Multivariate time series forecasting with multi-view hierarchical patching and lag-coupled channel attention

1 Introduction

Time series forecasting (TSF) seeks to learn predictive structure from historical observations in order to anticipate future trajectories, including trend, seasonality, and multi-scale temporal dynamics. It plays a central role in applications spanning financial markets [1], industrial monitoring [2], urban transportation systems [3], meteorological prediction [4], and energy dispatch and scheduling [5]. In these domains, TSF provides a core analytical foundation for the regulation of complex systems, efficient resource allocation, and risk-informed decision-making.

Early work on time-series modeling was dominated by classical statistical formulations, notably ARIMA [6] and vector autoregressions (VAR) [7]. These models are stable and interpretable in approximately stationary, linear environments, yet they are poorly equipped to accommodate the non-linearity and non-stationarity that frequently characterize empirical data. Later, machine-learning approaches such as support vector machines (SVM) [8] expanded the capacity for non-linear approximation. However, because they typically lack an explicit structural inductive bias for temporal dependence, they remain limited in expressing rich and evolving sequential dynamics.

With the rise of deep learning, TSF has shifted toward neural-network-based paradigms. Recurrent architectures and their gated variants, including LSTM [9] and GRU [10], leverage hidden-state transitions and have delivered strong performance in modeling short-horizon dynamics. Yet their inherently sequential computation constrains parallelization and makes them susceptible to vanishing gradients and progressive loss of stored information in long contexts. Transformers [11] address these limitations by using self-attention to model long-range dependencies in parallel, now constituting a dominant backbone for TSF. Despite recent advancements, much of the literature remains oriented toward global dependency modeling, lacking the capacity to represent the local motifs, multiscale interactions, and multi-view coupling structures ubiquitous in real-world systems.

As shown in Figure 1, the multivariate sequence from the ETTh1 dataset exhibits typical multi-view dependency structures. The vertical axis represents the Min-Max normalized values within the selected visualization window, which removes scale differences among variables and highlights their relative temporal variations. The blue solid line denotes the target channel OT, the gray dotted line denotes the leading covariate HUFL, the orange dashed line denotes HUFL shifted by the estimated lag, the red solid line denotes the smoothed trend of the target channel, and the light-blue shaded region indicates the local synchronization region between OT and HUFL. From the temporal perspective, the target sequence contains long-term smooth evolution, recurring cyclic structures, and short-range local fluctuations. In this paper, multi-scale trends refer to the coexistence of dynamic components at different temporal scales within the same sequence. The purple annotation “Recurring Cycle” highlights repeated oscillatory behavior in the target sequence, while the green annotation “Short-term Noise” indicates a local transient disturbance. From the channel-interaction perspective, HUFL denotes the high useful load variable in the ETTh1 dataset, while OT denotes the oil temperature target channel. Within the selected visualization window, the two channels exhibit similar local rising and falling patterns across different variation intervals, including high-value plateaus, transition phases, and low-value fluctuation regions. This observation indicates relatively stable local inter-channel synchrony between HUFL and OT. Moreover, the orange dashed curve shows that the shifted HUFL trajectory can be better aligned with the target OT under an appropriate lag, indicating that cross-channel influence may occur with a temporal offset rather than being strictly synchronous. The annotation “Adaptive Lag Alignment” denotes the phenomenon that the source channel HUFL better matches the target channel OT after temporal shifting.

From the perspective of existing studies, local dynamics in real-world multivariate time series should not be regarded as isolated phenomena. Prior work has investigated such dynamics from different angles, including recurring concept drift and regime recurrence, heterogeneous multivariate forecasting, local synchronization in coupled dynamical systems, and cross-dimensional dependency modeling [12–15]. These studies suggest that local disturbances, periodic variations, inter-channel synchrony, and cross-variable dependencies may jointly participate in the evolution of complex time series. Furthermore, studies on hierarchical correlation modeling, recursive multi-resolution interactions, trend-seasonal decomposition, and multi-granularity representation learning indicate that local structures may progressively interact and be organized into macroscopic temporal patterns through multi-scale and hierarchical coupling [16–19]. Therefore, effective modeling of complex non-stationary time series requires not only capturing individual temporal dependencies, but also jointly representing multi-view dependencies across the temporal domain, frequency domain, channel space, and cross-channel lag interactions. However, this high-dimensional coupling poses a significant modeling challenge. Effectively capturing such dynamics necessitates not only a unified treatment of multi-view dependencies across scales and variables, but also explicit information constraints. Absent these controls, local noise can mislead the model into confounding incidental co-movements with structural dependence, thereby proliferating spurious correlations and eroding out-of-sample generalization.

To address these challenges, we propose a hierarchical multi-view forecasting architecture, termed the Hierarchical Multi-View Transformer (HMV-Former). The proposed framework unifies multi-view representation learning with attention-logic regularization. Its progressively layered design enables the joint modeling of temporal patterns, frequency-domain characteristics, channel interactions, and adaptive cross-channel lag dependencies. This results in a structured representation of multi-scale dynamics under explicit information constraints, thereby improving predictive accuracy and robustness. Our main contributions are summarized as follows:

• 1) Multi-level patching scheme: we introduce a multi-level patching scheme that performs parallel feature extraction across complementary domains, including the time dimension, frequency domain, and channel space. By employing progressively deeper abstractions, this scheme explicitly captures a multi-scale hierarchy where macroscopic behavior emerges from localized components. This design facilitates the structured integration of multi-view information into a unified representation.
• 2) Lag-coupled channel attention: we propose a lag-coupled channel attention (LCCA) mechanism that incorporates adaptive cross-channel lag awareness directly into channel interaction modeling. This mechanism is tailored to capture time-varying, cross-view dependencies that arise in complex systems, providing a critical foundation for hierarchical interactions across layers.
• 3) Integrated multi-view encoder: we develop an integrated multi-view encoder as the central representation module, which unifies time and frequency processing with explicit modeling of cross-channel relations. Additionally, we introduce attention logic regularization (Attn-L-Reg) to suppress the propagation of spurious dependencies within attention maps, thereby enhancing out-of-sample generalization.
• 4) Empirical evaluations: empirical evaluations on 10 real-world benchmarks demonstrate that HMV-Former outperforms, or is competitive with, leading baselines in both complex dynamics characterization and long-horizon forecasting. These results substantiate the model's practical value across domains, including energy systems, transportation, and financial markets.

