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Neural Network-Based Piecewise Survival Models

Olov Holmer, Erik Frisk, Mattias Krysander

TL;DR

The paper tackles censored survival prediction by introducing a neural-network framework that defines four piecewise survival models on a time grid. It constructs the survival distribution using either piecewise constant or linear densities, or hazards, parameterized by neural network outputs and constrained to keep $S(0|x)=1$ and $S(t_{max}|x)>0$. Training is done via maximum likelihood on censored data, and the models are evaluated against an energy-based model on simulated Weibull data. The results show that piecewise linear hazard models offer the best accuracy, closely approaching the energy-based method but with significantly reduced computation, highlighting a practical trade-off between expressivity and efficiency.

Abstract

In this paper, a family of neural network-based survival models is presented. The models are specified based on piecewise definitions of the hazard function and the density function on a partitioning of the time; both constant and linear piecewise definitions are presented, resulting in a family of four models. The models can be seen as an extension of the commonly used discrete-time and piecewise exponential models and thereby add flexibility to this set of standard models. Using a simulated dataset the models are shown to perform well compared to the highly expressive, state-of-the-art energy-based model, while only requiring a fraction of the computation time.

Neural Network-Based Piecewise Survival Models

TL;DR

The paper tackles censored survival prediction by introducing a neural-network framework that defines four piecewise survival models on a time grid. It constructs the survival distribution using either piecewise constant or linear densities, or hazards, parameterized by neural network outputs and constrained to keep and . Training is done via maximum likelihood on censored data, and the models are evaluated against an energy-based model on simulated Weibull data. The results show that piecewise linear hazard models offer the best accuracy, closely approaching the energy-based method but with significantly reduced computation, highlighting a practical trade-off between expressivity and efficiency.

Abstract

In this paper, a family of neural network-based survival models is presented. The models are specified based on piecewise definitions of the hazard function and the density function on a partitioning of the time; both constant and linear piecewise definitions are presented, resulting in a family of four models. The models can be seen as an extension of the commonly used discrete-time and piecewise exponential models and thereby add flexibility to this set of standard models. Using a simulated dataset the models are shown to perform well compared to the highly expressive, state-of-the-art energy-based model, while only requiring a fraction of the computation time.
Paper Structure (14 sections, 28 equations, 2 figures, 1 table)

This paper contains 14 sections, 28 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Survival Models
  4. Censoring
  5. Maximum Likelihood Training
  6. Piecewise Survival Models
  7. Discretization Grid
  8. Piecewise Constant Density Model
  9. Piecewise Linear Density Model
  10. Piecewise Constant Hazard Model
  11. Piecewise Linear Hazard Model
  12. Some Notes on the Implementation of the Models
  13. Comparison of the Models
  14. Conclusion

Figures (2)

  • Figure 1: Comparison of the piecewise constant models for $\lambda=2$ and $k=3$. The number of grid points is only 3 to make the differences in appearance of the models more clear.
  • Figure 2: Comparison of the piecewise linear models for $\lambda=2$ and $k=3$. The number of grid points is only 3 to make the differences in appearance of the models more clear.