WT-RDF: Enhanced Structural Analysis

• WT-RDF is a physics-based method that employs the continuous wavelet transform to localize and enhance the features of radial distribution functions in disordered systems.
• WT-RDF⁺ integrates machine learning to tune key parameters, achieving sub-0.02 Å errors in atomic peak positions and improving coordination number accuracy.
• WT-RDF also provides a robust statistical signal analysis tool by connecting the zeros of the wavelet scalogram to closed-form pair correlation functions.

The wavelet-transform radial distribution function (WT-RDF) is a physics-based analytic method for capturing the local and medium-range structural features of irregular, non-periodic systems such as amorphous solids and complex glasses. WT-RDF applies the continuous wavelet transform to radial distribution functions (RDFs) or to the zero-point distributions in time–scale or distance–scale space, enhancing the identification of local ordering and resolving features that are poorly distinguished in conventional Fourier-based or histogram-based RDFs. WT-RDF and its machine-learning–tuned variant, WT-RDF⁺, have demonstrated superior performance in both predicting atomic peak positions and amplitudes (first- and second-neighbor shells) and enabling accurate extraction of coordination numbers, especially for Ge–Se and Ag–Ge–Se glass systems (Senjaya et al., 19 Dec 2025). An alternative formulation, in signal-processing contexts, connects the zeros of the wavelet scalogram to analytic zero processes, for which the pair correlation—the WT-RDF—admits closed-form statistical characterization (Abreu et al., 2018).

1. Theoretical Foundations

The classical radial distribution function $g(r)$ quantifies the probability density of finding an atom at a distance $r$ from a reference atom, formally defined as

$g(r) = \frac{1}{4\pi r^2 \rho_0}\frac{dN(r)}{dr}$

where $dN(r)$ is the number of atoms in a spherical shell at distance $r$, and $\rho_0$ the average atomic density. $g(r)$ arises directly from the Fourier transform of the structure factor $S(q)$. In amorphous or disordered materials, the limited $q$-resolution of experimental $S(q)$ leads to poorly resolved or broadened peaks in $g(r)$, particularly beyond the first shell.

WT-RDF introduces a continuous wavelet transform of $g(r)$:

$$\text{WT–RDF}(r, s) = \int_0^\infty g(r')\,\psi\left(\frac{r'-r}{s}\right)\frac{dr'}{\sqrt{s}}$$

with $\psi(x)$ the chosen mother wavelet and $s$ a positive scale parameter. The WT-RDF thus acts as a localized, multi-scale filter, enhancing features and resolving peaks more distinctly than Fourier-based approaches.

Specialized construction of $\psi(x)$ is fundamental: in the context of amorphous Ge–Se systems, $\psi$ is defined by a mean-field-inspired function in $q$-space involving polynomial/Bessel-type terms and a short-range repulsive–attractive envelope,

$$D(qr) = \sum_{s=0}^\infty \beta_{s, p}\left(\frac{qr-b}{2a}\right)^{2s+p} + K_f\,\Phi\left(\frac{qr-b}{a}\right) + \widetilde{C}$$

with $\Phi(x)=x^4\exp[-x^2]$ for short-range modulation (Senjaya et al., 19 Dec 2025). $\beta_{s, p}$ are constrained for orthogonality and zero-mean.

In the analytic signal-processing formulation, the WT-RDF describes the second-order (pair) statistics of the zero set of the wavelet transform applied to noise, with explicit closed-form equations for the pair correlation function $g(r)$ parameterized by the pseudo-hyperbolic distance in the scale–time plane (Abreu et al., 2018).

2. Parameterization and Model Structure

WT-RDF is characterized by a small set of physically interpretable parameters:

• $a$: scale/stretch of the wavelet argument (controls breadth of features)
• $b$: horizontal shift in $q$-space (aligns peak centers)
• $K_f$: amplitude of the short-range envelope ($\Phi$)
• $\widetilde{C}$: additive constant (vertical bias)
• $\Lambda$: global amplitude scaling
• $\{\beta_{s, p}\}$: coefficients in the polynomial expansion (subject to orthogonality)
• $s_\text{min}, s_\text{max}$: integration limits for wavelet scales (usually held fixed)

Amplitudes and peak positions in $\hat{G}(r)$, the model RDF, are sensitive to these parameters. Coordination number (CN) fidelity, important for physical interpretation, is determined by amplitude accuracy:

$$\mathrm{CN} = \int_{r_\text{lo}}^{r_\text{hi}} 4\pi r^2 g(r) dr$$

Errors are assessed globally (mean absolute error, MAE) and for local features (first- and second-peak errors, FPE and SPE). In conventional WT-RDF, peak alignment is high but amplitude accuracy is limited, leading to errors in quantitative metrics like CN (Senjaya et al., 19 Dec 2025).

