OPSurv: Orthogonal Polynomials Quadrature Algorithm for Survival Analysis

TL;DR

OPSurv presents a novel, time-continuous survival analysis method for single and competing risks by decomposing risk-specific densities into an orthogonal Hermite basis and linking them to cumulative incidence functions via Gauss–Legendre quadrature. The approach estimates coefficients with neural networks, enforcing the fundamental initial condition $F_e(0|x)=0$ and producing smooth, differentiable outputs that admit direct hazard and curvature analysis. It combines a likelihood objective with a ranking loss to handle noisy data and emphasizes a controllable degree of approximation to mitigate overfitting in competing risks settings. Empirical results show OPSurv achieving state-of-the-art or competitive performance on mortgage-default and medical datasets, while maintaining strong expressiveness and interpretability of the time-continuous survival outputs.

Abstract

This paper introduces the Orthogonal Polynomials Quadrature Algorithm for Survival Analysis (OPSurv), a new method providing time-continuous functional outputs for both single and competing risks scenarios in survival analysis. OPSurv utilizes the initial zero condition of the Cumulative Incidence function and a unique decomposition of probability densities using orthogonal polynomials, allowing it to learn functional approximation coefficients for each risk event and construct Cumulative Incidence Function estimates via Gauss--Legendre quadrature. This approach effectively counters overfitting, particularly in competing risks scenarios, enhancing model expressiveness and control. The paper further details empirical validations and theoretical justifications of OPSurv, highlighting its robust performance as an advancement in survival analysis with competing risks.

Figures (7)

• Figure 1: OPSurv method overview. Client attributes $x$ are passed into 2 neural networks, which output a set of weights and $E$ vectors of length $(J+1)$ respectively. They are coefficients of the probability densities, which can also be used to compute the CDFs and with the weights, the CIFs for each risk.
• Figure 2: METABRIC: survival functions of the six models for three random test patients who died. The x-axis is observation time (25 years). Dotted ertical lines indicate the times of death for each patient. Observe that the times of death happened near the inflection points of OPSurv survival functions that preceded a sudden drop.
• Figure 3: SEER (primary event): survival functions of four deep learning-based models for three random test patients who died from breast cancer. The x-axis is observation time (25 years). Dotted vertical lines indicate the times of death.
• Figure 4: SEER (primary event): cumulative incidence functions of six models for three random test patients who died from breast cancer. The x-axis is observation time (25 years). Dotted vertical lines indicate the times of death.
• Figure 5: SEER (competing risk): cumulative incidence functions of six models for three random test patients who died from cardiovascular disease. The x-axis is observation time (25 years). Dotted vertical lines indicate the times of death.