Original: T IMES N ET : T EMPORAL 2D-V ARIATION M ODELING FOR G ENERAL T IME S ERIES A NALYSIS
Revised: T IMES N ET : T EMPORAL 2D-V ARIATION M ODELING FOR G ENERAL T IME S ERIES A NALYSIS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Time series analysis is of immense importance in extensive applications, such as weather forecasting, anomaly detection, and action recognition.
Revised: Time series analysis is of immense importance in extensive applications, such as weather forecasting, anomaly detection, and action recognition.

Original: This paper focuses on temporal variation modeling, which is the common key problem of extensive analysis tasks.
Revised: This paper focuses on temporal variation modeling, which is the common key problem of extensive analysis tasks.

Original: Previous methods attempt to accomplish this directly from the 1D time series, which is extremely challenging due to the intricate temporal patterns.
Revised: Previous methods attempt to accomplish this directly from the 1D time series, which is extremely challenging due to the intricate temporal patterns.

Original: Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations .
Revised: Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations .

Original: To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods.
Revised: To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods.

Original: This transformation can embed the intraperiod- and interperiod-variations into the columns and rows of the 2D tensors respectively, making the 2D-variations to be easily modeled by 2D kernels.
Revised: This transformation can embed the intraperiod- and interperiod-variations into the columns and rows of the 2D tensors respectively, making the 2D-variations to be easily modeled by 2D kernels.

Original: Technically, we propose the TimesNet with TimesBlock as a task-general backbone for time series analysis.
Revised: Technically, we propose the TimesNet with TimesBlock as a task-general backbone for time series analysis.

Original: TimesBlock can discover the multi-periodicity adaptively and extract the complex temporal variations from transformed 2D tensors by a parameter-efficient inception block.
Revised: TimesBlock can discover the multi-periodicity adaptively and extract the complex temporal variations from transformed 2D tensors by a parameter-efficient inception block.

Original: Our proposed TimesNet achieves consistent state-of-the-art in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection.
Revised: Our proposed TimesNet achieves consistent state-of-the-art in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection.

Revised: Code is available at this repository: https://github.com/thuml/TimesNet.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Time series analysis is widely used in extensive real-world applications, such as the forecasting of meteorological factors for weather prediction (Wu et al., 2021), imputation of missing data for data mining (Friedman, 1962), anomaly detection of monitoring data for industrial maintenance (Xu et al., 2021) and classification of trajectories for action recognition (Franceschi et al., 2019).
Revised: Time series analysis is widely used in extensive real-world applications, such as the forecasting of meteorological factors for weather prediction (Wu et al., 2021), imputation of missing data for data mining (Friedman, 1962), anomaly detection of monitoring data for industrial maintenance (Xu et al., 2021) and classification of trajectories for action recognition (Franceschi et al., 2019).

Original: Because of its immense practical value, time series analysis has received great interest (Lim & Zohren, 2021).
Revised: Because of its immense practical value, time series analysis has received great interest (Lim & Zohren, 2021).

Original: Different from other types of sequential data, such as language or video, time series is recorded continuously and each time point only saves some scalars.
Revised: Different from other types of sequential data, such as language or video, time series is recorded continuously and each time point only saves some scalars.

Original: Since one single time point usually cannot provide sufficient semantic information for analysis, many works focus on the temporal variation, which is more informative and can reflect the inherent properties of time series, such as the continuity, periodicity, trend and etc.
Revised: Since one single time point usually cannot provide sufficient semantic information for analysis, many works focus on the temporal variation, which is more informative and can reflect the inherent properties of time series, such as the continuity, periodicity, trend and etc.

Original: However, the variations of real-world time series always involve intricate temporal patterns, where multiple variations (e.g. rising, falling, fluctuation and etc.) mix and overlap with each other, making the temporal variation modeling extremely challenging.
Revised: However, the variations of real-world time series always involve intricate temporal patterns, where multiple variations (e.g. rising, falling, fluctuation and etc.) mix and overlap with each other, making the temporal variation modeling extremely challenging.

Original: Especially in the deep learning communities, benefiting from the powerful non-linear modeling capacity of deep models, many works have been proposed to capture the complex temporal variations in real-world time series.
Revised: Especially in the deep learning communities, benefiting from the powerful non-linear modeling capacity of deep models, many works have been proposed to capture the complex temporal variations in real-world time series.

Original: One category of methods adopts recurrent neural networks (RNN) to model the successive time points based on the Markov assumption (Hochreiter & Schmidhuber, 1997; Lai et al., 2018;
Revised: One category of methods adopts recurrent neural networks (RNN) to model the successive time points based on the Markov assumption (Hochreiter & Schmidhuber, 1997; Lai et al., 2018;

Original: Shen et al., 2020).
Revised: Shen et al., 2020).

Original: However, these methods usually fail in capturing the longterm dependencies and their efficiencies suffer from the sequential computation paradigm.
Revised: However, these methods usually fail in capturing the longterm dependencies and their efficiencies suffer from the sequential computation paradigm.

Original: Another category of methods utilizes the convolutional neural network along the temporal dimension (TCN)to extract the variation information (Franceschi et al., 2019; He & Zhao, 2019).
Revised: Another category of methods utilizes the convolutional neural network along the temporal dimension (TCN)

Original: Also, because of the locality property of the one-dimension convolution kernels, they can only model the variations among adjacent time points, thereby still failing in long-term dependencies.

Original: Recently, Transformers with attention mechanism have been widely used in sequential modeling (Brown et al., 2020; Dosovitskiy et al., 2021; Liu et al., 2021b).

Original: In time series analysis, many Transformer-based models adopt the

Original: [Figure]
Revised: [Figure]

Original: Frequency V a l u e
Revised: V a l u e

Original: A m p li t u d e
Revised: A m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape to 2D Structure Period 1 Period 2 Period 1 Length Period 2 Length … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) Intraperiod Interperiod
Revised: Reshape to 2D Structure Period 1 Period 2 Period 1 Length Period 2 Length … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) Intraperiod Interperiod

Original: P e r i o d i c i t y Continuity Time V a l u e
Revised: P e r i o d i c i t y Continuity Time V a l u e

Original: Period 2 Period
Revised: Period 2 Period

Revised: [Figure]

Original: Intraperiod-variation Interperiod-variation Period 1 Period 2 Period 
Revised: Intraperiod-variation Interperiod-variation Period 1 Period 2 Period 3 Interperiod-variation

Original: [Figure]
Revised: [Figure]

Original:  attention mechanism or its variants to capture the pair-wise temporal dependencies among time points (Li et al., 2019; Kitaev et al., 2020;
Revised: to extract the variation information (Franceschi et al., 2019; He & Zhao, 2019). Also, because of the locality property of the one-dimension convolution kernels, they can only model the variations among adjacent time points, thereby still failing in long-term dependencies. Recently, Transformers with attention mechanism have been widely used in sequential modeling (Brown et al., 2020; Dosovitskiy et al., 2021; Liu et al., 2021b). In time series analysis, many Transformer-based models adopt the attention mechanism or its variants to capture the pair-wise temporal dependencies among time points (Li et al., 2019; Kitaev et al., 2020;

Original: Zhou et al., 2021; 2022).
Revised: Zhou et al., 2021; 2022).

Original: But it is hard for attention mechanism to find out reliable dependencies directly from scattered time points, since the temporal dependencies can be obscured deeply in intricate temporal patterns (Wu et al., 2021).
Revised: But it is hard for attention mechanism to find out reliable dependencies directly from scattered time points, since the temporal dependencies can be obscured deeply in intricate temporal patterns (Wu et al., 2021).

Original: In this paper, to tackle the intricate temporal variations, we analyze the time series from a new dimension of multi-periodicity.
Revised: In this paper, to tackle the intricate temporal variations, we analyze the time series from a new dimension of multi-periodicity.

Original: Firstly, we observe that real-world time series usually present multi-periodicity, such as daily and yearly variations for weather observations, weekly and quarterly variations for electricity consumption.
Revised: Firstly, we observe that real-world time series usually present multi-periodicity, such as daily and yearly variations for weather observations, weekly and quarterly variations for electricity consumption.

Original: These multiple periods overlap and interact with each other, making the variation modeling intractable.
Revised: These multiple periods overlap and interact with each other, making the variation modeling intractable.

Original: Secondly, for each period, we find out that the variation of each time point is not only affected by the temporal pattern of its adjacent area but also highly related to the variations of its adjacent periods.
Revised: Secondly, for each period, we find out that the variation of each time point is not only affected by the temporal pattern of its adjacent area but also highly related to the variations of its adjacent periods.

Original: For clearness, we name these two types of temporal variations as intraperiod-variation and interperiod-variation respectively.
Revised: For clearness, we name these two types of temporal variations as intraperiod-variation and interperiod-variation respectively.

Original: The former indicates short-term temporal patterns within a period.
Revised: The former indicates short-term temporal patterns within a period.

Original: The latter can reflect long-term trends of consecutive different periods.
Revised: The latter can reflect long-term trends of consecutive different periods.

Original: Note that for the time series without clear periodicity, the variations will be dominated by the intraperiod-variation and is equivalent to the ones with infinite period length.
Revised: Note that for the time series without clear periodicity, the variations will be dominated by the intraperiod-variation and is equivalent to the ones with infinite period length.

Original: Since different periods will lead to different intraperiod- and interperiod-variations, the multiperiodicity can naturally derive a modular architecture for temporal variation modeling, where we can capture the variations derived by a certain period in one module.
Revised: Since different periods will lead to different intraperiod- and interperiod-variations, the multiperiodicity can naturally derive a modular architecture for temporal variation modeling, where we can capture the variations derived by a certain period in one module.

Original: Besides, this design makes the intricate temporal patterns disentangled, benefiting the temporal variation modeling.
Revised: Besides, this design makes the intricate temporal patterns disentangled, benefiting the temporal variation modeling.

Original: However, it is notable that the 1D time series is hard to explicitly present two different types of variations simultaneously.
Revised: However, it is notable that the 1D time series is hard to explicitly present two different types of variations simultaneously.

Original: To tackle this obstacle, we extend the analysis of temporal variations into the 2D space.
Revised: To tackle this obstacle, we extend the analysis of temporal variations into the 2D space.

Original: Concretely, as shown in Figure 1, we can reshape the 1D time series into a 2D tensor, where each column contains the time points within a period and each row involves the time points at the same phase among different periods.
Revised: Concretely, as shown in Figure 1, we can reshape the 1D time series into a 2D tensor, where each column contains the time points within a period and each row involves the time points at the same phase among different periods.

Original: Thus, by transforming 1D time series into a set of 2D tensors, we can break the bottleneck of representation capability in the original 1D space and successfully unify the intraperiod- and interperiod-variations in 2D space, obtaining the temporal 2D-variations .
Revised: Thus, by transforming 1D time series into a set of 2D tensors, we can break the bottleneck of representation capability in the original 1D space and successfully unify the intraperiod- and interperiod-variations in 2D space, obtaining the temporal 2D-variations .

Original: Technically, based on above motivations, we go beyond previous backbones and propose the TimesNet as a new task-general model for time series analysis.
Revised: Technically, based on above motivations, we go beyond previous backbones and propose the TimesNet as a new task-general model for time series analysis.

Original: Empowering by TimesBlock , TimesNet can discover the multi-periodicity of time series and capture the corresponding temporal variations in a modular architecture.
Revised: Empowering by TimesBlock , TimesNet can discover the multi-periodicity of time series and capture the corresponding temporal variations in a modular architecture.

Original: Concretely, TimesBlock can adaptively transform the 1D time series into a set of 2D tensors based on learned periods and further capture intraperiod- and interperiod-variations in the 2D space by a parameter-efficient inception block.
Revised: Concretely, TimesBlock can adaptively transform the 1D time series into a set of 2D tensors based on learned periods and further capture intraperiod- and interperiod-variations in the 2D space by a parameter-efficient inception block.

Original: Experimentally, TimesNet achieves the consistent state-of-the-art in five mainstream analysis tasks, including short- and long-term forecasting, imputation, classification and anomaly detection.
Revised: Experimentally, TimesNet achieves the consistent state-of-the-art in five mainstream analysis tasks, including short- and long-term forecasting, imputation, classification and anomaly detection.

Original: Our contributions are summarized in three folds:
Revised: Our contributions are summarized in three folds:

Original: • Motivated by multi-periodicity and complex interactions within and between periods, we find out a modular way for temporal variation modeling.
Revised: • Motivated by multi-periodicity and complex interactions within and between periods, we find out a modular way for temporal variation modeling.

Original: By transforming the 1D time series into 2D space, we can present the intraperiod- and interperiod-variations simultaneously.
Revised: By transforming the 1D time series into 2D space, we can present the intraperiod- and interperiod-variations simultaneously.

Original: • We propose the TimesNet with TimesBlock to discover multiple periods and capture temporal 2D-variations from the transformed 2D tensors by a parameter-efficient inception block.
Revised: • We propose the TimesNet with TimesBlock to discover multiple periods and capture temporal 2D-variations from the transformed 2D tensors by a parameter-efficient inception block.

Original: • As a task-general foundation model, TimesNet achieves the consistent state-of-the-art in five mainstream time series analysis tasks.
Revised: • As a task-general foundation model, TimesNet achieves the consistent state-of-the-art in five mainstream time series analysis tasks.

