FreSH: Frequency-Segmented Hierarchical Multi-Expert Framework for Multivariate Time Series Classification

# Frequency-Segmented Hierarchical Multi-Expert Framework for Multivariate Time Series Classification
Source: [https://arxiv.org/html/2608.08207](https://arxiv.org/html/2608.08207)
Pingping Liu1, Muyao Wang1, Zijian Zhang1,∗, Tongshun Zhang1, Hao Miao2, Guorui Xie3, Qingliang Li4, Qiuzhan Zhou11Jilin University 2Hong Kong Polytechnic University 3Pengcheng Laboratory 4Changchun Normal University liupp@jlu\.edu\.cn, wangmy24@mails\.jlu\.edu\.cn, zhangzijian@jlu\.edu\.cn, tszhang23@mails\.jlu\.edu\.cn, hao\.miao@polyu\.edu\.hk, xiegrr@gmail\.com, liqingliang@ccsfu\.edu\.cn, zhouqz@jlu\.edu\.cn

###### Abstract

Multivariate Time Series Classification \(MTSC\) demands models that can effectively capture complex temporal patterns across multiple scales while remaining computationally efficient\. However, existing approaches generally struggle to reconcile fine\-grained representation learning, especially under class imbalance and real\-world constraints\. In this paper, we present FreSH, a Frequency\-Segmented Hierarchical Multi\-Expert Framework designed to address these challenges\. FreSH introduces a new perspective for MTSC by enabling adaptive, multi\-scale analysis of temporal signals, allowing different aspects of the data to be modeled in a complementary and coordinated manner\. By combining localized specialization with holistic context modeling, FreSH achieves strong representational capacity without incurring excessive computational overhead\. An adaptive fusion strategy further enhances flexibility, enabling the model to dynamically emphasize the most informative components of the input\. In addition, we incorporate a more robust optimization objective that improves learning stability across varying sample difficulties and class distributions\. Extensive evaluations on 30 UEA benchmark datasets and real\-world vibration data demonstrate that FreSH consistently outperforms state\-of\-the\-art methods in classification accuracy, while substantially reducing model size and efficiency\. The implementation code is publicly available at[https://github\.com/Wangmy2120/FreSH00](https://github.com/Wangmy2120/FreSH00)\.

## 1Introduction

![Refer to caption](https://arxiv.org/html/2608.08207v1/x1.png)Figure 1:Some models in MTSC cannot adapt to the diverse temporal patterns of different datasets, resulting in performance differences between datasets and across different categories\.Multivariate time series classification has attracted significant attention due to its broad applications in healthcareAnet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib10)\), industrial equipment fault diagnosisFarahaniet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib7)\), and human action recognitionLiet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib8)\)\. Accurate time series classification provides crucial support and insights for decision\-makers\. However, inherent properties of time series data, such as complex dynamics, noise, and class imbalance, make MTSC a particularly challenging taskIsmail Fawazet al\.\([2019](https://arxiv.org/html/2608.08207#bib.bib9)\)\.

Traditional MTSC algorithms, such as DTWWanget al\.\([2017](https://arxiv.org/html/2608.08207#bib.bib3)\), primarily rely on feature statistics or signal processing techniques\. As datasets become more complex, these methods struggle to scale to modern, high\-dimensional time series and fail to generalize across diverse application scenariosRuizet al\.\([2021](https://arxiv.org/html/2608.08207#bib.bib11)\)\. Recently, deep learning has emerged as the dominant paradigm for MTSC\. CNN\-based methods, such as OS\-CNNTanget al\.\([2020](https://arxiv.org/html/2608.08207#bib.bib44)\), excel in learning spatial hierarchical features through convolutional filters but are limited in comprehensively modeling global features, often requiring additional designs to compensate for this shortcoming\. RNN\-basedKarimet al\.\([2017](https://arxiv.org/html/2608.08207#bib.bib12)\)methods face challenges in capturing long\-term dependencies due to vanishing gradients\. Transformer\-based modelsWenet al\.\([2022](https://arxiv.org/html/2608.08207#bib.bib13)\); Zuoet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib14)\)propose to handle long\-range dependencies through self\-attention mechanisms but fall short in extracting local pattern features at adjacent time points\.

Despite the promising progress achieved by existing methods in time series classification, several key limitations hinder their performance and practicality\. First, existing models often struggle to effectively capture the intricate, multi\-scale nature of time series data\. They typically process either time\-domain data directlyHeet al\.\([2015](https://arxiv.org/html/2608.08207#bib.bib23)\)or a holistic frequency\-domainYiet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib15)\), thereby failing to distinguish and analyze the distinct information carried by different frequency bands\. The diversity of the MTSC dataset poses challenges for existing models in balancing differences between datasets and across categories\. As shown in Figure[1](https://arxiv.org/html/2608.08207#S1.F1), different models exhibit significant performance variations on different types of UEA datasets, and their accuracy is markedly affected by the number of samples in different categories, with performance improvements accompanied by considerable overhead\. Furthermore, recent complex models, particularly those leveraging global self\-attention mechanismsZhouet al\.\([2021](https://arxiv.org/html/2608.08207#bib.bib27)\), suffer from high computational costs and poor scalability, making them unsuitable for real\-time applications or large\-scale datasets\. Simultaneously, these models lack the adaptability to specialize their processing based on the local characteristics of the data\. Finally, conventional loss functions in MTSC, such as Cross\-EntropyWuet al\.\([2022a](https://arxiv.org/html/2608.08207#bib.bib19)\)and Focal LossLinet al\.\([2017](https://arxiv.org/html/2608.08207#bib.bib2)\), present a dilemma: the former is often dominated by majority classes, while the latter can over\-correct for difficult samples\. This makes them suboptimal for handling the joint challenges of class imbalance and varying sample difficulty, which are common in real\-world time series datasets\.