2 Related works

Originally designed for natural language processing, the standard Transformer architecture relies fundamentally on the Multi-Head Self-Attention mechanism (MHA). By calculating correlation weights between any two time steps, this mechanism excels at capturing global, long-range dependencies. However, directly applying this standard mechanism to time series forecasting presents two inherent limitations. First, the computational complexity of standard self-attention scales quadratically with the sequence length, making it computationally prohibitive for handling extensive historical observation windows. Second, early conventional applications of standard models typically process data pointwise (using a single time step as a token), which not only renders it highly vulnerable to high-frequency local noise but also limits its capacity to effectively extract the macro-level periodicities and multi-scale evolutionary features inherent in time series.

To systematically review how subsequent studies have sought to overcome the above limitations, this paper adopts architectural evolution as the main analytical lens and summarizes existing methods into three categories: variants of the standard mechanism, input patching-based paradigms, and explicit convolutional hybrid architectures. For instance, Informer [20] introduced sparse attention to reduce complexity; Autoformer [18] and FEDformer [21] incorporated trend-seasonal decomposition; and Crossformer [15] alongside iTransformer [22] shifted focus toward cross-dimensional interaction modeling. However, these methods typically model temporal dependencies, frequency structures, and channel relations in isolation. They overlook the “multi-view coupling” characteristics formed by the synergy of local perturbations, cross-scale structures, and cross-variable lags, thereby limiting their representational capacity for complex non-stationary series.

To mitigate the poor noise resistance of pointwise inputs, a second category of methods developed the patch-based multi-scale paradigm. PatchTST [23] and subsequent works (e.g., Pathformer [24], MSFformer [25], MultiPatchFormer [26], Sentinel [27]) pack continuous time steps into local patches, significantly enhancing local semantic modeling. Nevertheless, these optimizations are largely confined to the input side. Within the network, sub-sequence features are typically integrated through simple parallelization or flat concatenation, lacking a progressive, “local-to-macro” hierarchical abstraction mechanism. Consequently, their capacity for cross-scale evolution modeling remains insufficient.

The third category attempts to bridge the gap in multi-scale perception by introducing explicit convolutional structures. Models like MICN [28] and SCINet [17] have shown prominent performance in extracting local and global features. However, constrained by fixed receptive fields and static network topologies, they struggle to flexibly delineate complex, dynamic interactions across scales and channels. Furthermore, they may introduce redundant features and pseudo-dependencies during long-sequence alignment, which compromises efficiency and generalization performance.

In summary, existing methods still exhibit significant blind spots in multi-view synergistic modeling, hierarchical abstraction construction, and multi-scale information governance. Addressing these critical gaps, this paper proposes HMV-Former, a hierarchical multi-view modeling framework. By performing progressive abstraction and fusion across temporal, frequency, and channel spaces, and by introducing attention logic regularization to suppress the diffusion of pseudo-dependencies, HMV-Former aims to fundamentally improve the generalization and stability of forecasting complex non-stationary series.

3 Methodology

Formulating the multivariate time series forecasting problem, we consider a look-back window X ∈ ℝ^*L*×*C*. Here, L dictates the historical steps and signifies the variable dimensions, meaning X is assembled from L distinct C-dimensional temporal vectors, i.e., . The ultimate goal is to optimize a parameterized function f_θ capable of translating this past context into a future trajectory Ŷ ∈ ℝ^*S*×*C*, with S indicating the prediction range. Mathematically, this generative mechanism is expressed as Equation 1:

Ŷ=fθ(X)=(x^L+1,…,x^L+S)T(1)

Guided by a predefined loss function, the training procedure penalizes the prediction deviation, thereby forcing the generated sequence Ŷ to match the ground-truth future Y=(xL+1,…,xL+S)T as accurately as possible.

3.1 Overall architecture

The overall architecture of HMV-Former, as illustrated in Figure 2, consists of four functional stages: input preprocessing, integrated multi-view encoding, hierarchical abstraction, and information-constrained forecasting. In the input preprocessing stage, the raw multivariate time series is first processed by RevIN Normalization to reduce scale discrepancies across variables and samples. Then, Pre-Patch Time-Frequency Fusion is applied before patch construction to incorporate frequency-domain periodic information into the temporal representation. Subsequently, Patch Embedding converts the processed sequence into patch tokens with positional information, providing the initial input for the encoder.

In the integrated multi-view encoding stage, the embedded tokens are fed into the Integrated Multi-View Encoder, which contains parallel Time-domain Pathway and Frequency-domain Pathway branches. The time-domain pathway models global patch-level dependencies and cross-channel lag relationships through multi-head self-attention, LCCA, and feed-forward networks. In parallel, the frequency-domain pathway captures cross-patch spectral structures by applying the real-valued Fast Fourier Transform (rFFT) along the patch-token dimension, followed by ComplexLinear transformation and the inverse real-valued Fast Fourier Transform (irFFT) reconstruction. The outputs of the two paths are fused through residual connection and normalization to produce a unified high-dimensional contextual representation.

During the hierarchical abstraction stage, the encoded representation is progressively re-partitioned and re-encoded through the Hierarchical Patching Stack. At each hierarchy, the Integrated Multi-View Encoder is reused to extract higher-level representations from coarser patch structures. This recursive process enables the model to organize local patch-level information into multi-scale representations, which are then mapped to the future time horizon by the Forecasting Head.

To mitigate information overload and spurious dependencies induced by multi-layer and multi-view modeling, HMV-Former incorporates Attn-L-Reg during training. Specifically, the attention maps generated by the Integrated Multi-View Encoders in both the initial encoding stage and the hierarchical abstraction stage are collected and regularized with an L₁ sparsity constraint to suppress redundant dependencies. As indicated by the red dashed branch in Figure 2, this attention-map regularization term is combined with the forecasting loss to form the final training objective, thereby enhancing robustness and generalization capability.

In summary, HMV-Former establishes a hierarchical multi-view forecasting framework that integrates local fine-grained representation, progressive multi-scale abstraction, temporal-frequency modeling, and cross-channel dependency learning. This design enables effective characterization of complex dependencies across variables, scales, and temporal-frequency domains in multivariate time series.

3.2 Sequence representation and embedding

To mitigate distributional discrepancies among channels in multivariate time series and construct informative token sequences for the Transformer backbone, HMV-Former sequentially performs three key steps: Instance Normalization, Pre-Patch Time-Frequency Fusion, and Patch Embedding.

