Ray Regression Predictor Model

• Ray Regression Predictor is a regression methodology for positive real-valued responses that extends the Rayleigh distribution framework with generalized techniques.
• It leverages bias-adjusted maximum likelihood estimation using methods such as Firth’s and Cox & Snell corrections to improve parameter estimation.
• Applications include SAR imaging amplitude analysis and biomedical survival studies, using flexible link functions to capture covariate effects.

A Ray Regression Predictor is a regression-based methodology built on parameterizations and extensions of the Rayleigh and generalized Rayleigh (GR) distributions. It enables predictive modeling for positive real-valued responses, particularly in contexts such as amplitude data in synthetic aperture radar (SAR) imaging and time-to-event biomedical applications. Two principal frameworks are established: the standard Rayleigh regression model with bias-adjusted estimation, and an extended four-parameter Rayleigh regression based on the generalized odd log-logistic generalized Rayleigh (GOLLGR) law, with link functions mapping covariates to distribution parameters and enabling prediction analysis through maximum likelihood and advanced inference techniques (Cordeiro et al., 2022, &&&1&&&).

1. Foundations of Rayleigh Regression

The standard Rayleigh density for a response $Y$ with scale (mean) parameter $\mu > 0$ is

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with $\mathbb{E}[Y] = \mu$ and $$\operatorname{Var}[Y] = \mu^2 \left(\frac{4}{\pi} - 1\right)$$. Regression is constructed by linking $\mu_n$ for sample $n$ to covariates $x_n \in \mathbb{R}^k$ via a strictly monotone, twice-differentiable link function $g$, typically the log-link $g(\mu_n) = \log \mu_n$:

$$\eta_n = x_n^\top \beta, \qquad \mu_n = g^{-1}(\eta_n),$$

where $\beta \in \mathbb{R}^k$ is the regression coefficient vector. This regression structure captures systematic effects of covariates on the Rayleigh parameters, supporting modeling of various physical or biomedical phenomena (Palm et al., 2022).

2. Extended and Generalized Rayleigh Predictor: GOLLGR Model

The GOLLGR distribution generalizes the GR law by incorporating two additional shape parameters ($\alpha$, $\beta$) using the odd-log-logistic generator. The GR model of Vodă (1976) for $x>0$, $\delta > -1$, $\theta > 0$ has

$$G_{\mathrm{GR}}(x; \delta, \theta) = \gamma_1(\delta+1, \theta x^2),$$

where $\gamma_1(a, z)$ is the normalized lower-incomplete gamma ratio. The GOLLGR cumulative distribution function and density for $x>0$ are:

$$F(x; \alpha, \beta, \delta, \theta) = \frac{[G_{\mathrm{GR}}(x)]^{\alpha \beta}}{[G_{\mathrm{GR}}(x)]^{\alpha \beta} + [1-G_{\mathrm{GR}}(x)^{\beta}]^\alpha},$$

$$f(x; \alpha, \beta, \delta, \theta) = \frac{2 \alpha \beta \theta^{\delta+1} x^{2\delta+1} e^{-\theta x^2} \gamma_1(\delta+1, \theta x^2)^{\alpha\beta-1} [1 - \gamma_1(\delta+1, \theta x^2)^{\beta}]^{\alpha-1}}{\Gamma(\delta+1) \{ \gamma_1(\delta+1, \theta x^2)^{\alpha\beta} + [1 - \gamma_1(\delta+1, \theta x^2)^{\beta}]^\alpha \}^2 }.$$

Special cases include: GR ($\alpha=\beta=1$), odd-logistic GR ($\beta=1$), and exponentiated GR ($\alpha=1$). Parameters $\alpha$, $\beta$, $\delta_i$, $\theta_i$ can be linked to covariates via log-linear link functions,

$$\theta_i = \exp(v_i^\top \lambda_1), \qquad \delta_i = \exp(v_i^\top \lambda_2) - 1, \qquad \alpha = \exp(\eta), \qquad \beta = \exp(\zeta),$$

admitting a flexible regression structure for lifetime or amplitude data (Cordeiro et al., 2022).

3. Inference, Estimation, and Bias Correction

3.1 Maximum Likelihood Estimation

The log-likelihood for observed responses $y_n$ with corresponding covariates is:

$$\ell(\beta) = \sum_{n=1}^N \log \left( \frac{\pi}{2} \right) + \log y_n - 2 \log \mu_n - \frac{\pi y_n^2}{4 \mu_n^2}.$$

The score vector is

$$U(\beta) = \sum_{n=1}^N \left[ \frac{\pi y_n^2}{2 \mu_n^3} - \frac{2}{\mu_n} \right] \frac{1}{g'(\mu_n)} x_n,$$

and Fisher information is

$$I(\beta) = \sum_{n=1}^N \frac{4}{\mu_n^2} \left( \frac{d \mu_n}{d \eta_n} \right)^2 x_n x_n^\top.$$

Global maximization is achieved using iterative methods such as Fisher scoring and, for GOLLGR, Newton–Raphson or BFGS applied to the parameterized log-likelihood (Cordeiro et al., 2022, Palm et al., 2022).