Motivated by these challenges, we propose aFrequency\-SegmentedHierarchical Multi\-Expert Framework for time series classification,*i\.e\.,*FreSH\. Our main goal is to go beyond a single\-view approach by using the frequency domain and proposing a segmentation strategy that allows us to specifically analyze different frequency components\. To fix the lack of flexibility and high costs of current models, we design an efficient and adaptive hierarchical multi\-expert system\. This architecture uses dedicated local experts for specific data segments while a global expert provides overall context, all within a lightweight framework that avoids complex and slow mechanisms like global attention\. An adaptive gating mechanism fuses these outputs, dynamically weighting local frequency bands and global frequency spectrum\. We also aim for a better optimization strategy by introducing a polynomial Loss as an alternative to standard loss functions, which we believe can create a better balance when optimizing for both easy and difficult samples, as well as majority and minority classes\. Our major contributions can be summarized as follows:

- •We propose a frequency\-aware modeling paradigm that adaptively captures the multi\-scale and multi\-band characteristics of multivariate time series, offering a principled alternative to single\-view time\- or frequency\-domain approaches for time series classification\.
- •We design FreSH, a lightweight frequency\-segmented hierarchical multi\-expert framework that enables adaptive specialization across frequency components by integrating local and global experts, achieving effective multi\-scale representation with low computational overhead\.
- •Extensive experiments on 30 UEA benchmark datasets validate the effectiveness and generalization capability of our FreSH\. Our method outperforms diverse state\-of\-the\-art baselines while achieving advanced efficiency\.

## 2Methodology

### 2\.1Problem Formulation

Let𝒳=\{𝐗i\}1n\\mathcal\{X\}=\\\{\\mathbf\{X\}\_\{i\}\\\}\_\{1\}^\{n\}represent a multivariate time series dataset, where each sample𝐗i∈ℝd×l\\mathbf\{X\}\_\{i\}\\in\\mathbb\{R\}^\{d\\times l\}represents the observations ofddvariables overlltime steps, the goal of multivariate time series classification is to learn a classifierfθf\_\{\\theta\}to accurately predict the corresponding label,*i\.e\.,*𝐗i∈ℝd×l→fθ𝒴^i∈ℝc\\mathbf\{X\}\_\{i\}\\in\\mathbb\{R\}^\{d\\times l\}\\xrightarrow\{f\_\{\\theta\}\}\\hat\{\\mathcal\{Y\}\}\_\{i\}\\in\\mathbb\{R\}^\{c\}\.

### 2\.2Framework Overview

We proposeFreSH, a frequency\-domain expert framework for multivariate time series classification, which integrates localized and global modeling within a mixture\-of\-experts network\. The working pipeline is shown in Figure[2](https://arxiv.org/html/2608.08207#S2.F2)\(a\)\. We conduct mixup for different samples to enrich the data diversity first, and then transform the data into the frequency domain and process it by a Hierarchical Frequency\-Informed MoE \(HiFiMoE\) module, shown in Figure[2](https://arxiv.org/html/2608.08207#S2.F2)\(b\)\. The output is fed to the prediction layer for classification\.

![Refer to caption](https://arxiv.org/html/2608.08207v1/x2.png)Figure 2:Framework overview of FreSH\. After transforming time series into the frequency domain, theMulti\-Segment Expert Modulelearns segment\-wise patterns, theGlobal Expert Modulecaptures full\-spectrum dependencies, and theAdaptive Gating Mechanismadaptively fuses them for prediction\.
### 2\.3Data Preprocessing

Given an input multivariate time series sample𝐗i∈ℝd×l\\mathbf\{X\}\_\{i\}\\in\\mathbb\{R\}^\{d\\times l\}, whereddis the number of variables andllis the sequence length, we first conduct mixupZhanget al\.\([2017](https://arxiv.org/html/2608.08207#bib.bib25)\)to enrich data diversity\. We transform it into the frequency domain using the Fast Fourier Transform:

𝐅=FFT​\(𝐗i\)∈ℂd×s,\\mathbf\{F\}=\\mathrm\{FFT\}\(\\mathbf\{X\}\_\{i\}\)\\in\\mathbb\{C\}^\{d\\times s\},\(1\)wheres=⌊l2⌋\+1s=\\left\\lfloor\\frac\{l\}\{2\}\\right\\rfloor\+1\. This frequency\-domain representation allows the model to exploit periodic patterns and frequency\-specific information that are difficult to capture in the time domain\. The resulting signal𝐅\\mathbf\{F\}is then zero\-padded to a fixed lengthspaddeds\_\{\\text\{padded\}\}to ensure divisibility and consistency across samples\.

### 2\.4Frequency Segment Division

To capture the unevenly distributed information in the frequency domain, we divide the spectrum into multiple segments\. This segmentation allows specialized experts to focus on specific bands, enabling more targeted feature learning and preparing for adaptive fusion later\. Specifically, we divide the entire frequency spectrum intoKKequal\-length segments:

𝐅=\[𝐅\(1\),𝐅\(2\),…,𝐅\(K\)\],\\mathbf\{F\}=\[\\mathbf\{F\}^\{\(1\)\},\\mathbf\{F\}^\{\(2\)\},\\dots,\\mathbf\{F\}^\{\(K\)\}\],\(2\)where each segment𝐅\(k\)∈ℂd×lk\\mathbf\{F\}^\{\(k\)\}\\in\\mathbb\{C\}^\{d\\times l\_\{k\}\}corresponds to a specific frequency band, andlk×K=sp​a​d​d​e​dl\_\{k\}\\times K=s\_\{padded\}\. This segment division operation offers the opportunity for the specific utilization of individual frequency components that may carry unique patterns relevant for time series classification\.

### 2\.5Hierarchical Frequency\-Informed MoE

#### 2\.5\.1Multi\-Segment Expert Module

For each frequency segment𝐅\(k\)\\mathbf\{F\}^\{\(k\)\}, we design a dedicated multi\-segment expert module consisting ofMMparallel local experts\{𝐄m\(k\)​\(⋅\)\}m=1M\\mathbf\{\\\{E\}\_\{m\}^\{\(k\)\}\(\\cdot\)\\\}\_\{m=1\}^\{M\}, each implemented by a lightweight MLP\. These experts are intended to capture diverse, potentially complementary representations of the intra\-segment features\. Specifically, the generated hidden representation of themm\-th expert andkk\-th frequency segment𝐇m\(k\)\\mathbf\{H\}^\{\(k\)\}\_\{m\}is:

𝐇m\(k\)=𝐄m\(k\)​\(𝐅\(k\)\)\.\\mathbf\{H\}^\{\(k\)\}\_\{m\}=\\mathbf\{E\}\_\{m\}^\{\(k\)\}\\bigl\(\\mathbf\{F\}^\{\(k\)\}\\bigr\)\.\(3\)
WhenM=1M=1, the output of the single expert is directly used\. ForM\>1M\>1, the outputs of all experts are averaged to yield the representation of thekk\-th frequency segment𝐇\(k\)\\mathbf\{H\}^\{\(k\)\}:

𝐇\(k\)=1M​∑m=1M𝐇m\(k\)\.\\mathbf\{H\}^\{\(k\)\}=\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}\\mathbf\{H\}^\{\(k\)\}\_\{m\}\.\(4\)
𝐇\(k\)\\mathbf\{H\}^\{\(k\)\}processes a single frequency band𝐅\(k\)\\mathbf\{F\}^\{\(k\)\}and integrates the output results ofMMexpert networks by averaging, thereby fully leveraging the collaborative advantages of multiple expert networks to significantly enhance overall performance\. This design can balance expressiveness and computational efficiency, avoiding intra\-segment gating while still allowing for ensemble effects among experts\.

#### 2\.5\.2Global Expert Module

While segment\-wise experts focus on local frequency bands, it is equally important to model global dependencies that span the entire frequency spectrum\. To this end, we incorporateNNglobal experts\{𝐔i​\(⋅\)\}i=1N\\\{\\mathbf\{U\}\_\{i\}\(\\cdot\)\\\}\_\{i=1\}^\{N\}, each processing the complete frequency spectrum signal𝐅∈ℂd×spadded\\mathbf\{F\}\\in\\mathbb\{C\}^\{d\\times s\_\{\\text\{padded\}\}\}to extract holistic representations:

𝐇globali=𝐔i​\(𝐅\)\.\\mathbf\{H\}^\{i\}\_\{\\text\{global\}\}=\\mathbf\{U\}\_\{i\}\(\\mathbf\{F\}\)\.\(5\)
To adaptively weight the contributions of different global experts based on the input, we introduce a global gateσglobal​\(⋅\)\\sigma\_\{\\text\{global\}\}\(\\cdot\), which is implemented as a simple linear layer followed by a softmax:

λ=σglobal​\(𝐅\)∈ℝN,\\lambda\\ =\\sigma\_\{\\text\{global\}\}\\left\(\\mathbf\{F\}\\right\)\\in\\mathbb\{R\}^\{N\},\(6\)where𝐅\\mathbf\{F\}is the input full\-spectrum frequency representation, andλ\\lambdaprovides the normalized weights for allNNglobal experts\. The final global representation𝐇global\\mathbf\{H\}\_\{\\text\{global\}\}is then computed as a weighted sum of all the experts:

𝐇global=∑i=1Nλi⋅𝐇global,i\\mathbf\{H\}\_\{\\text\{global\}\}=\\sum\_\{i=1\}^\{N\}\\lambda\_\{i\}\\cdot\\mathbf\{H\}^\{\\text\{i\}\}\_\{\\text\{global\},\}\(7\)
where𝐇global∈ℂd×spadded\\mathbf\{H\}\_\{\\text\{global\}\}\\in\\mathbb\{C\}^\{d\\times s\_\{\\text\{padded\}\}\}\. This mechanism allows the model to dynamically adjust which global experts to emphasize, providing flexibility to adapt to varying signal characteristics\.

#### 2\.5\.3Adaptive Gating Mechanism

Beyond global expert fusion, we also introduce a segment\-level gating mechanism to adaptively combine segment\-wise representations\. The original full frequency domain information𝐅\\mathbf\{F\}is fed into a segment gateσsegment​\(⋅\)\\sigma\_\{\\text\{segment\}\}\(\\cdot\), which directly produces normalized weightsβ\\betafor each segment:

β=σsegment​\(𝐅\)∈ℝK\.\\beta=\\sigma\_\{\\text\{segment\}\}\\left\(\\mathbf\{F\}\\right\)\\in\\mathbb\{R\}^\{K\}\.\(8\)
The local representation is then computed as a weighted combination of segment outputs:

𝐇local=Concat​\(β1⋅𝐇\(1\),…,βK⋅𝐇\(K\)\),\\mathbf\{H\}\_\{\\text\{local\}\}=\\text\{Concat\}\\left\(\\beta\_\{1\}\\cdot\\mathbf\{H\}^\{\(1\)\},\\dots,\\beta\_\{K\}\\cdot\\mathbf\{H\}^\{\(K\)\}\\right\),\(9\)
where𝐇local∈ℂd×spadded\\mathbf\{H\}\_\{\\text\{local\}\}\\in\\mathbb\{C\}^\{d\\times s\_\{\\text\{padded\}\}\}andβk\\beta\_\{k\}provides the softmax\-normalized importance weights for segmentkk, enabling the model to dynamically balance contributions from different frequency bands\.

We have achieved global frequency spectrum representation𝐇global\\mathbf\{H\}\_\{\\text\{global\}\}and fused representation from individual frequency segments𝐇local\\mathbf\{H\}\_\{\\text\{local\}\}\. Then, we leverage a gate networkσr​e​w​e​i​g​h​t\\sigma\_\{reweight\}for global representation𝐇global\\mathbf\{H\}\_\{\\text\{global\}\}to adaptively modulate the channel in the complete frequency spectrum\. Finally, to yield a comprehensive representation, we combine the original frequency\-domain signal𝐅\\mathbf\{F\}, the adaptively fused local representation𝐇local\\mathbf\{H\}\_\{\\text\{local\}\}, and the adaptively fused global representation𝐇global\\mathbf\{H\}\_\{\\text\{global\}\}:

𝐇∗=𝐅\+𝐇local\+γ⋅α^⋅𝐇global\.\\mathbf\{H\}^\{\*\}=\\mathbf\{F\}\+\\mathbf\{H\}\_\{\\text\{local\}\}\+\\gamma\\cdot\\hat\{\\alpha\}\\cdot\\mathbf\{H\}\_\{\\text\{global\}\}\.\(10\)
Here,γ\\gammais a learnable scalar obtained through the local and global two\-path features, andα^\\hat\{\\alpha\}denotes the adaptive gating weight\.