3.2.1 Instance normalization

To process variables without cross-channel entanglement, our architecture integrates Reversible Instance Normalization (RevIN) [29] independently across each dimension. Assuming a multidimensional input X ∈ ℝ^*L*×*C*, the statistical moments—namely, the empirical mean and standard deviation σ_*c* for a specific channel c (1 ≤ c ≤ C)—are derived via Equation 2:

μc=1L∑i=1Lxi,c ; σc=1L∑i=1L(xi,c-μc)2+ϵ(2)

In this context, ϵ serves as a minuscule perturbation term to prevent division-by-zero errors. Following this, the channel-specific standardization is executed as Equation 3:

x′:,c=x:,c-μcσc(3)

The normalized sample is subsequently fed into the following Pre-Patch Time-Frequency Fusion Module. The essential feature of RevIN lies in its reversibility: the model retains the channel-specific statistics during normalization. After obtaining the prediction output, these statistics are used to precisely restore the results to the original data scale via inverse normalization as Equation 4:

x^:,c=x^′:,c×σc+μc(4)

3.2.2 Pre-patch time-frequency fusion

To explicitly incorporate global periodic structures, the model constructs parallel temporal and frequency-domain pathways on each individual channel. Through Pre-Patch Time-Frequency Fusion, the model injects globally structured features from the frequency domain while preserving local fine-grained information, thereby enhancing its capacity for unified characterization of multi-scale dynamic patterns. Since the input sequence is real-valued, this fusion is achieved via the rFFT as Equation 5:

xf=rFFT(x′:,c), xf∈ℂd, d=⌊L2⌋+1.(5)

where x′:,c∈ℝL denotes the input sequence of the -th channel before patching, and x_*f* = x_*r* + ix_*i* denotes the frequency-domain signal with x_*r* and x_*i* being its real and imaginary components, respectively. Subsequently, a ComplexLinear layer [30] is employed to jointly learn amplitude and phase features. Given learnable real-valued weight matrices Wr,Wi∈ℝd×d, the complex-valued operation is defined as Equation 6:

{yr=Wrxr-Wixiyi=Wrxi+Wixr(6)

yielding the complex-valued output as Equation 7:

Zc=yr+iyr=ComplexLinear(xft)(7)

This frequency-domain linear mapping enables adaptive modeling of energy spectrum distribution and phase coupling relationships. Subsequently, the frequency-domain features are transformed back to the time domain via the irFFT as Equation 8:

fc=iDFT(Zc), fc∈ℝL(8)

Finally, the frequency-enhanced representation is added back to the original time-domain sequence through an additive residual connection, followed by layer normalization to obtain a stable and scale-consistent output as Equation 9:

x″:,c=LayerNorm(x′:,c+fc)(9)

This residual fusion preserves fine-grained temporal details of the original sequence while incorporating global structural features from the frequency domain, thereby strengthening the model's ability to uniformly characterize multi-scale dynamic patterns.

3.2.3 Patching and embedding

To obtain a structured representation of local temporal semantics, the enhanced single-channel sequence x″c∈ℝL is independently divided into a set of patches. Before patch extraction, right-side replication padding is applied to the enhanced sequence by extending the sequence with its last observed value. This strategy allows the final patch to cover the tail region of the sequence when the input length is not exactly aligned with the stride. It also avoids introducing artificial zero values at the boundary, thereby preserving the local continuity and statistical characteristics near the end of the sequence.

The padded sequence is then reshaped into a patch matrix Xc(p)∈ℝNp×P, where is the number of patches and P is the patch length. The element at the i-th row and j-th column of the patch matrix is defined as Equation 10:

Xc(p)(i,j)=x̄″(i-1)s+j,c, i=1,2,…,Np, j=1,2,…,P(10)

Where x̄″ denotes the padded enhanced sequence, and s denotes the stride controlling the overlap between adjacent patches. Under the right-side padding strategy, the number of patches is given by: Np=⌊L-Ps⌋+2. Subsequently, each patch vector is mapped into a d_*model*-dimensional token space through a shared linear projection. For the i-th patch, this projection is formulated as Equation 11:

Zc(i,:)=Xc(p)(i,:)WE, i=1,2,…,Np(11)

where WE∈ℝP×dmodel and Zc∈ℝNp×dmodel. Here, “shared” indicates that the same projection matrix W_*E* is applied to all patch positions and all channels, rather than assigning separate projection parameters to different patches or variables. This design maps all patch representations into a unified embedding space while avoiding unnecessary parameter redundancy.

To encode the sequential order of patch tokens, a fixed sinusoidal positional embedding is added to the token matrix as Equation 12:

Z~c=Zc+PE, PE∈ℝNp×dmodel(12)

The positional embedding P_*E* assigns a position-dependent vector to each patch index, enabling the model to distinguish the temporal order of patch tokens. Finally, this process produces C independent token matrices{Z~1,...,Z~C}, which serve as the input representations for subsequent channel-wise Transformer encoding.

3.3 Integrated multi-view encoder

The embedded token sequences are stacked and fed in parallel into the integrated multi-view encoder. This module performs parallel encoding in both the temporal and frequency domains, followed by additive residual fusion, to collaboratively model temporal dependencies, spectral patterns, channel interactions, and adaptive cross-channel lag relationships. In this section, denotes the batch size, C denotes the number of input channels, N_*p* denotes the number of patch tokens, and d_model denotes the token embedding dimension.

3.3.1 Time domain layer overview

The temporal encoding layer operates through the coordinated work of MHA, LCCA, and a lightweight feed-forward network (FFN). Given the input of the ℓ -th layer as X(ℓ-1)∈ℝ(B·C)×Np×dmodel, the MHA first captures global dependencies among patch tokens, yielding output features X_attn and the corresponding attention map Ξ̄ℓ. Subsequently, the LCCA module performs channel-aware feature refinement and lag-aware cross-channel aggregation, producing refined features X_lcca. Finally, residual connection, normalization, and a lightweight FFN are applied to obtain the layer output X^(ℓ) as Equations 13 and 14:

Hffn =Conv1D2(ReLU(Conv1D1(Xlcca)))(13)

X(ℓ) =LayerNorm(Xlcca+Hffn)(14)

where Conv₁ and Conv₂ denote pointwise convolutions with a kernel size of 1, and Relu denotes the non-linear activation function. This lightweight FFN performs dimensional transformation and non-linear feature mapping with low computational overhead, while enhancing the representation of local temporal dynamics.

