Temporal 2D Variation Modeling

• Temporal 2D-Variation Modeling is a method that reshapes sequential data into 2D matrices to explicitly capture both intra-cycle (local) and inter-cycle (global) dependencies.
• It leverages advanced techniques like 2D convolutions, attention mechanisms, and graphical models to improve forecasting, anomaly detection, and image reconstruction.
• Empirical evaluations demonstrate improvements of 4–12% in error metrics while offering scalability for high-dimensional spatio-temporal and physiological data.

Temporal 2D-Variation Modeling is a paradigm that reformulates temporal data or spatio-temporal processes to expose, exploit, and model the interaction between two distinct temporal dimensions or axes. The transformation from inherently 1D or sequential temporal data to a 2D or matrix-tensor representation enables the use of richer operator classes (such as 2D convolutions, attention, and graphical models) to simultaneously capture both intra-cycle (local, within-period) and inter-cycle (global, cross-period) dependencies. This approach has impacted broad domains, including time series forecasting, spatio-temporal statistics, image reconstruction in dynamic systems, physiological signal analysis, and generative modeling of human motion. The unifying feature is the explicit factorization, reshaping, or functional decomposition of time into at least two interacting axes—commonly periodic structure (seasonality) and directional, linear progression. This article surveys the foundational mathematical models, algorithmic instantiations, and the empirical advances under the umbrella of Temporal 2D-Variation Modeling.

1. Mathematical Foundations and Motivations

Temporal 2D-Variation Modeling seeks to address the inadequacy of 1D models to disentangle or localize overlapping cyclic and non-cyclic temporal effects. Many real-world temporal signals are dominated by multiple periodicities coupled with non-periodic trends, abrupt changes, or local fluctuations. Canonical mathematical constructions include:

• Two-way period decomposition: A 1D sequence $x_t$ is mapped to a $p \times f$ array, where $p$ indexes intra-period (e.g., hour within a day), and $f$ indexes period (e.g., day number) (Wu et al., 2022, Nematirad et al., 31 Mar 2025, Li et al., 2024).
• Product temporal domains: Random fields on $T = \mathbb{R} \times S^1$ combine a linear axis with a circular/seasonal axis, supporting covariance structures of the form $C(h, u, \omega)$, where $u$ is linear time lag and $\omega$ is periodic-phase lag (Alegría et al., 2017).
• Matrix-variate time series: $X_t \in \mathbb{R}^{p_1 \times p_2}$ factorized as $A F_t B'$, where temporal dynamics are reduced to low-dimensional surfaces $F_t$ (Gao et al., 2020).
• Spatio-temporal variations: High-dimensional spatial-temporal data $Y(s, t)$ parameterized as $Y(s, t) = \Lambda(s) f(t) + \varepsilon(s, t)$, with $\Lambda(s)$ (2D spatial loading) and $f(t)$ (latent temporal factors) (Chen et al., 2020).

The primary motivation across these settings is to enable statistically and computationally efficient nonseparable modeling of temporally complex phenomena, allowing for the tailored use of 2D learnable operators and spectral analysis.

2. Temporal 2D-Variation in Sequence Modeling

In modern time series analysis, 2D-variation modeling is instantiated by constructing multiple 2D tensor representations for periodic components, leveraging Fast Fourier Transform (FFT)-based detection of dominant periods:

• Period extraction: FFT is applied to extract $k$ dominant frequencies $f_i$ (excluding DC), with period lengths $p_i = T / f_i$ (Wu et al., 2022, Li et al., 2024, Nematirad et al., 31 Mar 2025).
• Reshaping: The raw or embedded 1D time series $X_{1D} \in \mathbb{R}^{T \times C}$ is zero-padded as necessary and reshaped to $k$ tensors $$X^{(i)}_{2D} \in \mathbb{R}^{p_i \times f_i \times C}$$.
• 2D operator processing: Each 2D block is processed with shared, parameter-efficient Inception- or ResNet-style 2D CNNs, or Swin-transformer blocks, to extract intra- and inter-period patterns (Wu et al., 2022, Li et al., 2024).
• Aggregation: Frequency-detected amplitudes weight the outputs adaptively; softmax-weighted sum or concatenation is performed across all blocks.
• Extensions to derivatives: Higher-order difference heatmaps (e.g., 1st/2nd derivatives) are computed, stacked as a 2D “heatmap,” and fed to lightweight 2D convolutional modules to localize sharp transitions and turning points (Nematirad et al., 31 Mar 2025).