### 2\.6Prediction Layer

The final frequency\-domain representation𝐇∗\\mathbf\{H\}^\{\*\}is transformed back into the time domain using the inverse FFT:

𝐗∗=iFFT​\(𝐇∗\)\.\\mathbf\{X\}^\{\*\}=\\mathrm\{iFFT\}\(\\mathbf\{H\}^\{\*\}\)\.\(11\)
This reconstructed time\-domain signal is passed through a fully connected classification layer and a softmax activation to produce class probabilitiesy^\\hat\{y\}:

y^=Softmax​\(W⋅𝐗∗\+b\),\\hat\{y\}=\\text\{Softmax\}\(W\\cdot\\mathbf\{X\}^\{\*\}\+b\),\(12\)whereWWandbbrepresent the weight and bias of the linear layer\. This end\-to\-end pipeline enables the model to predict the class label based on frequency\-aware representations\.

Table 1:Overall experimental comparison results on 30 datasets\. The best results are highlighted inbold, and the second\-best results are marked with anunderline\. ↑ indicates higher is better, and ↓ indicates lower is better\.
### 2\.7Optimization Objective Function

Table 2:Experimental comparison on 10 UEA datasets\. The best results are highlighted inbold, and the second\-best results are marked with anunderline\. ↑ indicates higher is better, and ↓ indicates lower is better\.In MTSC tasks, data distributions are complex, class imbalance is common, and datasets differ significantly\. Cross Entropy\-Loss, which is widely used in prior work, struggles with class imbalance and fails to distinguish between easy and hard samples, leading to poor handling of minority classes and complex instances\. To address this, we propose adopting Polynomial LossLenget al\.\([2022](https://arxiv.org/html/2608.08207#bib.bib48)\)for time series classification\.

Polynomial Loss was initially proposed to improve optimization and calibration performance in general classification tasks\. However, the original Polynomial Loss introduces high\-order terms and multiple hyperparameters, which may complicate the optimization process in MTSC scenarios\. In this case, we simplify and customize it into an improved loss function calledP\-Loss\.

We propose P\-Loss to optimize the original formula by retaining only the core second\-order adjustment term while preserving the stability and robustness advantages\. The second\-order formulation is adopted because it provides nonlinear gradient corrections with negligible computational overhead, effectively optimizing the classification boundaries for multi\-channel data without introducing the noise associated with higher\-order terms\. Specifically, givenNNsamples, wherey^i\\hat\{y\}\_\{i\}represents the predicted probability of the true class for theii\-th sample, P\-Loss is defined as:

ℒP=−1N​∑i=1Nlog⁡y^i\+λP⋅1N​∑i=1N\(1−y^i\)2\\mathcal\{L\}\_\{P\}=\-\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\log\\hat\{y\}\_\{i\}\+\\lambda\_\{P\}\\cdot\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\(1\-\\hat\{y\}\_\{i\}\)^\{2\}\(13\)
The first term is the negative log\-likelihood, maximizing the predicted probabilityy^i\\hat\{y\}\_\{i\}, part of the standard cross\-entropy loss\. The second term is a second\-order adjustment that penalizes deviations from the true label, withλP\\lambda\_\{P\}controlling its weight\.

We also incorporate mixup with a simple adaptive adjustment\. If the loss stagnates after several training rounds, the mixing ratio is dynamically reduced to bring the mixed samples closer to the original ones, improving generalization without harming baseline performance\. Combined with P\-Loss, this design simplifies optimization while preserving model stability and robustness, ultimately demonstrating stronger generalization on diverse MTSC benchmarks\.

### 2\.8Computational Complexity Analysis

Most operations in FreSH, except for the frequency\-domain transformation, are linear in the sequence length, which makes the framework computationally efficient\. The overall computational complexity of a single forward pass through FreSH can be expressed asO​\(d​n​log⁡n\+m​h​n\+g​h​n\)O\(dn\\log n\+mhn\+ghn\), whereddis the number of channels \(variables\) in the multivariate time series,nnis the frequency\-domain vector length of the input,mmis the number of experts per frequency segment,ggis the number of global experts, andhhis the hidden dimension of each expert\. TheO​\(d​n​log⁡n\)O\(dn\\log n\)term comes from the FFT and inverse FFT transformations applied acrossddchannels, which dominate for largenn\. The segment\-wise expert networks contributeO​\(m​h​n\)O\(mhn\), as each of thekksegments independently appliesmmsmall MLPs onn/kn/kelements\. Similarly, the global experts contributeO​\(g​h​n\)O\(ghn\)by processing the fullnn\-dimensional vector throughggMLPs\. Sincedd,mm,gg, andhhare small constants compared tonnin practice, the computational complexity grows approximately asO​\(d​n​log⁡n\)O\(dn\\log n\), making the model efficient and scalable for long multivariate time series\.

## 3Experiments

### 3\.1Experimental Setups

We conduct experimental comparison on the UEA MTSC benchmarksBagnallet al\.\([2018](https://arxiv.org/html/2608.08207#bib.bib33)\), which span applications such as human activity recognition, speech processing, medical EEG, and audio analysis\. The datasets differ substantially in sequence length, dimensionality, and train/test sizes, enabling a robust assessment of generalization\.

To systematically evaluate the effectiveness of our proposed FreSH, we conduct comprehensive comparisons against a wide range of state\-of\-the\-art baselines\.

On one hand, our main experiments focus on comparisons with dedicated multivariate time series classification methods\. Specifically, we consider both traditional and recent MTSC approaches, including DTWD, ShapeNetLiet al\.\([2021](https://arxiv.org/html/2608.08207#bib.bib40)\), TapNetZhanget al\.\([2020](https://arxiv.org/html/2608.08207#bib.bib41)\), TNCTonekaboniet al\.\([2021](https://arxiv.org/html/2608.08207#bib.bib6)\), TS2VecYueet al\.\([2022](https://arxiv.org/html/2608.08207#bib.bib43)\), InfoMinTianet al\.\([2020](https://arxiv.org/html/2608.08207#bib.bib45)\), InfoTSLuoet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib46)\), AutoTCLZhenget al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib47)\), MPTSNetMuet al\.\([2025](https://arxiv.org/html/2608.08207#bib.bib22)\), and FreRATianet al\.\([2025](https://arxiv.org/html/2608.08207#bib.bib30)\)\. These methods are evaluated on 30 UEA multivariate time series datasets, providing a thorough and fair comparison with existing MTSC models\.

