MMFNet: Multi-Scale Frequency Masking Neural Network for Multivariate Time Series Forecasting

## Abstract

Abstract Long-term Time Series Forecasting (LTSF) is critical for numerous real-world applications, such as electricity consumption planning, financial forecasting, and disease propagation analysis. LTSF requires capturing long-range dependencies between inputs and outputs, which poses significant challenges due to complex temporal dynamics and high computational demands. While linear models reduce model complexity by employing frequency domain decomposition, current approaches often assume stationarity and filter out high-frequency components that may contain crucial short-term fluctuations. In this paper, we introduce MMFNet, a novel model designed to enhance long-term multivariate forecasting by leveraging a multi-scale masked frequency decomposition approach. MMFNet captures fine, intermediate, and coarse-grained temporal patterns by converting time series into frequency segments at varying scales while employing a learnable mask to filter out irrelevant components adaptively.
Extensive experimentation with benchmark datasets shows that MMFNet not only addresses the limitations of the existing methods but also consistently achieves good performance. Specifically, MMFNet achieves up to 6.0 % percent 6.0 6.0\% 6.0 % reductions in the Mean Squared Error (MSE) compared to state-of-the-art models designed for multivariate forecasting tasks.

## 1 Introduction

Time series forecasting is pivotal in a wide range of domains, such as environmental monitoring (Bhandari et al., ), electrical grid management (Zufferey et al., ), financial analysis (Sezer et al., ), and healthcare (Zeroual et al., ). Accurate long-term forecasting is essential for informed decision-making and strategic planning. Traditional methods, such as autoregressive (AR) models (Nassar et al., ), exponential smoothing (Hyndman & Athanasopoulos, ), and structural time series models (Harvey, ), have provided a robust foundation for time series analysis by leveraging historical data to predict future values. However, real-world systems frequently exhibit complex, non-stationary behavior, with time series characterized by intricate patterns such as trends, fluctuations, and cycles. Those complexities pose significant challenges to achieving accurate forecasts (Makridakis et al., ; Box et al., ).

Long-term Time Series Forecasting (LTSF) has seen significant advancements in recent years, driven by the development of sophisticated models, such as Transformer-based models (Zhou et al., ; Wu et al., ; Nie et al., ) and linear models (Zeng et al., ; Xu et al., ; Lin et al., ). Transformer-based architectures have demonstrated exceptional capacity in capturing complex temporal patterns by effectively modeling long-range dependencies through self-attention mechanisms at the cost of heavy computation workload, particularly when facing large-scale time series data, which significantly limits their practicality in real-time applications.

In contrast, the linear models provide a lightweight alternative for real-time forecasting. In particular, FITS demonstrates superior predictive performance across a wide range of scenarios with only $10K$ parameters by utilizing a single-scale frequency domain decomposition method combined with a low-pass filter employing a fixed cutoff frequency (Xu et al., ). While single-scale frequency domain decomposition offers a global perspective of time series data in the frequency domain, it lacks the ability to localize specific frequency components within the sequence. Furthermore, such methods assume that frequency components remain constant throughout the entire sequence, thereby failing to account for the non-stationary behavior frequently observed in real-world time series. Additionally, the low-pass filter employed by FITS may inadvertently smooth out crucial short-term fluctuations necessary for accurate predictions. The fixed cutoff frequency of the low-pass filter may not be universally optimal for diverse time series datasets, further limiting its adaptability.

In this paper, we present MMFNet, a novel model designed to enhance LTSF through a multi-scale masked frequency decomposition approach. MMFNet captures fine, intermediate, and coarse-grained patterns in the frequency domain by segmenting the time series at multiple scales. At each scale, MMFNet employs a learnable mask that adaptively filters out irrelevant frequency components based on the segment’s spectral characteristics.
MMFNet offers two key advantages: (i) the multi-scale frequency decomposition enables MMFNet to effectively capture both short-term fluctuations and broader trends in the data, and (ii) the learnable frequency mask adaptively filters irrelevant frequency components, allowing the model to focus on the most informative signals. These features make MMFNet well-suited to capturing both short-term and long-term dependencies in complex time series, positioning it as an effective solution for various LTSF tasks.

In summary, the contributions of this paper are as follows:

- •
To our knowledge, MMFNet is the first model that employs multi-scale frequency domain decomposition to capture the dynamic variations in the frequency domain;
- •
MMFNet introduces a novel learnable masking mechanism that adaptively filters out irrelevant frequency components;
- •
Extensive experiments show that MMFNet consistently achieves good performance in a variety of multivariate time series forecasting tasks, with up to a $6.0\%$ reduction in the Mean Squared Error (MSE) compared to the existing models.

## 2 Preliminaries

#### Long-term Time Series Forecasting.

LTSF involves predicting future values over an extended time horizon based on previously observed multivariate time series data. The LTSF problem can be formulated as:

$$ $\hat{x}_{t+1:t+H}=f(x_{t-L+1:t}),$ (1) $$

where $x_{t-L+1:t}\in\mathbb{R}^{L\times C}$ denotes the historical observation window, and $\hat{x}_{t+1:t+H}\in\mathbb{R}^{H\times C}$ represents the predicted future values. In this formulation, $L$ is the length of the historical window, $H$ is the forecast horizon, and $C$ denotes the number of features or channels. As the forecast horizon $H$ increases, the models face challenges to accurately capture both long-term and short-term dependencies within the time series.

#### Single-Scale Frequency Transformation (SFT).

SFT refers to the process of converting the time-domain data into the frequency domain at a single, global scale without segmenting the time series. Such a transformation is typically performed using methods, such as the Fast Fourier Transform (FFT), which efficiently computes the Discrete Fourier Transform (DFT). SFT decomposes the entire signal into sinusoidal components, enabling the analysis of its frequency content. Each frequency component can be expressed as:

$$ $X_{k}=|X_{k}|e^{j\phi_{k}},$ (2) $$

where $|X_{k}|$ represents the amplitude and $\phi_{k}$ the phase of the $k$-th frequency component. While the frequency decomposition provides valuable insights into periodic patterns and trends, traditional approaches assume stationarity and operate on a global scale, limiting their capacity to capture the complex, non-stationary characteristics frequently observed in real-world time series. Current frequency-based LTSF models, such as FITS (Xu et al., ), implement this method by performing frequency domain interpolation at a single scale, which can be formulated as:

$$ $\tilde{x}_{t+1:t+H}=g(\mathcal{F}(x_{t-L+1:t})),$ (3) $$

where $\mathcal{F}$ denotes the Fourier transform, and $g$ represents the filtering operation applied uniformly across the signal. Although SFT is capable of capturing broad temporal patterns, such as long-term trends through low-pass filtering or short-term fluctuations through high-pass filtering, its global application treats the entire signal uniformly. This uniform treatment may result in the loss of important local temporal variations and non-stationary behaviors occurring at different scales.

## 3 Method

### 3.1 Overview

To overcome the limitations of SFT, we propose the Multi-scale Masked Frequency Transformation (MMFT). MMFT performs frequency decomposition across multiple temporal scales, enabling the model to capture both global and local temporal patterns. Formally, the MMFT problem can be expressed as:

$$ $\tilde{x}_{t+1:t+H}=h(\{\mathcal{F}_{s}(x_{t-L+1:t})\}_{s=1}^{S}),$ (4) $$

where $\mathcal{F}_{s}$ denotes the frequency transformation at scale $s$, and $h$ represents the aggregation and filtering operation applied to the learnable frequency masks at various scales. Unlike SFT, which applies a single transformation to the entire time series, MMFT divides the signal into multiple scales, each subjected to frequency decomposition. At each scale, a learnable frequency mask is applied to retain the most informative frequency components while selectively discarding noise.
This multi-scale approach allows the model to adapt to non-stationary signals, capturing complex dependencies that span different temporal ranges. By leveraging frequency decomposition at multiple scales and applying adaptive masks, MMFT enhances long-term forecasting accuracy by focusing on both short-term fluctuations and long-term trends within the data. This method increases the model’s flexibility and robustness, particularly for non-stationary and multivariate time series. Further analysis of the differences between SFT and MMFT can be found in Appendix B.