This approach has enabled multitask backbones (forecasting, imputation, anomaly detection, and classification) that outperform specialized 1D-convolutional, recurrent, and attention-based baselines by capturing nuanced, multi-frequency temporal variation (Wu et al., 2022, Nematirad et al., 31 Mar 2025, Li et al., 2024).

3. Statistical and Geostatistical Formulations

Temporal 2D-variation has a parallel statistical theory in covariance modeling and high-dimensional time series:

• Space-time random fields: The domain is expanded to $\mathcal{X} \times (\mathbb{R} \times S^1)$, encompassing space, linear time, and periodic time (Alegría et al., 2017).
• Nonseparable covariance classes:
  • Gneiting-type extension:

  $$C_{1}(h,u,\omega) = \frac{\sigma^2}{\psi(|u|)^{d/2}} \varphi \left( \frac{\|h\|}{\psi(|u|)^{1/2}} \right) \exp\{\kappa\cos\omega\}$$ - Lagrangian framework:

  $$C_{2}(h,u,\omega) = \mathbb{E}_{V,W}\left[\varphi(\|h - uV - \omega W\|)\right]$$

  where $V$ (velocity) and $W$ (circular drift) are random vectors (Alegría et al., 2017).

• Estimation: Maximum likelihood (for moderate sample sizes); composite or pairwise likelihoods for $n \gtrsim 10^4$; empirical variogram fitting for initialization. Covariance tapering and circulant embedding can amortize computational cost (Alegría et al., 2017).

Empirical gains of 7–12% RMSE reduction were observed when incorporating both linear and seasonal lags, compared to single-dimension models. Extensions to global domains exploit intrinsic geometry ($\mathbb{S}^2$) and corresponding positive-definite kernels.

4. Temporal 2D-Variation in Inverse Problems and Image Reconstruction

Variational inverse problems in dynamic imaging explicitly penalize spatial and temporal variation interactions:

• Continuous Models: Objective functions consist of three terms:

$$\int_0^T \Big( \mathcal{D}(\mathcal{A}(t, f(t)), g(t)) + \mathcal{R}_{\text{spat}}(f(t)) + \mathcal{R}_{\text{temp}}(\partial_t f) \Big) dt$$

where $\mathcal{D}$ is a data fidelity term, and $\mathcal{R}$ are spatial and temporal regularizers (Hauptmann et al., 2020, Chen et al., 2018).

• Motion-constrained Models: Impose PDE constraints such as the continuity equation $\partial_t f + \nabla \cdot (v f) = 0$ or use LDDMM-based diffeomorphic flows for template deformation over time. Sequential indirect registration minimizes:

$$E[u_0, v] = \int_0^1 \left( D_t(\varphi_{0,t}\cdot u_0) + \mu_2 R_2[\varphi_{0,t}] \right) dt + \mu_1 R_1(u_0)$$

where $u_0$ is the template and $\varphi_{0,t}$ is the flow (Chen et al., 2018).

• Numerical strategies: Alternating minimization between image and motion fields; linearized deformation for computational efficiency.

This framework yields sharper recovery, significant improvements in SSIM/PSNR metrics under extreme data sparsity or noise, and robust performance to hyperparameter variation (Hauptmann et al., 2020, Chen et al., 2018).

5. Temporal 2D-Variation in Physiological Signal Analysis and Generative Modeling

Beyond classical time series and images, temporal 2D-variation has been leveraged in:

• Physiological variability analysis: Heart rate sequences are mapped via second-order difference plots to $(X(n), Y(n))$ points. Temporal Variation Measure (TVM) further augments these to 3D via a local acceleration-magnitude coordinate $Z_{TVM}$:

$$Z_{TVM}(n) = \left|Y(n)\right| - \left|X(n)\right| \cdot \mathrm{sigmoid}(L_E(n)/\mu_E)$$

A weighted Shannon entropy over this 3D cloud (E_{Tv}) achieves superior discrimination among physiological classes relative to standard central tendency or distance metrics (Diao et al., 2022).