Original: Detailed and insightful visualizations are included.
Revised: Detailed and insightful visualizations are included.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: As a key problem of time series analysis, temporal variation modeling has been well explored.
Revised: As a key problem of time series analysis, temporal variation modeling has been well explored.

Original: Many classical methods assume that the temporal variations follow the pre-defined patterns, such as
Revised: Many classical methods assume that the temporal variations follow the pre-defined patterns, such as

Original: ARIMA (Anderson & Kendall, 1976), Holt-Winter (Hyndman & Athanasopoulos, 2018) and Prophet (Taylor & Letham, 2018).
Revised: ARIMA (Anderson & Kendall, 1976), Holt-Winter (Hyndman & Athanasopoulos, 2018) and Prophet (Taylor & Letham, 2018).

Original: However, the variations of real-world time series are usually too complex to be covered by pre-defined patterns, limiting the practical applicability of these classical methods.
Revised: However, the variations of real-world time series are usually too complex to be covered by pre-defined patterns, limiting the practical applicability of these classical methods.

Original: In recent years, many deep models have been proposed for temporal modeling, such as MLP, TCN,
Revised: In recent years, many deep models have been proposed for temporal modeling, such as MLP, TCN,

Original: RNN-based models (Hochreiter & Schmidhuber, 1997; Lai et al., 2018; Franceschi et al., 2019).
Revised: RNN-based models (Hochreiter & Schmidhuber, 1997; Lai et al., 2018; Franceschi et al., 2019).

Original: Technically, MLP-based methods (Oreshkin et al., 2019; Challu et al., 2022; Zeng et al., 2022; Zhang et al., 2022) adopt the MLP along the temporal dimension and encode the temporal dependencies into the fixed parameter of MLP layers.
Revised: Technically, MLP-based methods (Oreshkin et al., 2019; Challu et al., 2022; Zeng et al., 2023; Zhang et al., 2022) adopt the MLP along the temporal dimension and encode the temporal dependencies into the fixed parameter of MLP layers.

Original: The TCN-based (2019) methods capture the temporal variations by convolutional kernels that slide along the temporal dimension.
Revised: The TCN-based (2019) methods capture the temporal variations by convolutional kernels that slide along the temporal dimension.

Original: The RNN-based methods (Hochreiter & Schmidhuber, 1997; Lai et al., 2018; Gu et al., 2022) utilize the recurrent structure and capture temporal variations implicitly by state transitions among time steps.
Revised: The RNN-based methods (Hochreiter & Schmidhuber, 1997; Lai et al., 2018; Gu et al., 2022) utilize the recurrent structure and capture temporal variations implicitly by state transitions among time steps.

Original: Note that none of these methods consider the temporal 2D-variations derived by periodicity, which is proposed in this paper.
Revised: Note that none of these methods consider the temporal 2D-variations derived by periodicity, which is proposed in this paper.

Original: Besides, Transformers have shown great performance in time series forecasting (Zhou et al., 2021;
Revised: Besides, Transformers have shown great performance in time series forecasting (Zhou et al., 2021;

Original: Liu et al., 2021a; Wu et al., 2021;
Revised: Liu et al., 2021a; Wu et al., 2021;

Original: Zhou et al., 2022).
Revised: Zhou et al., 2022).

Original: With attention mechanism, they can discover the temporal dependencies among time points.
Revised: With attention mechanism, they can discover the temporal dependencies among time points.

Original: Especially, Wu et al.
Revised: Especially, Wu et al.

Original: present the Autoformer with AutoCorrelation mechanism to capture the series-wise temporal dependencies based on the learned periods.
Revised: present the Autoformer with AutoCorrelation mechanism to capture the series-wise temporal dependencies based on the learned periods.

Original: In addition, to tackle the intricate temporal patterns, Autoformer also presents a deep decomposition architecture to obtain the seasonal and trend parts of input series.
Revised: In addition, to tackle the intricate temporal patterns, Autoformer also presents a deep decomposition architecture to obtain the seasonal and trend parts of input series.

Original: Afterward, FEDformer (Zhou et al., 2022) employs the mixture-of-expert design to enhance the seasonal-trend decomposition and presents a sparse attention within the frequency domain.
Revised: Afterward, FEDformer (Zhou et al., 2022) employs the mixture-of-expert design to enhance the seasonal-trend decomposition and presents a sparse attention within the frequency domain.

Original: Unlike previous methods, we ravel out the intricate temporal patterns by exploring the multi-periodicity of time series and capture the temporal 2D-variations in 2D space by well-acknowledged computer vision backbones for the first time.
Revised: Unlike previous methods, we ravel out the intricate temporal patterns by exploring the multi-periodicity of time series and capture the temporal 2D-variations in 2D space by well-acknowledged computer vision backbones for the first time.

Original: It is also notable that, different from previous methods, we no longer limit to a specific analysis task and attempt to propose a task-general foundation model for time series analysis.
Revised: It is also notable that, different from previous methods, we no longer limit to a specific analysis task and attempt to propose a task-general foundation model for time series analysis.

Original:  T IMES N ET
Revised:  T IMES N ET

Original: As aforementioned, based on the multi-periodicity of time series, we propose the TimesNet with a modular architecture to capture the temporal patterns derived from different periods.
Revised: As aforementioned, based on the multi-periodicity of time series, we propose the TimesNet with a modular architecture to capture the temporal patterns derived from different periods.

Original: For each period, to capture the corresponding intraperiod- and interperiod-variations, we design a TimesBlock within the TimesNet, which can transform the 1D time series into 2D space and simultaneously model the two types of variations by a parameter-efficient inception block.
Revised: For each period, to capture the corresponding intraperiod- and interperiod-variations, we design a TimesBlock within the TimesNet, which can transform the 1D time series into 2D space and simultaneously model the two types of variations by a parameter-efficient inception block.

Original: 3. T RANSFORM 1D- VARIATIONS INTO 2D- VARIATIONS
Revised: 3. T RANSFORM 1D- VARIATIONS INTO 2D- VARIATIONS

Original: As shown in Figure 1, each time point involves two types of temporal variations with its adjacent area and with the same phase among different periods simultaneously, namely intraperiod- and interperiodvariations .
Revised: As shown in Figure 1, each time point involves two types of temporal variations with its adjacent area and with the same phase among different periods simultaneously, namely intraperiod- and interperiodvariations .

Original: However, this original 1D structure of time series can only present the variations among adjacent time points.
Revised: However, this original 1D structure of time series can only present the variations among adjacent time points.

Original: To tackle this limitation, we explore the two-dimension structure for temporal variations, which can explicitly present variations within and between periods, thereby with more advantages in representation capability and benefiting the subsequent representation learning.
Revised: To tackle this limitation, we explore the two-dimension structure for temporal variations, which can explicitly present variations within and between periods, thereby with more advantages in representation capability and benefiting the subsequent representation learning.

Original: Concretely, for the length- T time series with C recorded variates, the original 1D organization is
Revised: Concretely, for the length- T time series with C recorded variates, the original 1D organization is

Original: X 1D ∈ R T × C .
Revised: X 1D ∈ R T × C .

Original: To represent the interperiod-variation, we need to discover periods first.
Revised: To represent the interperiod-variation, we need to discover periods first.

Original: Technically, we analyze the time series in the frequency domain by Fast Fourier Transform (FFT) as follows:
Revised: Technically, we analyze the time series in the frequency domain by Fast Fourier Transform (FFT) as follows:

Original: [Equation]
Revised: [Equation]

Original: Time V a l u e
Revised: Time V a l u e

Original: Time Frequency V a l u e A
Revised: Time Frequency V a l u e A

Original: m p li t u d e
Revised: m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape to 2D Structure Period 1 Period 2 Period 1 Length Period 2 Length … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) Continuity Periodicity
Revised: Reshape to 2D Structure Period 1 Period 2 Period 1 Length Period 2 Length … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) Continuity Periodicity

Original: P e r i o d i c i t y
Revised: P e r i o d i c i t y

Original: Capture
Revised: Capture

Original: [Figure]

Original: 2D

Original: [Figure]
Revised: [Figure]

Original: Interperiod-variation
Revised: Interperiod-variation

Original: Continuity
Revised: Continuity

Original: [Figure]
Revised: [Figure]

Original: Frequency
Revised: Frequency

Original: A m p li t u d e Frequency 1 Frequency 2 Frequency 3 …
Revised: A m p li t u d e Frequency 1 Frequency 2 Frequency 3 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154)
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154)

Original: Reshaped Time Series (cid:1153)(cid:22)(cid:1154)
Revised: Reshaped Time Series (cid:1153)(cid:22)(cid:1154)

Original: Reshape to 2D Space FFT for Periods
Revised: Reshape to 2D Space FFT for Periods

Original: Frequency
Revised: Frequency

Original: P e r i o d
Revised: P e r i o d

Original: [Figure]
Revised: [Figure]

Original: Here, FFT( · ) and Amp( · ) denote the FFT and the calculation of amplitude values.
Revised: Here, FFT( · ) and Amp( · ) denote the FFT and the calculation of amplitude values.

Original: A ∈ R T represents the calculated amplitude of each frequency, which is averaged from C dimensions by
Revised: A ∈ R T represents the calculated amplitude of each frequency, which is averaged from C dimensions by

Original: Avg( · ) .
Revised: Avg( · ) .

Original: Note that the j -th value A j represents the intensity of the frequency- j periodic basis function, corresponding to the period length ⌈
Revised: Note that the j -th value A j represents the intensity of the frequency- j periodic basis function, corresponding to the period length ⌈

Original: Tj ⌉ .
Revised: Tj ⌉ .

Original: Considering the sparsity of frequency domain and to avoid the noises brought by meaningless high frequencies (Chatfield, 1981; Zhou et al., 2022), we only select the top- k amplitude values and obtain the most significant frequencies { f 1 , · · · , f k } with the unnormalized amplitudes { A f 1 , · · · , A f k } , where k is the hyper-parameter.
Revised: Considering the sparsity of frequency domain and to avoid the noises brought by meaningless high frequencies (Chatfield, 1981; Zhou et al., 2022), we only select the top- k amplitude values and obtain the most significant frequencies { f 1 , · · · , f k } with the unnormalized amplitudes { A f 1 , · · · , A f k } , where k is the hyper-parameter.

Original: These selected frequencies also correspond to k period lengths { p 1 , · · · , p k } .
Revised: These selected frequencies also correspond to k period lengths { p 1 , · · · , p k } .

Original: Due to the conjugacy of frequency domain, we only consider the frequencies within { 1 , · · · , [ T 2 ] } .
Revised: Due to the conjugacy of frequency domain, we only consider the frequencies within { 1 , · · · , [ T 2 ] } .

Original: We summarize Equation 1 as follows:
Revised: We summarize Equation 1 as follows:

Original: [Equation]
Revised: [Equation]

Original: Based on the selected frequencies { f 1 , · · · , f k } and corresponding period lengths { p 1 , · · · , p k } , we can reshape the 1D time series X 1D ∈ R T × C into multiple 2D tensors by the following equations:
Revised: Based on the selected frequencies { f 1 , · · · , f k } and corresponding period lengths { p 1 , · · · , p k } , we can reshape the 1D time series X 1D ∈ R T × C into multiple 2D tensors by the following equations:

Original: [Equation]
Revised: [Equation]

Original: where Padding( · ) is to extend the time series by zeros along temporal dimension to make it compatible for Reshape p
Revised: where Padding( · ) is to extend the time series by zeros along temporal dimension to make it compatible for Reshape p

Original: i ,f i ( · ) , where p i and f i represent the number of rows and columns of the transformed 2D tensors respectively.
Revised: i ,f i ( · ) , where p i and f i represent the number of rows and columns of the transformed 2D tensors respectively.

Original: Note that X
Revised: Note that X

Original: i 2D ∈ R p i ×
Revised: i 2D ∈ R p i ×

Original: f i × C denotes the i -th reshaped time series based on frequency- f i , whose columns and rows represent the intraperiod-variation and interperiod-variation under the corresponding period length p i respectively.
Revised: f i × C denotes the i -th reshaped time series based on frequency- f i , whose columns and rows represent the intraperiod-variation and interperiod-variation under the corresponding period length p i respectively.

Original: Eventually, as shown in Figure 2, based on the selected frequencies and estimated periods, we obtain a set of 2D tensors { X 1 2D , · · · , X k 2D } , which indicates k different temporal 2D-variations derived by different periods.
Revised: Eventually, as shown in Figure 2, based on the selected frequencies and estimated periods, we obtain a set of 2D tensors { X 1 2D , · · · , X k 2D } , which indicates k different temporal 2D-variations derived by different periods.

Original: It is also notable that, this transformation brings two types of localities to the transformed 2D tensors, that is localities among adjacent time points (columns, intraperiod-variation) and adjacent periods (rows, interperiod-variation).
Revised: It is also notable that, this transformation brings two types of localities to the transformed 2D tensors, that is localities among adjacent time points (columns, intraperiod-variation) and adjacent periods (rows, interperiod-variation).

Original: Thus, the temporal 2D-variations can be easily processed by 2D kernels.
Revised: Thus, the temporal 2D-variations can be easily processed by 2D kernels.