3.3.2 Lag-coupled channel attention

To integrate synchronous channel dependency modeling and asynchronous lag propagation within a unified framework, we design the LCCA module by integrating and extending the complementary mechanisms from [31, 32]. Specifically, LCCA consists of a Temporal Channel Adapter (TCA) branch and a Lag Mixing branch, which jointly model channel-wise importance and lag-aware cross-channel interactions. Figure 3 illustrates the overall processing flow of LCCA.

The output from the MHA, Xattn∈ℝ(B·C)×Np×dmodel, is first reshaped intoE∈ℝB×C×Np×dmodel, where the explicit channel dimension is restored for subsequent channel dependency and lag-aware relation modeling. The representation E is then fed into two complementary pathways.

In the TCA branch, global average pooling is performed over both the patch-token and feature dimensions of E, resulting in channel-wise contextual statistics s ∈ ℝ^*B*×*C*. These statistics are passed through a bottleneck structure to generate channel gating weights w_*c* as Equations 15 and 16:

wc=Sigmoid(W2ReLU(W1s))(15)

Etca=E⊙wc(16)

Where W1∈ℝ(C/r)×C, W2∈ℝC×(C/r), r denotes the channel reduction factor, and ⊙ denotes element-wise multiplication with broadcasting. This branch adaptively scales channel features according to global context, thereby emphasizing informative channels and suppressing less relevant responses.

In the Lag Mixing branch, average pooling is performed only over the patch-token dimension of E, while the feature dimension is preserved to obtain channel embeddings H∈ℝB×C×dmodel. Although both branches contain an average pooling operation, they serve different purposes: The TCA branch produces scalar channel statistics for adaptive gating; The Lag Mixing branch constructs feature-preserving channel embeddings for relation modeling. For any two channels i and j, their embeddings H_*i* and H_*j* are used to compute the cosine similarity score as Equation 17:

Scos(i,j)=Hi·Hj‖Hi||2‖Hj||2+ϵ,(17)

where ϵ is a small positive constant introduced for numerical stability, preventing division by zero when the norm of a channel embedding is zero or close to zero. To reduce interference from weakly related channels, a sparse top-k selection mechanism is introduced. For each target channel, source channels are ranked according to their similarity scores, and only the top-k most relevant source channels are retained. The filtered channel pairs are further processed to extract high-dimensional relational features. These features are then aggregated and fused with the original channel embeddings to obtain the lag-aware channel representation H_lag.

Finally, a dynamic affinity matrix is generated from the normalized similarity between lag-aware channel representations as Equation 18:

Ĥlag=Hi·Hj‖Hlag||2+ϵ,(18)

Saff=ĤlagĤlagT, Adyn=Softmax(Saff)

where the same ϵ is used to ensure numerical stability during the normalization of H_lag. S_aff denotes the affinity score matrix used as the input to the Softmax operation, and A_dyn denotes the normalized dynamic affinity matrix. The Softmax operation is applied along the source-channel dimension to obtain channel mixing weights.

The dynamic affinity matrix is then applied to the original representation E for cross-channel weighted aggregation. The aggregated lag-aware representation is injected into the TCA output through a learnable coefficient α, yielding the refined feature representation as Equation 19:

Xlcca=Etca+α·(Adyn⊗E)(19)

The obtained features X_lcca are reshaped back to ℝ(B·C)×Np×dmodel, restoring the channel-independent format for subsequent layers. Through the coefficient, LCCA balances synchronous channel gating and asynchronous lag-aware mixing, enabling robust characterization of multi-level dynamic dependencies.

3.3.3 Frequency domain layer processing

In the integrated multi-view encoder, the frequency-domain pathway operates as a lightweight spectral branch. It applies rFFT along the patch-token dimension N_*p*, uses a complex-valued linear transformation to jointly model amplitude- and phase-related patterns, and restores the representation through irFFT. This branch is designed to capture cross-patch spectral structures and medium- to long-term periodic patterns, thereby providing a spectral prior that complements the temporal abstraction learned by the time-domain pathway.

Finally, the output of the time-domain branch Z_time and the output of the frequency-domain branch Z_freq are fused through an additive residual connection and then passed through layer normalization to obtain a unified context representation Z_*context* as Equation 20:

Zcontext=LayerNorm(Ztime+Zfreq)(20)

This design leverages the complementarity between the time-domain and frequency-domain pathways to jointly model local temporal dynamics and global periodicity, thereby strengthening the structural coupling of multi-scale features.

3.4 Hierarchical patching stack

To enable progressive modeling of the hierarchical structure whereby local components aggregate into global behavior, we propose a multi-level patching mechanism, termed the Hierarchical Patching Stack. The stack cascades K modules; the -th module takes as input the preceding encoder output Z(k-1)∈ℝB′×Np(k-1)×dmodel (B′ = B × C) and Z⁽⁰⁾ denoting the initial encoder representation.

Within each module, we first perform re-patching: using a sliding window with stride S_*h* and window size P_*h*, we extract subsequences along the patch dimension to form a meta-patch tensor U(k)∈ℝB′×Np(k)×dmodel×Ph. Next, to aggregate the local information within each window into a single feature unit at the current level, we apply a dimension permutation followed by flattening, reshaping U(k)∈ℝB′×Np(k)×dmodel×Ph into F(k)∈ℝB′×Np(k)×(Lh·dmodel). This operation explicitly concatenates, along the channel dimension, the features of the P_*h* adjacent patches covered by the window, thereby producing a high-dimensional representation that encodes the complete local context as Equation 21:

F(k)=Flatten(Permute(U(k)))(21)

Next, a linear projection layer maps the high-dimensional vectors F^(*k*) back to a dimensional space. By effectively enlarging the receptive field, this step consolidates fine-grained patterns within each local window into higher-level semantic tokens. In doing so, it constructs a hierarchical abstraction while preserving temporal coherence. We then inject a new positional encoding into the resulting representation E^(*k*). Subsequently, we apply a Dropout operation to the position-augmented representation. This regularization prevents the model from overfitting to local high-dimensional features and specific positional patterns, thereby improving the robustness and generalization capabilities of the hierarchical representations as Equation 22:

E(k)=Dropout(Proj(k)(F(k))+P(k))(22)

The resulting E^(*k*) is then fed into a new integrated multi-view encoder—sharing the same architecture as in Section 3.3 (including the LCCA module and the joint time–frequency processing)—to extract dynamic features at this coarser scale. The encoder's final output, Z^(*k*), serves as the context representation at the k-th level, as Equation 23:

Z(k)=Encoder(k)(E(k))(23)

After K recursive stages, the model consolidates multi-granularity features and passes the final representation, Z^(*K*), to the prediction head.

