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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Parametric Survival Analysis: Weibull and Exponential Methods01:14

Parametric Survival Analysis: Weibull and Exponential Methods

Parametric survival analysis models survival data by assuming a specific probability distribution for the time until an event occurs. The Weibull and exponential distributions are two of the most commonly used methods in this context, due to their versatility and relatively straightforward application.
Weibull Distribution
The Weibull distribution is a flexible model used in parametric survival analysis. It can handle both increasing and decreasing hazard rates, depending on its shape parameter...
One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation01:24

One-Compartment Open Model: Wagner-Nelson and Loo Riegelman Method for ka Estimation

This lesson introduces two critical methods in pharmacokinetics, the Wagner-Nelson and Loo-Riegelman methods, used for estimating the absorption rate constant (ka) for drugs administered via non-intravenous routes. The Wagner-Nelson method relates ka to the plasma concentration derived from the slope of a semilog percent unabsorbed time plot. However, it is limited to drugs with one-compartment kinetics and can be impacted by factors like gastrointestinal motility or enzymatic degradation.
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Related Experiment Video

Updated: Jun 25, 2026

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data
10:46

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

Published on: December 9, 2015

Scalable nonlinear Cox modeling via random Fourier features with analytic uncertainty.

Fahrettin Kaya1

  • 1Computer Technology Department, Andırın Vocational School, Sütçü İmam University, Kahramanmaraş, 46050, Türkiye. fkaya@ksu.edu.tr.

BMC Medical Research Methodology
|June 24, 2026
PubMed
Summary

A new Random Fourier Features-based Cox regression (RFF-Cox) model effectively captures complex, non-linear survival data relationships, outperforming traditional methods in accuracy and offering scalable, interpretable predictions for biomedical research.

Keywords:
Cox modelKernel methodsRandom fourier featuresScalabilitySurvival analysisUncertainty quantification

Related Experiment Videos

Last Updated: Jun 25, 2026

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data
10:46

A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

Published on: December 9, 2015

Area of Science:

  • Biostatistics
  • Machine Learning
  • Survival Analysis

Background:

  • The Cox proportional hazards model assumes linearity, limiting its ability to model complex biomedical risk structures like U-shaped associations.
  • Existing flexible kernel-based methods have high computational costs, restricting their use in large cohorts.
  • Many non-linear machine learning approaches lack analytical uncertainty measures for individual predictions.

Purpose of the Study:

  • To introduce a scalable Random Fourier Features-based Cox regression (RFF-Cox) approach for modeling non-linear risk relationships.
  • To enable uncertainty quantification and formal inference, including covariate-level interpretation and interaction detection.
  • To provide a computationally efficient alternative to existing methods for large-scale survival analysis.

Main Methods:

  • Developed RFF-Cox by mapping stationary kernels to a finite-dimensional feature space, reducing computational complexity.
  • Estimated model parameters using Newton-Raphson on a ridge-regularized partial likelihood with automatic bandwidth optimization.
  • Quantified uncertainty via the Fisher information matrix and a multivariate Delta method, enabling Taylor-expansion-based inference.

Main Results:

  • RFF-Cox accurately modeled non-linear associations (RMSE: 0.137 vs. 0.314) and recovered U-shaped risk functions, while performing comparably to classical Cox under linearity.
  • Detected a significant SBP × Smoking interaction (estimate -1.093, p < 0.001) with good analytical 95% confidence interval coverage (81.9% in non-linear scenarios).
  • Demonstrated competitive discriminatory power and computational efficiency against Random Survival Forests and XGBoost, with favorable calibration on the METABRIC dataset.

Conclusions:

  • RFF-Cox offers a practical survival analysis framework that handles non-linear relationships efficiently.
  • The method provides formal inference tools, including covariate-level interpretation and interaction detection, surpassing limitations of tree-based or deep learning survival models.
  • RFF-Cox degenerates to the classical Cox model under linearity, ensuring broad applicability.