**Leonid Kalachev**

**Department of Mathematical Sciences**

**University of Montana**

**Reduction and Identification of (Simple) Dynamic Models**

Classical statistical methods are usually applied for the analysis of experimental data of time series type in various applied fields (e.g., in chemical kinetics, microbiology, pharmacokinetics, etc.). Statistical approach assumes the presence of a known function with unknown parameters (statistical model) that is fitted to data, e.g., using linear or non-linear regression. The confidence intervals for estimated parameters are usually constructed using approach based on linearization of the original model function near the set of fitted parameter values. The classical statistics usually does not address the question: Where do the model functions come from? Incidentally, most of the functions currently used in regression analyses of scientific experimental data are the explicit solutions of dynamic models of various phenomena formulated in terms of differential equations that are derived using some basic laws of physics, chemistry, biology, etc. However, not every differential equation or system of differential equations has an explicit solution expressed as a function that can be directly used in a standard statistical analysis. With advances in scientific computing, estimation of a large number of parameters in complex dynamic models, whose solutions are not explicitly known, became possible. The statistical analysis of reliability regions for estimated parameters in such, usually strongly nonlinear, models still remains an important issue that is not adequately addressed in the earlier studies. Markov Chain Monte Carlo (MCMC) simulation approach is one of the powerful methods for estimating reliability regions for dynamic models’ parameters.

At the same time a lot of scientific modeling that does not rely on particular experimental data is done using purely deterministic approach: the dynamics models are analyzed qualitatively and conclusions on their behavior are made based on running heuristic “what if” scenarios. If such models are eventually fitted to real data, the fits produce estimates for means of parameter values, without analysis of their reliability regions. Deterministic approach usually does not address the question: How do we know that the model contains the optimal number of parameters (supported by available data)? If the number of parameters is too small, the data fit will not be sufficiently accurate. On the other hand, if the number of parameters is too large, some of the parameters will be correlated, not uniquely defined, etc.

The perturbation (asymptotic reduction) techniques provide a method for reduction of a number of equations and parameters in dynamic models under certain conditions. Application of the procedure based on consecutive repetition of a combination of steps including model fit, model reduction, and reliability regions estimation, leads to simplified models containing the smallest possible number of parameters reliably identifiable from the available data. We illustrate the ideas related to models’ reduction, models’ identification, and their combination using simple examples.

Prior to the colloquium, coffee will be served in FO 2.404

Sponsored by the Department of Mathematical Sciences

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