Original: 3. T IMES B LOCK
Revised: 3. T IMES B LOCK

Original: As shown in Figure 3, we organize the TimesBlock in a residual way (He et al., 2016).
Revised: As shown in Figure 3, we organize the TimesBlock in a residual way (He et al., 2016).

Original: Concretely, for the length- T 1D input time series X 1D ∈ R T × C , we project the raw inputs into the deep features
Revised: Concretely, for the length- T 1D input time series X 1D ∈ R T × C , we project the raw inputs into the deep features

Original: X 0 1D ∈ R T × d model by the embedding layer X 0 1D =
Revised: X 0 1D ∈ R T × d model by the embedding layer X 0 1D =

Original: Embed( X 1D ) at the very beginning.
Revised: Embed( X 1D ) at the very beginning.

Original: For the l -th layer of TimesNet, the input is X l − 1 1D ∈ R T × d model and the process can be formalized as:
Revised: For the l -th layer of TimesNet, the input is X l − 1 1D ∈ R T × d model and the process can be formalized as:

Original: [Equation]
Revised: [Equation]

Original: As shown in Figure 3, for the l -th TimesBlock, the whole process involves two successive parts: capturing temporal 2D-variations and adaptively aggregating representations from different periods.
Revised: As shown in Figure 3, for the l -th TimesBlock, the whole process involves two successive parts: capturing temporal 2D-variations and adaptively aggregating representations from different periods.

Original: Capturing temporal 2D-variations Similar to Equation 1, we can estimate period lengths for deep features X l − 1 1D by Period( · ) .
Revised: Capturing temporal 2D-variations Similar to Equation 1, we can estimate period lengths for deep features X l − 1 1D by Period( · ) .

Original: Based on estimated period lengths, we can transform the 1D time series
Revised: Based on estimated period lengths, we can transform the 1D time series

Original: Reshape Back Q K V FFT x FFT Inverse FFT Time Delay Agg Linear Linear Linear Linear Concat Conjugate Resize Resize Top k Reshape FFT for Periods X Process for Period i Kernel 1 Kernel 2 … Kernel M Dynamic Kernel Bank Conv & Pooling ReshapeBack … + + N Softmax X’
Revised: Reshape Back Q K V FFT x FFT Inverse FFT Time Delay Agg Linear Linear Linear Linear Concat Conjugate Resize Resize Top k Reshape FFT for Periods X Process for Period i Kernel 1 Kernel 2 … Kernel M Dynamic Kernel Bank Conv & Pooling ReshapeBack … + + N Softmax X’

Original: Time Domain Frequency Domain V a l u e A
Revised: Time Domain Frequency Domain V a l u e A

Original: m p li t u d e Period 1 Period 2 …
Revised: m p li t u d e Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape Period 1 Period 2 Period 1 Length Period 2 Length
Revised: Reshape Period 1 Period 2 Period 1 Length Period 2 Length

Original: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X Process for Period
Revised: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X Process for Period

Original: i Kernel 1 Kernel 2 … Kernel M Dynamic Kernel Bank Conv & Pooling ReshapeBack …
Revised: i Kernel 1 Kernel 2 … Kernel M Dynamic Kernel Bank Conv & Pooling ReshapeBack …

Original: + + N Softmax X’
Revised: + + N Softmax X’

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k Dynamic Model + TimesBlock
Revised: ReshapeBack k Dynamic Model + TimesBlock

Original: Dynamic Model
Revised: Dynamic Model

Original: Block m Block 2 Block 1 Conv2D & AvgPool * * * + …
Revised: Block m Block 2 Block 1 Conv2D & AvgPool * * * + …

Original: Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +
Revised: Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k + TimesBlock
Revised: ReshapeBack k + TimesBlock

Original: MSUnit 3x3 Conv + 5x

Revised: MSUnit 3x3 Conv + 5x

Original: Conv 7x7 Conv 1x

Revised: Conv 7x7 Conv 1x

Original: Conv
Revised: Conv

Original: Reshaped Time Series … MSUnit
Revised: Reshaped Time Series … MSUnit

Original: TimesBlock TimesBlock
Revised: TimesBlock TimesBlock

Original: TimesBlock + + +
Revised: TimesBlock + + +

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax
Revised: Softmax

Original: Reshape
Revised: Reshape

Original: ReshapeBack k + TimesBlock Parameter-efﬁcient
Revised: ReshapeBack k + TimesBlock Parameter-efﬁcient

Original: Inception block 1D Space 2D Space 1D Space …
Revised: Inception block 1D Space 2D Space 1D Space …

Original: Time
Revised: Time

Original: Inception Block
Revised: Inception Block

Original: Adaptive Aggregation
Revised: Adaptive Aggregation

Original: (cid:335)
Revised: (cid:335)

Original: Reshape Back Q K V FFT x FFT Inverse FFT Time Delay Agg Linear Linear Linear Linear Concat Conjugate Resize Resize Top k
Revised: Reshape Back Q K V FFT x FFT Inverse FFT Time Delay Agg Linear Linear Linear Linear Concat Conjugate Resize Resize Top k

Original: Reshape FFT for Periods X
Revised: Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: Time Domain Frequency Domain V a l u e A
Revised: Time Domain Frequency Domain V a l u e A

Original: m p li t u d e
Revised: m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape Period 1 Period 2 Period 1 Length Period 2 Length
Revised: Reshape Period 1 Period 2 Period 1 Length Period 2 Length

Original: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154)
Revised: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154)

Original: TimesBlock
Revised: TimesBlock

Original: TimesBlock TimesBlock + + + Reshape FFT for Periods
Revised: TimesBlock TimesBlock + + + Reshape FFT for Periods

Original: X
Revised: X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax
Revised: Softmax

Original: Reshape
Revised: Reshape

Original: ReshapeBack k Dynamic Model + TimesBlock
Revised: ReshapeBack k Dynamic Model + TimesBlock

Original: Dynamic Model
Revised: Dynamic Model

Original: Block m Block 2 Block 1 Conv2D & AvgPool * * * + …
Revised: Block m [Figure] 2 Block 1 Conv2D & AvgPool * * * + …

Original: Block Choices Conv2D ResNet ConvNext …
Revised: Block Choices Conv2D ResNet ConvNext …

Original: [Figure]
Revised: [Figure]

Original: + ++
Revised: + +

Original: FFT

Original: [Figure]
Revised: [Figure]

Original: Reshape

Original: [Figure]

Original: k +
Revised: k +

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Equation]
Revised: [Equation]

Original: where X l,i 2D 2 R ( f i ⇥ p i ) ⇥
Revised: where X l,i 2D [Equation] f [Equation]

Revised: ⇥ p i ) ⇥

Original: d model is the transformed 2D tensor.
Revised: d model is the transformed 2D tensor.

Original: After the transformation, we process the 2D tensor by a parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbone.
Revised: After the transformation, we process the 2D tensor by a parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbone.

Original: Then we transform the processed 2D feature into b
Revised: Then we transform the processed 2D feature into b

Original: X l,i 2D back to 1D tensor b
Revised: X l,i 2D back to 1D tensor b

Original: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.
Revised: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.

Original: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.
Revised: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.

Original: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .
Revised: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .

Original: { X l, 1 2D , ··· , X l,k 2D } { b
Revised: { X l, 1 2D , ··· , X l,k 2D } { b

Original: [Equation]
Revised: [Equation]

Original: b
Revised: b

Original: [Equation]
Revised: [Equation]

Revised: 2D

Revised: [Equation]

Original: { b
Revised: { b

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: b
Revised: b

Original: X
Revised: X

Original: [Equation]
Revised: [Equation]

Original: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:
Revised: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:

Original: [Equation]
Revised: [Equation]

Original: 1 b A l  1 f i ⇥ b
Revised: b A l  1 f i ⇥ b

Original: X l,i 1D .
Revised: X [Equation]

Original:  Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.
Revised: [Equation] Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.

Original: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).
Revised: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).

Original: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.
Revised: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.

Original: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.
Revised: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.

Original: Under review as a conference paper at ICLR(cid:335) Reshape Back Q
Revised: Under review as a conference paper at ICLR(cid:335) Reshape Back Q

Original: K
Revised: K

Original: V FFT x FFT Inverse FFT Time Delay Agg Linear
Revised: V FFT x FFT Inverse FFT Time Delay Agg Linear

Original: Linear
Revised: Linear

Original: Linear Linear Concat Conjugate Resize
Revised: Linear Linear Concat Conjugate Resize

Original: Resize Top k
Revised: Resize Top k

Original: Reshape FFT for Periods X
Revised: Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: Time Domain Frequency Domain V a l u e A
Revised: Time Domain Frequency Domain V a l u e A

Original: m p li t u d e
Revised: m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape Period 1 Period 2 Period 1 Length Period 2 Length
Revised: Reshape Period 1 Period 2 Period 1 Length Period 2 Length

Original: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X
Revised: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k Dynamic Model + TimesBlock
Revised: ReshapeBack k Dynamic Model + TimesBlock

Original: Dynamic Model Block m Block 2 Block 1 Conv2D & AvgPool * * * + … Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +
Revised: Dynamic Model Block m Block 2 Block 1 Conv2D & AvgPool * * * + … Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k + TimesBlock
Revised: ReshapeBack k + TimesBlock

Original: MSUnit 3x3 Conv + 5x

Revised: MSUnit 3x3 Conv + 5x

Original: Conv 7x7 Conv 1x1 Conv Reshaped Time Series , (cid:335) , … MSUnit
Revised: Conv 7x7 Conv 1x1 Conv Reshaped Time Series , (cid:335) , … MSUnit

Original: A l  1 , { f 1 , ··· ,f k } , { p 1 , ··· ,p k } = Period 
Revised: A l  1 , { f 1 , ··· ,f k } , { p 1 , ··· ,p k } = Period

Original: X l  1 1D  ,
Revised: X l  1 1D  ,

Original: X l,i 2D =
Revised: X l,i 2D =

Original: Reshape f i ⇥ p
Revised: Reshape f i ⇥ p

Original: i 
Revised: i

Original: Padding( X l  1 1D )  , i 2 { 1 , ··· ,k } b
Revised: Padding( X l  1 1D )  , i 2 { 1 , ··· ,k } b

Original: X l,i 2D =
Revised: X l,i 2D =

Original: Inception ⇣
Revised: Inception ⇣

Original: X l,i 2D ⌘ , i 2 { 1 , ··· ,k } b
Revised: X l,i 2D ⌘ , i 2 { 1 , ··· ,k } b

Original: X l,i 1D =
Revised: X l,i 1D =

Original: Trunc ⇣
Revised: Trunc ⇣

Original: Reshape 1 ⇥ ( f i ⇥ p i )
Revised: Reshape 1 ⇥ ( f i ⇥ p i )

Original: ⇣b
Revised: ⇣b

Original: X l,i 2D ⌘⌘ , i 2 { 1 , ··· ,k } (5) where X l,i 2D 2 R ( f i ⇥ p i ) ⇥
Revised: X l,i 2D ⌘⌘ , i 2 { 1 , ··· ,k } (5) where X l,i 2D 2 R ( f i ⇥ p i ) ⇥

Original: d model is the transformed 2D tensor.
Revised: d model is the transformed 2D tensor.

Original: After the transformation, we process the 2D tensor bya parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2Dkernels and is one of the most well-acknowledged vision backbone.
Revised: After the transformation, we process the 2D tensor by parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D and is one of the most well-acknowledged vision backbone.

Original: Then we transform the processed 2D feature into b
Revised: Then we transform the processed 2D feature into b

Original: X l,i 2D back to 1D tensor b
Revised: X l,i 2D back to 1D tensor b

Original: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.
Revised: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.

Original: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.
Revised: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.

Original: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .
Revised: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .

Original: { X l, 1 2D , ··· , X l,k 2D } { b
Revised: { X l, 1 2D , ··· , X l,k 2D } { b

Original: X l, 1 2D , ··· , b
Revised: X l, 1 2D , ··· , b

Original: X l,k 2D } { b
Revised: X l,k 2D } { b

Original: X l, 1 1D , ··· , b
Revised: X l, 1 1D , ··· , b

Original: X l,k 1D }
Revised: X l,k 1D }

Original: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:
Revised: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:

Original: b
Revised: b

Original: [Equation]
Revised: A l  1 f 1 , ··· , b A l  1 f k = SoftMax ⇣ A l  1 f 1 , ··· , A l  1 f k ⌘ X l 1D = k

Original:  Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.
Revised: X i =1 b A l  1 f i ⇥ b X l,i 1D . Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.

Original: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).
Revised: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).

Original: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.
Revised: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.

Original: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.
Revised: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.

