# Non linear curve fit - parameter accuracy

1. Oct 13, 2004

### arwelbath

Hi,
Suppose i fit some data with a curve, using Levenberg-Marquardt (or equivalent). How do I estimate the accuracy of the parameters (by this i mean p plus or minus something). I've read somewhere that its done using the jacobian and covariance matricies I think, but not sure. Anyone know if there's a 'standard' way of doing this?
Cheers

2. Oct 13, 2004

### gerben

most usually you determine the difference between each point you fit and the fitted point and you square all these difference and sum them, you then divide this sum by the number of points you have summed and call this "the mean squared error" (tip: google this).

3. Oct 13, 2004

### ehild

As I read in an old Origin Help, the standard error of the i-th parameter of a given parameter set p is

$$\sigma_i=\sqrt{\chi^2(\bold{p})C_{ii}}$$

where C is the variance-covariance matrix. It is calculated from the Jacobian F ($$F_{i,j}=\partial f(\bold{p},x_j)/\partial p_i)$$ as
$$\bold{C}=(\bold{F}^{'} \cdot \bold{F})^{-1}$$

If you understand this, explain me, please!

ehild

4. Oct 15, 2004

### arwelbath

Erm. Thanks gerben for the definition of chi squared. Not quite what I was after.

5. Oct 15, 2004

### chronon

The "Numerical recipes in .." books have a discussion of this. You can find the C book online at

http://www.library.cornell.edu/nr/bookcpdf.html [Broken]

The second half of chapter 15 discusses non-linear fitting and uncertainty of the estimated parameters

Last edited by a moderator: May 1, 2017