On the other hand, to comprehensively position our proposed FreSH, we compare FreSH with representative time series representation models, including LSTNetLaiet al\.\([2018](https://arxiv.org/html/2608.08207#bib.bib39)\), LSSLGuet al\.\([2021](https://arxiv.org/html/2608.08207#bib.bib24)\), TimesNetWuet al\.\([2022a](https://arxiv.org/html/2608.08207#bib.bib19)\), PatchTSTNieet al\.\([2022](https://arxiv.org/html/2608.08207#bib.bib21)\), FlowFormerWuet al\.\([2022b](https://arxiv.org/html/2608.08207#bib.bib37)\), FEDformerZhouet al\.\([2022b](https://arxiv.org/html/2608.08207#bib.bib38)\), SCINetLiuet al\.\([2022](https://arxiv.org/html/2608.08207#bib.bib32)\), DLinearZenget al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib36)\), CrossformerZhang and Yan \([2023](https://arxiv.org/html/2608.08207#bib.bib29)\), MICNWanget al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib35)\), and ModernTCNLuo and Wang \([2024](https://arxiv.org/html/2608.08207#bib.bib34)\)\. Following the general settingMuet al\.\([2025](https://arxiv.org/html/2608.08207#bib.bib22)\), we conduct the comparison on 10 UEA datasets\. All experiments are conducted on an NVIDIA A100\. We report average accuracy \(Avg\. Acc\.\), average rank \(Avg\. Rank\), and the number of best\-accuracy datasets \(Top\-1 count\) to evaluate MTSC performance and enable fair comparisons with baseline methods\.

### 3\.2Experimental Results

Table[1](https://arxiv.org/html/2608.08207#S2.T1)presents an extensive empirical evaluation of FreSH against 10 state\-of\-the\-art Multivariate Time Series Classification \(MTSC\) baselines across a diverse collection of 30 UEA datasets\. The quantitative results demonstrate the overwhelming superiority and robustness of our proposed method\.

In terms of overall performance across all 30 datasets, FreSH establishes a new state\-of\-the\-art benchmark\. It achieves the highest average accuracy of 76\.1% and the best \(lowest\) average rank of 3\.4 among all compared methods\. When compared to the second\-best performer, FreRA, FreSH not only improves the average accuracy by 0\.7% \(76\.1% vs\. 75\.4%\) but also demonstrates greater consistency, surpassing FreRA by a margin of 0\.6 in average ranking \(3\.2 vs\. 3\.8\)\. This indicates that FreSH maintains high performance stability across varying data domains\.

A deeper analysis of the specific 10 MTSC datasets \(Table[2](https://arxiv.org/html/2608.08207#S2.T2)\) further highlights the model’s adaptability\. In this challenging subset, the performance gap between FreSH and existing methods becomes even more pronounced\. Compared to the strong baseline ModernTCN, which ranks second in both metrics, FreSH delivers a substantial improvement: it boosts the average accuracy by 1\.3% and advances the average ranking by a full position \(1\.0\)\.

Beyond the average metrics, the ranking distribution across datasets further confirms the core advantages of our method\. In the evaluation of the complete UEA dataset, FreSH achieved the highest number of Top\-1 results, significantly surpassing the next best model\. Furthermore, in the evaluation setting across 10 datasets, FreSH achieved best accuracy on 5 datasets and second best accuracy on 4 datasets\.

Overall, this outstanding performance across multiple metrics and settings demonstrates that FreSH is an excellent model capable of handling complex temporal patterns, significantly outperforming both traditional and recently proposed MTSC models, with remarkable adaptability and effectiveness\.

### 3\.3Ablation Study

Table 3:Ablation experiments on UEA datasets\.We conduct a comprehensive ablation study to investigate the contributions of the key components in our model\.

- •w/o\-GlobalExperts: Remove the global experts and only rely on segment\-wise experts for feature extraction\.
- •w/o\-SegmentExperts: Remove the segment\-wise experts, relying solely on global experts\.
- •w/o\-P\-Loss: Replace the P\-loss with cross\-entropy loss\.
- •w/o\-Mixup: Disable the adaptive mixup data augmentation\.
- •w/o\-β\\beta: Disable adaptive gating fusion for Segment experts\.
- •w/o\-λ\\lambda: Disable adaptive gating fusion for Global experts\.

The ablation study confirms that all components of the FreSH framework are critical to its performance\. The largest drop in accuracy occurs when removing P\-Loss \(76\.1% to 72\.9%\), a strong indicator of its central role in achieving robust optimization for imbalanced and difficult samples\. The hierarchical multi\-expert framework is also essential, as removing either the global experts \(76\.1% to 73\.3%\) or the segment experts \(76\.1% to 73\.1%\) significantly degrades performance, validating the synergy between local feature analysis and global information integration\. While the mixup augmentation also contributes, its removal leads to a comparatively smaller drop \(76\.1% to 74\.6%\), showing it is a valuable but supplementary component\. Under the premise of retaining the linear expert, ablation experiments were conducted on the adaptive gated fusion of local \(β\\beta\) and global \(λ\\lambda\) experts\. Although the removal of this fusion module led to a decrease in average accuracy \(2\.7% and 1\.8%\), its performance still significantly outperformed the setting where all experts were removed simultaneously, with dual comparisons validating that the linear expert provides a stable baseline and the adaptive fusion further achieves effective collaboration among experts\.

### 3\.4Efficiency Comparison

To assess the prediction precision and computational efficiency in realistic settings, we benchmark all models on a real\-world vibration dataset\. We report model parameter volume, average batch latency, total test time, and classification accuracy to characterize the trade\-off between computational cost and predictive performance\.

Table 4:Efficiency comparison on a real\-world vibration dataset\. Certain models adopt a classification adaptation scheme identical to FreSH\.Table[4](https://arxiv.org/html/2608.08207#S3.T4)shows large efficiency gaps\. DLinear is extremely fast \(0\.66 ms/batch; 0\.187 s total\) but underfits vibration signals, yielding low accuracy \(44\.89%\)\. In contrast, TimesNet, Crossformer, and other Transformer\-based models achieve competitive accuracy at much higher cost \(25\.3–63\.9 ms/batch\)\. MPTSNet further demonstrates diminishing returns: despite strong capacity, its\>\>88M parameters lead to prohibitive latency \(101\.4 ms/batch;\>\>123 s total\), limiting practicality\. By contrast, our proposed FreSH achieves the best balance between performance and efficiency\. With only 54,243 parameters, it remains fast \(1\.2 ms/batch; 0\.344 s total\) while achieving 94\.37% accuracy\.