Original: Under review as a conference paper at ICLR(cid:335) Reshape Back Q
Revised: Under review as a conference paper at ICLR(cid:335) Reshape Back Q

Original: K
Revised: K

Original: V FFT x FFT Inverse FFT Time Delay Agg Linear
Revised: V FFT x FFT Inverse FFT Time Delay Agg Linear

Original: Linear
Revised: Linear

Original: Linear Linear Concat Conjugate Resize
Revised: Linear Linear Concat Conjugate Resize

Original: Resize Top k
Revised: Resize Top k

Original: Reshape FFT for Periods X
Revised: Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: Time Domain Frequency Domain V a l u e A
Revised: Time Domain Frequency Domain V a l u e A

Original: m p li t u d e
Revised: m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape Period 1 Period 2 Period 1 Length Period 2 Length
Revised: Reshape Period 1 Period 2 Period 1 Length Period 2 Length

Original: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X
Revised: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k Dynamic Model + TimesBlock
Revised: ReshapeBack k Dynamic Model + TimesBlock

Original: Dynamic Model Block m Block 2 Block 1 Conv2D & AvgPool * * * + … Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +
Revised: Dynamic Model Block m Block 2 Block 1 Conv2D & AvgPool * * * + … Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k + TimesBlock
Revised: ReshapeBack k + TimesBlock

Original: 3x3 Conv + 5x

Revised: 3x3 Conv + 5x

Original: Conv 7x7 Conv 1x1{+1x
+} Conv Reshaped Time SeriesA l  1 , { f 1 , ··· ,f k } , { p 1 , ··· ,p k } = Period  X l  1 1D  , X l, i 2D =
Revised: Conv 7x7 Conv {+1x
+} Conv Reshaped Time Series l  1 , { f 1 , ··· ,f k } , { p 1 , ··· ,p k } = Period  X l  1 1D  , X l,i 2D =

Original: Reshape f i ⇥ p
Revised: Reshape f i ⇥ p

Original: i 
Revised: i

Original: Padding( X l  1 1D )  , i 2 { 1 , ··· ,k } b
Revised: Padding( X l  1 1D )  , i 2 { 1 , ··· ,k } b

Original: X l,i 2D =
Revised: X l,i 2D =

Original: Inception ⇣
Revised: Inception ⇣

Original: X l,i 2D ⌘ , i 2 { 1 , ··· ,k } b
Revised: X l,i 2D ⌘ , i 2 { 1 , ··· ,k } b

Original: X l,i 1D =
Revised: X l,i 1D =

Original: Trunc ⇣
Revised: Trunc ⇣

Original: Reshape 1 ⇥ ( f i ⇥ p i )
Revised: Reshape 1 ⇥ ( f i ⇥ p i )

Original: ⇣b
Revised: ⇣b

Original: X l,i 2D ⌘⌘ , i 2 { 1 , ··· ,k } (5) where X l,i 2D 2 R ( f i ⇥ p i ) ⇥
Revised: X l,i 2D ⌘⌘ , i 2 { 1 , ··· ,k } (5) where X l,i 2D 2 R ( f i ⇥ p i ) ⇥

Original: d model is the transformed 2D tensor.
Revised: d model is the transformed 2D tensor.

Original: After the transformation, we process the 2D tensor by a parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbone.
Revised: After the transformation, we process the 2D tensor by a parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbone.

Original: Then we transform the processed 2D feature into b
Revised: Then we transform the processed 2D feature into b

Original: X l,i 2D back to 1D tensor b
Revised: X l,i 2D back to 1D tensor b

Original: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.
Revised: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.

Original: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.
Revised: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.

Original: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .
Revised: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .

Original: { X l, 1 2D , ··· , X l,k 2D } { b
Revised: { X l, 1 2D , ··· , X l,k 2D } { b

Original: X l, 1 2D , ··· , b
Revised: X l, 1 2D , ··· , b

Original: X l,k 2D } { b
Revised: X l,k 2D } { b

Original: X l, 1 1D , ··· , b
Revised: X l, 1 1D , ··· , b

Original: X l,k 1D }
Revised: X l,k 1D }

Original: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:
Revised: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:

Original: b
Revised: b

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: A l  1 f k ⌘
Revised: l  1 f k ⌘

Original: X l 1D = k
Revised: X l 1D = k

Original: X i =1 b
Revised: X i =1 b

Original: A l  1 f i ⇥ b
Revised: A l  1 f i ⇥ b

Original: X l,i 1D .
Revised: X l,i 1D .

Original: Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.
Revised: Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.

Original: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).
Revised: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).

Original: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.
Revised: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.

Original: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.
Revised: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.

Original: Under review as a conference paper at ICLR(cid:335) Reshape Back Q
Revised: Under review as a conference paper at ICLR(cid:335) Reshape Back Q

Original: K
Revised: K

Original: V FFT x FFT Inverse FFT Time Delay Agg Linear
Revised: V FFT x FFT Inverse FFT Time Delay Agg Linear

Original: Linear
Revised: Linear

Original: Linear Linear Concat Conjugate Resize
Revised: Linear Linear Concat Conjugate Resize

Original: Resize Top k
Revised: Resize Top k

Original: Reshape FFT for Periods X
Revised: Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: Time Domain Frequency Domain V a l u e A
Revised: Time Domain Frequency Domain V a l u e A

Original: m p li t u d e
Revised: m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape Period 1 Period 2 Period 1 Length Period 2 Length
Revised: Reshape Period 1 Period 2 Period 1 Length Period 2 Length

Original: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X
Revised: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k Dynamic Model + TimesBlock
Revised: ReshapeBack k Dynamic Model + TimesBlock

Original: Dynamic Model Block m Block 2 Block 1 Conv2D & AvgPool * * * + … Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +
Revised: Dynamic Model Block m Block 2 Block 1 Conv2D & AvgPool * * * + … Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k + TimesBlock
Revised: ReshapeBack k + TimesBlock

Original: MSUnit 3x3 Conv + 5x

Revised: MSUnit 3x3 Conv + 5x

Original: Conv 7x7 Conv 1x1 Conv Reshaped Time Series , (cid:335) , … l 
Revised: Conv 7x7 Conv 1x1 Conv Reshaped Time Series , (cid:335) , … l

Original:  X l 
Revised:  X l

Original:  
Revised:  , l i  l  )  , b

Original:  l  b
Revised: 2D b

Original:  Trunc
Revised: = Trunc

Original: ( f
Revised: ( f

Original: ⇥ p i )
Revised: ⇥ p i )

Original: ⇣b X l,i 2D
Revised: X l,i 2D

Original: , i 2 { 1 , ··· ,k } (5) where X l,i 2D 2 R ( f i ⇥ p i ) ⇥
Revised: , i 2 { 1 , ··· ,k } (5) where X l,i 2D 2 R ( f i ⇥ p i ) ⇥

Original: d model is the transformed 2D tensor.
Revised: d model is the transformed 2D tensor.

Original: After the transformation, we process the 2D tensor by a parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbone.
Revised: After the transformation, we process the 2D tensor by a parameter-efﬁcient inception block (Szegedy et al., 2015), which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbone.

Original: Then we transform the processed 2D feature into b
Revised: Then we transform the processed 2D feature into b

Original: X l,i 2D back to 1D tensor b
Revised: X l,i 2D back to 1D tensor b

Original: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.
Revised: X l,i 1D 2 R T ⇥ d model for aggregation by Trunc( · ) to truncate the padding series into original length.

Original: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.
Revised: Note that beneﬁting from the transformation of 1D time series, the 2D kernels in inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously.

Original: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .
Revised: Besides, the parameter sharing deign makes the model size invariant to the selection of the number of periods k .

Original: { X l, 1 2D , ··· , X l,k 2D } { b
Revised: { X l, 1 2D , ··· , X l,k 2D } { b

Original: X l, 1 2D , ··· , b
Revised: X l, 1 2D , ··· , b

Original: X l,k 2D } { b
Revised: X l,k 2D } { b

Original: X l, 1 1D , ··· , b
Revised: X l, 1 1D , ··· , b

Original: X l,k 1D }
Revised: X l,k 1D }

Original: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:
Revised: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:

Original: b
Revised: b

Original: A l  1 f 1 , ··· , b
Revised: A l  1 f 1 , ··· , b

Original: A l  1 f k = SoftMax ⇣
Revised: A l  1 f k = SoftMax ⇣

Original: A l  1 f 1 , ··· , A l  1 f k ⌘
Revised: A l  1 f 1 , ··· , A l  1 f k ⌘

Original: X l 1D = k
Revised: X l 1D = k

Original: X i =1 b
Revised: X i =1 b

Original: A l  1 f i ⇥ b
Revised: A l  1 f i ⇥ b

Original: X l,i 1D .
Revised: X l,i 1D .

Original: Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.
Revised: Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.

Original: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).
Revised: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).

Original: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.
Revised: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysistakeadvantageofthedevelopmentofvisioncommunity.

Original: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.
Revised: Forefﬁciency,weconductthemain experiments based on the parameter-efﬁcient inception block as Equation 5.

Original: Under review as a conference paper at ICLR(cid:335)
Revised: Under review as a conference paper at ICLR(cid:335)

Original: Reshape Back
Revised: Reshape Back

Original: Q K V FFT x FFT Inverse FFT Time Delay Agg Linear Linear Linear Linear Concat Conjugate Resize Resize Top k
Revised: Q K V FFT x FFT Inverse FFT Time Delay Agg Linear Linear Linear Linear Concat Conjugate Resize Resize Top k

Original: Reshape FFT for Periods X
Revised: Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: Time Domain Frequency Domain V a l u e A
Revised: Time Domain Frequency Domain V a l u e A

Original: m p li t u d e
Revised: m p li t u d e

Original: Period 1 Period 2 …
Revised: Period 1 Period 2 …

Original: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods
Revised: ReshapedTimeSeries (cid:1153)(cid:20)(cid:1154) FFT for Periods

Original: Reshape Period 1 Period 2 Period 1 Length Period 2 Length
Revised: Reshape Period 1 Period 2 Period 1 Length Period 2 Length

Original: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X
Revised: Kernel 1 Kernel 2 Kernel M Conv2D … * * * + … ReshapedTimeSeries (cid:1153)(cid:21)(cid:1154) (cid:1153)(cid:20)(cid:1154) (cid:1153)(cid:21)(cid:1154) TimesBlock TimesBlock TimesBlock + + + Reshape FFT for Periods X

Original: Process for Period i Kernel 1 Kernel 2 … Kernel M
Revised: Process for Period i Kernel 1 Kernel 2 … Kernel M

Original: Dynamic Kernel Bank Conv & Pooling
Revised: Dynamic Kernel Bank Conv & Pooling

Original: ReshapeBack … + + N Softmax
Revised: ReshapeBack … + + N Softmax

Original: X’
Revised: X’

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k Dynamic Model + TimesBlock
Revised: ReshapeBack k Dynamic Model + TimesBlock

Original: Dynamic Model
Revised: Dynamic Model

Original: Block m Block 2 Block 1 Conv2D & AvgPool * * * + …
Revised: Block m Block 2 Block 1 Conv2D & AvgPool * * * + …

Original: Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +
Revised: Block Choices Conv2D ResNet ConvNext … Attention TimesBlock TimesBlock TimesBlock + + +

Original: FFT for Periods
Revised: FFT for Periods

Original: Softmax Reshape
Revised: Softmax Reshape

Original: ReshapeBack k + TimesBlock
Revised: ReshapeBack k + TimesBlock

Original: MSUnit 3x3 Conv + 5x

Revised: MSUnit 3x3 Conv + 5x

Original: Conv 7x7 Conv 1x

Revised: Conv 7x7 Conv 1x

Original: Conv
Revised: Conv

Original: Reshaped Time Series , (cid:335) , …
Revised: Reshaped Time Series , (cid:335) , …

Original: A l  1 , {f
Revised: A l  1 , {

Original: [Table]
Revised: [Table]

Original: { b
Revised: { b

Original: X l, 1 2D , ··· , b
Revised: X l, 1 2D , ··· , b

Original: X l,k 2D } { b
Revised: X l,k 2D } { b

Original: X l, 1 1D , ··· , b
Revised: X l, 1 1D , ··· , b

Original: X l,k 1D }
Revised: X l,k 1D }

Original: Adaptive aggregation Finally, we fuse the k different processed features w.r.t their corresponding amplitudes of the estimated periods as follows:
Revised: Adaptive aggregation Finally, we fuse the k different processed features w.r [Table] t their corresponding amplitudes of the estimated periods as follows:

Original: [Equation]
Revised: [Equation]

Original: ,i 1D .
Revised: i 1D .

Original: Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.
Revised: Since the temporal variations are involved in the multiple highly-structured 2D tensors, the de- sign of TimesBlock can fully capture the multi-scale temporal 2D-variations from multiple views simultaneously, making the representatin learning more effective.

Original: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).
Revised: Generality in 2D vision backbones Since we transform the 1D time series into 2D space, we can also choose various vision backbones to replace the inception module for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNext (Xie et al., 2017), advanced ConvNext (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).

Original: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysis take advantage of the development of vision community.
Revised: Thus, our temporal 2D-variation design also bridges the 1D time series and the booming 2D vision backbones, making the time series analysis take advantage of the development of vision community.

Original: For efﬁciency, we conduct the main experiments based on the parameter-efﬁcient inception block as Equation 5. into 2D space and obtain a set of 2D tensors, from which we can obtain informative representations by parameter-efficient inception block conveniently.
Revised: For efﬁciency, we conduct the main experiments based on the parameter-efﬁcient inception block as Equation 5. into 2D space and obtain a set of 2D tensors, from which we can obtain informative representations by parameter-efficient inception block conveniently.