3.5 Prediction head

To project the hierarchical representation onto the future time horizon T_*pred*, the model adopts a lightweight shared prediction head. It first receives the final output Z^(*K*) and reshapes it into Efinal∈ℝB×C×Np(K)×dmodel to support channel-wise independent forecasting. The core of this module lies in applying a feature flattening operation to each channel, which merges the patch sequence Np(K) with the feature dimension d_*model* into a unified high-dimensional contextual vector (Zc∈(Np(K)·dmodel)), thereby encapsulating the full-scale historical information of that channel as Equation 24:

Zflat=Flatten(Efinal) (Zflat∈ℝB×C×(Np(K)·dmodel))(24)

Subsequently, we employ a shared linear layer (W_*head*, b_*head*) to map the context vectors directly into the prediction space, and apply dropout as a regularizer to mitigate overfitting, as given by Equation 25:

Ŷraw=Dropout(Linear(Zflat)) (Ŷraw∈ℝB×C×Tpred) (25)

To facilitate the computation of the final empirical loss, a denormalization procedure is first applied to the preliminary predictions, projecting them into the unnormalized domain.

3.6 Loss function

The overall training objective, denoted as L_*total*, is formulated by assigning weights to two distinct components: a forecasting fidelity term L_*pred* and an information constraint L_*reg*. To evaluate the former, we utilize the Mean Squared Error (MSE) criterion, calculating the squared Frobenius distance separating the unnormalized forecasts Ŷ from the actual targets Y as Equation 26:

Lpred=‖Ŷ-Y‖F2(26)

To suppress spurious dependencies in multi-view modeling, an Attn-L-Reg mechanism [33] is introduced. This approach imposes an L1 sparsity constraint on the attention maps Ξ̄ℓ from the initial encoder and the encoders at the ℓ -th layer within the multi-level patch stacking of the HMV-Former. By using L1 regularization to make the attention map as sparse as possible, this mechanism encourages the model to focus on a minimal set of dependencies. This constraint forces the network to discard spurious dependencies and strictly focus on fewer but more effective core temporal patterns, thereby effectively enhancing the model's generalization capability as Equation 27:

Lreg(Ξ̄ℓ)=‖Ξ̄ℓ‖1(27)

Finally, the overall optimization objective is defined as the weighted sum of the prediction loss and the regularization losses from all layers as Equation 28:

Ltotal=Lpred+λreg∑ℓ=1Nreg‖Ξ̄ℓ‖1(28)

where N_reg represents the total number of encoder layers, and λ_*reg* is a unified regularization coefficient that encourages the model to form robust yet sparse attention pathways across all hierarchical levels.

4 Experiments

4.1 Datasets

For a thorough evaluation of HMV-Former's predictive capabilities, our empirical study utilizes 10 practical benchmarks from established application areas. Following common practice in time-series forecasting, we adopt the established dataset-specific partition protocol when a standard setting¹ is available; otherwise, the data are chronologically partitioned into training, validation, and testing sets using a 7:1:2 split ratio. For each dataset, all compared methods are evaluated using exactly the same data partition to ensure a fair comparison. These chosen collections include:

• 1) Electricity Transformer Temperature (ETT)² the ETT dataset is an open-source benchmark for long-sequence time-series forecasting, collected from regional power systems in China over a continuous 2-year period. It records transformer-related operational variables, including oil temperature and multiple load features. Following the standard ETT setting, we evaluate four subsets: ETTh1, ETTh2, ETTm1, and ETTm2. The suffixes “h” and “m” indicate the sampling granularity, where ETTh1 and ETTh2 are recorded at 1-h intervals, while ETTm1 and ETTm2 are recorded at 15-min intervals. The suffixes “1” and “2” indicate two different transformer stations or regional sources. Therefore, ETTh1 and ETTm1 correspond to the same first source under different temporal resolutions, whereas ETTh2 and ETTm2 correspond to the second source under the corresponding resolutions. This setting allows the experiments to assess both station-dependent operating patterns and the effect of temporal granularity on forecasting performance. The historical sequences are chronologically partitioned into training, validation, and testing sets using a 6:2:2 split ratio.
• 2) Exchange³ [34]: spanning the period from 1990 to 2016, this collection provides daily currency conversion rates across eight distinct nations. The temporal data is partitioned chronologically into training, validation, and testing segments following a 7:1:2 split ratio, respectively.
• 3) Tokyo Foreign Exchange (TKFE)⁴ this dataset consists of 14 different daily time series recording the exchange-rate fluctuations of major currency pairs in the Tokyo market from September 28, 2005, to September 9, 2025. The time series are chronologically partitioned into training, validation, and testing sets using a 7:1:2 split ratio.
• 4) Austin Bergstrom International Airport Operational Data (AUS)⁵ this dataset records seven operational indicators for Austin–Bergstrom International Airport in the United States over the period from January 12, 2022, to March 3, 2025, including key metrics such as the number of departing flights and the on-time performance rate. The time series are chronologically partitioned into training, validation, and testing sets using a 7:1:2 split ratio.
• 5) Flight Operation Volume (FlightVol)⁶ this dataset comprises daily time-series observations from China's civil aviation passenger-transport sector over the period from February 13, 2019, to January 13, 2025. Capacity is summarized along two salient dimensions: route scope, including national, domestic, regional, and international routes; and operating carrier. The principal variables include flight volume and the corresponding operation rate for each route category and each airline. The time series are chronologically partitioned into training, validation, and testing sets using a 7:1:2 split ratio.
• 6) Tetouan Power Consumption (TPC)⁷ [35]: gathered throughout 2017 in Tetouan, Morocco, this dataset records power consumption across three zones at 10-min intervals, together with relevant weather features. The sequences are chronologically partitioned into training, validation, and testing sets using a 7:1:2 split ratio.
• 7) Ant Group's Yu'ebao Fund Yield (YBE)⁸ chronicling the period from January 2017 to September 2025, this dataset captures the daily yields of the Yu'e Bao money-market fund. By consolidating the returns of 31 constituent funds, it represents the holistic portfolio performance. The records are chronologically partitioned into training, validation, and testing sets using a 7:1:2split ratio.

The detailed characteristics of the datasets are summarized in Table 1.

Table 1. Datasets information.

