# Propogating Error when Fitting Polynomial to Data Set [on hold]

I have a data set, and I am fitting it with an nth order polynomial. My fit has associated standard error with the constants, eg.:

$y = a + bx + cx^2$

a +/- da

b +/- db

c +/- dc

Using my polynomial I am finding the maximum of my data set, and getting the associated x value. I'm trying to find the error associated with that x value due to the uncertainty of the fit.

Any help would be most appreciated!

-
This seems like a problem better suited for one of the sites in the FAQ. Voting to close. –  Steve Huntsman May 31 '10 at 19:47
If I have interpreted your question right, it's quite easy. You just use the fact that var(ax + by) = a^2 var x + b^2\var y + 2ab \cov(x,y). In your case the scalars are the various powers of x and the random variables are the constants. You will need the estimated covariances between the constants as well as the standard errors (which as I expect you know are the square roots of the estimated variances). –  Tom Smith May 31 '10 at 22:20
You could also just look at where the maxima of various polynomials within your errors bars lie. Anything fancier than this, and I'd want to be really sure that "fitting with an n-th order polynomial" was actually sensible to begin with. –  Scott Morrison Jun 1 '10 at 0:00
Are you using ordinary least squares? Suppose $$\begin{pmatrix} Y_1 \\ \vdots \\ Y_n \end{pmatrix} = \begin{pmatrix} 1 & x_1 & x_1^2 \\ \vdots & \vdots & \vdots \\ 1 & x_n & x_n^2 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} + \text{error} = XA + E.$$ And the "errors" are i.i.d. normal with expectation 0 and all the same variance $\sigma^2$. In that case the estimated $A$-values are $(X^T X)^{-1}X^T Y$, and so the matrix of their variances and covariances is $\sigma^2 (X^T X)^{-1}$. The location of the maximum depends on those coefficients. To be continued.... –  Michael Hardy Jun 1 '10 at 14:02