PowerPoint Presentation

Nonlinear regression

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Review of Linear Regression

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Basic equations

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Example Linear vs. Nonlinear Regression

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Estimating uncertainty in coefficients

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Model 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

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Rational function example

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Application 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

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

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All 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.10

Problems