These results indicate that FreSH effectively reconciles efficiency and accuracy, making it well\-suited for real\-world vibration analysis where both are critical\.

### 3\.5Hyper\-Parameter Analysis

We conduct quantitative experiments to evaluate the contributions of different components of FreSH\. Figure[3](https://arxiv.org/html/2608.08207#S3.F3)shows average accuracy on 30 UEA datasets under various hyperparameter settings\. Figure[3](https://arxiv.org/html/2608.08207#S3.F3)\(a\) tests the number of frequency segments\. Figure[3](https://arxiv.org/html/2608.08207#S3.F3)\(b\) examines experts per segment and analyzes global experts\.

![Refer to caption](https://arxiv.org/html/2608.08207v1/figures/SuperCompare.jpg)Figure 3:Experiments on 30 UEA datasets to evaluate different structural configurations, we report the average accuracy\.Experimental results shown in Figure[3](https://arxiv.org/html/2608.08207#S3.F3)highlight the critical role of structural design in our framework\. Specifically, segmenting the frequency spectrum into 3 bands strikes a balance between capturing fine\-grained local patterns and preserving sufficient global context\. Likewise, assigning 3 experts enables diverse feature extraction within each band without introducing unnecessary redundancy\.

In contrast, overly fine segmentation fragments the spectrum and dilutes useful information, while too many experts per segment increases the risk of overfitting and leads to unstable representations\.

These findings confirm that carefully calibrating the segmentation granularity and expert allocation is essential for effectively modeling both local and global dependencies in multivariate time series classification\.

## 4Related Work

### 4\.1Time\-Domain MTSC Methods

A substantial portion of multivariate time series classification research has concentrated on modeling signals directly in the time domain\. Notably, the MC\-DCNNZhenget al\.\([2014](https://arxiv.org/html/2608.08207#bib.bib16)\)applies one\-dimensional convolutions to capture inter\-variable relations, pairing them with fully connected layers for classification purposes\. Building on this, the Multiscale Convolutional Neural Network \(MSCNN\) designs convolution kernels of multiple sizes to extract multiscale featuresCuiet al\.\([2016](https://arxiv.org/html/2608.08207#bib.bib17)\)\. Hybrid architectures have also been explored, such as LSTM\-FCNKarimet al\.\([2017](https://arxiv.org/html/2608.08207#bib.bib12)\), which leverages LSTM layers for short\- and long\-term dependency capture while CNN layers extract salient time series patterns\. Its enhanced version MLSTM\-FCNKarimet al\.\([2019](https://arxiv.org/html/2608.08207#bib.bib18)\)further refines this combination for performance gains\. More recent developments like TimesNetWuet al\.\([2022a](https://arxiv.org/html/2608.08207#bib.bib19)\)disentangle complex temporal variations into intra\- and inter\-period components for improved local and global feature modeling, whereas MS\-GNetCaiet al\.\([2024](https://arxiv.org/html/2608.08207#bib.bib20)\)integrates graph convolution for inter\-series correlation alongside multi\-head attention for intra\-series feature learning\. Likewise, PatchTSTNieet al\.\([2022](https://arxiv.org/html/2608.08207#bib.bib21)\)partitions sequences into local patches to capture hierarchical patterns, while AutoformerWuet al\.\([2021](https://arxiv.org/html/2608.08207#bib.bib26)\)proposes a novel auto\-correlation mechanism that captures long\-range dependencies by calculating the periodic similarity of time series\.

Compared with this line of methods, our FreSH’s hierarchical multi\-expert architecture allows us to specifically extract the crucial frequency characteristics, achieving superior computational efficiency and adaptability\.

### 4\.2Frequency\-Domain MTSC Methods

In parallel, a growing body of work has sought to harness frequency\-domain representations for MTSC, leveraging spectral analysis to uncover new optimization pathways\. For instance, the Frequency\-improved Legendre Memory ModelZhouet al\.\([2022a](https://arxiv.org/html/2608.08207#bib.bib28)\)augments Legendre memory structures with spectral components, markedly improving long\-term sequence classification\. CrossFormerZhang and Yan \([2023](https://arxiv.org/html/2608.08207#bib.bib29)\)unifies frequency\-domain decomposition with Transformer architectures to jointly refine local and global feature extraction, while MPTSNetMuet al\.\([2025](https://arxiv.org/html/2608.08207#bib.bib22)\)explicitly utilizes amplitude information to detect salient periodicities for rapid feature localization\. Other methods bypass the time domain entirely: FreTSYiet al\.\([2023](https://arxiv.org/html/2608.08207#bib.bib15)\)demonstrates the compactness of spectral information and constructs a frequency\-domain MLP to achieve state\-of\-the\-art performance, and FreRATianet al\.\([2025](https://arxiv.org/html/2608.08207#bib.bib30)\)proposes a parameterized augmentation strategy with time\-frequency consistency constraints for robust training\.

Existing frequency\-domain methods typically perform a single, global analysis of the entire spectrum, which overlooks the unique information contained in different frequency bands\. Our FreSH addresses this by segmenting the spectrum and using an adaptive expert system to process each segment individually, enabling a fine\-grained and specialized analysis of each frequency band\.

## 5Conclusion

To advance multivariate time series classification, we propose FreSH, which combines frequency\-domain analysis with an adaptive expert system\.

FreSH transforms time series into the frequency domain and applies spectral segmentation: segment experts capture band\-specific features, global experts model full\-spectrum context, and adaptive gating fuses them into a more balanced representation\.

FreSH further introduces P\-Loss to address the limitations of cross\-entropy and focal loss, and incorporates an adaptive mixup strategy to improve robustness and generalization\.

Extensive experiments on a broad range of UEA benchmarks and a real\-world vibration dataset demonstrate that FreSH consistently achieves superior or competitive accuracy compared to state\-of\-the\-art methods, while maintaining high computational efficiency and a compact parameter footprint\. The strong performance across diverse datasets underscores the framework’s practical applicability and robustness, highlighting its potential for real\-world deployment scenarios where both accuracy and efficiency are essential\.

