Enhanced Rayleigh Parametrisation Advances

• Enhanced Rayleigh parametrisation is a refined suite of estimation and modeling techniques that extends the classical Rayleigh model by integrating nonlinear subspace and median-based methods.
• It utilizes Lyapunov-stable gradient flows and manifold projection techniques to optimally extract parameters from noisy and limited data, reducing bias and variance.
• Applications span wireless communications, fluid instabilities, and turbulent convection, where models like fdRLoS and slip-enhanced Rayleigh–Plateau improve practical outcomes.

Enhanced Rayleigh parametrisation refers to a suite of modern estimation, transformation, and modeling frameworks that extend or refine the classical Rayleigh distribution, its role in physical and engineering models, and its statistical parameterisation in data analysis. These enhancements target both the estimation of Rayleigh parameters from noisy or small datasets as well as the physical generalisation in systems where standard Rayleigh assumptions fail—particularly in wireless communications, fluid instability, and spectral turbulence modeling.

1. Classical Rayleigh Parameterisation: Foundations and Limitations

The Rayleigh distribution arises when the magnitude of a two-dimensional gaussian vector is measured, leading to the probability density function (PDF)

$$p_X(x;\sigma) = \frac{x}{\sigma^2} \exp \left( -\frac{x^2}{2\sigma^2} \right), \quad x\geq 0.$$

The canonical parameter is the scale $\sigma>0$. Standard parameter estimation in the i.i.d. sample setting is achieved via maximum likelihood (ML),

$$\hat{\sigma}_{\text{ML}} = \sqrt{\frac{1}{2K} \sum_{i=1}^K x_i^2},$$

and via moment or Bayesian methods, which are closed-form and well behaved for large $K$. These estimators, however, break down or exhibit high bias and variance for small sample sizes ($K<50$) or in the presence of adverse noise, prompting advanced approaches.

2. Nonlinear Subspace Estimation: Manifold Projection Techniques

An essential advance is the nonlinear subspace estimator introduced for Rayleigh parameters from short data records (Masoud, 2022). The approach recognises that the entire family of Rayleigh PDFs with varying $\sigma$ forms a one-dimensional nonlinear submanifold in the high-dimensional space of measurement histograms.

Let $Y(\sigma) \in \mathbb{R}^N$ be the model histogram vector evaluated on a fixed grid. The observed histogram $Y_{\text{hist}}$ decomposes as

$$Y_{\text{hist}} = Y(\sigma_0) + Y_{\text{tangential}} + Y_{\text{normal}},$$

where $Y(\sigma_0)$ is the true Rayleigh model, and the other terms represent histogram noise. The estimator seeks the parameter $\hat{\sigma}$ such that the error $D(\sigma) = Y_{\text{hist}} - Y(\sigma)$ is orthogonal (in the tangent-space sense) to the Rayleigh manifold: $J(\sigma)^{\mathsf{T}} D(\sigma) = 0,$ $J(\sigma)$ being the Jacobian of $Y$ with respect to $\sigma$. Update is performed via

$$\sigma_{n+1} = \sigma_n - \eta\, J(\sigma_n)^{\mathsf{T}} D(\sigma_n),$$

a Lyapunov-stable gradient flow that eliminates histogram noise orthogonal to the model subspace.

Performance and Advantages

• For $K=50$, subspace estimator produces mean=0.9956 and var=0.017 (with $\sigma_0=1$), compared to MLE mean=1.0004, var=0.0081, and moment method mean=0.9884, var=0.0193.
• The subspace estimator maintains low bias and variance for $K$ as small as 30, while $L^2$ norm minimisation suffers heavy bias (up to 25% underestimation).
• Computational complexity per iteration is $O(N)$, suitable for DSP/FPGAs; robustness to arbitrary histogram noise is intrinsic, with no need for explicit noise modeling.

3. Generalisations: Median-based and Odd-parameter Extensions

Certain applications require reparametrisation of Rayleigh into alternative domains, notably for bounded random variables on $(0,1)$. The Median-Based Unit-Rayleigh (MBUR) and the Generalized Odd MBUR (GOMBUR) (Attia, 12 Mar 2025) are notable developments.

