Nonlinear regression Review of Linear Regression Basic equations

Nonlinear regression •

Review of Linear Regression •

Basic equations •

Example – Linear vs. Nonlinear Regression •

Estimating uncertainty in coefficients •

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

Rational function example •

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

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.

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

Problems •