## Acknowledgements

This work was supported by Jilin Province Industrial Key Core Technology Tackling Project \(20230201085GX\)\. Zijian Zhang is supported by the China Postdoctoral Science Foundation \(2025M771587\) and the Open Funding Programs of State Key Laboratory of AI Safety \(2025\-09\)\. Hao Miao is supported by SCRI, The Hong Kong Polytechnic University \(No\. Q\-CDDG\)\. Qingliang Li is supported by the National Natural Science Foundation of China\(42575159, 42275155, 62206028\)\.

## References

- A comprehensive review on machine learning in healthcare industry: classification, restrictions, opportunities and challenges\.Sensors23\(9\),pp\. 4178\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p1.1)\.
- A\. Bagnall, H\. A\. Dau, J\. Lines, M\. Flynn, J\. Large, A\. Bostrom, P\. Southam, and E\. Keogh \(2018\)The uea multivariate time series classification archive, 2018\.arXiv preprint arXiv:1811\.00075\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p1.1)\.
- W\. Cai, Y\. Liang, X\. Liu, J\. Feng, and Y\. Wu \(2024\)Msgnet: learning multi\-scale inter\-series correlations for multivariate time series forecasting\.InProceedings of the AAAI conference on artificial intelligence,Vol\.38,pp\. 11141–11149\.Cited by:[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- Z\. Cui, W\. Chen, and Y\. Chen \(2016\)Multi\-scale convolutional neural networks for time series classification\.arXiv preprint arXiv:1603\.06995\.Cited by:[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- M\. A\. Farahani, M\. McCormick, R\. Gianinny, F\. Hudacheck, R\. Harik, Z\. Liu, and T\. Wuest \(2023\)Time\-series pattern recognition in smart manufacturing systems: a literature review and ontology\.Journal of Manufacturing Systems69,pp\. 208–241\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p1.1)\.
- A\. Gu, K\. Goel, and C\. Ré \(2021\)Efficiently modeling long sequences with structured state spaces\.arXiv preprint arXiv:2111\.00396\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- G\. He, Y\. Duan, R\. Peng, X\. Jing, T\. Qian, and L\. Wang \(2015\)Early classification on multivariate time series\.Neurocomputing149,pp\. 777–787\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p3.1)\.
- H\. Ismail Fawaz, G\. Forestier, J\. Weber, L\. Idoumghar, and P\. Muller \(2019\)Deep learning for time series classification: a review\.Data mining and knowledge discovery33\(4\),pp\. 917–963\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p1.1)\.
- F\. Karim, S\. Majumdar, H\. Darabi, and S\. Chen \(2017\)LSTM fully convolutional networks for time series classification\.IEEE access6,pp\. 1662–1669\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p2.1),[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- F\. Karim, S\. Majumdar, H\. Darabi, and S\. Harford \(2019\)Multivariate lstm\-fcns for time series classification\.Neural networks116,pp\. 237–245\.Cited by:[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- G\. Lai, W\. Chang, Y\. Yang, and H\. Liu \(2018\)Modeling long\-and short\-term temporal patterns with deep neural networks\.InThe 41st international ACM SIGIR conference on research & development in information retrieval,pp\. 95–104\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- Z\. Leng, M\. Tan, C\. Liu, E\. D\. Cubuk, X\. Shi, S\. Cheng, and D\. Anguelov \(2022\)Polyloss: a polynomial expansion perspective of classification loss functions\.arXiv preprint arXiv:2204\.12511\.Cited by:[§2\.7](https://arxiv.org/html/2608.08207#S2.SS7.p1.1)\.
- G\. Li, B\. Choi, J\. Xu, S\. S\. Bhowmick, K\. Chun, and G\. L\. Wong \(2021\)Shapenet: a shapelet\-neural network approach for multivariate time series classification\.InProceedings of the AAAI conference on artificial intelligence,Vol\.35,pp\. 8375–8383\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- Y\. Li, G\. Yang, Z\. Su, S\. Li, and Y\. Wang \(2023\)Human activity recognition based on multienvironment sensor data\.Information Fusion91,pp\. 47–63\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p1.1)\.
- T\. Lin, P\. Goyal, R\. Girshick, K\. He, and P\. Dollár \(2017\)Focal loss for dense object detection\.InProceedings of the IEEE international conference on computer vision,pp\. 2980–2988\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p3.1)\.
- M\. Liu, A\. Zeng, M\. Chen, Z\. Xu, Q\. Lai, L\. Ma, and Q\. Xu \(2022\)Scinet: time series modeling and forecasting with sample convolution and interaction\.Advances in Neural Information Processing Systems35,pp\. 5816–5828\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- D\. Luo and X\. Wang \(2024\)Moderntcn: a modern pure convolution structure for general time series analysis\.InThe twelfth international conference on learning representations,pp\. 1–43\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- D\. Luo, W\. Cheng, Y\. Wang, D\. Xu, J\. Ni, W\. Yu, X\. Zhang, Y\. Liu, Y\. Chen, H\. Chen,et al\.\(2023\)Time series contrastive learning with information\-aware augmentations\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.37,pp\. 4534–4542\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- Y\. Mu, M\. Shahzad, and X\. X\. Zhu \(2025\)MPTSNet: integrating multiscale periodic local patterns and global dependencies for multivariate time series classification\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.39,pp\. 19572–19580\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1),[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1),[§4\.2](https://arxiv.org/html/2608.08207#S4.SS2.p1.1)\.
- Y\. Nie, N\. H\. Nguyen, P\. Sinthong, and J\. Kalagnanam \(2022\)A time series is worth 64 words: long\-term forecasting with transformers\.arXiv preprint arXiv:2211\.14730\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1),[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- A\. P\. Ruiz, M\. Flynn, J\. Large, M\. Middlehurst, and A\. Bagnall \(2021\)The great multivariate time series classification bake off: a review and experimental evaluation of recent algorithmic advances\.Data mining and knowledge discovery35\(2\),pp\. 401–449\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p2.1)\.
