling Block needs to be further explained.\n2)Were the experimental data in the article averaged over multiple runs with different random seeds? This can enhance the stability of the experimental results.\n3)The experimental section mentions that the significant performance gain of TiDE [2] in traffic largely stems from static covariates. It would be preferable to provide experiments to substantiate this claim.\n\n[1] Wu, Haixu, et al. “Timesnet: Temporal 2d-variation modeling for general time series analysis.”, 2022,  https://arxiv.org/pdf/2210.02186.\n[2] Das, Abhimanyu, et al. “Long-term Forecasting with TiDE: Time-series Dense Encoder.”, 2023, https://arxiv.org/pdf/2304.08424.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)