Original: The process is formalized as follows:
Revised: The process is formalized as follows:

Original: [Equation]
Revised: [Equation]

Revised: f

Revised: [Equation]

Revised: } , { p

Revised: [Equation]

Revised: · · · , p k } = Period

Revised: [Equation]

Original: where X l,i 2D ∈
Revised: where X l,i 2D ∈

Original: R p i × f i × d model is the i -th transformed 2D tensor.
Revised: R p i × f i × d model is the i -th transformed 2D tensor.

Original: After the transformation, we process the 2D tensor by a parameter-efficient inception block (Szegedy et al., 2015) as Inception( · ) , which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbones.
Revised: After the transformation, we process the 2D tensor by a parameter-efficient inception block (Szegedy et al., 2015) as Inception( · ) , which involves multi-scale 2D kernels and is one of the most well-acknowledged vision backbones.

Original: Then we transform the learned 2D representations
Revised: Then we transform the learned 2D representations

Original: [Equation]
Revised: [Equation]

Original: l,i 2D back to 1D space
Revised: l,i 2D back to 1D space

Original: [Equation]
Revised: [Equation]

Original: for aggregation, where we employ Trunc( · ) to truncate the padded series with length ( p i × f i ) into original length T .
Revised: for aggregation, where we employ Trunc( · ) to truncate the padded series with length ( p i × f i ) into original length T .

Original: Note that benefiting from the transformation of 1D time series, the 2D kernels in the inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously, covering both adjacent time points and adjacent periods.
Revised: Note that benefiting from the transformation of 1D time series, the 2D kernels in the inception block can aggregate the multi-scale intraperiod-variation (columns) and interperiod-variation (rows) simultaneously, covering both adjacent time points and adjacent periods.

Original: Besides, we adopt a shared inception block for different reshaped 2D tensors { X l, 1 2D , · · · , X l,k 2D } to improve parameter efficiency, which can make the model size invariant to the selection of hyper-parameter k .
Revised: Besides, we adopt a shared inception block for different reshaped 2D tensors { X l, 1 2D , · · · , X l,k 2D } to improve parameter efficiency, which can make the model size invariant to the selection of hyper-parameter k .

Original: Adaptive aggregation Finally, we need to fuse k different 1D-representations {
Revised: Adaptive aggregation Finally, we need to fuse k different 1D-representations {

Original: [Equation]
Revised: [Equation]

Original: for the next layer.
Revised: for the next layer.

Original: Inspired by Auto-Correlation (Wu et al., 2021), the amplitudes A can reflect the relative importance of selected frequencies and periods, thereby corresponding to the importance of each transformed 2D tensor.
Revised: Inspired by Auto-Correlation (Wu et al., 2021), the amplitudes A can reflect the relative importance of selected frequencies and periods, thereby corresponding to the importance of each transformed 2D tensor.

Original: Thus, we aggregate the 1D-representations based on the amplitudes:
Revised: Thus, we aggregate the 1D-representations based on the amplitudes:

Original: [Equation]
Revised: [Equation]

Original: Since the variations within and between periods are already involved in multiple highly-structured 2D tensors, TimesBlock can fully capture multi-scale temporal 2D-variations simultaneously.
Revised: Since the variations within and between periods are already involved in multiple highly-structured 2D tensors, TimesBlock can fully capture multi-scale temporal 2D-variations simultaneously.

Original: Thus,
Revised: Thus,

Original: TimesNet can achieve a more effective representation learning than directly from 1D time series.
Revised: TimesNet can achieve a more effective representation learning than directly from 1D time series.

Original: Generality in 2D vision backbones Benefiting from the transformation of 1D time series into temporal 2D-variations, we can choose various computer vision backbones to replace the inception block for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNeXt (Xie et al., 2017), advanced ConvNeXt (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).
Revised: Generality in 2D vision backbones Benefiting from the transformation of 1D time series into temporal 2D-variations, we can choose various computer vision backbones to replace the inception block for representation learning, such as the widely-used ResNet (He et al., 2016) and ResNeXt (Xie et al., 2017), advanced ConvNeXt (Liu et al., 2022b) and attention-based models (Liu et al., 2021b).

Original: Thus, our temporal 2D-variation design also bridges the 1D time series to the booming 2D vision backbones, making the time series analysis take advantage of the development of computer vision community.
Revised: Thus, our temporal 2D-variation design also bridges the 1D time series to the booming 2D vision backbones, making the time series analysis take advantage of the development of computer vision community.

Original: In general, more powerful 2D backbones for representation learning will bring better performance.
Revised: In general, more powerful 2D backbones for representation learning will bring better performance.

Original: Considering both performance and efficiency (Figure 4 right), we conduct the main experiments based on the parameter-efficient inception block as shown in Equation 5.
Revised: Considering both performance and efficiency (Figure 4 right), we conduct the main experiments based on the parameter-efficient inception block as shown in Equation 5.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: To verify the generality of TimesNet, we extensively experiment on five mainstream analysis tasks, including short- and long-term forecasting, imputation, classification and anomaly detection.
Revised: To verify the generality of TimesNet, we extensively experiment on five mainstream analysis tasks, including short- and long-term forecasting, imputation, classification and anomaly detection.

Original: Implementation Table 1 is a summary of benchmarks.
Revised: Implementation Table 1 is a summary of benchmarks.

Original: More details about the dataset, experiment implementation and model configuration can be found in Appendix A.
Revised: More details about the dataset, experiment implementation and model configuration can be found in Appendix A.

Original: [Table]
Revised: [Table]

Original: Baselines Since we attempt to propose a foundation model for time series analysis, we extensively compare the well-acknowledged and advanced models in all five tasks, including the RNN-based models: LSTM (1997), LSTNet (2018) and LSSL (2022); CNN-based Model: TCN (2019); MLP-based models: LightTS (2022) and DLinear (2022); Transformer-based models: Reformer (2020), Informer (2021), Pyraformer (2021a), Autoformer (2021), FEDformer (2022), Non-stationary Transformer (2022a) and ETSformer (2022).
Revised: Baselines Since we attempt to propose a foundation model for time series analysis, we extensively compare the well-acknowledged and advanced models in all five tasks, including the RNN-based models: LSTM (1997), LSTNet (2018) and LSSL (2022); CNN-based Model: TCN (2019); MLP-based models: LightTS (2022) and DLinear (2023); Transformer-based models: Reformer (2020), Informer (2021), Pyraformer (2021a), Autoformer (2021), FEDformer (2022), Non-stationary Transformer (2022a) and ETSformer (2022).

Original: Besides, we also compare the state-of-the-art models for each specific task, such as N-HiTS (2022) and N-BEATS (2019) for short-term forecasting, Anomaly
Revised: Besides, we also compare the state-of-the-art models for each specific task, such as N-HiTS (2022) and N-BEATS (2019) for short-term forecasting, Anomaly

Original: Transformer (2021) for anomaly detection, Rocket (2020) and Flowformer (2022) for classification and etc.
Revised: Transformer (2021) for anomaly detection, Rocket (2020) and Flowformer (2022) for classification and etc.

Original: Overall, more than 15 baselines are included for a comprehensive comparison.
Revised: Overall, more than 15 baselines are included for a comprehensive comparison.

Original: [Figure]
Revised: [Figure]

Original: 4. M AIN R ESULTS
Revised: 4. M AIN R ESULTS

Original: As a foundation model, TimesNet achieves consistent state-of-the-art performance on five mainstream analysis tasks compared with other customized models (Figure 4 left).
Revised: As a foundation model, TimesNet achieves consistent state-of-the-art performance on five mainstream analysis tasks compared with other customized models (Figure 4 left).

Original: The full efficiency comparison is provided in Table 11 of Appendix.
Revised: The full efficiency comparison is provided in Table 11 of Appendix.

Original: Besides, by replacing the inception block with more powerful vision backbones, we can further promote the performance of TimesNet (Figure 4 right), confirming that our design can make time series analysis take advantage of booming vision backbones
Revised: Besides, by replacing the inception block with more powerful vision backbones, we can further promote the performance of TimesNet (Figure 4 right), confirming that our design can make time series analysis take advantage of booming vision backbones

Original: [Figure]
Revised: [Figure]

Original: 4. S 
Revised: S HORT - AND

Original: [Equation]

Original: AND L ONG - TERM F ORECASTING
Revised: L ONG - TERM F ORECASTING

Original: Setups Time series forecasting is essential in weather forecasting, traffic and energy consumption planning.
Revised: Setups Time series forecasting is essential in weather forecasting, traffic and energy consumption planning.

Original: To fully evaluate the model performance in forecasting, we adopt two types of benchmarks, including long-term and short-term forecasting.
Revised: To fully evaluate the model performance in forecasting, we adopt two types of benchmarks, including long-term and short-term forecasting.

Original: Especially for the long-term setting, we follow the benchmarks used in Autoformer (2021), including ETT (Zhou et al., 2021), Electricity (UCI), Traffic (PeMS), Weather (Wetterstation), Exchange (Lai et al., 2018) and ILI (CDC), covering five real-world applications.
Revised: Especially for the long-term setting, we follow the benchmarks used in Autoformer (2021), including ETT (Zhou et al., 2021), Electricity (UCI), Traffic (PeMS), Weather (Wetterstation), Exchange (Lai et al., 2018) and ILI (CDC), covering five real-world applications.

Original: For the short-term dataset, we adopt the M4 (Spyros Makridakis, 2018), which contains the yearly, quarterly and monthly collected univariate marketing data.
Revised: For the short-term dataset, we adopt the M4 (Spyros Makridakis, 2018), which contains the yearly, quarterly and monthly collected univariate marketing data.

Original: Note that each dataset in the long-term setting only contains one continuous time series, where we obtain samples by sliding window, while M4 involves 100,000 different time series collected in different frequencies.
Revised: Note that each dataset in the long-term setting only contains one continuous time series, where we obtain samples by sliding window, while M4 involves 100,000 different time series collected in different frequencies.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: Results TimesNet shows great performance in both long-term and short-term settings (Table 2–3)
Revised: Results TimesNet shows great performance in both long-term and short-term settings (Table 2–3)

Original: [Table]
Revised: [Table]

Original: Concretely, TimesNet achieves state-of-the-art in more than 80% of cases in long-term forecasting (Table 13).
Revised: Concretely [Table] TimesNet achieves state-of-the-art in more than 80% of cases in long-term forecasting (Table 13).

Original: For the M4 dataset, since the time series are collected from different sources, the temporal variations can be quite diverse, making forecasting much more challenging.
Revised: For the M4 dataset, since the time series are collected from different sources, the temporal variations can be quite diverse, making forecasting much more challenging.

Original: Our model still performs best in this task, surpassing extensive advanced MLP-based and Transformer-based models.
Revised: Our model still performs best in this task, surpassing extensive advanced MLP-based and Transformer-based models.

Original: 4. I MPUTATION
Revised: 4. I MPUTATION

Original: Setups Real-world systems always work continuously and are monitored by automatic observation equipment.
Revised: Setups Real-world systems always work continuously and are monitored by automatic observation equipment.

Original: However, due to malfunctions, the collected time series can be partially missing, making the downstream analysis difficult.
Revised: However, due to malfunctions, the collected time series can be partially missing, making the downstream analysis difficult.

Original: Thus, imputation is widely-used in practical applications.
Revised: Thus, imputation is widely-used in practical applications.

Original: In this paper, we select the datasets from the electricity and weather scenarios as our benchmarks, including
Revised: In this paper, we select the datasets from the electricity and weather scenarios as our benchmarks, including

Original: ETT (Zhou et al., 2021), Electricity (UCI) and Weather (Wetterstation), where the data-missing problem happens commonly.
Revised: ETT (Zhou et al., 2021), Electricity (UCI) and Weather (Wetterstation), where the data-missing problem happens commonly.

Original: To compare the model capacity under different proportions of missing data, we randomly mask the time points in the ratio of { 12 . 5% , 25% , 37 .
Revised: To compare the model capacity under different proportions of missing data, we randomly mask the time points in the ratio of { 12 . 5% , 25% , 37 .

Original: Results Due to the missing time points, the imputation task requires the model to discover underlying temporal patterns from the irregular and partially observed time series.
Revised: Results Due to the missing time points, the imputation task requires the model to discover underlying temporal patterns from the irregular and partially observed time series.

Original: As shown in Table 4, our proposed TimesNet still achieves the consistent state-of-the-art in this difficult task, verifying the model capacity in capturing temporal variation from extremely complicated time series.
Revised: As shown in Table 4, our proposed TimesNet still achieves the consistent state-of-the-art in this difficult task, verifying the model capacity in capturing temporal variation from extremely complicated time series.

Original: 4. C LASSIFICATION
Revised: 4. C LASSIFICATION

Original: Setups Time series classification can be used in recognition and medical diagnosis (Moody et al., 2011).
Revised: Setups Time series classification can be used in recognition and medical diagnosis (Moody et al., 2011).

Original: We adopt the sequence-level classification to verify the model capacity in high-level representation learning.
Revised: We adopt the sequence-level classification to verify the model capacity in high-level representation learning.

Original: Concretely, we select 10 multivariate datasets from UEA Time Series Classification
Revised: Concretely, we select 10 multivariate datasets from UEA Time Series Classification

Original: Archive (Bagnall et al., 2018), covering the gesture, action and audio recognition, medical diagnosis by heartbeat monitoring and other practical tasks.
Revised: Archive (Bagnall et al., 2018), covering the gesture, action and audio recognition, medical diagnosis by heartbeat monitoring and other practical tasks.