• Motion generative models: Human motion sequences (frames $\times$ joints) are vector-quantized into $T \times J$ 2D token grids, processed by joint-level VQ-VAE and 2D transformers, employing spatial-temporal masking and multi-head attention. This preserves spatial dependencies while capturing temporal dynamics, yielding state-of-the-art FID and MPJPE scores (Yuan et al., 2024).

This suggests that 2D-variation modeling enables both higher discrimination (for physiological status) and better synthesis (for motion) by incorporating short- and long-range temporal interactions in a 2D lattice.

6. Architectural Paradigms and Computational Considerations

Temporal 2D-variation modeling facilitates the migration of computer vision tools to time-series and spatiotemporal data:

• Parameter sharing: Multi-periodicity is handled without parameter blow-up through shared 2D backbone weights across all period/frequency branches (Wu et al., 2022, Nematirad et al., 31 Mar 2025).
• Adaptive fusion: The importance of each periodic block is inferred from signal energy and used in softmax-based fusion; similar schemes apply to derivative-based modules (Wu et al., 2022, Nematirad et al., 31 Mar 2025).
• Alternative backbones: 2D vision architectures (ConvNeXt, Swin, ResNeXt, etc.) can be substituted, delivering up to 1% additional improvement in downstream metrics (Wu et al., 2022).
• Computational efficiency: 2D representation typically reduces receptive field size, enabling hierarchical modeling at lower memory overhead compared to 1D attention/transformer models.
• Scalability: Composite-likelihood and circulant embedding extend tractability to $n>10^4$ for statistical models (Alegría et al., 2017, Chen et al., 2020).

A plausible implication is that this approach will be further leveraged in foundational, pre-trained backbones for time-series and dynamic spatial data.

7. Empirical Evaluation and Predictive Gains

Temporal 2D-variation modeling delivers measurable improvements across extensive empirical evaluations:

Domain	Task	Typical Gains	Reference
Time series	Forecasting	4–12% lower MSE/MAE, best OWA/SMAPE in >80% of cases	(Wu et al., 2022, Nematirad et al., 31 Mar 2025, Li et al., 2024)
Anomaly detection	Reconstruction	+1.0% F1 over Transformer/Autoformer	(Wu et al., 2022)
Classification	UEA datasets	+1.1–6.1% vs strong 1D/MLP baselines	(Wu et al., 2022)
Spatio-temporal statistics	RMSE (OOS)	7–12% relative reduction with 2D temporal covariance	(Alegría et al., 2017)
Dynamic medical imaging	SSIM, PSNR	Up to +8 dB PSNR, +0.2 SSIM relative to 1D/TV	(Chen et al., 2018, Hauptmann et al., 2020)
Motion generation	FID, MPJPE	26–30% FID drop, 50% MPJPE drop	(Yuan et al., 2024)
Physiological signal	Clustering (RI)	+0.06–0.36 in RI over classical SODP	(Diao et al., 2022)

Tasks encompassed both short-term and long-term forecasting, missing value imputation, probabilistic classification, and high-dimensional generative synthesis.

8. Extensions and Future Directions

Research on temporal 2D-variation modeling is advancing on several theoretical and applied axes:

• Pretraining and foundation models: TimesNet’s parameter-sharing and efficiency are seen as enabling pre-trained models for universal time-series tasks (Wu et al., 2022).
• Irregular, asynchronous, or continuous-time sampling: Extensions of 2D reshaping to non-uniform or event-based data are being investigated (Wu et al., 2022).
• Generalization to spherical and non-Euclidean domains: Spherical expansions (Schoenberg, Gegenbauer) allow 2D-variation principles to model global, geodesic spatio-temporal fields (Alegría et al., 2017).
• Integration with exogenous and multimodal inputs: Contextual/covariate fusion modules demonstrate robust performance in settings with auxiliary information (Li et al., 2024).
• Self-supervised representation learning: 2D-masked modeling and 2D-variation contrastive objectives are a direction for learning universal representations that generalize beyond single-task objectives (Wu et al., 2022, Yuan et al., 2024).

In sum, Temporal 2D-Variation Modeling establishes a robust theoretical and algorithmic basis for disentangling, processing, and forecasting in domains dominated by coupled or multi-scale temporal structures. Its integration with classical statistical theory, deep learning architectures, and dynamic systems positions it as a scalable foundation for future advances in spatio-temporal analytics, generative modeling, and real-time sequence understanding.