PowerPoint Presentation
Nonlinear regression
1Review of Linear Regression
2Basic equations
3Example Linear vs. Nonlinear Regression
4Estimating uncertainty in coefficients
5Model based error for linear regressionThe common assumptions
for linear regression Surrogate is in functional form of true
functionThe 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
6Rational function example
7Application to crack propagationParis law and its solution
Coppe, A. ,Haftka, R.T., and Kim, N.H. (2011) " Uncertainty
Identication of Damage Growth Parameters Using Nonlinear
Regression" AIAA Journal ,Vol 49(12), 28182621 Properties to be
identified from measurements
8Example with only m unknownSimulation with b=0 v=[-1,1]mm,
m=3.8Excellent agreement between Monte Carlo (1,000 repetitions)
simulation and linearization.
9All three unknownDifficult to differentiate between initial
crack size and bias
When the simulation was repeated with all three unknowns, the
results as shown in the figure were much poorer. Let us first
consider the uncertainty in m. The standard deviation is larger
than it was before, but it is still small considering that the true
value of m is 3.8. However, the value obtained from the linear
regression (standard error) is larger by two orders of magnitude to
begin with, and the two agree well after 1500 cycles (15
measurements). For the other two parameters we see similar
behavior.
The reason for the poor performance is that with a small number
of measurements, the linearized equations are ill conditioned
because it is difficult to distinguish between the effect of the
initial crack size and the bias. If we increase the initial crack
size from 10mm to 10.1mm, and reduce the bias from zero to -0.1mm,
the calculated crack size will grow a bit faster, but when the
crack is small, the difference will be miniscule. So it is only
when the crack grows large and grows fast, the difference is
appreciable.10Problems