Probability density function evolution. Application to breast cancer

What is it about?

In this work we show why using models with randomness, or stochasticity, is crucial for the correct understanding of mathematical models representing real-world data. To do so, we use a non-standard approach: The Liouville-Gibbs equation. This partial differential equation appears in many areas in science and engineering. It allows us to study the evolution of uncertainty as if the corresponding probability density function was a fluid. We show that we can couple this approach with evolutionary calibration techniques. Finally, we use all of this to obtain a good fit for a dataset corresponding to breast cancer. We find that using this approach we obtain better results than with Montecarlo sampling-based simulations.

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Photo by David Ballew on Unsplash

Photo by David Ballew on Unsplash

Why is it important?

This work shows that the simulation of the evolution of probability density functions via the Liouville-Gibbs equation is a promising technique. Despite the fact that this equation has been known for some time, recent advances in numerical methods for partial differential equations allows us to build a general, efficient and accurate method from which all statistical information about a mathematical model can be obtained at a given time. Comparing to other methods, the Liouville-Gibbs equation gives more information than Polynomial Chaos Expansions. Also, it is more efficient and robust than Montecarlo sampling because of its deterministic nature. Finally, we can obtain more information about the dynamics of the probability density function than with the Random Variable Transformation theorem both from a theoretical and numerical point of view.

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