A novel partially functional linear regression model with random effects is proposed to address the case of Euclidean covariates and functional covariates. Specifically, the model assumes that the random effects follow a Gaussian process prior to establish the linkage structure between Euclidean covariates and scalar responses. For functional covariates, a linear relationship with scalar responses is assumed, and the functional covariates are approximated using the Karhunen–Loève expansion. To enhance the robustness of the predictive model, a cross-validation-based ensemble strategy is employed to optimize the proposed method. Results from both simulation studies and real-world data analyses demonstrate the superior performance and competitiveness of the proposed approach in terms of prediction accuracy and model stability.

This paper proposes a computational procedure to resolve the temporal fractional financial option pricing partial differential equation (PDE) using a localized meshless approach via the multiquadric radial basis function (RBF). Given that financial market information is best characterized within a martingale framework, the resulting option pricing model follows a modified Black–Sholes (BS) equation, requiring efficient numerical techniques for practical implementation. The key innovation in this study is the derivation of analytical weights for approximating first and second derivatives, ensuring improved numerical stability and accuracy. The construction of these weights is grounded in the second integration of a variant of the multiquadric RBF, which enhances smoothness and convergence properties. The performance of the presented solver is analyzed through computational tests, where the analytical weights exhibit superior accuracy and stability in comparison to conventional numerical weights. The results confirm that the new approach reduces absolute errors, demonstrating its effectiveness for financial option pricing problems.

This study introduces an intrinsic Gaussian Process Regression (iGPR) model for the first time, which incorporates non-Euclidean spatial covariates via a Gaussian process prior to analyzing the relationship between digitalization and carbon emission efficiency. The iGPR model’s hierarchical design embeds a Gaussian process as a flexible spatial random effect with a heat-kernel-based covariance function to capture the manifold geometry of spatial features. To enable tractable inference, we employ a penalized maximum-likelihood estimation (PMLE) approach to jointly estimate regression coefficients and covariance hyperparameters. Using a panel dataset linking a national digitalization (modernization) index to carbon emission efficiency, the empirical analysis demonstrates that digitalization has a significantly positive impact on carbon emission efficiency while accounting for spatial heterogeneity. The iGPR model also exhibits superior predictive accuracy compared to state-of-the-art machine learning methods (including XGBoost, random forest, support vector regression, ElasticNet, and a standard Gaussian process regression), achieving the lowest mean squared error (MSE = 0.0047) and an average prediction error near zero. Robustness checks include instrumental-variable GMM estimation to address potential endogeneity across the efficiency distribution and confirm the stability of the estimated positive effect of digitalization.

Poisson regression is a statistical method specifically designed for analyzing count data. Considering the case where the functional and vector-valued covariates exhibit a linear relationship with the log-transformed Poisson mean, while the covariates in complex domains act as nonlinear random effects, an intrinsic functional partially linear Poisson regression model is proposed. This model flexibly integrates predictors from different spaces, including functional covariates, vector-valued covariates, and other non-Euclidean covariates taking values in complex domains. A truncation scheme is applied to approximate the functional covariates, and the random effects related to non-Euclidean covariates are modeled based on the reproducing kernel method. A quasi-Newton iterative algorithm is employed to optimize the parameters of the proposed model. Furthermore, to capture the intrinsic geometric structure of the covariates in complex domains, the heat kernel is employed as the kernel function, estimated via Brownian motion simulations. Both simulation studies and real data analysis demonstrate that the proposed method offers significant advantages over the classical Poisson regression model.

Clustered proportion responses often exhibit bounded support, skewness, heterogeneous dispersion, and within-cluster dependence. We propose a correlated mean–precision random-effects beta regression model that jointly represents cluster-level heterogeneity in the conditional mean and conditional precision. Its main innovation is to treat the cross-submodel random-effect correlation as a scientific estimand. The same frequentist Beta mixed model estimates and tests this correlation while allowing nonlinear adjustment and cluster-level interpretation. Penalized B-splines allow nonlinear effects in both submodels, and estimation is performed by maximizing a Laplace-approximated penalized marginal likelihood. The fitted model provides likelihood-based inference for the mean–stability association and empirical Bayes estimates for cluster ranking and quadrant classification. Among M1–M5, M5 gives the lowest average errors for conditional-mean and conditional-precision recovery and the best average AIC, cluster-level BIC, and full-data NLPD across the Monte Carlo settings considered. It is also the only model compared here that estimates and tests the latent association while retaining paired cluster effects for interpretation. All M5 fits succeeded in the enlarged 30-replication stress suite, and the five aggregate mean M5–M4 criteria favored M5 across 299 successful pairs; quadrature checks indicate where safeguards are needed under weak information. In both CDC PLACES and the representative World Bank panel of 128 eligible countries, M5 has the largest marginal likelihood, the smallest AIC and cluster-level BIC, a significant M4–M5 likelihood-ratio test, and the lowest application-specific point-prediction errors. For the World Bank data, $\widehat{\rho }=-0.500$ (profile 95% interval $-0.621$ to $-0.295$), the likelihood-ratio statistic is $17.370$ ($p=3.08\times {10}^{-5}$), and M5 has the lowest rolling-origin MSPE, RMSE, and MAE. On the combined evidence from fit, prediction, and correlation inference, M5 is the best overall model evaluated in both applications.