Estimating second order system parameters from noise power spectra using nonlinear regression
… the safe use of regression requires a good deal of thought and a good dose of skepticism
Overview
In the limits of detection lab, you will use regression to estimate the parameters of second order system models of the optical trap and AFM. MATLAB includes a powerful nonlinear regression function called nlinfit. Like all sharp tools, nlinfit must be used with extreme care. Regression is a delicate business.
This page illustrates several flawed regressions with the hope of helping you navigate the concealed hazards of fitting power spectra. Many common regression missteps spawn subtle errors rather than catastrophic failures. Because of this, bad regressions easily masquerade as good ones — hiding their insidious flaws behind a lovely, plausible pair of overlapping curves.
Remain vigilant at all times when you regress.
Regression review
Regression assumes that you are trying to find the relationship between a response variable $ y_n $ and an explanatory varialbe $ x_n $. The two quantities are related by a model function $ f(\beta, x_n) $. $ \beta $ is a vector of model parameters. $ y_n $ is measured in the presence of random noise, which is represented mathematically by a random variable $ \epsilon_n $. In equation form:
$ y_n=f(\beta, x_n)+\epsilon_n $.
Basic regression assumes that:
- the error terms have a mean value of zero,
- the error terms are independent and identically distributed (i.i.d.), and
- the explanatory variable is known exactly.
These assumptions are almost never perfectly met.
