Gramian Angular Difference Field

• Gramian Angular Difference Field is a time series transformation that converts 1D signals into skew-symmetric 2D matrices preserving phase differences and temporal order.
• It leverages trigonometric relationships to map normalized data, enabling convolutional feature extraction in applications such as biomedicine, autonomous systems, and process analytics.
• Optimized implementations using buffer precomputation, incremental windowing, and fixed-point arithmetic facilitate its effective integration into deep learning pipelines.

The Gramian Angular Difference Field (GADF) is a structured, bijective 2D representation of a univariate or multivariate time series, encoding the pairwise angular difference between normalized values in a matrix format suitable for convolutional feature extraction. By leveraging trigonometric relationships, the GADF maps 1D signal dynamics into a spatial domain where temporal order and directional relationships are preserved. This transformation has been adopted in deep learning pipelines for biomedicine, autonomous systems, process analytics, and stochastic trajectory analysis, regularly yielding improvements over direct sequence or basic spectro-temporal embeddings.

1. Mathematical Definition and Construction

Given a real-valued time series $X = \{x_1, x_2, \ldots, x_n\}$, the GADF construction follows three steps:

1. Normalization: The original sequence is linearly scaled into $[-1, 1]$ so that trigonometric maps are well-defined:

   $$\tilde{x}_i = 2 \cdot \frac{x_i - \min(x)}{\max(x) - \min(x)} - 1$$

   or, equivalently, as in some sources,

   $$\tilde{x}_i = \frac{(x_i - \max X) + (x_i - \min X)}{\max X - \min X}$$

2. Polar Encoding: Each normalized value $\tilde{x}_i$ is mapped to an angle $\phi_i = \arccos(\tilde{x}_i) \in [0, \pi]$, optionally paired with a radius $r_i = i/n$ to encode the chronological order.
3. Gramian Matrix Construction: The $n \times n$ GADF matrix is defined as

   $\mathrm{GADF}_{i,j} = \sin(\phi_i - \phi_j)$

   With trigonometric expansion, this can be computed using only the normalized values:

   $$\mathrm{GADF}_{i,j} = \tilde{x}_j \sqrt{1 - \tilde{x}_i^2} - \tilde{x}_i \sqrt{1 - \tilde{x}_j^2}$$

The resulting matrix is skew-symmetric, with zeros along the diagonal. This procedure is universally employed in the cited literature and underpins all practical applications of the GADF (Elmir et al., 4 Nov 2025, Yousuf et al., 2023, Elmir et al., 2023, Kothari et al., 2021, Qin et al., 7 Dec 2024, Garibo-i-Orts et al., 2023, You et al., 2023, Pfeiffer et al., 2021).

2. Distinction from Gramian Angular Summation Field (GASF)

The GASF provides a symmetric alternative by encoding pairwise angle sums: $$\mathrm{GASF}_{i,j} = \cos(\phi_i + \phi_j) = \tilde{x}_i \tilde{x}_j - \sqrt{1-\tilde{x}_i^2}\sqrt{1 - \tilde{x}_j^2}$$ Key differences are:

• Symmetry: GASF is symmetric, GADF is skew-symmetric (i.e., antisymmetric: $\mathrm{GADF}_{i,j} = -\mathrm{GADF}_{j,i}$).
• Interpretation: GASF encodes absolute correlations (alignment), while GADF encodes directional, phase-difference relationships, making it especially sensitive to transitions and temporal orientation.
• Diagonal Structure: GASF's main diagonal contains ones (perfect auto-correlation); GADF's main diagonal is zero (no phase difference).
• Information Content: GADF representation preserves both fine-grained fluctuations (local transitions) and global structure; GASF emphasizes aggregate pairwise phase (Yousuf et al., 2023, Elmir et al., 4 Nov 2025, Garibo-i-Orts et al., 2023, Qin et al., 7 Dec 2024).

3. Algorithmic Implementation and Computational Aspects

For practical deployment, particularly on edge/IoT devices or large datasets, several implementation strategies have emerged:

• Buffer Precomputation: Compute and cache all $\phi_i$ and $\sqrt{1-\tilde{x}_i^2}$ to avoid repeated expensive operations.
• Symmetry Exploitation: Only compute the upper (or lower) triangle, reflecting the skew-symmetry to fill the matrix.
• Fixed-Point Arithmetic: Employ lookup tables or integer arithmetic for trigonometric computations in resource-constrained settings (Elmir et al., 4 Nov 2025).
• Incremental Windowing: For long signals, apply GADF on sliding windows to control memory and enable streaming computation (Elmir et al., 2023).
• Downsampling: Resize high-dimensional GADF matrices (e.g., $N \times N$ for $N > 100$) to lower resolutions (e.g., $32 \times 32$ or $128 \times 128$) for ingestion by CNNs or ViTs (Yousuf et al., 2023, Elmir et al., 4 Nov 2025).
• Multichannel Extension: For multivariate input, stack GADF images (one per channel/lead/perspective) as parallel channels, forming 3D tensors suitable for CNNs or Vision Transformers (Kothari et al., 2021, You et al., 2023, Pfeiffer et al., 2021).