3. Machine Learning–Aided Parameter Optimization: WT-RDF⁺

WT-RDF⁺ extends WT-RDF by treating the key physical parameters $\theta = \{a, b, K_f, \widetilde{C}, \Lambda\}$ as learnable and optimizing them via gradient-based machine learning using experimental RDF datasets. The training procedure includes:

• Input: Z-score normalized $r$ values; output: target $g(r)$
• Two-stage optimization: unconstrained training (100 epochs, higher learning rate), followed by fine-tuning with tight bounds on critical parameters (e.g., $a\in[0.60, 0.61]$, $K_f\in[0.01, 0.30]$)
• Loss: weighted combination of overall curve MAE and selective peak loss focused on bins containing key peak regions.

The procedure enables robust sharpening of predicted peaks and corrects amplitude scaling, resolving the limitations of the static WT-RDF. Notably, WT-RDF⁺ maintains strong performance (low FPE and SPE) even when trained on only 25% of available data, outperforming standard ML regressors such as radial basis function (RBF) networks and long short-term memory (LSTM) models in both amplitude and peak accuracy. Ablation studies confirm that parameter bounding and selective peak loss are critical for peak fidelity (Senjaya et al., 19 Dec 2025).

Data Ratio	Method	MAE	FPE	SPE
25 %	RBF	0.8522	4.9250	1.2345
25 %	LSTM	0.8365	5.3931	0.6486
25 %	WT-RDF⁺	0.7296	0.0620	0.0444

WT-RDF⁺ also generalizes: parameters fit on a binary Ge–Se dataset transferred to Ag-doped ternaries (with x = 5–25%), with consistently lower errors compared to untuned WT-RDF.

4. Applications in Structural Characterization

WT-RDF and WT-RDF⁺ have been applied extensively to chalcogenide glasses, specifically Ge₀.₂₅Se₀.₇₅ binary systems and Ag-doped ternaries $$\mathrm{Ag}_x(\mathrm{Ge}_{0.25}\mathrm{Se}_{0.75})_{100-x}$$ for $x=5,10,15,20,25$. Key outcomes include:

• Accurate reproduction of first- and second-neighbor distances (~2.35 Å, ~3.85 Å)
• Precise tracking of compositional shifts and broadening of RDF peaks under varying Ag content (2–5% position error)
• Enhanced sensitivity for network fragmentation and medium-range order, previously masked in Fourier-based RDFs

In the context of phase-change materials (PCM), such as Ge–Se-based thin films, WT-RDF⁺ provides rapid, in-line capable structural metrology, supporting parameter extraction (bond lengths, coordination numbers) critical for modeling electronic and thermal properties in device applications (Senjaya et al., 19 Dec 2025).

5. WT-RDF for Statistical Signal Analysis

In parallel, the WT-RDF emerges in the study of signal filtering, especially as the pair-correlation function of the zero set of the continuous wavelet transform of white Gaussian noise. For the analytic Cauchy mother wavelet $\psi_\alpha$, the zeros in the upper half-plane are distributed as those of a hyperbolic Gaussian analytic function. Explicit forms for the first intensity and pair-correlation function $g(r)$ of the zero point process enable local estimators for deviation detection:

• First intensity: $\rho(z) = \alpha/(4\pi y^2)$
• Pair-correlation (WT–RDF proper): closed form in terms of $r$, $\alpha$, and $s=1-r^2$

Empirical estimators for $g(r)$ constructed from observed zeros can detect structured signal components against background noise in acoustic time–frequency representations (Abreu et al., 2018).

6. Advantages, Limitations, and Outlook

WT-RDF and WT-RDF⁺ provide a highly interpretable, physics-guided parametrization (five learnable parameters) that remains robust under significant data scarcity and exhibits strong transferability across compositionally related amorphous systems. WT-RDF⁺ specifically achieves sub-0.02 Å errors in peak positions and amplitudes, supporting quantitative coordination statistics within typical experimental uncertainties (±0.1 neighbors) (Senjaya et al., 19 Dec 2025).

Limitations include:

• Chemistry-specific construction of the wavelet kernel $D(qr)$; adaptation to new systems may require rederivation or reparameterization of $\psi$
• Computational cost in evaluating the wavelet integral with highly sampled $S(q)$ or extended $q$-ranges
• Selective loss functions may underemphasize features beyond the first two RDF peaks, and performance for crystalline or strongly ionic systems remains untested.

In statistical signal analysis, WT-RDF is positioned as a rigorous, phase-less, and amplitude-free filtering tool, grounded in the theory of analytic zero sets and exhibiting robust match between empirical and theoretical statistics under null (noise) conditions (Abreu et al., 2018).

WT-RDF and its ML-enhanced variant, WT-RDF⁺, constitute an advanced framework for resolving atomic-scale order in amorphous materials and for signal analysis, revealing new avenues for efficient materials design and robust feature extraction in noisy environments.