Figure: Figure 1: MMFNet Architecture. MMFNet consists of the following key components: ① The input time series is first normalized to have zero mean using Reversible Instance-wise Normalization (RIN) (Lai et al., ). The multi-scale frequency decomposition process then divides the time series instance $X$ into fine, intermediate, and coarse-scale segments, which are subsequently transformed into the frequency domain via the Discrete Cosine Transform (DCT). ② A learnable mask is applied to the frequency segments, followed by a linear layer that predicts the transformed frequency components. ③ Finally, the predicted frequency segments from each scale are transformed back into the time domain, merged, and denormalized using inverse RIN (iRIN).
Refer to caption: x1.png

MMFNet enhances time series forecasting by incorporating the proposed MMFT method to capture intricate frequency features across different scales. The overall architecture of MMFNet is depicted in Figure 1. The model comprises three key components: Multi-scale Frequency Decomposition, Masked Frequency Interpolation, and Spectral Inversion. Multi-scale Frequency Decomposition normalizes the input time series, divides it into segments of varying scales, and transforms these segments into the frequency domain using the DCT. Masked Frequency Interpolation applies a self-adaptive, learnable mask to filter out irrelevant frequency components, followed by a linear transformation of the filtered frequency domain segments. Finally, Spectral Inversion converts the processed frequency components back into the time domain via the Inverse Discrete Cosine Transform (iDCT) (Ahmed et al., ). The outputs from different scales are then aggregated, resulting in a refined signal that preserves the essential characteristics of the original input.

### 3.2 Multi-scale Frequency Decomposition

The core concept of Multi-scale Frequency Decomposition lies in applying frequency domain transformations to time series sequences at multiple scales. This approach enables the model to capture both global patterns and fine-grained temporal dynamics by analyzing the data across various segment levels. Multi-scale Frequency Decomposition consists of two fundamental steps: fragmentation and decomposition. Details about the overall workflow can be seen in Appendix A.1.

#### Fragmentation.

This step decomposes the time series data into segments of varying lengths to capture features across multiple scales. Specifically, the input sequence $X$ is first normalized using RIN (Lai et al., ) and then partitioned into three sets of segments: fine-scale, intermediate-scale, and coarse-scale segments. Fine-scale segments (${\bm{X}}^{fine}$) consist of shorter segments that capture detailed, high-frequency components of the time series, enabling the detection of intricate patterns and anomalies that may be missed in longer segments. Intermediate-scale segments (${\bm{X}}^{intermediate}$) are of moderate length and are designed to capture intermediate-level patterns and trends, striking a balance between the fine and coarse segments. Coarse-scale segments (${\bm{X}}^{coarse}$) comprise longer segments that capture broader, low-frequency trends and overarching patterns within the data. This multi-scale fragmentation allows the model to effectively capture and leverage patterns across different temporal scales.

#### Decomposition.

This step converts the multi-scale time-domain segments into their corresponding frequency components to capture frequency patterns across various temporal scales. For each segment, the DCT is applied to extract frequency domain representations. Specifically, the fine-scale segments in ${\bm{X}}^{fine}$ are transformed into ${\bm{X}}^{fine}_{DCT}$, the intermediate-scale segments in ${\bm{X}}^{intermediate}$ are converted into ${\bm{X}}^{intermediate}_{DCT}$, and the coarse-scale segments in ${\bm{X}}^{coarse}$ are transformed into ${\bm{X}}^{coarse}_{DCT}$.

The DCT for each segment is computed using the following formula:

$$ $X_{k}=\sum_{n=0}^{N-1}x_{n}\cos\left(\frac{\pi}{N}\left(n+\frac{1}{2}\right)k \right),$ (5) $$

where $x_{n}$ represents the time-domain signal values, $N$ is the segment length, and $k$ denotes the frequency component. The resulting coefficients $X_{k}$ represent the frequency components of the segment. This transformation enables MMFNet to capture and analyze patterns at multiple temporal scales in the frequency domain, thereby enhancing its ability to recognize and interpret complex patterns in time series data.

### 3.3 Masked Frequency Interpolation

Masked Frequency Interpolation leverages a learnable mask to adaptively filter frequency components across different scales in the frequency domain, followed by reconstruction through a linear layer neural network. This approach enables the model to learn and apply scale-specific filtering strategies tailored to diverse datasets. The process consists of two primary steps: Masking and Interpolation.

#### Masking.

Traditional methods often employ fixed low-pass filters with a predefined cutoff frequency to filter frequency components. These approaches assume that certain frequencies are universally important or irrelevant across the entire time series, an assumption that may not hold for non-stationary data where the relevance of frequency components varies over time. Moreover, over-filtering can lead to the loss of critical details, resulting in oversimplified representations and diminished model performance in tasks such as forecasting and signal analysis. To address these limitations, MMFNet employs an adaptive masking technique to capture dynamic behaviors in the frequency domain. Given the frequency segments ${\bm{X}}_{DCT}$, a learnable mask is generated to adaptively filter the frequency components. The mask adjusts the significance of different frequency components by attenuating or emphasizing them based on their relevance to the task. This filtering process is applied via element-wise multiplication, represented as:

$$ ${\bm{X}}_{mask_DCT}={\bm{X}}_{DCT}\odot M,$ (6) $$

where $\odot$ denotes element-wise multiplication, $M$ represents the learnable mask, and ${\bm{X}}_{mask_DCT}$ is the resulting masked frequency representation. During training, the mask is iteratively updated based on the loss function, allowing MMFNet to focus on the most relevant aspects of the frequency domain representation. This adaptive mechanism improves the model’s capacity to capture meaningful patterns while minimizing the influence of irrelevant or noisy information.

#### Interpolation.

In this step, the masked frequency segments ${\bm{X}}_{mask_DCT}$ are transformed into predicted frequency domain segments ${\bm{X}}_{pred_DCT}$ through a linear layer. This linear transformation maps the filtered frequency components to the target frequency representations aligned with the model’s forecasting objectives. Specifically, a fully connected (dense) layer is applied to the masked frequency components, and this operation can be expressed as:

$$ ${\bm{X}}_{pred_DCT}=W\cdot{\bm{X}}_{mask_DCT}+b,$ (7) $$

where $W$ denotes the weight matrix of the linear layer, and $b$ is the bias term. The linear layer is designed to learn a projection that aligns the filtered frequency components with the target prediction space. This transformation further refines the frequency domain information, producing ${\bm{X}}_{pred_DCT}$, which is essential for reconstructing accurate time-domain predictions. By leveraging the refined frequency information and reducing the influence of irrelevant frequency components, this step improves the overall prediction accuracy.

### 3.4 Spectral Inversion

The final process, Spectral Inversion, transforms the interpolated frequency components back into the time domain using the iDCT, reversing the earlier DCT process. The iDCT is applied individually to the predicted frequency domain segments ${\bm{X}}^{fine}_{pred_DCT}$, ${\bm{X}}^{intermediate}_{pred_DCT}$, and ${\bm{X}}^{coarse}_{pred_DCT}$. The iDCT for a segment is given by the following formula:

$$ $x_{n}=\frac{1}{2}x_{0}+\sum_{k=1}^{N-1}X_{k}\cos\left(\frac{\pi}{N}\left(n+ \frac{1}{2}\right)k\right),$ (8) $$

where $x_{n}$ represents the time-domain signal values, $X_{k}$ are the frequency components, and $N$ denotes the segment length. This equation reconstructs the time-domain signal by summing the contributions of each frequency component (Davis & Marsaglia, ).

Once the iDCT is performed separately for each scale, the resulting time-domain signals are combined. This integration step merges the multi-scale frequency information by combining the outputs from the fine, intermediate, and coarse scales. The final reconstructed signal preserves the key characteristics of the original input while incorporating the enhanced interpolation achieved through the masked frequency filtering.