4.2 Comparison of models and evaluation metrics

4.2.1 Comparison of models

For a rigorous assessment of HMV-Former's predictive efficacy, we compare our framework with twelve representative baselines that span the dominant architectural paradigms and modeling strategies in contemporary time-series forecasting. The comparison set includes Autoformer [18], which combines series decomposition with an autocorrelation mechanism; FEDformer [21], which integrates frequency-domain transforms with sparse attention; TimesNet [36] which performs generic temporal analysis via multi-period modeling; and MICN [28], which exploits multi-resolution convolutions to extract both short-term and long-range dependencies. We also consider methods that emphasize multivariate dependency modeling, including Crossformer [15] and iTransformer [22]; SOFTS [37], which enables efficient variable interaction through centralized representations; PatchTST [23] and PAttn [38], which strengthen long-horizon modeling via patching and channel-independent designs; as well as lightweight MLP- or linear-based baselines such as TiDE [39] and DLinear [40]. Collectively, these approaches cover the principal methodological spectrum, ranging from architectures guided by strong structural priors to data-driven representation learning.

Given the large number of reference models and the complexity of the forecasting settings, presenting all experimental results in a single table would make the table overly large and reduce its readability. Therefore, we divide the comparison results into two groups and report them in two separate tables. Detailed results are provided in Section 4.4.

4.2.2 Evaluation metrics

For a rigorous assessment of forecasting efficacy, Mean Squared Error (MSE) and Mean Absolute Error (MAE) are adopted as the primary benchmarking criteria. These indices characterize the residual gaps separating the generated sequences from the empirical targets, capturing quadratic penalties and absolute variations, respectively. Consequently, diminished metric readouts denote a tighter alignment with real-world observations, thereby confirming the architecture's superior precision.

Assuming the test set contains N evaluated samples, let y_*i* denote the ground truth and ŷ_*i* denote the predicted value for the i-th sample. The overall MSE and MAE for the model are respectively defined as Equations 29 and 30:

MSE=1N∑i=1N(yi-ŷi)2(29)

MAE=1N∑i=1N|yi-ŷi|(30)

4.3 Experiment setup

4.3.1 Experimental platform

All model implementations and evaluations were programmed utilizing the PyTorch framework. The underlying hardware infrastructure consisted of a single NVIDIA GeForce RTX 4080 GPU and an Intel Xeon Silver 4310 CPU.

4.3.2 Experimental implementation details and hyperparameter settings

Regarding the optimization setup, network weights are updated via the Adam algorithm. We assign an initial learning step of 0.0005 specifically for the AUS and TKFE collections, while adopting 0.0001 for the remaining benchmarks. All experimental trials undergo a maximum of 10 training epochs. To circumvent overfitting, a premature termination protocol is triggered if the validation metrics remain stagnant across three successive training cycles. Architecturally, the base patch dimension and sliding window step are uniformly held at 16 and 8. Within the Hierarchical Patching Stack, the layer depth L_*h*, meta-patch size P_*h*, and meta-stride S_*h* are configured to 3, 2, and 1. Finally, a reduction ratio of 4 is applied to the bottleneck design of the TCA module.

To accommodate the differences in scale and sampling frequency across datasets, this study adopts two sets of input and forecast length configurations. For large-scale datasets such as ETT, Exchange, and TPC, the input window is set to S_*L* = 96 and the model is evaluated over the corresponding forecast horizons P_*L* ∈ {96, 192, 336, 720}. For smaller-scale datasets including TKFE, PHX, FlightVol, and YBE, shorter input windows S_*L* = 36 and forecast lengths are used.

Following the canonical Transformer design paradigm, the FFN dimension is uniformly set to four times the model embedding dimension, and the strength of the attention logic regularization is fixed at 0.001 across all experiments.

To determine the optimal configuration of the model, all remaining hyperparameters were systematically optimized using a grid search strategy. The search space for the general architectural parameters is defined as follows: batch size B_*s* ∈ {16, 32}, the number of temporal processing layers in the integrated multi-view encoder L_enc ∈ {1, 2, 3}, the model's hidden dimension d_model ∈ {128, 256, 512}, and the attention logic regularization coefficient λreg∈{0.1,0.01,10-3,10-4,10-5}.

For the specific parameters of the LCCA module, the bottleneck dimension D_rel is selected from the set {32, 64, 128, 256}, while the adaptive cross-channel lag dependencies mixing coefficient a is tuned via a fine-grained search within the interval [0.1, 1.0] with a step size of 0.1.

The hyperparameter k for sparse channel selection is dynamically adjusted according to the total number of channels C in each dataset, with its value constrained to the range {0, 2, …, ≤ C}. This parameter is used to select the k most relevant source channels for each target channel based on the cosine similarity between channel embeddings. Notably, when k = C or k = 0, the sparse mechanism reduces to a fully-connected mode, i.e., all inter-channel interactions are considered.

As a regularization strategy to improve generalizability, we explored dropout probabilities ranging from 0.05 to 0.3 (in steps of 0.05). Interestingly, the response to this constraint was highly dataset-dependent. While a fixed dropout value of 0.1 provided the best compromise between learning speed and prediction quality for the majority of the datasets, the AUS records required specific dynamic adjustments—namely {0.1, 0.05, 0.2, 0.1}—to maximize accuracy across the respective forecasting horizons.

Table 2 details the optimal configurations for each dataset and forecasting window, derived from an exhaustive grid search. We applied these standardized settings uniformly throughout the benchmarking process, including all baseline comparisons, sensitivity tests, and ablation analyses. To mitigate variance and guarantee empirical robustness, the metrics reported herein represent the average performance over three independent trials using distinct random seeds.

Table 2. Hyperparameter configurations for different datasets across forecast horizons.

4.4 Experiment results

Table 3 presents the comparative results between HMV-Former and models including Autoformer, FEDformer, TimesNet, MICN, TiDE, and DLinear; Table 4 shows the comparison of HMV-Former against PatchTST, Crossformer, iTransformer, MultiPatchFormer, PAttn, and SOFTS. In both tables, bold values indicate the best-performing results within each group.

Table 3. Comparison results of HMV-former with Autoformer, FEDformer, TimesNet, MICN, TiDE, and DLinear.

Bold values indicate the best performance.

Table 4. Comparison results of HMV-Former vs. PatchTST, Crossformer, iTransformer, MultiPatchFormer, PAttn, and SOFTS.

Bold values indicate the best performance.

