Radial Basis Function Approximations of Bayesian Parameter Posterior Densities for Uncertainty Analysis
Dynamical models are widely used in systems biology to describe biological processes ranging from single cell transcription of genes to the tissue scale formation of gradients for cell guidance. One of the key issues for this class of models is the estimation of kinetic parameters from given measurement data, the so called parameter estimation. Measurement noise and the limited amount of data, give rise to uncertainty in estimates which can be captured in a probability density over the parameter space. Unfortunately, studying this probability density, using e.g. Markov chain Monte-Carlo, is often computationally demanding as it requires the repeated simulation of the underlying model. In the case of highly complex models, such as PDE models, this can render the study intractable. In this paper, we will present novel methods for analysis of such probability densities using networks of radial basis functions. We employed lattice generation algorithms, adaptive interacting particle sampling schemes as well as classical sampling schemes for the generation of approximation nodes coupled to the respective weighting scheme and compared their efficiency on different application examples. Our analysis showed that the novel method can yield an expected L 2 approximation error in marginals that is several orders of magnitude lower compared to classical approximations. This allows for a drastic reduction of the number of model evaluations. This facilitates the analysis of uncertainty for problems with high computational complexity. Finally, we successfully applied our method to a complex partial differential equation model for guided cell migration of dendritic cells.
KeywordsMigration Covariance Eter
Unable to display preview. Download preview PDF.
- 4.MacKay, D.: Information Theory, Inference and Learning Algorithms (2003)Google Scholar
- 6.Fasshauer, G.: Meshfree Approximation Methods with MATLAB. World Scientific (2007)Google Scholar
- 7.Sarra, S.A., Kansa, E.J.: Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations. Advances in Computational Mechanics 2 (2009)Google Scholar
- 8.Müller, J.: Surrogate Model Algorithms for Computationally Expensive Black-Box Global Optimization Problems. PhD thesis (2012)Google Scholar
- 10.Silverman, B.: Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC (1986)Google Scholar
- 11.Ramsay, P.H., Scott, D.W.: Multivariate Density Estimation, Theory, Practice, and Visualization. Wiley (1993)Google Scholar
- 13.Fröhlich, F.: Approximation and Analysis of Probability Densities using Radial Basis Functions. Master’s thesis, Technische Universität München, GermanyGoogle Scholar
- 15.Torquato, S.: Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles. Physical Review E 82(5), 1–52 (2010)Google Scholar
- 17.Hock, S., Hasenauer, J., Theis, F.J.: Modeling of 2D Diffusion Processes based on Microscopy Data: Parameter Estimation and Practical Identifiability Analysis. BMC Bioinformatics 14(suppl.7) (2013)Google Scholar
- 18.Blizniouk, N., Ruppert, D., Shoemaker, C., Regis, R.: Bayesian Calibration of Computationally Expensive Models using Optimization and Radial Basis Function Approximation. Journal of Computational and Graphical Statistics 17(2) (2008)Google Scholar