## 4 Experiment

In this section, we evaluate MMFNet with several LTSF benchmark datasets across a range of forecast horizons. We also conduct ablation studies to assess the impact of MMFT and our frequency masking techniques. Finally, we evaluate MMFNet’s performance in ultra-long-term forecasting scenarios.

### 4.1 Experimental Setup

#### Datasets.

We perform experiments with seven widely-used LTSF datasets: ETTh1, ETTh2, ETTm1, ETTm2, Weather, Electricity, and Traffic. More details on those datasets can be found in Appendix A.2.

#### Baselines.

We compare MMFNet against several state-of-the-art models, including FEDformer (Zhou et al., ), TimesNet (Wu et al., ), TimeMixer (Wang et al., ), and PatchTST (Nie et al., ). In addition, we compare MMFNet against several lightweight models, including DLinear (Zeng et al., ), FITS (Xu et al., ), and SparseTSF (Lin et al., ). More details on our baseline models can be found in Appendix A.3.

#### Environment.

All experiments are implemented using PyTorch (Paszke et al., ) and run on a single NVIDIA GeForce RTX 4090 GPU with $24$GB of memory.

### 4.2 Performance on LTSF Benchmarks

**Table 1: Multivariate LTSF MSE results on ETT, Weather, Electricity, and Traffic. The best result is emphasized in bold, while the second-best is underlined. “Imp.” represents the improvement between MMFNet and either the best or second-best result, with a higher “Imp.” indicating greater improvement.**
| Models | MMFNet | FITS | SparseTSF | DLinear | PatchTST | TimeMixer | TimesNet | FEDformer | Imp. |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Data | Horizon | (ours) | (2024) | (2024) | (2023) | (2023) | (2024) | (2023) | (2022) |  |
| ETTh1 | 96 | 0.359 | 0.372 | 0.362 | 0.384 | 0.385 | 0.380 | 0.384 | 0.375 | +0.003 |
| 192 | 0.396 | 0.404 | 0.403 | 0.443 | 0.413 | 0.413 | 0.436 | 0.427 | +0.006 |  |
| 336 | 0.409 | 0.427 | 0.434 | 0.446 | 0.440 | 0.445 | 0.491 | 0.459 | +0.018 |  |
| 720 | 0.419 | 0.424 | 0.426 | 0.504 | 0.456 | 0.491 | 0.521 | 0.484 | +0.005 |  |
| ETTh2 | 96 | 0.263 | 0.271 | 0.294 | 0.282 | 0.274 | 0.281 | 0.340 | 0.340 | +0.008 |
| 192 | 0.317 | 0.331 | 0.339 | 0.340 | 0.338 | 0.356 | 0.402 | 0.433 | +0.014 |  |
| 336 | 0.336 | 0.354 | 0.359 | 0.414 | 0.367 | 0.371 | 0.452 | 0.508 | +0.018 |  |
| 720 | 0.376 | 0.377 | 0.383 | 0.588 | 0.391 | 0.403 | 0.462 | 0.480 | +0.001 |  |
| ETTm1 | 96 | 0.307 | 0.303 | 0.314 | 0.301 | 0.292 | 0.315 | 0.338 | 0.362 | -0.015 |
| 192 | 0.334 | 0.337 | 0.343 | 0.335 | 0.330 | 0.339 | 0.374 | 0.393 | -0.004 |  |
| 336 | 0.358 | 0.366 | 0.369 | 0.371 | 0.365 | 0.366 | 0.410 | 0.442 | +0.007 |  |
| 720 | 0.396 | 0.415 | 0.418 | 0.426 | 0.419 | 0.423 | 0.478 | 0.483 | +0.019 |  |
| ETTm2 | 96 | 0.160 | 0.162 | 0.165 | 0.171 | 0.163 | 0.176 | 0.187 | 0.189 | +0.002 |
| 192 | 0.212 | 0.216 | 0.218 | 0.237 | 0.219 | 0.226 | 0.249 | 0.256 | +0.004 |  |
| 336 | 0.259 | 0.268 | 0.272 | 0.294 | 0.276 | 0.276 | 0.321 | 0.326 | +0.009 |  |
| 720 | 0.327 | 0.348 | 0.352 | 0.426 | 0.368 | 0.372 | 0.408 | 0.437 | +0.021 |  |
| Weather | 96 | 0.153 | 0.143 | 0.172 | 0.174 | 0.151 | 0.159 | 0.172 | 0.246 | -0.010 |
| 192 | 0.194 | 0.186 | 0.215 | 0.217 | 0.195 | 0.202 | 0.219 | 0.292 | -0.008 |  |
| 336 | 0.241 | 0.236 | 0.263 | 0.262 | 0.249 | 0.281 | 0.280 | 0.378 | -0.005 |  |
| 720 | 0.302 | 0.307 | 0.318 | 0.332 | 0.321 | 0.335 | 0.365 | 0.447 | +0.005 |  |
| Electricity | 96 | 0.131 | 0.134 | 0.138 | 0.140 | 0.129 | 0.158 | 0.168 | 0.188 | -0.002 |
| 192 | 0.146 | 0.149 | 0.151 | 0.153 | 0.149 | 0.174 | 0.184 | 0.197 | +0.003 |  |
| 336 | 0.162 | 0.165 | 0.166 | 0.169 | 0.166 | 0.190 | 0.198 | 0.212 | +0.003 |  |
| 720 | 0.199 | 0.203 | 0.205 | 0.204 | 0.210 | 0.229 | 0.220 | 0.244 | +0.004 |  |
| Traffic | 96 | 0.381 | 0.385 | 0.389 | 0.413 | 0.366 | 0.380 | 0.593 | 0.573 | -0.015 |
| 192 | 0.394 | 0.397 | 0.398 | 0.423 | 0.388 | 0.397 | 0.617 | 0.611 | -0.006 |  |
| 336 | 0.408 | 0.410 | 0.411 | 0.437 | 0.398 | 0.418 | 0.629 | 0.621 | -0.010 |  |
| 720 | 0.446 | 0.448 | 0.448 | 0.466 | 0.457 | 0.436 | 0.640 | 0.630 | -0.010 |  |

The experimental results offer several key insights into MMFNet’s performance across a range of datasets and forecast horizons. As Table 1 shows, MMFNet demonstrates superior performance on the ETT dataset and consistently achieves the best results even at extended forecasting horizons. Additionally, it maintains strong performance across a range of channel numbers and sampling rates.

#### Performance on the ETT Dataset.

As Table 1 shows, MMFNet consistently outperforms other models across all forecast horizons on the ETTh1, ETTh2, and ETTm2 datasets.
For example, on ETTh1, compared with other baseline models, MMFNet achieves the best MSE results of $0.359$, $0.396$, $0.409$, and $0.419$ at forecast horizons of $96$, $192$, $336$, and $720$, respectively. Moreover, it demonstrates a $4.2\%$ MSE reduction (+0.018) at the forecast horizon of $336$ on ETTh1 and a $5.1\%$ MSE reduction (+0.018) at the forecast horizon of $336$ on ETTh2. This consistent performance highlights MMFNet’s ability to effectively capture both short-term fluctuations and long-term dependencies in time series data, positioning it as a versatile model for a wide variety of LTSF tasks.

#### Performance at the Extended Horizon.

As Table 1 shows, at the extended forecast horizon of $720$, MMFNet consistently achieves the highest predictive accuracy across all datasets, except for Traffic where it ranks second. Notably, MMFNet demonstrates significant improvements over baseline models, achieving MSE reductions of $4.6\%$ ($+0.019$) on ETTm1 and $6.0\%$ ($+0.021$) on ETTm2 at forecast horizon $720$ compared to the second-best models. These results highlight the robustness of MMFNet in addressing long-term forecasting tasks.

