2. Nonlinear regression

3. Review of Linear Regression

4. Basic equations

5. Example – Linear vs. Nonlinear Regression

6. Estimating uncertainty in coefficients

7. Model based error for linear regression The common assumptions for linear regression –Surrogate is in functional form of true function –The data is contaminated with normally distributed error with the same standard deviation at every point. –The errors at different points are not correlated. Under these assumptions, the noise standard deviation (called standard error) is estimated as. Similarly, the standard error in the coefficients is

8. Rational function example

9. Application to crack propagation Paris law and its solution Coppe, A.,Haftka, R.T., and Kim, N.H. (2011) " Uncertainty Identification of Damage Growth Parameters Using Nonlinear Regression" AIAA Journal,Vol 49(12), 2818–2621 Properties to be identified from measurements

10. Example with only m unknown Simulation with b=0 v=[-1,1]mm, m=3.8 Excellent agreement between Monte Carlo (1,000 repetitions) simulation and linearization.

11. All three unknown Difficult to differentiate between initial crack size and bias

12. Problems