• MBUR maps the Rayleigh via its median $m=\sigma\sqrt{2\ln2}$ and transforms $X$ as $U = X/(m+X)$, yielding a $\mathrm{Beta}(2,2)$ distribution on $(0,1)$ with a shape parameter $a>0$ introduced: $f(y;a) = 6a\, y^{2a-1}(1-y^a), \qquad 0<y<1.$
• GOMBUR introduces a second "odd median-based" parameter $n$ for enhanced tail and skewness control: $$f(y;a,n) = \frac{\Gamma(2n+2)}{[\Gamma(n+1)]^2} a\, y^{(n+1)a-1}(1-y^a)^n, \quad n\geq0,\, a>0.$$ MLE via Nelder-Mead yields lower asymptotic variance, improved goodness-of-fit indices, and superior empirical fit in diverse datasets compared to MBUR and standard unit-interval competitors.

4. Physical Model Extensions: fdRLoS and Slipped Instabilities

fdRLoS: Fluctuating Double-Rayleigh with LoS

In wireless communications, Rayleigh fading is insufficient where line-of-sight and multi-bounce diffuse scattering coexist and fluctuate (Lopez-Fernandez et al., 2021). The fdRLoS model parameterises the received signal as

$$S = \omega_0\sqrt{\xi}\,e^{j\phi} + \omega_2 G_2 G_3$$

with $\xi\sim\Gamma(m,m)$, $G_2, G_3$ i.i.d. complex Gaussians, $K = \omega_0^2/\omega_2^2$, and $m$ controlling LoS fluctuation. Closed-form expressions are available for PDF, CDF, MGF (involving Kummer confluent and Tricomi functions).

• Tuning $m$ and $K$ smoothly interpolates between pure double-Rayleigh, Rician-shadowed, and classical Rayleigh fading.
• Performance metrics (outage probability, BER) are analytic for arbitrary $K$ and $m$.
• Facilitates direct inference from field-measured distributions.

Slip-enhanced Rayleigh–Plateau Instability

For liquid films on fibres, the traditional Rayleigh-Plateau model is extended by introducing a slip length $l_s$ at the solid interface (Zhao et al., 2022), yielding modified growth rates: $$\omega(k, l_s) = \frac{k^2-1}{2} \frac{K_0(k)\Delta_1 - K_1(k)\Delta_2 + I_0(k)\Delta_3 - I_1(k)\Delta_4}{kK_1(k)\Delta_1 - [kK_0(k)+K_1(k)]\Delta_2 - kI_1(k)\Delta_3 + [kI_0(k)-I_1(k)]\Delta_4}$$ with $I_n, K_n$ modified Bessel functions. The full Stokes solution predicts a slip-dependent dominant wavelength and more accurate growth rates than the classical lubrication model, crucial for interpreting experimental drop formation and designing fibre coating processes.

5. Spectral Parametrisation in Turbulent Convection

Enhanced Rayleigh parametrisation is further exemplified in data-driven models for Rayleigh-Bénard convection (Ephrati et al., 2023). Energy spectra and bulk heat fluxes are enforced via Ornstein-Uhlenbeck mode-by-mode nudging: $$d |c_{k,l}| = L_r(c,k,l) dt + \frac{1}{\tau_{k,l}}[\mu_{k,l} - |c_{k,l}|] dt + \sigma_{k,l} dW_{k,l}(t)$$ where $\mu_{k,l}$, $\sigma_{k,l}$, $\tau_{k,l}$ are empirically inferred parameters, ensuring surrogate models capture reference DNS spectra and Nusselt numbers robustly across several decades of Rayleigh numbers.

6. Implications for Statistical Estimation and Model Design

Enhanced Rayleigh parametrisation frameworks share several features:

Approach	Parameter Control	Typical Use Case
Nonlinear subspace estimator	$\sigma$ only	Short/incomplete records
MBUR/GOMBUR	$a$, ($n$)	Bounded proportions/flexible tails
fdRLoS	$K$, $m$	Wireless multi-path/fading
Slip-Plateau	$l_s$	Fluid instabilities/fibre coating
Spectral O-U forcing	$\mu,\sigma,\tau$	Turbulent flows/convection

These advances offer improved robustness, parameter identifiability, and physical fidelity in scenarios where the classic Rayleigh law is insufficient—from noisy radio data and liquid coatings to turbulent flow fields. They allow refined inference, flexible fitting, and direct inclusion of physics-based or data-driven constraints, often with closed-form solutions and low computational cost. The explicit handling of manifold structure, multi-parameter extensions, and physical boundary effects underpin current best practices in enhanced Rayleigh parametrisation across disciplines.