Original: Then, we pre-process the datasets following the descriptions in (Zerveas et al., 2021), where different subsets have different sequence lengths.
Revised: Then, we pre-process the datasets following the descriptions in (Zerveas et al., 2021), where different subsets have different sequence lengths.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: A v e ra g e A cc u r a c y ( % ) [Figure] [Figure]
Revised: r

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original: T
Revised: A

Original: [Figure]

Original: X G

Original: [Figure]
Revised: [Figure]

Original: o o s t R o c 
Revised: u r a c y

Original: k e t L S T M

Original: [Figure]
Revised: [Figure]

Original: e

Original: [Figure]

Original: Results As shown in Figure 5,
Revised: Results As shown in Figure 5,

Original: TimesNet achieves the best performance with an average accuracy of 73.6%, surpassing the previous state-of-the-art classical method
Revised: TimesNet achieves the best performance with an average accuracy of 73.6%, surpassing the previous state-of-the-art classical method

Original: Rocket (72.5%) and deep model
Revised: Rocket (72.5%) and deep model

Original: Flowformer (73.0%).
Revised: Flowformer (73.0%).

Original: It is also notable that the MLP-based model
Revised: It is also notable that the MLP-based model

Original: DLinear fails in this classification task (67.5%), which performs well in some time series forecasting datasets.
Revised: DLinear fails in this classification task (67.5%), which performs well in some time series forecasting datasets.

Original: This is because DLinear only adopts a one-layer MLP model on the temporal dimension, which might be suitable for some autoregressive tasks with fixed temporal dependencies but will degenerate a lot in learning high-level representations.
Revised: This is because DLinear only adopts a one-layer MLP model on the temporal dimension, which might be suitable for some autoregressive tasks with fixed temporal dependencies but will degenerate a lot in learning high-level representations.

Original: In contrast, TimesNet unifies the temporal 2D-variation in 2D space, which is convenient to learn informative representation by 2D kernels, thereby benefiting the classification task that requires hierarchical representations.
Revised: In contrast, TimesNet unifies the temporal 2D-variation in 2D space, which is convenient to learn informative representation by 2D kernels, thereby benefiting the classification task that requires hierarchical representations.

Original: 4. A NOMALY D ETECTION
Revised: 4. A NOMALY D ETECTION

Original: Setups Detecting anomalies from monitoring data is vital to industrial maintenance.
Revised: Setups Detecting anomalies from monitoring data is vital to industrial maintenance.

Original: Since the anomalies are usually hidden in the large-scale data, making the data labeling hard, we focus on unsupervised time series anomaly detection, which is to detect the abnormal time points.
Revised: Since the anomalies are usually hidden in the large-scale data, making the data labeling hard, we focus on unsupervised time series anomaly detection, which is to detect the abnormal time points.

Original: We compare models on five widely-used anomaly detection benchmarks: SMD (Su et al., 2019), MSL (Hundman et al., 2018), SMAP (Hundman et al., 2018), SWaT (Mathur & Tippenhauer, 2016), PSM (Abdulaal et al., 2021), covering service monitoring, space & earth exploration, and water treatment applications.
Revised: We compare models on five widely-used anomaly detection benchmarks: SMD (Su et al., 2019), MSL (Hundman et al., 2018), SMAP (Hundman et al., 2018), SWaT (Mathur & Tippenhauer, 2016), PSM (Abdulaal et al., 2021), covering service monitoring, space & earth exploration, and water treatment applications.

Original: Following the pre-processing methods in Anomaly Transformer (2021), we split the dataset into consecutive non-overlapping segments by sliding window.
Revised: Following the pre-processing methods in Anomaly Transformer (2021), we split the dataset into consecutive non-overlapping segments by sliding window.

Original: In previous works, the reconstruction is a classical task for unsupervised point-wise representation learning, where the reconstruction error is a natural anomaly criterion.
Revised: In previous works, the reconstruction is a classical task for unsupervised point-wise representation learning, where the reconstruction error is a natural anomaly criterion.

Original: For a fair comparison, we only change the base models for reconstruction and use the classical reconstruction error as the shared anomaly criterion for all experiments.
Revised: For a fair comparison, we only change the base models for reconstruction and use the classical reconstruction error as the shared anomaly criterion for all experiments.

Original: Results Table 5 demonstrates that TimesNet still achieves the best performance in anomaly detection, outperforming the advanced Transformer-based models FEDformer (2022) and Autoformer (2021).
Revised: Results Table 5 demonstrates that TimesNet still achieves the best performance in anomaly detection, outperforming the advanced Transformer-based models FEDformer (2022) and Autoformer (2021).

Original: The canonical Transformer performs worse in this task (averaged F1-score 76.88%).
Revised: The canonical Transformer performs worse in this task (averaged F1-score 76.88%).

Original: This may come from that anomaly detection requires the model to find out the rare abnormal temporal patterns (Lai et al., 2021), while the vanilla attention mechanism calculates the similarity between each pair of time points, which can be distracted by the dominant normal time points.
Revised: This may come from that anomaly detection requires the model to find out the rare abnormal temporal patterns (Lai et al., 2021), while the vanilla attention mechanism calculates the similarity between each pair of time points, which can be distracted by the dominant normal time points.

Original: Besides, by taking the periodicity into consideration, TimesNet, FEDformer and Autoformer all achieve great performance.
Revised: Besides, by taking the periodicity into consideration, TimesNet, FEDformer and Autoformer all achieve great performance.

Original: Thus, these results also demonstrate the importance of periodicity analysis, which can highlight variations that violate the periodicity implicitly, further benefiting the anomaly detection.
Revised: Thus, these results also demonstrate the importance of periodicity analysis, which can highlight variations that violate the periodicity implicitly, further benefiting the anomaly detection.

Original: [Table]
Revised: [Table]

Original: 4. M ODEL A NALYSIS
Revised: 4. M ODEL A NALYSIS

Original: Representation analysis We attempt to explain model performance from the representation learning aspect.
Revised: Representation analysis We attempt to explain model performance from the representation learning aspect.

Original: From Figure 6, we can find that the better performance in forecasting and anomaly detection corresponds to the higher CKA similarity (2019), which is opposite to the imputation and classification tasks.
Revised: From Figure 6, we can find that the better performance in forecasting and anomaly detection corresponds to the higher CKA similarity (2019), which is opposite to the imputation and classification tasks.

Original: Note that the lower CKA similarity means that the representations are distinguishing among different layers, namely hierarchical representations.
Revised: Note that the lower CKA similarity means that the representations are distinguishing among different layers, namely hierarchical representations.

Original: Thus, these results also indicate the property of representations that each task requires.
Revised: Thus, these results also indicate the property of representations that each task requires.

Original: As shown in Figure 6, TimesNet can learn appropriate representations for different tasks, such as low-level representations for forecasting and reconstruction in anomaly detection, and hierarchical representations for imputation and classification.
Revised: As shown in Figure 6, TimesNet can learn appropriate representations for different tasks, such as low-level representations for forecasting and reconstruction in anomaly detection, and hierarchical representations for imputation and classification.

Original: In contrast,
Revised: In contrast,

Original: FEDformer (2022) performs well in forecasting and anomaly detection tasks but fails in learning hierarchical representations, resulting in poor performance in imputation and classification.
Revised: FEDformer (2022) performs well in forecasting and anomaly detection tasks but fails in learning hierarchical representations, resulting in poor performance in imputation and classification.

Original: These results also verify the task-generality of our proposed TimesNet as a foundation model.
Revised: These results also verify the task-generality of our proposed TimesNet as a foundation model.

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Table]

Original: = 0 5 10 0 2 4 6 8 10 121 °

Revised: 0 2 4 6 8 10 121 °

Original: 50 100 150 200 250 300°°°°  0 50 100 150 200 250 300 350 1 .1 .
Revised: 50 100 150 200 250 300°°°°  0 50 100 150 200 250 300 350 1 .1 .

Original: 50 100 150 200 250 300 350 ° 1 . 0 ° 0 .
Revised: 50 100 150 200 250 300 350 ° 1 . 0 ° 0 .

Original: 8 ° 0 .
Revised: 8 ° 0 .

Original: 6 ° 0 .
Revised: 6 ° 0 .

Original: 4 ° 0 .0 . 

Revised: 4 ° 0 .0 .

Original: 300 3  °
Revised: 300 3  °

Original: ID Series 2D-variations
Revised: ID Series 2D-variations

Original: E TT h
Revised: E TT h

Original: E l e c t r i c t y
Revised: E l e c t r i c t y

Original: E x
Revised: E x

Original: c h a n g e
Revised: c h a n g e

Original: W e a t
Revised: W e a t

Original: h e r 0 20 40 60 80 1001 °0 5 10 15 20 25 30 35   °0 10 20 30 40   
Revised: h e r 0 20 40 60 80 1001 °0 5 10 15 20 25 30 35   °0 10 20 30 40

Original: ° 1 0 5  0 26 812 °0 10 20 30 40 50 60 70  °

Revised: ° 1 0 5  0 26 812 °0 10 20 30 40 50 60 70  °

Original: 5 ° 1 .0 5 10 15 20 25 30 35 5°0 10 20 50 01 °

Revised: 5 ° 1 .0 5 10 15 20 25 30 35 5°0 10 20 50 01 °

Original: 50 100 150 200 250 300 350 ° 1 .
Revised: 50 100 150 200 250 300 350 ° 1 .

Original: ° 0 .0 .0 .1 .1 .2 .

Revised: ° 0 .0 .0 .1 .1 .2 .

Original: 5 10 1504812 ° 1 0 5 10051015  °00102030 °0 5 10 15048120 .
Revised: 5 10 1504812 ° 1 0 5 10051015  °00102030 °0 5 10 15048120 .

Original: ° 1 . 0 0 5 10  0 26 80 .
Revised: ° 1 . 0 0 5 10  0 26 80 .

Original: ° 1 .0 5 10 15 20 25 30 3500 .
Revised: ° 1 .0 5 10 15 20 25 30 3500 .

Original: ° 1 .
Revised: ° 1 .

Original: [Figure]

Original: Period
Revised: Period

Original: F r e q u e n c y
Revised: F r e q u e n c y

Original: [Figure]
Revised: [Figure]

Original: Temporal 2D-variations We provide a case study of temporal 2D-variations in Figure 7.
Revised: Temporal 2D-variations We provide a case study of temporal 2D-variations in Figure 7.

Original: We can find that TimesNet can capture the multi-periodicities precisely.
Revised: We can find that TimesNet can capture the multi-periodicities precisely.

Original: Besides, the transformed 2D tensor is highly structured and informative, where the columns and rows can reflect the localities between time points and periods respectively, supporting our motivation in adopting 2D kernels for representation learning.
Revised: Besides, the transformed 2D tensor is highly structured and informative, where the columns and rows can reflect the localities between time points and periods respectively, supporting our motivation in adopting 2D kernels for representation learning [Figure]

Original: See Appendix D for more visualizations.
Revised: See Appendix D for more visualizations.

Original:  C ONCLUSION AND F UTURE W ORK
Revised:  C ONCLUSION AND F UTURE W ORK

Original: This paper presents the TimesNet as a task-general foundation model for time series analysis.
Revised: This paper presents the TimesNet as a task-general foundation model for time series analysis.

Original: Motivated by the multi-periodicity, TimesNet can ravel out intricate temporal variations by a modular architecture and capture intraperiod- and interperiod-variations in 2D space by a parameter-efficient inception block.
Revised: Motivated by the multi-periodicity, TimesNet can ravel out intricate temporal variations by a modular architecture and capture intraperiod- and interperiod-variations in 2D space by a parameter-efficient inception block.

Original: Experimentally, TimesNet shows great generality and performance in five mainstream analysis tasks.
Revised: Experimentally, TimesNet shows great generality and performance in five mainstream analysis tasks.

Original: In the future, we will further explore large-scale pre-training methods in time series, which utilize TimesNet as the backbone and can generally benefit extensive downstream tasks.
Revised: In the future, we will further explore large-scale pre-training methods in time series, which utilize TimesNet as the backbone and can generally benefit extensive downstream tasks.

Original: A I MPLEMENTATION D ETAILS
Revised: A I MPLEMENTATION D ETAILS

Original: We provide the dataset descriptions and experiment configurations in Table 6 and 7.
Revised: We provide the dataset descriptions and experiment configurations in Table 6 and 7.

Original: All experiments are repeated three times, implemented in PyTorch (Paszke et al., 2019) and conducted on a single NVIDIA TITAN RTX 24GB GPU.
Revised: All experiments are repeated three times, implemented in PyTorch (Paszke et al., 2019) and conducted on a single NVIDIA TITAN RTX 24GB GPU.

Original: To make the model handle various dimensions of input series from different datasets, we select the d model based on the input series dimension C by min { max { 2 ⌈ log C ⌉ , d min } , d max } (see Table 7 for details of d min and d max ).
Revised: To make the model handle various dimensions of input series from different datasets, we select the d model based on the input series dimension C by min { max { 2 ⌈ log C ⌉ , d min } , d max } (see Table 7 for details of d min and d max ).

Original: This protocol can make the model powerful enough for multiple variates and also keep the model size compact.
Revised: This protocol can make the model powerful enough for multiple variates and also keep the model size compact.