Overall, HMV-Former demonstrates leading performance across the majority of evaluation scenarios. Among the 40 experimental settings compared in Table 3, it achieves the best results in 32 cases in terms of MSE and in 33 cases in terms of MAE. In the comparably scaled tests presented in Table 4, HMV-Former attains the top performance in 36 and 32 experiments, respectively. This overwhelming and consistent advantage across diverse datasets and multiple forecast horizons indicates that the model maintains superior and robust generalization capability when confronted with dynamically complex time series.

To trace the origin of this performance gap, we first examine models rooted in traditional paradigms. In comparison to HMV-Former, methods such as Autoformer, FEDformer, and DLinear primarily rely on explicit trend or seasonal decomposition, while TimesNet and MICN lean toward modeling with fixed-scale or convolutional receptive fields. TiDE is based on a simplified linear and covariate mapping framework. Although these approaches possess certain modeling capacities under specific stationarity assumptions, they typically decouple the modeling of time-domain dynamics, frequency-domain structures, and inter-channel relationships. This fragmented treatment prevents them from establishing a unified representation that bridges local perturbations, cross-scale structures, and long-term trends, leading to limited performance in high-dimensional, non-stationary scenarios.

To bypass the limitations of these global architectures, recent mainstream research has pivoted toward patch-based paradigms. Methods such as PatchTST successfully extract local information, yet they operate solely at a single scale, lacking complementary validation across multi-view dimensions. Although MultiPatchFormer attempts to extend the patch mechanism, it still relies on pre-defined resolutions and feature concatenation, requiring multi-step decoding to mitigate error accumulation. Furthermore, PAttn and SOFTS are hindered by shallow network depths and flattened global interactions, respectively. Consequently, while existing patching methods heavily optimize the input stage, they systematically lack a deep, hierarchical abstraction mechanism capable of synthesizing macroscopic semantic structures from local granular features.

In contrast, HMV-Former is fundamentally designed to overcome the deficiencies of both aforementioned trajectories. The model employs a hierarchical progressive patching mechanism that dynamically aggregates local dependencies across varying scales, variables, and temporal perspectives, thereby constructing semantically enriched macro-scale representations layer by layer. Unlike flattened or single-dimensional aggregations, this synergistic multi-view abstraction process—empowered by deep time-frequency fusion and lag-aware cross-channel alignment—works in tandem with attention-logit regularization. This structure not only enhances the model's active denoising capability but also effectively suppresses the propagation of spurious dependencies and the error accumulation inherent in multi-step decoding. Ultimately, it is this robust governance of multi-scale information that allows HMV-Former to exhibit significantly higher robustness and predictive accuracy in complex, long-horizon forecasting tasks.

4.5 Ablation study

To assess the contributions of the core components in HMV-Former, this study constructs six ablation variants: HMV-Former/H removes the hierarchical patching stack module, retaining only the initial patching at the input stage; HMV-Former/P omits the pre-patch temporal-frequency fusion and relies solely on patching in the temporal domain; HMV-Former/F discards the frequency-domain encoding branch, preserving only the temporal branch; HMV-Former/C excludes the LCCA module; HMV-Former/L removes the lag-mixing within LCCA, keeping only the channel-wise adaptive scaling; HMV-Former/R eliminates the attention logic regularization, applying no sparsity constraint to the attention maps.

Table 5 presents the ablation results (with bold values indicating the best performance). Across 40 experimental settings, covering 10 datasets and four forecast horizons, the complete HMV-Former achieves the best results in 31 cases for both MSE and MAE, demonstrating a clear and consistent advantage over all ablated variants. This outcome confirms that its superior performance stems from the synergistic and complementary interplay of multiple modules, rather than any single component. Specifically, the temporal-frequency fusion provides global structural priors from the frequency domain; the hierarchical patch-stacking module progressively refines multi-scale semantics in the time domain; the LCCA captures cross-variable dependencies; and the attention logic regularization further suppresses redundant correlations, enhancing generalization. Each module functionally supports and mutually constrains the others—removing any single component disrupts this integrated synergy, leading to systematic degradation in overall performance.

Table 5. Ablation experiment comparison results.

Bold values indicate the best performance.

Despite the overall superior performance, a “removal-induced improvement” is observed on a few specific datasets. For instance, in the high-sampling-density TPC dataset, removing regularization helps retain high-frequency informative signals; in the financial series YBE, which lacks stable periodic structures, discarding frequency-domain fusion avoids imposing mismatched spectral priors. These exceptions do not contradict the global optimality of the complete model; rather, they reflect how well the underlying assumptions of different modules align with particular data distributions, further demonstrating the model's tunable adaptability.

Furthermore, the hierarchical patching stack module proves to be critical across all datasets—its removal (HMV-Former/H) consistently leads to significant performance degradation, indicating that multi-scale progressive abstraction plays a foundational role in bridging local variations and global trends.

In summary, through the synergistic design of multi-scale architecture, temporal-frequency fusion, and channel-wise interaction, HMV-Former maintains optimal performance in general scenarios while retaining the flexibility to adapt to specific data characteristics when needed.

4.6 Hyperparameter sensitivity analysis

To evaluate the robustness of HMV-Former and identify its optimal configuration, this section presents a systematic analysis of two key hyperparameter categories. First, a full-range assessment is conducted for the attention logic regularization coefficient λ_reg, which controls the strength of the sparsity constraint, across its entire value domain λreg∈{0.1,0.01,10-3,10-4,10-5}. As this coefficient directly acts on the loss function and impacts the model's global generalization capacity, its optimal value is highly correlated with the data's signal-to-noise ratio and distributional characteristics. Therefore, validation is performed on all four ETT datasets—ETTh1, ETTh2, ETTm1, and ETTm2—to examine the stability of this sparsity prior under varying temporal resolutions and noise conditions.

Secondly, regarding the hierarchical patching stack parameters, a search space comprising the recursive patch layer count L_*h* ∈ {1, 2, 3, 4, 5}, the base patch lengthP_*h* ∈ {2, 3, 4}, and the base stride S_*h* ∈ {1, 2} is constructed to evaluate the model's capability to progressively abstract information from local details toward global semantics. Given the vast combinatorial space of these structural parameters and the high computational cost of exhaustive training, this study selects ETTh1 and ETTm1 as representative cases, corresponding to low-frequency smooth sequences and high-frequency volatile sequences, respectively. This anchor-based feature analysis not only ensures the reliability of conclusions regarding structural adaptability but also significantly reduces computational overhead, thereby achieving a balance between experimental rigor and efficiency.