3.2 Analytical and Numerical Bias Adjustment

MLE estimates of $\beta$ exhibit $O(1/N)$ bias, especially in small-sample SAR windows. Three correction schemes are implemented:

• Cox & Snell’s Analytical Correction:

$$\text{Bias}[\hat{\beta}] = B(\hat{\beta}) = I(\hat{\beta})^{-1} X^\top \widetilde{W} \delta,$$

and the bias-adjusted estimator is $$\tilde{\beta}_{CS} = \hat{\beta} - \widehat{B}(\hat{\beta})$$.

• Firth’s Penalized-Likelihood Correction:

The score is augmented by $A(\beta) = - I(\beta) B(\beta)$, with estimation via Fisher scoring for reduced bias.

• Parametric Bootstrap:

Multiple Rayleigh samples are simulated at $\hat{\beta}$, yielding bootstrap-corrected estimates $$\tilde{\beta}_{boot} = 2 \hat{\beta} - \bar{\beta}^*$$.

Bias correction is crucial for SAR pixel windows ($9$–$49$ pixels), with Firth’s method offering optimal bias–variance trade-off in empirical tests (Palm et al., 2022).

4. Prediction, Confidence, and Interval Construction

4.1 Quantile and Mean Prediction

For any new covariate vector $x_{new}$, construct $\eta_{new} = x_{new}^\top \tilde{\beta}$, then predict:

• Mean: $\hat{\mu}_{new} = g^{-1}(\eta_{new})$
• Median: In Rayleigh regression, $$\operatorname{med}[Y_{new}] = \hat{\mu}_{new} \sqrt{4 \ln 2 / \pi}$$; in GOLLGR, the quantile function inverts the CDF:

$$Q(u | v_i) = \left[ \theta_i^{-1} \gamma^{-1} \left( \delta_i + 1 ; \left\{ \frac{(u/(1-u))^{1/\alpha}}{1 + (u/(1-u))^{1/\alpha}} \right\}^{1/\beta} \right) \right]^{1/2},$$

for arbitrary quantiles $u$ (Cordeiro et al., 2022).

4.2 Confidence and Prediction Intervals

• Confidence Intervals: For $\hat{\mu}_{new}$, delta-method yields $$\operatorname{Var}(\hat{\mu}_{new}) \approx (d\mu/d\eta)^2 x_{new}^\top \mathrm{Cov}(\tilde{\beta}) x_{new}$$; standard 95% intervals are constructed as $$\hat{\mu}_{new} \pm 1.96 \sqrt{\operatorname{Var}(\hat{\mu}_{new})}$$.
• Prediction Intervals: Combine the fitted variance with model variance, i.e., for Rayleigh, $$\operatorname{Var}[Y_{new}|x_{new}] = \hat{\mu}_{new}^2 (4/\pi-1)$$. For GOLLGR, intervals can be constructed using plug-in quantiles or empirical percentiles from bootstrap simulations of $(\theta_i^*, \delta_i^*, \alpha^*, \beta^*)$ (Cordeiro et al., 2022, Palm et al., 2022).

5. Applications and Empirical Results

Rayleigh regression predictors have been applied to amplitude modeling in SAR images and survival analysis in COVID-19 patient datasets. In SAR applications, the regression structure with covariate-dependent mean provides a model for pixel intensity variation; bias corrections are particularly effective in small pixel windows, with Firth’s correction demonstrating favorable properties (Palm et al., 2022). In biomedical survival analysis, the GOLLGR extension enables nuanced modeling of lifetimes using covariates such as age and comorbidity indicators. Implementation in R’s gamlss framework reports highly significant covariate effects, strong model selection preference (AIC/BIC), and accurate quantile residual diagnostics. Individual prediction for new cases (e.g., given age and diabetes status) is supported using plug-in median and interval estimates, enabling patient-level prognostication (Cordeiro et al., 2022).

6. Algorithmic Implementation and Practical Guidance

Implementation of Ray Regression Predictors proceeds via iterative estimation algorithms:

• Fisher scoring for MLE: $$\beta^{(t+1)} = \beta^{(t)} + I(\beta^{(t)})^{-1} U(\beta^{(t)})$$
• Penalized scoring for Firth’s method
• Convergence criterion: $$\| \beta^{(t+1)} - \beta^{(t)} \|_\infty < \mathrm{tol}$$, for example, $\mathrm{tol} = 10^{-6}$
• Regression fitting: R’s nlm(), gamlss(), SAS PROC NLMIXED accept the required likelihood and gradients
• Bias correction selection: plain MLE for $N > 50$; Cox–Snell or Firth for $N < 30$; bootstrap when a fully simulation-based adjustment is preferable

This methodology offers flexible and accurate predictive modeling for positive-valued data, robust under both substantial and limited sample regimes, with full support for model diagnostics, confidence assessment, and empirical application (Cordeiro et al., 2022, Palm et al., 2022).