- W\. Tang, G\. Long, L\. Liu, T\. Zhou, M\. Blumenstein, and J\. Jiang \(2020\)Omni\-scale cnns: a simple and effective kernel size configuration for time series classification\.arXiv preprint arXiv:2002\.10061\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p2.1)\.
- T\. Tian, C\. Miao, and H\. Qian \(2025\)FreRA: a frequency\-refined augmentation for contrastive learning on time series classification\.arXiv preprint arXiv:2505\.23181\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1),[§4\.2](https://arxiv.org/html/2608.08207#S4.SS2.p1.1)\.
- Y\. Tian, C\. Sun, B\. Poole, D\. Krishnan, C\. Schmid, and P\. Isola \(2020\)What makes for good views for contrastive learning?\.Advances in neural information processing systems33,pp\. 6827–6839\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- S\. Tonekaboni, D\. Eytan, and A\. Goldenberg \(2021\)Unsupervised representation learning for time series with temporal neighborhood coding\.arXiv preprint arXiv:2106\.00750\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- H\. Wang, J\. Peng, F\. Huang, J\. Wang, J\. Chen, and Y\. Xiao \(2023\)Micn: multi\-scale local and global context modeling for long\-term series forecasting\.InThe eleventh international conference on learning representations,Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- Z\. Wang, W\. Yan, and T\. Oates \(2017\)Time series classification from scratch with deep neural networks: a strong baseline\.In2017 International joint conference on neural networks \(IJCNN\),pp\. 1578–1585\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p2.1)\.
- Q\. Wen, T\. Zhou, C\. Zhang, W\. Chen, Z\. Ma, J\. Yan, and L\. Sun \(2022\)Transformers in time series: a survey\.arXiv preprint arXiv:2202\.07125\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p2.1)\.
- H\. Wu, T\. Hu, Y\. Liu, H\. Zhou, J\. Wang, and M\. Long \(2022a\)Timesnet: temporal 2d\-variation modeling for general time series analysis\.arXiv preprint arXiv:2210\.02186\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p3.1),[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1),[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- H\. Wu, J\. Wu, J\. Xu, J\. Wang, and M\. Long \(2022b\)Flowformer: linearizing transformers with conservation flows\.arXiv preprint arXiv:2202\.06258\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- H\. Wu, J\. Xu, J\. Wang, and M\. Long \(2021\)Autoformer: decomposition transformers with auto\-correlation for long\-term series forecasting\.Advances in neural information processing systems34,pp\. 22419–22430\.Cited by:[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- K\. Yi, Q\. Zhang, W\. Fan, S\. Wang, P\. Wang, H\. He, N\. An, D\. Lian, L\. Cao, and Z\. Niu \(2023\)Frequency\-domain mlps are more effective learners in time series forecasting\.Advances in Neural Information Processing Systems36,pp\. 76656–76679\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p3.1),[§4\.2](https://arxiv.org/html/2608.08207#S4.SS2.p1.1)\.
- Z\. Yue, Y\. Wang, J\. Duan, T\. Yang, C\. Huang, Y\. Tong, and B\. Xu \(2022\)Ts2vec: towards universal representation of time series\.InProceedings of the AAAI conference on artificial intelligence,Vol\.36,pp\. 8980–8987\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- A\. Zeng, M\. Chen, L\. Zhang, and Q\. Xu \(2023\)Are transformers effective for time series forecasting?\.InProceedings of the AAAI conference on artificial intelligence,Vol\.37,pp\. 11121–11128\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- H\. Zhang, M\. Cisse, Y\. N\. Dauphin, and D\. Lopez\-Paz \(2017\)Mixup: beyond empirical risk minimization\.arXiv preprint arXiv:1710\.09412\.Cited by:[§2\.3](https://arxiv.org/html/2608.08207#S2.SS3.p1.3)\.
- X\. Zhang, Y\. Gao, J\. Lin, and C\. Lu \(2020\)Tapnet: multivariate time series classification with attentional prototypical network\.InProceedings of the AAAI conference on artificial intelligence,Vol\.34,pp\. 6845–6852\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- Y\. Zhang and J\. Yan \(2023\)Crossformer: transformer utilizing cross\-dimension dependency for multivariate time series forecasting\.InThe eleventh international conference on learning representations,Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1),[§4\.2](https://arxiv.org/html/2608.08207#S4.SS2.p1.1)\.
- X\. Zheng, T\. Wang, W\. Cheng, A\. Ma, H\. Chen, M\. Sha, and D\. Luo \(2023\)Auto tcl: automated time series contrastive learning with adaptive augmentations\.InProc\. 32nd Int\. Joint Conf\. Artif\. Intell\.\(IJCAI\),pp\. 1–19\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p3.1)\.
- Y\. Zheng, Q\. Liu, E\. Chen, Y\. Ge, and J\. L\. Zhao \(2014\)Time series classification using multi\-channels deep convolutional neural networks\.InInternational conference on web\-age information management,pp\. 298–310\.Cited by:[§4\.1](https://arxiv.org/html/2608.08207#S4.SS1.p1.1)\.
- H\. Zhou, S\. Zhang, J\. Peng, S\. Zhang, J\. Li, H\. Xiong, and W\. Zhang \(2021\)Informer: beyond efficient transformer for long sequence time\-series forecasting\.InProceedings of the AAAI conference on artificial intelligence,Vol\.35,pp\. 11106–11115\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p3.1)\.
- T\. Zhou, Z\. Ma, Q\. Wen, L\. Sun, T\. Yao, W\. Yin, R\. Jin,et al\.\(2022a\)Film: frequency improved legendre memory model for long\-term time series forecasting\.Advances in neural information processing systems35,pp\. 12677–12690\.Cited by:[§4\.2](https://arxiv.org/html/2608.08207#S4.SS2.p1.1)\.
- T\. Zhou, Z\. Ma, Q\. Wen, X\. Wang, L\. Sun, and R\. Jin \(2022b\)Fedformer: frequency enhanced decomposed transformer for long\-term series forecasting\.InInternational conference on machine learning,pp\. 27268–27286\.Cited by:[§3\.1](https://arxiv.org/html/2608.08207#S3.SS1.p4.1)\.
- R\. Zuo, G\. Li, B\. Choi, S\. S\. Bhowmick, D\. N\. Mah, and G\. L\. Wong \(2023\)SVP\-t: a shape\-level variable\-position transformer for multivariate time series classification\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.37,pp\. 11497–11505\.Cited by:[§1](https://arxiv.org/html/2608.08207#S1.p2.1)\.