Background The Cox proportional hazards model often fails to capture complex biomedical risk structures, such as U-shaped biomarker associations, due to its assumption of linearity between the log-hazard and covariates. While existing kernel-based generalizations offer the necessary flexibility, their <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(\:O\left({n}^{3}\right)\)</EquationSource></InlineEquation> computational complexity limits applicability in large-scale cohort studies. Furthermore, most non-linear machine learning methods lack closed-form analytical measures of uncertainty for individual predictions.. Methods A novel Random Fourier Features-based Cox regression approach (RFF-Cox) is presented to model non-linear risk relationships within a scalable framework. By mapping stationary kernels into a finite-dimensional explicit feature space, the method reduces computational complexity to <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(\:O\left(n{m}^{2}\right)\)</EquationSource></InlineEquation>. Model parameters are estimated via the Newton–Raphson algorithm on a ridge-regularized partial likelihood, while the bandwidth parameter (<InlineEquation ID="IEq3"><EquationSource Format="TEX">\(\:\sigma\:\)</EquationSource></InlineEquation>) is automatically optimized using a marginal likelihood criterion based on the Laplace approximation. Uncertainty quantification is performed via the Fisher information matrix, with a multivariate Delta method propagating variance from both parameter estimation and baseline hazard estimation. A Taylor-expansion-based inference framework enables covariate-level hazard ratio estimation, Wald-type significance testing, and formal interaction detection. Performance was evaluated using controlled simulations and four real-world datasets with sample sizes ranging from 500 to 9,105. Results In simulation scenarios, RFF-Cox degenerated to classical Cox estimates under linearity (RMSE: 0.111 vs. 0.110) while demonstrating a marked accuracy advantage under non-linearity (RMSE: 0.137 vs. 0.314), including the recovery of U-shaped risk functions. The true SBP × Smoking interaction was detected (estimate −1.093, p < 0.001). Analytical 95% CI coverage reached 95.9% in the linear scenario and 81.9% in the non-linear scenario. In real-world applications, the model exhibited discriminatory power competitive with Random Survival Forests and XGBoost with favorable computational efficiency. IPCW-weighted calibration analyses on the METABRIC dataset yielded low Integrated Calibration Error (ICI < 0.05) across most time horizons, though calibration slope deviations were noted at longer follow-up periods. Moreover, uncertainty in individual predictions, quantified via analytical confidence intervals, varied meaningfully across risk groups. Conclusions RFF-Cox provides a practical survival analysis framework that degenerates to the classical Cox model under linearity while offering non-linear modelling capabilities, computational efficiency, and formal inference tools — including covariate-level interpretation and interaction detection — that are not readily available in tree-based or deep learning survival methods.

Scalable nonlinear Cox modeling via random Fourier features with analytic uncertainty

期刊论文
出版模式：

开放获取
发表日期：

2026年6月23日

查看全文

Fahrettin Kaya

摘要

Background

The Cox proportional hazards model often fails to capture complex biomedical risk structures, such as U-shaped biomarker associations, due to its assumption of linearity between the log-hazard and covariates. While existing kernel-based generalizations offer the necessary flexibility, their \(\:O\left({n}^{3}\right)\) computational complexity limits applicability in large-scale cohort studies. Furthermore, most non-linear machine learning methods lack closed-form analytical measures of uncertainty for individual predictions..

Methods

A novel Random Fourier Features-based Cox regression approach (RFF-Cox) is presented to model non-linear risk relationships within a scalable framework. By mapping stationary kernels into a finite-dimensional explicit feature space, the method reduces computational complexity to \(\:O\left(n{m}^{2}\right)\). Model parameters are estimated via the Newton–Raphson algorithm on a ridge-regularized partial likelihood, while the bandwidth parameter (\(\:\sigma\:\)) is automatically optimized using a marginal likelihood criterion based on the Laplace approximation. Uncertainty quantification is performed via the Fisher information matrix, with a multivariate Delta method propagating variance from both parameter estimation and baseline hazard estimation. A Taylor-expansion-based inference framework enables covariate-level hazard ratio estimation, Wald-type significance testing, and formal interaction detection. Performance was evaluated using controlled simulations and four real-world datasets with sample sizes ranging from 500 to 9,105.

Results

In simulation scenarios, RFF-Cox degenerated to classical Cox estimates under linearity (RMSE: 0.111 vs. 0.110) while demonstrating a marked accuracy advantage under non-linearity (RMSE: 0.137 vs. 0.314), including the recovery of U-shaped risk functions. The true SBP × Smoking interaction was detected (estimate −1.093, p < 0.001). Analytical 95% CI coverage reached 95.9% in the linear scenario and 81.9% in the non-linear scenario. In real-world applications, the model exhibited discriminatory power competitive with Random Survival Forests and XGBoost with favorable computational efficiency. IPCW-weighted calibration analyses on the METABRIC dataset yielded low Integrated Calibration Error (ICI < 0.05) across most time horizons, though calibration slope deviations were noted at longer follow-up periods. Moreover, uncertainty in individual predictions, quantified via analytical confidence intervals, varied meaningfully across risk groups.

Conclusions

RFF-Cox provides a practical survival analysis framework that degenerates to the classical Cox model under linearity while offering non-linear modelling capabilities, computational efficiency, and formal inference tools — including covariate-level interpretation and interaction detection — that are not readily available in tree-based or deep learning survival methods.