Original: All the baselines that we reproduced are implemented based on configurations of the original paper or official code.
Revised: All the baselines that we reproduced are implemented based on configurations of the original paper or official code.

Original: It is also notable that none of the previous methods are proposed for general time series analysis.
Revised: It is also notable that none of the previous methods are proposed for general time series analysis.

Original: For a fair comparison, we keep the input embedding and the final projection layer the same among different base models and only evaluate the capability of base models.
Revised: For a fair comparison, we keep the input embedding and the final projection layer the same among different base models and only evaluate the capability of base models.

Original: Especially for the forecasting task, we use a MLP on temporal dimension to get the initialization of predicted future.
Revised: Especially for the forecasting task, we use a MLP on temporal dimension to get the initialization of predicted future.

Revised: Since we focus on the temporal variation modeling, we also adopt the Series Stationarization from

Revised: Non-stationary Transformer (Liu et al., 2022a) to eliminate the affect the distribution shift.

Original: For the metrics, we adopt the mean square error (MSE) and mean absolute error (MAE) for long-term forecasting and imputations.
Revised: For the metrics, we adopt the mean square error (MSE) and mean absolute error (MAE) for long-term forecasting and imputations.

Original: For anomaly detection, we adopt the F1-score, which is the harmonic mean of precision and recall.
Revised: For anomaly detection, we adopt the F1-score, which is the harmonic mean of precision and recall.

Original: For the short-term forecasting, following the N-BEATS (Oreshkin et al., 2019), we adopt the symmetric mean absolute percentage error (SMAPE), mean absolute scaled error (MASE) and overall weighted average (OWA) as the metrics, where OWA is a special metric used in
Revised: For the short-term forecasting, following the N-BEATS (Oreshkin et al., 2019), we adopt the symmetric mean absolute percentage error (SMAPE), mean absolute scaled error (MASE) and overall weighted average (OWA) as the metrics, where OWA is a special metric used in

Original: M4 competition.
Revised: M4 competition.

Original: These metrics can be calculated as follows:
Revised: These metrics can be calculated as follows:

Original: [Equation]
Revised: [Equation]

Original: where m is the periodicity of the data.
Revised: where m is the periodicity of the data.

Original: X ,
Revised: X ,

Original: [Equation]
Revised: [Equation]

Original: X ∈ R H
Revised: X [Equation]

Original:  × C are the ground truth and prediction results of the future with H time pints and C dimensions.
Revised: R H × C are the ground truth and prediction results of the future with H time pints and C dimensions.

Original: X i means the i -th future time point.
Revised: X i means the i -th future time point.

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: Weather (10 mins) Exchange { 96, 192, 336, 720 } (5120, 665, 1422)

Original: Exchange rate (Daily)
Revised: Exchange rate (Daily)

Original: ILI { 24, 36, 48, 60 } (617, 74, 170)
Revised: ILI { 24, 36, 48, 60 } (617, 74, 170)

Original: Illness (Weekly)
Revised: Illness (Weekly)

Original: M [Table] Yearly  (23000, 0, 23000)
Revised: M4-Yearly  (23000, 0, 23000)

Original: Demographic
Revised: Demographic

Original: M4-Quarterly  (24000, 0, 24000)
Revised: M4-Quarterly  (24000, 0, 24000)

Original: Finance
Revised: Finance

Original: Forecasting
Revised: Forecasting

Original: M4-Monthly  (48000, 0, 48000)
Revised: M4-Monthly  (48000, 0, 48000)

Original: Industry (short-term)
Revised: Industry (short-term)

Original: M4-Weakly  (359, 0, 359)
Revised: M4-Weakly  (359, 0, 359)

Original: Macro
Revised: Macro

Original: M4-Daily  (4227, 0, 4227)
Revised: M4-Daily  (4227, 0, 4227)

Original: Micro
Revised: Micro

Original: M4-Hourly  (414, 0, 414)
Revised: M4-Hourly  (414, 0, 414)

Original: Other
Revised: Other

Original: Imputation
Revised: Imputation

Original: ETTm1, ETTm   (34465, 11521, 11521)
Revised: ETTm1, ETTm   (34465, 11521, 11521)

Original: Electricity (15 mins)
Revised: Electricity (15 mins)

Original: ETTh1, ETTh   (8545, 2881, 2881)
Revised: ETTh1, ETTh   (8545, 2881, 2881)

Original: Electricity (15 mins)
Revised: Electricity (15 mins)

Original: Electricity  (18317, 2633, 5261)
Revised: Electricity  (18317, 2633, 5261)

Original: Electricity (15 mins)
Revised: Electricity (15 mins)

Original: Weather  (36792, 5271, 10540)
Revised: Weather  (36792, 5271, 10540)

Original: Weather (10 mins)
Revised: Weather (10 mins)

Original: EthanolConcentration  (261, 0, 263)
Revised: EthanolConcentration  (261, 0, 263)

Original: Alcohol Industry
Revised: Alcohol Industry

Original: FaceDetection  (5890, 0, 3524)
Revised: FaceDetection  (5890, 0, 3524)

Original: Face (250Hz)
Revised: Face (250Hz)

Original: Handwriting  (150, 0, 850)
Revised: Handwriting  (150, 0, 850)

Original: Handwriting
Revised: Handwriting

Original: Heartbeat  (204, 0, 205)
Revised: Heartbeat  (204, 0, 205)

Original: Heart Beat
Revised: Heart Beat

Original: Classification JapaneseVowels  (270, 0, 370)
Revised: Classification JapaneseVowels  (270, 0, 370)

Original: Voice (UEA)
Revised: Voice (UEA)

Original: PEMS-SF  (267, 0, 173)
Revised: PEMS-SF  (267, 0, 173)

Original: Transportation (Daily)
Revised: Transportation (Daily)

Original: SelfRegulationSCP   (268, 0, 293)
Revised: SelfRegulationSCP   (268, 0, 293)

Original: Health (256Hz)
Revised: Health (256Hz)

Original: SelfRegulationSCP   (200, 0, 180)
Revised: SelfRegulationSCP   (200, 0, 180)

Original: Health (256Hz)
Revised: Health (256Hz)

Original: SpokenArabicDigits(6599, 0, 2199)
Revised: SpokenArabicDigits

Original: Voice (11025Hz)
Revised: [Table]

Original: UWaveGestureLibrary
Revised: UWaveGestureLibrary

Original: [Table]
Revised: [Table]

Revised: Gesture

Revised: [Table]

Revised: Server Machine

Revised: Anomaly

Revised: [Table]

Revised: [Table]

Original: Spacecraft
Revised: Spacecraft

Revised: Detection

Revised: SMAP  (

Original: [Table]
Revised: [Table]

Original: Spacecraft
Revised: Spacecraft

Revised: [Table]

Revised: [Table]

Revised: Infrastructure

Revised: PSM  (

Original: [Table]
Revised: [Table]

Original: Server 
Revised: Server Machine

Original: [Table] [Table]
Revised: [Table] [Table]

Original: B H YPER - PARAMETER S ENSITIVITY
Revised: B H YPER - PARAMETER S ENSITIVITY

Original: We introduce a hyper-parameter in Equation 1 to select the most significant frequencies.
Revised: We introduce a hyper-parameter in Equation 1 to select the most significant frequencies.

Original: We provide the sensitivity analysis for these two hyper-parameters in Figure 8.
Revised: We provide the sensitivity analysis for these two hyper-parameters in Figure 8.

Original: We can find that our proposed
Revised: We can find that our proposed

Original: TimesNet can present a stable performance under different choices of k in all four tasks
Revised: TimesNet can present a stable performance under different choices of k in all four tasks

Original: [Table]
Revised: [Table]

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original: SE

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure] [Figure] [Figure] Especially, we select the 37.5% mask ratio for hyper-parameter experiments.
Revised: Especially, we select the 37.5% mask ratio for hyper-parameter experiments.

Original: For classification, we choose the two largest subsets: SpokenArobicDigits and FaceDetection for evaluation.
Revised: For classification, we choose the two largest subsets: SpokenArobicDigits and FaceDetection for evaluation.

Original: For short-term forecasting, we adopt the weighted average for sensitivity analysis.
Revised: For short-term forecasting, we adopt the weighted average for sensitivity analysis.

Original: Besides, from Figure 8, we can also find out the following observations:
Revised: Besides, from Figure 8, we can also find out the following observations:

Original: • For the low-level modeling tasks, such as forecasting and anomaly detection, the selection of k will affect the final performance more.
Revised: • For the low-level modeling tasks, such as forecasting and anomaly detection, the selection of k will affect the final performance more.

Original: This may come from that k will directly affect the amount of information in deep representations.
Revised: This may come from that k will directly affect the amount of information in deep representations.

Original: • For the high-level modeling tasks, such as classification and imputation, the model performance will be more robust to the selection of k , since the key to these tasks is to extract hierarchical representations.
Revised: • For the high-level modeling tasks, such as classification and imputation, the model performance will be more robust to the selection of k , since the key to these tasks is to extract hierarchical representations.

Original: Giving consideration to both efficiency and performance
Revised: Giving consideration to both efficiency and performance

Original: [Table]
Revised: [Table]

Original: we set k = 3 for imputation, classification and anomaly detection and k = 5 for short-term forecasting.
Revised: we set k = 3 for imputation, classification and anomaly detection and k = 5 for short-term forecasting.

Original: C A BLATION S TUDIES
Revised: C A BLATION S TUDIES

Original: To elaborate the property of our proposed TimesNet
Revised: To elaborate the property of our proposed TimesNet

Original: [Table]
Revised: [Table]

Original: we conduct detailed ablations on the representation leaning in 2D space, model architecture and adaptive aggregation.
Revised: we conduct detailed ablations on the representation leaning in 2D space, model architecture and adaptive aggregation.

Original: 2D space As shown in Table 8, replacing the inception block with more powerful blocks will bring further performance promotion, such as ResNeXt (Xie et al., 2017), Swin Transformer
Revised: 2D space As shown in Table 8, replacing the inception block with more powerful blocks will bring further performance promotion, such as ResNeXt (Xie et al., 2017), Swin Transformer

Original: (Liu et al., 2021b) and ConvNeXt (Liu et al., 2022b).
Revised: (Liu et al., 2021b) and ConvNeXt (Liu et al., 2022b).

Original: It is also notable that using the independent parameters will also bring improvement, while this will cause the model size related to the selection of hyperparameter k .
Revised: It is also notable that using the independent parameters will also bring improvement, while this will cause the model size related to the selection of hyperparameter k .

Original: Considering the efficiency and model performance, we choose the parameter-efficient inception block as our final solution.
Revised: Considering the efficiency and model performance, we choose the parameter-efficient inception block as our final solution.

Original: These results also verify that our design bridges the 1D time series analysis with 2D computer vision backbones.
Revised: These results also verify that our design bridges the 1D time series analysis with 2D computer vision backbones.

Original: [Table]
Revised: [Table]

Original: Model architecture We also conduct experiments on different architectures.
Revised: Model architecture We also conduct experiments on different architectures.

Original: Surprisingly, as shown in Table 9, we find that combining with the deep decomposition architecture in Autoformer (Wu et al., 2021) cannot bring further promotion.
Revised: Surprisingly, as shown in Table 9, we find that combining with the deep decomposition architecture in Autoformer (Wu et al., 2021) cannot bring further promotion.

Original: These results may come from that in the case that the input series already present clear periodicity, our design can capture the 2D-variations effectively.
Revised: These results may come from that in the case that the input series already present clear periodicity, our design can capture the 2D-variations effectively.

Original: As for the case that they are without clear periodicity, the model will learn the most significant frequency as 1, where the trend of time series be covered by the intraperiod-variation modeling.
Revised: As for the case that they are without clear periodicity, the model will learn the most significant frequency as 1, where the trend of time series be covered by the intraperiod-variation modeling.

Original: These results also verify that our proposed TimesNet can handle the analysis for time series without clear periodicity.
Revised: These results also verify that our proposed TimesNet can handle the analysis for time series without clear periodicity.

Original: Besides, in this paper, to take advantages of the deep representations, we place the transformation from 1D-variations to 2D-variations in every TimesBlock.
Revised: Besides, in this paper, to take advantages of the deep representations, we place the transformation from 1D-variations to 2D-variations in every TimesBlock.

Original: Here, we compare our design with the case that only conducts the transformation on the raw input series.
Revised: Here, we compare our design with the case that only conducts the transformation on the raw input series.

Original: From Table 9, we can find that the performance of Transform raw data degenerates a lot (Avg F1-score: 85.49% → 84.85%), indicating the advantages of our design.
Revised: From Table 9, we can find that the performance of Transform raw data degenerates a lot (Avg F1-score: 85.49% → 84.85%), indicating the advantages of our design.

Original: architecture proposed by Autoformer (Wu et al., 2021) with TimesNet.
Revised: proposed by Autoformer (Wu et al., 2021) with TimesNet.

Original: Transform raw data refers conducting the transformation on the original time series, instead of deep features.
Revised: Transform raw data refers conducting the transformation on the original time series, instead of deep features.

