Model identification from noisy data: solving ill-posed inverse problems using regularization
by
Stefan Mueller
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences
Coauthors: James Lu (Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science) Rainer Machne (Theoretical Biochemistry Group, University of Vienna) Lukas Endler (Theoretical Biochemistry Group, University of Vienna)

The quality of a mathematical model for a biological system depends - aside from its explanatory value - on its consistency with the data available. From a data-driven viewpoint, modeling is an "inverse problem": given a certain class of models, one tries to identify unknown parameters or even functions which give rise to the observed data or a desired qualitative dynamics. In the presence of data noise, however, model identification is an ill-posed inverse problem in the sense that its solution lacks stability properties: a small amount of data noise can be considerably amplified and may lead to unreliable solutions. To overcome this problem, we suggest the use of so-called regularization methods.

One of the systems we study is an ODE model of a metabolic pathway, which has been used as a benchmark problem for parameter identification. The ODE model contains 36 parameters all of which are identified from noisy data. Using simple least squares minimization (without regularization), a few percent of data noise leads to more than 100% relative error in some of the identified parameters, thus highlighting the ill-posedness of the inverse problem. Using regularization, the relative parameter error is comparable with the data noise. More specifically, we use Tikhonov regularization to counter the instability of the problem. By choosing the regularization parameter appropriately (based on the knowledge of the data noise), the model parameters can be identified in a stable and accurate manner.

Date received: May 15, 2008