4.6.1 Analysis of attention logic regularization parameter

Figure 4 illustrates the impact of the attention logic regularization coefficient at varying magnitudes on the model's forecasting performance. The error trends across datasets reveal that model performance does not improve monotonically with the strength of regularization; instead, it exhibits a distinct pattern of initial decrease followed by an increase, indicating the existence of an optimal range of regularization intensity. When the coefficient is excessively large (λ_*reg* = 0.1), overly stringent sparsity constraints suppress the expression of informative features, resulting in elevated errors. Conversely, when the coefficient is too small (λreg=10-5), the model faces overfitting risks due to insufficient noise suppression. Notably, at a coefficient value of 0.001, the model consistently achieves the best overall performance across all benchmark datasets. This empirical finding not only reveals the crucial role of moderate sparsity in balancing information retention and noise filtering, but also strongly validates the correctness and robustness of the selected parameter (λ_*reg* = 0.001), demonstrating that this configuration can effectively maximize the model's generalization potential.

4.6.2 Analysis of hierarchical patching-stack structure parameters

To systematically evaluate the impact of hierarchical patching parameters, we construct a parameter space comprising the recursive patch layer count (L_*h* ∈ {1, 2, 3, 4, 5}), base patch length (P_*h* ∈ {2, 3, 4}), and base stride (S_*h* ∈ {1, 2}), and conduct our analysis on the ETTh1 and ETTm1 datasets. As the hierarchical patching operation progressively downsamples the sequence length layer by layer, employing large patches (patch lengths of 3 or 4) or large strides (a stride of 2) causes the sequence data to be completely exhausted by the time it reaches the 4th and 5th layers, thereby preventing further model construction. Consequently, this inherent architectural constraint effectively precludes the feasibility of combining deep networks with large patch configurations.

Having filtered out the invalid configurations, we visualize the average forecasting errors across four prediction horizons for various parameter combinations via the heatmaps on the left of Figure 5. It is observable that the model error exhibits a “U-shaped” trend—initially decreasing and subsequently increasing—as both the layer count and patch length grow. Specifically, excessively shallow architectures (one or two layers) or overly coarse patches (patch lengths of 3 or 4) invariably lead to an escalation in error. On both the ETTh1 and ETTm1 datasets, the configuration comprising a 3-layer network and a patch length of 2 achieves the global minimum error within the evaluation matrix, thereby demonstrating robust cross-dataset adaptability.

To investigate the influence of the stride parameter, the middle column of Figure 5 compares the performance of strides 1 and 2 across 1-, 2-, and 3-layer architectures, with the patch length fixed at 2. The empirical results indicate that the discrepancy in average error between the two strides is relatively marginal across all evaluated depths, although a stride of 1 generally delivers superior overall performance. Coupled with the practical observation that a sliding window with a stride of 1 captures the high-frequency temporal continuity more meticulously—and yields a comparatively lower error under the 3-layer configuration on the ETTm1 dataset—we set the base stride to 1 in our experiments.

Furthermore, the line charts on the right of Figure 5 illustrate the error trajectories of 1-, 3-, and 5-layer networks across forecasting horizons of 96, 192, 336, and 720. On the ETTh1 dataset, the 5-layer architecture suffers a drastic spike in error at the 720 horizon, indicative of severe overfitting. Conversely, the error of the 1-layer network remains consistently higher than that of its 3-layer counterpart. Notably, the 3-layer network achieves the lowest error metrics across all prediction horizons. A similar pattern is observed on the ETTm1 dataset, where the 5-layer network fails to yield further error reductions in long-sequence forecasting, while the 3-layer network establishes a sustained performance advantage.

Overall, the 3-layer configuration consistently delivers not only the lowest absolute errors but also the most gradual rate of error accumulation across all evaluated horizons. Synthesizing these empirical findings, we conclude that excessively deep networks are highly susceptible to overfitting in long-horizon forecasting tasks, while coarse patch sizes lead to premature information loss and fail to sustain deep architectural construction. Consequently, we establish an architecture comprising a recursive patch layer count of three, a base patch length of 2, and a base stride of 1 as the default configuration for the proposed model.

4.7 Visual analysis

Figure 6 presents a visual comparison of HMV-Former with TimesNet, DLinear, and PAttn on the AUS dataset, which exhibits high volatility, and the ETTm2 dataset, which shows strong periodicity. The visualized curves are representative samples selected from the test set. In each subplot, the x-axis denotes the relative time-step index within the displayed test sample, and the y-axis denotes the normalized value of the target variable used for visualization.

On the AUS dataset, although DLinear and PAttn can capture the overall trend, their responses to high-frequency fluctuations are relatively weak, leading to a noticeable smoothing effect. On the ETTm2 dataset, the baseline models tend to show amplitude compression or phase deviation around peaks and troughs.

In contrast, HMV-Former better preserves sharp variations in AUS and achieves closer alignment with the phase and amplitude of the periodic patterns in ETTm2. These visual results indicate that HMV-Former has strong fitting ability in both non-stationary and periodic scenarios, which is consistent with its quantitative performance.

5 Conclusion

This paper introduces HMV-Former, a model that facilitates the synergistic modeling of multi-scale and multi-perspective temporal patterns via an integrated multi-view encoder. The multi-level patching scheme explicitly captures the hierarchical “local-to-global” temporal structure, while the embedded LCCA effectively models time-varying cross-view dependencies during channel interactions. Additionally, the Attn-L-Reg mechanism suppresses the propagation of spurious dependencies, thereby substantially enhancing the framework's generalizability capability.

Empirical validations on 10 practical benchmarks spanning sectors like power, transit, and economics show that HMV-Former outperforms current cutting-edge baselines across both complex dynamic modeling and extended-horizon prediction scenarios. Future research will expand its applicability into diverse sequential data applications, including categorization and outlier identification, while concurrently focusing on its lightweight deployment.

Generative AI statement

The author(s) declared that Generative AI was not used in the creation of this manuscript.

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Footnotes

Edited by: Qiang Wu, The University of Tennessee, Knoxville, TN, United States

Reviewed by: Peng Li, Lanzhou University, China

Lucia Maddalena, National Research Council (CNR), Italy

Footnotes

https://github.com/thuml/Time-Series-Library

https://github.com/zhouhaoyi/ETDataset

https://github.com/laiguokun/multivariate-time-series-data

https://www.boj.or.jp/en/statistics

https://www.faa.gov/

https://data.variflight.com/

https://archive.ics.uci.edu/dataset/849/power+consumption+of+tetouan+city

https://www.thfund.com.cn/notice_list