4. Applications Across Signal Modalities and Tasks

The GADF transformation supports a range of deep learning and representation learning tasks, including:

Domain	GADF Role	Reference
ECG Classification	Encoding 1D ECG signals as 2D images for high-accuracy arrhythmia and MI detection via CNN; boosting federated learning on IoT devices	(Elmir et al., 4 Nov 2025, Yousuf et al., 2023, Qin et al., 7 Dec 2024, Elmir et al., 2023)
EEG Regression	Converting multi-channel EEG to stacked GADF tensors for attention estimation with 2D/3D CNNs	(Kothari et al., 2021)
Diffusion Analysis	Transforming single-particle trajectories to GADF images, enabling regime/exponent inference using pretrained vision models	(Garibo-i-Orts et al., 2023)
AV Behavior	Stacking GADF images for multivariate driving features, utilizing ViT backbones for behavior classification	(You et al., 2023)
Business Analytics	Mapping multivariate process traces to GADF images for representation learning and downstream predictive analytics	(Pfeiffer et al., 2021)

ECG-centric studies consistently report GADF as marginally outperforming GASF in single-modality ablation (accuracy improvements of $0.1{-}1.0\%$) (Yousuf et al., 2023, Qin et al., 7 Dec 2024). In multimodal pipelines (e.g., GAF-FusionNet (Qin et al., 7 Dec 2024)), GADF channels provide complementary discriminative information, especially for subtle waveform variations, leading to state-of-the-art results in clinical and commercial tasks.

5. Mathematical and Representational Properties

GADF matrices inherit several important mathematical properties, traceable to their trigonometric definition:

• Skew-Symmetry: $\mathrm{GADF}_{i,j} = -\mathrm{GADF}_{j,i}$; main diagonal strictly zero.
• Range: Each entry has $|\mathrm{GADF}_{i,j}| \leq 1$ since $\phi_i \in [0, \pi]$.
• Temporal Dependency: The $(i,j)$ entry encodes both the temporal distance and phase-relationship, preserving local and long-range correlations.
• Bijectivity: The transformation is invertible (modulo floating-point and boundary effects) since the angular mapping is monotonic on $[-1,1]$ (Elmir et al., 2023).
• Sensitivity to Directionality: By capturing phase-differences, GADF highlights upward vs. downward transitions—crucial for discriminating physiological events (QRS complexes, arrhythmia morphology) or stochastic regime shifts (Yousuf et al., 2023, Garibo-i-Orts et al., 2023).

6. Integration with Deep Learning Architectures

GADF images are used as direct 2D inputs for convolutional architectures:

• 2D CNNs: These exploit texture, symmetry, and spatial patterns native to GADF matrices (Yousuf et al., 2023, Elmir et al., 2023, Kothari et al., 2021).
• Vision Transformers (ViT): GADF images can be patchified and combined with channel attention for multivariate sequence classification (You et al., 2023).
• Cross-Modal Fusion: In frameworks like GAF-FusionNet (Qin et al., 7 Dec 2024), GADF and GASF features are fused with temporal representations via dual-layer split attention, leading to enhanced discriminative power.
• Self-Supervised Representation Learning: GADF-based process representations enable self-supervised pre-training (e.g., via predicting next-events), producing dense, generalizable embeddings for downstream tasks (Pfeiffer et al., 2021).

Empirical findings show that GADF-based pipelines consistently match or outperform traditional 1D DL models and alternative 2D mappings, with state-of-the-art results on diverse datasets (ECG, EEG, stochastic trajectories, business logs), as evidenced by superior accuracy, F1, and recall metrics (Qin et al., 7 Dec 2024, Yousuf et al., 2023, Garibo-i-Orts et al., 2023, Pfeiffer et al., 2021).

7. Limitations and Operational Considerations

GADF transformation presents several practical challenges:

• Quadratic Scaling: The $n \times n$ matrix is costly for very long sequences ($n \gg 1000$), mandating windowing or resizing (Elmir et al., 2023, Garibo-i-Orts et al., 2023).
• Padding/Truncation: To form consistent image inputs, fixed-length normalization may be required, possibly truncating relevant context (Pfeiffer et al., 2021).
• Computational Overhead: While highly parallelizable, GADF computation can tax resource-constrained edge devices, necessitating algorithmic optimizations (lookup tables, upper-triangle calculation, etc.) (Elmir et al., 4 Nov 2025).
• Comparison to GASF/Alternatives: Though often slightly superior, GADF does not universally outperform GASF for all tasks/domains (Yousuf et al., 2023, Qin et al., 7 Dec 2024). For certain global-pattern inputs, GASF can be optimal; hybrid or fused approaches are common.
• Interpretability: While pattern-localizing, GADF image features are nontrivial to interpret directly, relying on model-driven saliency for diagnostic confirmation.

This suggests ongoing advances may target computational efficiency, domain-specific adaptation, and improved fusion with other representation paradigms for both interpretability and robustness.