#### Performance in Low-Channel, Low-Sampling Rate Scenarios.

As Table 1 shows, in scenarios involving datasets with fewer channels (7 channels) and lower sampling rates (1-hour intervals), such as in the ETTh1 and ETTh2 datasets, linear models like FITS, SparseTSF, and DLinear exhibit strong performance. For example, on ETTh2, FITS achieves the MSE results of $0.271$, $0.331$, $0.354$, and $0.377$ at forecast horizons of $96$, $192$, $336$, and $720$, respectively. MMFNet continues to surpass these models on ETTh2 by achieving the MSE results of $0.263$, $0.317$, $0.336$, and $0.376$ at forecast horizons of $96$, $192$, $336$, and $720$, respectively. This suggests that multi-scale frequency decomposition methods are particularly well-suited for datasets with fewer channels and broader time intervals between measurements.

#### Performance in High-Channel Scenarios.

As Table 1 shows, for datasets with larger numbers of channels, such as Electricity ($321$ channels, $1$-hour sampling rate) and Traffic ($862$ channels, $1$-hour sampling rate), MMFNet and FITS consistently demonstrate strong performance. Despite the increased complexity that arises from higher channel counts.
For example, on Electricity, MMFnet achieves the best MSE results of $0.146$, $0.162$, and $0.199$ at forecast horizons of $192$, $336$, and $720$, respectively.
MMFNet’s multi-scale frequency decomposition enables it to effectively model complex temporal dependencies while maintaining high predictive accuracy. While PatchTST performs better on the traffic dataset, it leverages a patching transformer mechanism rather than a purely linear frequency-based approach, distinguishing it from MMFNet and FITS in terms of the model architecture. This further indicates that more sophisticated decomposition methods are required for lightweight models to handle high-channel scenarios effectively.

#### Performance in High-Sampling Rate Scenarios.

As Table 1 shows, for datasets with higher sampling rates, such as Weather ($21$ channels, $10$-minute sampling rate), ETTm1 and ETTm2 ($7$ channels, $15$-minute sampling rate), MMFNet and FITS consistently demonstrate strong performance. For example, on ETTm2, MMFnet achieves the best MSE results of $0.160$, $0.212$, $0.259$, and $0.327$ at forecast horizons of $96$, $192$, $336$, and $720$, respectively. Despite the increased complexity that arises from a faster sampling rate, MMFNet’s multi-scale frequency decomposition enables it to effectively model complex temporal dependencies while maintaining high predictive accuracy.

### 4.3 Comparisons between MMFT and SFT

**Table 2: MSE values of MMFNet when it uses SFT and MMFT on the ETT dataset. SFT denotes the standard single-scale frequency decomposition approach. MFT refers to the masked frequency transformation with fragmentation applied at a single scale, where $N_{seg}$ specifies the segment length. MMFT denotes the full MMFT method, which performs frequency decomposition with multi-scale fragmentation. “Imp.” indicates the improvement of MMFT over SFT.**
| Dataset |  | ETTh1 | ETTh2 |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Horizon |  | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 |
| SFT |  | 0.372 | 0.404 | 0.427 | 0.424 | 0.271 | 0.331 | 0.354 | 0.377 |
| MFT ($N_{seg}=24$) |  | 0.362 | 0.400 | 0.412 | 0.421 | 0.264 | 0.317 | 0.336 | 0.376 |
| MFT ($N_{seg}=120$) |  | 0.366 | 0.401 | 0.426 | 0.423 | 0.265 | 0.317 | 0.336 | 0.376 |
| MFT ($N_{seg}=360$) |  | 0.366 | 0.403 | 0.418 | 0.425 | 0.265 | 0.317 | 0.340 | 0.376 |
| MMFT |  | 0.359 | 0.396 | 0.409 | 0.419 | 0.263 | 0.317 | 0.336 | 0.376 |
| Imp.(MMFT over SFT) |  | +0.013 | +0.008 | +0.018 | +0.005 | +0.008 | +0.014 | +0.018 | +0.001 |

To evaluate the effectiveness of the MMFT method (see Section 3.1), we perform experiments using the ETT dataset. Both SFT and MMFT incorporate the same adaptive masking strategy to ensure fair and consistent comparisons. SFT applies FFT to the entire time series without fragmentation, while MFT introduces a single-scale fragmentation, and MMFT performs a multi-scale fragmentation. The results presented in Table 2 reveal two important insights.

First, fragmentation consistently enhances frequency domain decomposition. On the ETTh1 dataset, MFT ($N_{seg}=360$) achives the MSE results of $0.160$, $0.212$, $0.259$, and $0.327$ at forecast horizons of $96$, $192$, $336$, and $720$, respectively. MFT delivers the most significant gains observed at a segment length of $24$ with a $4.2\%$ MSE reduction (+0.018) at then forecast horizon of $336$. This improvement suggests that segmenting the time series into smaller segments enables MFT to capture localized frequency features more effectively.

Second, MMFT, leveraging multi-scale decomposition, consistently delivers superior results compared to both SFT and single-scale MFT.
On the ETTh2 dataset, MMFT achives the MSE results of $0.263$, $0.317$, $0.336$, and $0.376$ at forecast horizons of $96$, $192$, $336$, and $720$, respectively. At the forecast horizon of $336$, MMFT achieves substantial reductions in MSE, including a $0.018$ improvement over SFT. These results suggest that the multi-scale decomposition employed by MMFT allows for the capture of a broader range of frequency patterns, leading to more accurate predictions, particularly in long-term forecasting scenarios.

### 4.4 Effectiveness of Masking

**Table 3: MSE results for multivariate LTSF with MMFNet on the ETT dataset with or without the masking module. “Mask” refers to results with the masking module, while “w/o Mask” refers to results without it. “Imp.” denotes the improvement enabled by the masking module.**
| Dataset |  | ETTh1 | ETTh2 | Electricity | Traffic |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Horizon |  | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 |
| w/o Mask |  | 0.372 | 0.405 | 0.410 | 0.420 | 0.269 | 0.319 | 0.339 | 0.376 | 0.312 | 0.338 | 0.360 | 0.397 | 0.166 | 0.218 | 0.264 | 0.330 |
| Mask |  | 0.359 | 0.396 | 0.409 | 0.419 | 0.263 | 0.317 | 0.336 | 0.376 | 0.307 | 0.334 | 0.358 | 0.396 | 0.160 | 0.212 | 0.259 | 0.327 |
| Imp. |  | +0.013 | +0.009 | +0.001 | +0.001 | +0.006 | +0.002 | +0.003 | +0.000 | +0.005 | +0.003 | +0.002 | +0.001 | +0.006 | +0.006 | +0.005 | +0.003 |

To evaluate the effectiveness of the self-adaptive masking mechanism, we compare MMFNet’s performance on the ETT dataset with and without the masking module across four forecast horizons: $96$, $192$, $336$, and $720$.
As Table 3 lists, MMFNet with masking consistently outperforms the version without masking across all horizons. The most notable improvements occur at the horizon $96$ with a $3.5\%$ MSE reduction on ETTh1 (+0.013) and a $2.2\%$ MSE reduction on ETTh2 (+0.006). With the Electricity dataset, the largest improvement is at horizon $96$ with an improvement of $+0.005$. Similarly, the largest improvement is at horizon $192$ with an improvement of $+0.006$ on the Traffic dataset. The results show that the self-adaptive masking mechanism which filters out frequency noise at different scales consistently enhances forecasting accuracy across various datasets and forecast horizons. A more detailed analysis of the mask output is provided in Appendix C.