Original: [Table]
Revised: [Table]

Original: Adaptive aggregation As shown in Equation 6, following the design in Autoformer (2021), we adopt the amplitudes after the Softmax function as aggregation weights of processed tensors { (cid:98)
Revised: Adaptive aggregation As shown in Equation 6, following the design in Autoformer (2021), we adopt the amplitudes after the Softmax function as aggregation weights of processed tensors { (cid:98)

Original: Xl,
Revised: X

Original: [Equation]
Revised: [Equation]

Original: 1D , · · · ,

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: X l,k 1D } .
Revised: X l,k 1D } .

Original: Here we include two variants for comparison.
Revised: Here we include two variants for comparison.

Original: The first is directly-sum, namely (cid:80) ki
Revised: The first is directly-sum, namely (cid:80) ki

Original: [Equation]
Revised: [Equation]

Original: X l,i 1D .
Revised: X l,i 1D .

Original: The second is to remove the Softmax function, that is
Revised: The second is to remove the Softmax function, that is

Original: [Equation]
Revised: [Equation]

Original: X
Revised: X

Original: [Equation]
Revised: [Equation]

Original: Tabledemonstrates that our design in adaptive aggregation performs the best.
Revised: Tabledemonstrates that our design in adaptive aggregation performs the best.

Original: [Table]
Revised: [Table]

Original: D M ORE R EPRESENTATION A NALYSIS
Revised: D M ORE R EPRESENTATION A NALYSIS

Original: To give an intuitive understanding of 2D-variations, we visualize the transformed 2D tensors in Figure 9.
Revised: To give an intuitive understanding of 2D-variations, we visualize the transformed 2D tensors in Figure 9.

Original: From the visualization, we can obtain the following observations:
Revised: From the visualization, we can obtain the following observations:

Original: • The interperiod-variation can present the long-term trends of time series.
Revised: • The interperiod-variation can present the long-term trends of time series.

Original: For example, in the first case of Exchange, the values in each row decrease from left to right, indicating the downtrend of the raw series.
Revised: For example, in the first case of Exchange, the values in each row decrease from left to right, indicating the downtrend of the raw series.

Original: And for the ETTh1 dataset, the values in each row are similar to each other, reflecting the global stable variation of the raw series.
Revised: And for the ETTh1 dataset, the values in each row are similar to each other, reflecting the global stable variation of the raw series.

Original: • For the time series without clear periodicity, the temporal 2D-variations can still present informative 2D structure.
Revised: • For the time series without clear periodicity, the temporal 2D-variations can still present informative 2D structure.

Original: If the frequency is one, the intraperiod-variation is just the original variation of raw series.
Revised: If the frequency is one, the intraperiod-variation is just the original variation of raw series.

Original: Besides, the interperiod-variation can also present the long-term trend, benefiting the temporal variation modeling.
Revised: Besides, the interperiod-variation can also present the long-term trend, benefiting the temporal variation modeling.

Original: • The transformed 2D-variations demonstrate two types of localities.
Revised: • The transformed 2D-variations demonstrate two types of localities.

Original: Firstly, for each column (intraperiod-variation), the adjacent values are close to each other, presenting the locality among adjacent time points.
Revised: Firstly, for each column (intraperiod-variation), the adjacent values are close to each other, presenting the locality among adjacent time points.

Original: Secondly, for each row (interperiod-variation), the adjacent values are also close, corresponding to the locality among adjacent periods.
Revised: Secondly, for each row (interperiod-variation), the adjacent values are also close, corresponding to the locality among adjacent periods.

Original: Note that the non-adjacent periods can be quite different from each other, which can be caused by global trend, such as the case from the Exchange dataset.
Revised: Note that the non-adjacent periods can be quite different from each other, which can be caused by global trend, such as the case from the Exchange dataset.

Original: These observations of localities also motivate us to adopt the 2D kernel for representation learning.
Revised: These observations of localities also motivate us to adopt the 2D kernel for representation learning.

Original: [Table]
Revised: [Table]

Original: 50 100 150 200 250 300 350 ° 1 . 0 ° 0 . 8 ° 0 . 6 ° 0 . 4 ° 0 .
Revised: [Table]

Original: [-6 0 5 10 15 20 25 30 35 0 3 5 7 1 ° 1 ID Series 2D Variations E TT h 1 E l e c t r i c t y E x c h a n g e W e a t h e r 
-]
Revised: [-6 0 5 10 15 20 25 30 35 0 3 5 7 1 ° 1 ID Series 2D Variations E TT h 1 E l e c t r i c t y E x c h a n g e W e a t h e r 
-][Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Revised: ID Series

Original: [Table]
Revised: [Table]

Revised: E TT

Original: [Table]
Revised: [Table]

Revised: 1 E l e c t r i c t

Original: [Table]
Revised: [Table]

Original: [Table] [Table] [Table]
Revised: E

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: c

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: n

Original: [Table]
Revised: [Table]

Original: 0 5 10 15 0 3 5 7 10 13 15 18 20 1 ° 1 0 5 [-
-] 30 35 1 ° 1 Period F r e q u e n c y Weight 0 5 10 15 20 25 0 2 4 6 8 10 0 . {+
+}
Revised: 5 10 15 0 3 5 7 10 13 15 18 20 1 ° 1 0 5 [-
-]0 5 10 15 20 25 30 35 1 ° 1 Period F r e q u e n c y Weight 0 5 10 15 20 25 0 2 4 6 8 10 0 . {+
+}

Original: 8 ° 1 . 0 0 5 10 0 5 10 15 20 0 .
Revised: ° 1 . 0 0 5 10 0 5 10 15 20 0 .

Original: 8 ° 1 . 0 0 5 10 15 20 25 30 35 40 0 5 0 . 8 ° 1 .00 5 10 15 20 25 30 35 0 .
Revised: 8 ° 1 . 0 0 5 10 15 20 25 30 35 40 0 5 0 . 8 ° 1 .00 5 10 15 20 25 30 35 0 .

Original: ° 1 .0 50 5  0 .
Revised: ° 1 .0 50 5  0 .

Original: ° 1 .0 . ° 1 .0 .
Revised: ° 1 .0 . ° 1 .0 .

Original: ° 1 .0 . ° 1 .0         0 .
Revised: ° 1 .0 . ° 1 .0         0 .

Original: ° 1 .
Revised: ° 1 .

Original: [Figure]
Revised: [Figure]

Original: E TT h
Revised: E TT h

Original: E l e c t r i c i t y
Revised: E l e c t r i c i t y

Original: E x
Revised: E x

Original: c h a n g e
Revised: c h a n g e

Original: ID Series 2D Variations
Revised: ID Series 2D Variations

Original: [Figure]
Revised: [Figure]

Original: [Figure]

Revised: ° 1 .

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: ° 
Revised: ° 1 .

Original: [Figure]
Revised: [Figure]

Original: ° 1 .0        0 . ° 1 .0        0 .
Revised: ° 1 .0        0 . ° 1 .0        0 .

Original: ° 1 .{+.
+}
Revised: ° 1 {+.
+}

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: E M ULTI - PERIODICITY OF T IME S ERIES
Revised: E M ULTI - PERIODICITY OF T IME S ERIES

Original: As shown in Figure
Revised: As shown in Figure

Original: [Table]
Revised: [Table]

Original: we calculate the density of each period length for different datasets.
Revised: we calculate the density of each period length for different datasets.

Original: We can find that real-world time series present multi-periodicity to some extent.
Revised: We can find that real-world time series present multi-periodicity to some extent.

Original: For example, the Electricity dataset contains the periods with length-12 and length-24.
Revised: For example, the Electricity dataset contains the periods with length-12 and length- [Table]

Original: D e n s it y
Revised: D e n s it y

Original: [Table]
Revised: [Table]

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Original: [Table]
Revised: [Table]

Revised: =</latexit

Original: [Table]
Revised: [Table]

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Revised: <latexit

Original: [Table]
Revised: [Table]

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Original: [Figure]
Revised: [Figure]

Original: F S HOWCASES
Revised: F S HOWCASES

Original: To provide a clear comparison among different models, we provide showcases to the regression tasks, including the imputation (Figure 11), long-term forecasting (Figure 12) and short-term forecasting (Figure 13).
Revised: To provide a clear comparison among different models, we provide showcases to the regression tasks, including the imputation (Figure 11), long-term forecasting (Figure 12) and short-term forecasting (Figure 13).

Original: Especially in the imputation task, the MLP-based models degenerate a lot.
Revised: Especially in the imputation task, the MLP-based models degenerate a lot.

Original: This is because the input series has been randomly masked.
Revised: This is because the input series has been randomly masked.

Original: However, the MLP-based models adopt the fixed model parameter to model the temporal dependencies among time points, thereby failing in this task
Revised: However, the MLP-based models adopt the fixed model parameter to model the temporal dependencies among time points, thereby failing in this task

Original: [Figure]
Revised: [Figure]

Original: G M ODEL E FFICIENCY A NALYSIS
Revised: G M ODEL E FFICIENCY A NALYSIS

Original: To summarize the model performance and efficiency, we calculate the relative performance rankings for comparing baselines.
Revised: To summarize the model performance and efficiency, we calculate the relative performance rankings for comparing baselines.

Original: The rankings are compared from the common models used in all five tasks: LSTM (1997) and LSSL (2022); TCN (2019); LightTS (2022) and DLinear (2022); Reformer (2020), Informer (2021), Pyraformer (2021a), Autoformer (2021), FEDformer (2022)
Revised: The rankings are compared from the common models used in all five tasks: LSTM (1997) and LSSL (2022); TCN (2019); LightTS (2022) and DLinear (2023); Reformer (2020), Informer (2021), Pyraformer (2021a), Autoformer (2021), FEDformer (2022), Non-stationary

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: As shown in Table 11, our proposed TimesNet achieves the best performance in all five tasks.
Revised: As shown in Table 11, our proposed TimesNet achieves the best performance in all five tasks.

Original: Among the top three models, TimesNet also achieves the greatest efficiency.
Revised: Among the top three models, TimesNet also achieves the greatest efficiency.

Original: Compared to MLP-based models, our proposed TimesNet shows a significant advantage in performance.
Revised: Compared to MLP-based models, our proposed TimesNet shows a significant advantage in performance.

Original: And benefiting from the utilization of 2D kernels and parameter-efficient design, the parameter size is invariant when the input series changes.
Revised: And benefiting from the utilization of 2D kernels and parameter-efficient design, the parameter size is invariant when the input series changes.

Original: Compared to Transformer-based models, TimesNet is with great efficiency in GPU memory, which is essential in long sequence modeling. 
Revised: Compared to Transformer-based models, TimesNet is with great efficiency in GPU memory, which is essential in long sequence modeling. [Table] [Table] [Table]

Revised: [Table]

Original: [Table]
Revised: [Table]

Original: H M ODEL PERFORMANCE IN M IXED D ATASET
Revised: H M ODEL PERFORMANCE IN M IXED D ATASET

Original: To verify the model capacity in large-scale pre-training
Revised: To verify the model capacity in large-scale pre-training

Original: [Table]
Revised: [Table]

Original: we evaluate the model performance when it is trained from a mixed dataset.
Revised: we evaluate the model performance when it is trained from a mixed dataset.

Original: Concretely, we mixed the hourly-collected ETTh1, ETTh2 and the 15-minute collected ETTm1, ETTm2 as the mixed dataset.
Revised: Concretely, we mixed the hourly-collected ETTh1, ETTh2 and the 15-minute collected ETTm1, ETTm2 as the mixed dataset.

Original: Note that this mixed dataset contains diverse temporal patterns and periodicities in different data instances, making the unified training challenging.
Revised: Note that this mixed dataset contains diverse temporal patterns and periodicities in different data instances, making the unified training challenging.

Original: From Table 12, we can find that TimesNet can handle this mixed dataset well and generally promote the model performance in four independent subsets.
Revised: From Table 12, we can find that TimesNet can handle this mixed dataset well and generally promote the model performance in four independent subsets.

Original: Besides
Revised: Besides

Original: [Table]
Revised: [Table]

Original: we can also find that except TimesNet, for other baselines
Revised: we can also find that except TimesNet [Table] for other baselines

Original: [Table]
Revised: [Table]

Original: the mixed training may decrease the model performance in some subsets, indicating that other baselines cannot handle the complex periodicities in the mixed dataset.
Revised: the mixed training may decrease the model performance in some subsets, indicating that other baselines cannot handle the complex periodicities in the mixed dataset.

Original: These results also verify the potential of TimesNet in performing as the general-purpose backbone for large-scale pre-training in time series
Revised: These results also verify the potential of TimesNet in performing as the general-purpose backbone for large-scale pre-training in time series

Original: [Table] [Table]
Revised: [Table] [Table]

Original: I F ULL R ESULTS
Revised: I F ULL R ESULTS

Original: Due to the space limitation of the main text, we place the full results of all experiments in the following: long-term forecasting in Table 13, short-term forecasting in Table 14, imputation in Table 16, classification in Table 17 and anomaly detection in Table 15.
Revised: Due to the space limitation of the main text, we place the full results of all experiments in the following: long-term forecasting in Table 13, short-term forecasting in Table 14, imputation in Table 16, classification in Table 17 and anomaly detection in Table 15.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: ETT
Revised: ETT

Original: [Table]
Revised: [Table]

Original: ETT
Revised: ETT

Original: [Table] [Table]
Revised: [Table] [Table]