1. Bounded Error Parameter Estimation for Delay Differential Equations Adam Childers and John Burns Interdisciplinary Center for Applied Mathematics Virginia Tech 2-21-2009

2. The Modeling Process Ask a question about a system. Describe the system mathematically. Fit the model to experimental data. Validate the model. Use the model to answer the question.

3. The Modeling Process Ask a question about a system. Describe the system mathematically. Fit the model to experimental data. Validate the model. Use the model to answer the question. Parameter Estimation Parameter Validation

4. Our Research Question If data can be collected at only a small number to times, can we validate a parameter estimate for a model described by a delay differential equation? Using traditional statistical techniques? Using dynamics of the model? NO YES!

5. The Mathematical Model Hutchison’s Equation The model is the solution to the differential equation,. Bacteria in a Petri dish

6. “All models are wrong, some are useful.” George Box Interpreting a Model George Box Born 1919 in Kent, England

7. The Statistical Model Population Time

8. Parameter Identification Population Time

9. Parameter Identification Population Time x(t,[1,1]) What can we say statistically about the parameter estimate?

10. The Statistical Model We assume the error is bounded

11. Bounded Error Parameter Identification data point the true output of the system will fall in an interval centered at the data point. Assuming that the errors are bounded

12. Bounded Error Parameter Identification Population Time Data points become bounded sets

13. Bounded Error Parameter Identification t=3.589 Bounded setsPossible Parameters Population Time a (growth rate) r (the delay)

14. Bounded Error Parameter Identification t=3.589 t=6.136 Bounded setsPossible Parameters Population Time a (growth rate) r (the delay)

15. Bounded Error Parameter Identification t=3.589 t=6.136 t=2.000 Bounded sets Possible Parameters Population Time a (growth rate) r (the delay)

16. Bounded Error Parameter Identification t=3.589 t=6.136 t=2.000 Bounded sets Possible Parameters Population Time a (growth rate) r (the delay) x(t,[1,1]) [1,1]

17. Bounded Error Parameter Identification and DOE A design is said to be V-optimal if the measure of the membership set is minimized. a (growth rate) r (the delay)

18. Questions?