### 4.5 Performance on Ultra-long-term Time Series Forecasting

**Table 4: MSE results for multivariate ultra long-term time series forecasting with MMFNet. The best result is emphasized in bold, while the second-best is underlined. “Imp.” represents the improvement between MMFNet and either the best or second-best result, with a higher “Imp.” value indicating greater improvement.**
| Dataset |  | ETTm1 | ETTm2 | Electricity | Weather |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Horizon |  | 960 | 1200 | 1440 | 1680 | 960 | 1200 | 1440 | 1680 | 960 | 1200 | 1440 | 1680 | 960 | 1200 | 1440 | 1680 |
| DLinear |  | 0.429 | 0.440 | 0.463 | 0.481 | 0.412 | 0.398 | 0.430 | 0.478 | 0.238 | 0.267 | 0.277 | 0.296 | 0.330 | 0.341 | 0.345 | 0.356 |
| FITS |  | 0.413 | 0.422 | 0.425 | 0.427 | 0.347 | 0.358 | 0.355 | 0.350 | 0.238 | 0.268 | 0.293 | 0.311 | 0.333 | 0.343 | 0.353 | 0.360 |
| SparseTSF |  | 0.415 | 0.422 | 0.424 | 0.425 | 0.353 | 0.367 | 0.357 | 0.353 | 0.228 | 0.256 | 0.281 | 0.298 | 0.329 | 0.339 | 0.347 | 0.353 |
| MMFNet(ours) |  | 0.411 | 0.419 | 0.423 | 0.424 | 0.346 | 0.357 | 0.356 | 0.349 | 0.224 | 0.255 | 0.280 | 0.292 | 0.318 | 0.331 | 0.340 | 0.349 |
| Imp. |  | +0.002 | +0.003 | +0.001 | +0.001 | +0.001 | +0.001 | -0.001 | +0.001 | +0.004 | +0.001 | +0.001 | +0.004 | +0.011 | +0.008 | +0.005 | +0.004 |

We evaluate MMFNet’s performance in ultra-long-term time series forecasting scenarios. Table 4 presents the MSE results for various models applied to multivariate ultra-long-term time series forecasting across four datasets at forecast horizons of $960$, $1200$, $1440$, and $1680$. Due to the significant memory requirements of models such as FEDformer, TimesNet, TimeMixer, and PatchTST when forecast horizons are extended, these models exceed GPU memory limitations. Consequently, in this context, we limit the comparison to more lightweight models: DLinear, FITS, SparseTSF, and the proposed MMFNet.

The results show that MMFNet consistently outperforms the existing models across most datasets and forecast horizons. For example, with the ETTh1 dataset, MMFNet achieves the MSE values of $0.411$, $0.419$, $0.423$, and $0.424$ at horizons of $960$, $1200$, $1440$, and $1680$, respectively. With the Electricity dataset, MMFNet delivers very good performance, particularly at longer horizons, with the MSE values of $0.255$ at $1200$ and $0.292$ at $1680$.On the Weather dataset, MMFNet demonstrates superior performance, achieving MSE values of $0.318$ at the $960$ horizon and $0.331$ at the $1200$ horizon, representing a $3.3\%$ ($+0.011$) and $2.4\%$ ($+0.008$) reduction in MSE compared to the second-best baseline. The results demonstrate the robustness of MMFNet in forecasting multivariate ultra-long-term time series data across various datasets and extended forecast horizons by effectively capturing frequency variations at different scales.

## 5 Related Work

### 5.1 Long-term Time Series Forecasting

LTSF is a critical area in data science and machine learning and focuses on predicting future values over extended periods. Such a task is challenging due to the inherent seasonality, trends, and noise in time series data. In addition, time series data is often complex and high-dimensional Zheng et al. (; ). Traditional statistical methods, such as ARIMA (Contreras et al., ) and Holt-Winters (Chatfield & Yar, ), are effective for short-term forecasting but frequently fall short for longer horizons. Machine learning models, such as SVM (Wang & Hu, ), Random Forests Breiman (), and Gradient Boosting Machines (Natekin & Knoll, ), offer improved performance by capturing non-linear relationships but typically require extensive feature engineering. Recently, deep learning models, such as RNNs, LSTMs, GRUs, and Transformer-based models (Informer and Autoformer), have demonstrated notable efficiency in modeling long-term dependencies. Furthermore, the hybrid models that combine statistical methods with machine learning or deep learning techniques have shown improved accuracy. State-of-the-art models, such as FEDformer (Zhou et al., ), FiLM (Zhou et al., ), PatchTST Nie et al. (), and SparseTSF, leverage frequency domain transformations and efficient self-attention to improve prediction performance.

### 5.2 Time Series Forecasting in the Frequency Domain

Recent advancements in time series analysis have increasingly utilized frequency domain information to reveal underlying patterns. For instance, FNet (Lee-Thorp et al., ) adopts an attention-based approach to capture temporal dependencies within the frequency domain, thereby eliminating the need for convolutional or recurrent layers. Models such as FEDformer (Zhou et al., ) and FiLM (Zhou et al., ) improve predictive performance by incorporating frequency domain information as auxiliary features. FITS (Xu et al., ) also demonstrates strong predictive capabilities by converting time-domain forecasting tasks into the frequency domain and utilizing low-pass filters to reduce the number of parameters required. However, many of these techniques rely on manual feature engineering to identify dominant periods, which can constrain the amount of information captured and introduce inefficiencies or risks of overfitting.

### 5.3 Multiscaling Model

In the field of computer vision, several multi-scale Vision Transformers (ViTs) have leveraged hierarchical architectures to generate progressively down-sampled pyramid features. For instance, Multi-Scale Vision Transformers (Fan et al., ) enhance the standard Vision Transformer architecture by incorporating multi-scale processing, allowing for improved detail capture across varying spatial resolutions. Pyramid Vision Transformer (Wang et al., ) integrates a pyramid structure within ViTs to facilitate multi-scale feature extraction, while Twins (Dai et al., ) combines local and global attention to effectively model multi-scale representations. SegFormer (Xie et al., ) introduces an efficient hierarchical encoder that captures both coarse and fine features, and CSWin (Dong et al., ) further improves performance by using multi-scale cross-shaped local attention mechanisms.
In the context of time series forecasting, TimeMixer (Wang et al., ) represents a significant advancement with its fully MLP-based architecture, which employs Past-Decomposable-Mixing and Future-Multipredictor-Mixing blocks. This architecture enables TimeMixer to effectively leverage disentangled multi-scale time series data during both past extraction and future prediction phases.

### 5.4 Masked Modeling

Masked language modeling and its autoregressive variants have emerged as dominant self-supervised learning approaches in natural language processing. These techniques enable large-scale language models to excel in both language understanding and generation by predicting masked or hidden tokens within sentences (Devlin et al., ; Radford et al., ). In computer vision, early approaches, such as the context encoder (Pathak et al., ), involve masking specific regions of an image and predicting the missing pixels, while Contrastive Predictive Coding (van den Oord et al., ) uses contrastive learning to improve feature representations. Recent innovations in MIM include models like iGPT (Chen et al., ), ViT (Dosovitskiy et al., ), and BEiT (Bao et al., ), which leverage Vision Transformers and techniques, such as pixel clustering, mean color prediction, and block-wise masking. In the realm of multivariate time series forecasting, masked encoders have recently been employed with notable success in classification and regression tasks (Zerveas et al., ). For example, PatchTST uses a masked self-supervised representation learning method to reconstruct the masked patches and showcases its effectiveness in time series data (Nie et al., ). However, the application of masked modeling techniques in linear time series forecasting remains relatively under-explored.

## 6 Conclusion

MMFNet significantly advances long-term multivariate forecasting by employing the MMFT approach. Through comprehensive evaluations on benchmark datasets, we have demonstrated that MMFNet consistently outperforms state-of-the-art models in forecasting accuracy, highlighting its robustness in capturing complex data patterns. By effectively integrating multi-scale decomposition with a learnable masked filter, MMFNet captures intricate temporal details while adaptively mitigating noise, making it a versatile and reliable solution for a wide range of LTSF tasks.

## Appendix A More on MMFNet

### A.1 Overall Workflow

The overall workflow of MMFNet is presented in Algorithm 1. The algorithm takes a univariate historical look-back window as input, $x_{t-L+1:t}$, and produces the corresponding forecast, $\hat{x}_{t+1:t+H}$. By incorporating the channel-independent strategy, in which multiple channels are modeled using a shared set of parameters, MMFNet can efficiently extend to multivariate time series forecasting tasks. Such an approach enables the model to leverage its multi-scale frequency decomposition and adaptive masking framework across various input channels to enhance its predictive capabilities in complex multivariate settings.

Figure: Algorithm 1 Overall Pseudocode of MMFNet

### A.2 Detailed Dataset Description

**Table 5: Statistics of the datasets.**
| Dataset | Traffic | Electricity | Weather | ETTh1 | ETTh2 | ETTm1 | ETTm2 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Channels | 862 | 321 | 21 | 7 | 7 | 7 | 7 |
| Sampling Rate | 1 hour | 1 hour | 10 min | 1 hour | 1 hour | 15 min | 15 min |
| Total Timesteps | 17,544 | 26,304 | 52,696 | 17,420 | 17,420 | 69,680 | 69,680 |

Here is a brief description of the datasets used in our experiments.

- •
The ETT dataset(^1^11https://github.com/zhouhaoyi/ETDataset) comprises data originally collected for Informer (Zhou et al., ), including load and oil temperature measurements recorded at $15$-minute intervals between July 2016 and July 2018. The ETTh1 and ETTh2 subsets are sampled at $1$-hour intervals, while ETTm1 and ETTm2 are sampled at $15$-minute intervals.
- •
The Electricity dataset(^2^22https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014) contains hourly electricity consumption data for $321$ customers from 2012 to 2014.
- •
The Traffic dataset(^3^33http://pems.dot.ca.gov) consists of hourly road occupancy rates, collected by various sensors deployed on freeways in the San Francisco Bay area, sourced from the California Department of Transportation.
- •
The Weather dataset(^4^44https://www.bgc-jena.mpg.de/wetter/) includes local climatological data from nearly $1,600$ locations across the United States, covering a period of four years (2010 to 2013), with data points recorded at $1$-hour intervals.

### A.3 Baseline Models

Here is a brief description of the baseline models used in this paper.

- •
FEDformer (Zhou et al., ) is a Transformer-based model proposing seasonal-trend decomposition and
exploiting the sparsity of time series in the frequency domain. The source code is available at [https://github.com/DAMO-DI-ML/ICML2022-FEDformer](https://github.com/DAMO-DI-ML/ICML2022-FEDformer).
- •
TimesNet (Wu et al., ) is a CNN-based model with TimesBlock as a task-general backbone. It transforms 1D time series into 2D tensors to capture intraperiod and interperiod variations. The source code is available at [https://github.com/thuml/TimesNet](https://github.com/thuml/TimesNet).
- •
TimeMixer (Wang et al., ) is a fully MLP-based architecture with PDM and FMM blocks to take full advantage of disentangled multiscale series in both past extraction and future prediction phases. The source code is available at [https://github.com/kwuking/TimeMixer](https://github.com/kwuking/TimeMixer).
- •
PatchTST (Nie et al., ) is a transformer-based model utilizing patching and CI technique. It also enables effective pre-training and transfer learning across datasets. The source code is available at [https://github.com/yuqinie98/PatchTST](https://github.com/yuqinie98/PatchTST).
- •
DLinear (Zeng et al., ) is an MLP-based model with just one linear layer, which outperforms Transformer-based models in LTSF tasks. The source code is available at [https://github.com/cure-lab/LTSF-Linear](https://github.com/cure-lab/LTSF-Linear).
- •
FITS (Xu et al., ) is a linear model that manipulates time series data through interpolation in the complex frequency domain. The source code is available at [https://github.com/VEWOXIC/FITS](https://github.com/VEWOXIC/FITS).
- •
SparseTSF (Lin et al., ) a novel, extremely lightweight model for LTSF, designed to address the challenges of modeling complex temporal dependencies over extended horizons with minimal computational resources. The source code is available at [https://github.com/lss-1138/SparseTSF](https://github.com/lss-1138/SparseTSF).

## Appendix B Advantages of MMFT

MMFT leverages a multi-scale approach to address the limitations of SFT. By operating across multiple scales, MMFT offers several key features:

- •
Good Adaptability to Non-Stationarity. Non-stationarity in time series, where statistical properties such as trends or seasonality evolve over time, presents a challenge for SFT, which assumes stationarity. MMFT mitigates this limitation by decomposing the time series into multiple frequency components, each of which captures specific temporal patterns (e.g., short-term fluctuations or long-term trends). By adapting to non-stationary characteristics that SFT may overlook, MMFT effectively reduces bias. For instance, in financial datasets with shifting trends, MMFT can simultaneously analyze long-term patterns and short-term variations, enhancing predictive performance.
- •
Effectively Capturing of Local and Global Patterns. Time series data often contain both short-term (local) and long-term (global) patterns. SFT’s reliance on a single global scale may fail to capture local variations, leading to higher prediction variance. MMFT addresses this issue by operating at multiple scales, allowing it to capture local patterns at finer resolutions and global trends at coarser ones. This enables MMFT to better adapt to the varying characteristics of the data, reducing both overfitting to local noise and underfitting of broader trends. For example, MMFT can effectively model daily temperature fluctuations alongside longer seasonal cycles, leading to improved predictive accuracy across different time horizons.
- •
Employing Learnable Frequency Masks. MMFT introduces learnable frequency masks that selectively filter out irrelevant frequency components while retaining the key frequencies necessary for accurate prediction. Unlike the static filters used in SFT, these masks are optimized during training, enabling the model to focus on informative frequency components while discarding noise. This adaptive filtering process reduces both bias and variance, further enhancing model performance.

## Appendix C More Analysis on Mask Output

Figure: Fine-scale
Refer to caption: x2.png

Figure: Fine-scale
Refer to caption: x5.png

To analyze the learned masks at different scales, we visualize the mask outputs. Figures 2 and 3 illustrate the mask outputs for various frequency decompositions applied to the ETTh1 and ETTh2 datasets. These figures reveal a consistent pattern across time segments and show that the masks predominantly target high-frequency components with larger mask values indicating more aggressive attenuation of these components compared to lower-frequency ones.

A more detailed examination shows that the degree of frequency attenuation varies across time segments and scales. Specifically, in the fine-scale and intermediate-scale scenarios, high-frequency components exhibit a greater attenuation ratio in the earlier time segments. This observation suggests that at finer temporal resolutions, earlier time points experience a more significant reduction in high-frequency information. In contrast, for the coarse-scale scenario, the most recent time segments display a higher attenuation ratio for high-frequency components.

This pattern suggests that, as the temporal resolution increases, the model increasingly focuses on masking high-frequency components more aggressively in earlier time segments. Conversely, at coarser scales, more recent data points are more heavily filtered. This behavior likely reflects the model’s adaptive strategy to prioritize different temporal patterns or noise levels depending on the scale of the analysis. The variation in attenuation ratios across scales and time segments indicates a nuanced approach to frequency masking, which may optimize the model’s performance by selectively emphasizing or de-emphasizing specific temporal features based on their relevance at each scale.

The consistency of the Mask outputs across these scales suggests that the frequency decomposition method is both robust and effective in isolating different aspects of the time series data. Fine-scale outputs are particularly useful for identifying rapid fluctuations and short-term patterns, while coarse-scale outputs are essential for understanding broader trends and long-term behavior in the data. This multi-scale approach is highly beneficial for time series forecasting, as it allows the model to leverage both the fine details and the overarching trends, leading to more accurate and comprehensive predictions.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 3
presentation: 3
questions: In conclusion, while the paper proposes a novel approach to long-term time series forecasting through multi-scale frequency decomposition, the necessity of this complex approach is not fully justified. Although the motivation to capture both short-term fluctuations and long-term trends is valid, the rationale behind key design choices, such as the adaptive masking mechanism, is underexplained. Furthermore, the model’s empirical gains over existing baselines are modest and inconsistent across datasets, reducing the impact of the experimental findings. Nonetheless, the paper is well-organized, and the architecture shows potential, especially if the authors can clarify the interpretability of the frequency masking mechanism and strengthen the model's practical applicability through more targeted benchmarks.
soundness: 2
strengths: - The paper is well-organized, with a logical flow from the introduction to the proposed method and experiments
- The motivation section highlights the limitations of current LTSF models in capturing both short-term fluctuations and long-term trends, emphasizing the need for a model that can balance these aspects. By framing MMFNet as a solution to this gap, the paper sets a compelling context for its multi-scale approach.
- The experiments cover a range of popular LTSF datasets (e.g. ETT, Weather, Traffic, and Electricity), and inclusion of both high-channel and low-channel datasets helps validate the model’s versatility in different settings.
summary: This paper introduces MMFNet, a model aimed at enhancing long-term time series forecasting (LTSF) by employing a multi-scale frequency decomposition technique. Addressing limitations of existing models that assume stationarity and filter out high-frequency components—thus potentially overlooking short-term fluctuations—MMFNet captures both short-term variations and long-term trends by decomposing time series data into fine, intermediate, and coarse segments. These segments are selectively filtered through a learnable mask to focus on relevant frequencies.
weaknesses: Despite the performance gains across multiple datasets, the motivation is not fully substantiated: 

- The rationale for multi-scale frequency decomposition lacks empirical grounding, as the manuscript does not convincingly demonstrate why a single-scale approach falls short. While the authors reference assumptions of stationarity, they do not provide empirical evidence or citations that highlight the limitations of single-scale models in specific LTSF scenarios.

Additionally, several design choices require further explanation:

- The learnable mask's role in frequency filtering is not well-clarified in terms of its operational benefits over traditional fixed-frequency filters. While described as "self-adaptive," the paper does not detail how the mask parameters are optimized or discuss their consistent effectiveness across datasets. For instance, the appendix includes some masking results, but without analysis on how these masks contribute to the model’s forecasting accuracy.

Improving the experimental design to highlight robustness in non-stationary settings would enhance the study's validity:

- Although MMFNet achieves moderate performance improvements on benchmark datasets, its robustness to non-stationary data remains unclear. The experiments do not include clearly non-stationary datasets or demonstrate superior performance under such conditions. The chosen datasets, like ETTh1 and Electricity, mainly display seasonal patterns, making it uncertain if MMFNet is truly resilient to non-stationary series.

Lastly, computational analysis and comparisons to related works [1][2] would be beneficial:

- The multi-scale decomposition and frequency masking likely introduce computational overhead, but the paper does not quantify these costs or evaluate MMFNet’s efficiency in practical applications. A detailed comparison of MMFNet’s computational complexity against simpler models would clarify the trade-offs between overhead and performance. Additionally, recent relevant baselines, such as [2], have not been addressed in the manuscript.

[1] FITS: Modeling Time Series with 10k Parameters (ICLR 2024)

[2] DAM: TOWARDS A FOUNDATION MODEL FOR TIME SERIES FORECASTING (ICLR 2024)

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 3
questions: 1. How does the proposed Multi-scale Frequency Decomposition specifically address non-stationarity?
Further explanation of how multi-scale decomposition effectively handles non-stationary characteristics would help clarify this point.

2. What are the limitations of the multi-scale approach, and are there any suggested improvements?
Every multi-scale approach may have limitations, such as increased noise or computational complexity. It would be valuable to discuss any potential drawbacks of this method and ways to mitigate them.

3. How does the proposed method compare in terms of computational cost or time complexity to existing models?
Providing a comparison of computational efficiency would help readers understand the practicality of the proposed model.

4. Are there experimental results on the National Illness dataset?
Results on such datasets would help demonstrate the relevance and applicability of the method, particularly for health-related time series data.
soundness: 3
strengths: 1. Clear Identification of Limitations: The paper effectively identifies the limitations of existing methods like FITS, particularly in handling non-stationary data and capturing short-term fluctuations. This acknowledgment establishes a strong rationale and relevance for the proposed research.
2. Well-Organized Presentation: The methodology is well-organized and explained step-by-step, making the approach accessible and easier for readers to understand.
summary: MMFNet is a predictive model designed for non-stationary time series, addressing the limitations of Single-scale Frequency Decomposition methods like FITS, which struggle to capture non-stationary patterns and may miss short-term fluctuations. To overcome these limitations, MMFNet introduces a Multi-scale Frequency Decomposition approach. Additionally, it incorporates a learnable mask that selectively filters out irrelevant frequency components at each scale, thereby enhancing prediction accuracy and effectively capturing both short-term variations and long-term trends.
weaknesses: 1. Unclear Motivation: The paper aims to analyze time series data across multiple scales, an approach that has also been explored by methods like TimeMixer, PITS, and FITS. However, it lacks a clear motivation for why this particular approach is necessary and does not sufficiently explain how MMFNet differs from or improves upon these existing methods. It would help if the authors more explicitly justified their approach by highlighting the specific advantages or unique aspects of MMFNet that set it apart from similar multi-scale methods.

2. Limited Novelty: The proposed method feels more like a variation on established approaches rather than a fundamentally new concept. Both multi-scale analysis and frequency masking techniques have been explored previously in time series research, making it challenging to see where MMFNet offers substantial innovation. The paper could benefit from a clearer articulation of its novel aspects, such as whether MMFNet provides specific advancements in frequency decomposition or interpretability that distinguish it from other multi-scale approaches.

3. Insufficient Comparative Experiments with Existing Models: The paper lacks enough experimental comparisons with other prominent models (e.g., TimeMixer) to effectively demonstrate any performance advantages or unique capabilities of MMFNet. Direct comparisons on standard datasets with similar metrics could better showcase the proposed model’s strengths and allow readers to more easily assess its advantages over existing approaches.

4. Limited Persuasiveness Due to Lack of Novelty: Given the number of studies on multi-scale time series analysis, the paper does not provide sufficient differentiation to persuade readers of MMFNet's unique contributions. Expanding the explanation of MMFNet’s specific innovations and benefits could strengthen its positioning as a valuable addition to the existing research.

---

confidence: 4
contribution: 1
initial_rating: 3
presentation: 2
questions: More discussions: 
- FITS can handle the anomaly detection task, does MMFNet also have such capabilities? Since the MMFNet adopts a masking and interpolation approach, it seems more suitable for reconstruction tasks?
- Is the number of model parameters large? From my understanding, each segment has its own learnable mask and linear layer. Consequently, it appears that increasing the number of segments could lead to a significant expansion in the model's parameter count.
soundness: 2
strengths: - The paper is well-organized and highly readable, and the figures makes it easy to understand the model designs.
- Authors provide ablation analysis to demonstrate the effectiveness of the proposed techniques, i.e,  the multi-scale frequency decomposition and learnable masking.
- I especially appreciated the exploration of  frequency domain for time series modeling, and the experiments show its promising performance.
summary: This paper presents MMFNet, a multi-scale frequency masking model for long-term time series forecasting. Unlike most existing models that operate in the time domain, MMFNet follows the light-weight model, FITS (ICLR’24), to learn and capture the dynamic variations in the frequency domain. Compared with FITS, MMFNet has made the following improvements: (1) the multi-scale frequency decomposition to effectively capture both local and global patterns in data; (2) the learnable frequency mask to focus on the most informative frequency components. MMFNet achieves competitive performance to SOTA baselines on several widely-used long-term time series forecasting (LTSF) datasets.
weaknesses: - MMFNet seems to be an incremental improvement over FITS, and it appears that the motivation and claim are not well-supported. Specifically, MMFNet may not adequately address the“non-stationary characteristics that SFT may "overlook". Since learnable masks are trained and shared across the whole time series dataset, different time series will be masked out in the same way (via the shared masks).  Therefore, the learnable mask is not so “adaptive”.

- The paper lacks some details. For methodology, it is unclear how to interpolate $X_{DCT}$ of various-length segments and merge them together, some formulas are needed. For experiment settings, the segment num of different temporal scales and the look-back window lengths for various forecasting horizons are not mentioned. I am especially curious about input window length for ultra-long-term forecasting. 

- The experimental evaluation seems insufficient. For ablation study, in section 4.3, the impact of different segment num combinations on MMFT should be explored; in section 4.4, the results comparing “Mask” vs. “low pass filter” should be provided to support the paper’s claim. Additionally, I would greatly appreciate it if more comprehensive comparisons could be included in the paper. For examples, the results on different look-back window lengths and visualization of prediction results.

---

confidence: 4
contribution: 3
initial_rating: 5
presentation: 3
questions: Questions:
1.What is the length of the historical look-back window used in the experiments reported in Table 1? Were the baselines’ results in Table 1 obtained from other literature or tested by the authors?
2.Can the Multi-Scale Frequency Transformation and Masked Frequency Interpolation methods proposed in the paper be used as plug-and-play components with other time series models?
soundness: 2
strengths: Strengths: 
1. MMFNet employs multi-scale frequency domain decomposition, exhibiting a certain degree of innovation.
2.MMFNet achieves good performance in a variety of multivariate time series forecasting tasks.
3.The paper is largely understandable.
summary: Summary: 
This paper introduces MMFNet, a novel model designed to enhance long-term multivariate forecasting by leveraging a multi-scale masked frequency decomposition approach. Experimentation with benchmark datasets shows that MMFNet achieves good performance.
weaknesses: Weaknesses:
1.The motivation behind the paper is not clearly articulated. The author mentions: “While single-scale frequency domain decomposition offers a global perspective of time series data in the frequency domain, it lacks the ability to localize specific frequency components within the sequence.” This statement requires further explanation—why is it unable to localize specific frequencies? Similarly, the author needs to further elaborate on why MMFNet can handle non-stationary series (along with detailed examples). These are key points and contributions of the paper.
2.Using only the MSE metric is insufficient. Does the model show similar performance when evaluated using other metrics such as MAE?
3.The visualization work in the paper is not particularly sufficient. The authors should include more comparative figures between Single-Scale Frequency Transformation and Multi-Scale Frequency Transformation, as well as visualizations of the mask effects, and prediction showcases of different models.
4.The authors should compare the models across more dimensions, such as parameter count, training and inference time.
5.The baseline models are not very rich; the current research primarily focuses on analysis at the temporal scale. Recently, there has been some work exploring the variable perspective, yielding good results, such as with the iTransformer and Client models. It is recommended to include these comparisons as well.
6.The authors claim that "Transformer-based architectures have demonstrated exceptional capacity in capturing complex temporal patterns by effectively modeling long-range dependencies through self-attention mechanisms at the cost of heavy computational workload, particularly when facing large-scale time series data, which significantly limits their practicality in real-time applications." This claim is not comprehensive; a more significant limitation of Transformer-based models is their tendency to overfit.

---

confidence: 5
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. In the paper,  three specific scales were designed. Why were these particular sizes (2, 24, 720) chosen? Are there other choices for small and medium scales (such as 4, 48)? Is the model's performance sensitive to the choice of these scales?
2. In Figures 2 and 3 of the appendix, the actual output of the mask is shown, but the shape of the mask seems inconsistent with what was described in the text. The segment lengths for fine-scale and intermediate-scale are 2 and 24, respectively, so the mask lengths should correspond to 2 and 24. However, the mask length in the figure seems to be 720. Was there an intermediate step involved? Can the authors clarify this?
3. The mask output seems to exhibit certain low-pass characteristics at fine-scale and intermediate-scale, while showing high-pass characteristics at coarse-scale. Can you further explain this phenomenon?
4. Can you provide some visualizations of the features learned at different scales? For example, the weights of the linear interpolation layer corresponding to different scales and the final output, etc.
5. During the inversion step, the 3 output were first applied with the inverse DCT, then summed up. Is there any reason for this? Because from a theoretical perspective, the inverse DCT is a linear operation, so the sum of the inverse DCT of the 3 outputs is equivalent to the inverse DCT of the sum of the 3 outputs. By swapping the order of the sum and the inverse DCT, the computational complexity can be greatly reduced. Can you explain this design choice?
soundness: 3
strengths: 1. The model design appears very simple and achieves good results, surpassing existing mainstream models.
2. The authors validate the effectiveness of multi-scale frequency domain feature extraction and the masking mechanism on the model's performance through a series of visual experiments and ablation studies, providing some explanations for these.
3. The experiments on extending the horizon sufficiently show that the model has excellent long-term sequence prediction capabilities.
summary: This paper introduces a simple time-series prediction model, MMFNet, based on multi-scale feature fusion using DCT combined with a masking mechanism. The model first divides the input sequence into three different scales, applies DCT transformation to each scale, and obtains frequency domain features at different scales. Then, adaptive filtering is applied to these frequency domain features using a masking operation. A simple linear interpolation layer is used to obtain predictions at each scale, and the final prediction is obtained by applying the inverse DCT and summing these results. The authors validate the model's effectiveness through a series of experiments and demonstrate the importance of the masking mechanism and multi-scale feature fusion to the model's performance through ablation studies.
weaknesses: 1. The authors did not explain why DCT was chosen rather than other transformations, such as the more common Fourier transform. What's the advantage of DCT? Are there other transformation methods that could be used instead?
2. The authors argued that this is the first model to use multi-scale frequency domain feature fusion, but there is no discussion on the advantage of frequency domain processing compared to time domain multi-scale feature fusion.
3. The paper claimed that multi-scale frequency decomposition can capture short-term fluctuation characteristics, but there is no intuitive result that demonstrates this. Moreover, what's different information captured by the features at different scales?

---

confidence: 4
contribution: 2
initial_rating: 3
presentation: 3
questions: Does MMFNet employ the “channel independent” strategy for training?
soundness: 2
strengths: This paper is overall well written.

The experiments and ablations are relatively comprehensive.

The idea of decomposing time series into multi-frequency components is reasonable and effective.
summary: This paper focuses on the multivariate time series forecasting task and proposes the MMFNet for decomposing the input series into multi-scale segments and further capturing the multi-frequency components. A learnable masking and frequency interpolation strategy is employed for prediction. The authors have tested MMFNet on typical time-series forecasting benchmarks.
weaknesses: 1.	About the method design.

As shown in Figure 1, the authors implement the frequency decomposition by segmenting the input series into multiscale subseries. 

I am curious about does this design perform like directly segmenting the transformed frequency domain. I mean that we do not need to split the input series at first for multifrequency components. Directly applying FFT to the original input series and then splitting the frequency domain can obtain the same results.

Going further, I think without multifrequency decomposition, a learnable mask strategy along with frequency interpolation can achieve a similar performance. Thus, I am doubtful about the actual novelty of this paper.

2.	About the claim of “multivariate time series forecasting”.

I am not sure if MMFNet employs the same “channel independent” training strategy as PatchTST. I believe that channel-independent training is not “multivariate series modeling”, which does not consider the correlation among different series.

Besides, I think the current design can be extended to univariate series (as presented in Figure 1). The experiments on M4 are necessary. Actually, as discussed in FITS, these linear models may fail in M4, which contains much more complex temporal variations than the widely used LTSF benchmarks.

With the experiments on complex datasets, I cannot confirm whether the proposed method is really effective.

3.	How about the efficiency of MMFNet w.r.t. other linear models, including GPU memory, parameter size, and running time?

4.	Since the absolution promotion is kind of limited, the standard deviation or confidence level is expected.

Meta-review: The authors propose to capture multi-scale patterns in the frequency domain for time series forecasting. One of their motivation is existing methods are overlooking the importance of the harmonic, periodic patterns in the frequency domain. As I know, however, this claim is not true. One example is TimeMixer [Wang et al., ICLR 2024] which uses the multi-scale frequency decomposition. In this regard, this paper's motivation fails to reflect recent research outcomes for time series forecasting. I recommend that the authors carefully revise the motivation and the model design. Reviewers also initially raised issues regarding the motivation and the novelty of the paper. Even after the rebuttal discussion, they were not all addressed. We strongly recommend that the authors significantly improve the quality of the paper before resubmission.

Decision: Reject