Sebastien,
Hi Dimitri, I'm obviously missing something in my litterature review. I did a new MC simulation, with a much smaller number of observation points (namely 3, to fit a straight line!!!). It turns out that the formula you are advocating for is the best estimate of the standard deviation of the parameters. Could you please explain why this fomula differs from formulas (34) and (35) in http://mathworld.wolfram.com/LeastSquaresFitting.html?
First thing worth noting is Worlfram is wise enough to call 34 and 35 standard error ... and not standard deviation! As Gilles and you have shown with your MC simulations, the standard deviation (sigma_i=sqrt(cov[i][i])) approximates by how much the fitted parameter can vary when several sets of 'observations' are sampled with the same error distribution. I wrote 'approximate' because the true standard deviation is not accessible, instead it is approximated as the inverse of Fisher information matrix which is directly related to the Hessian matrix. The relation between Fisher and the variance of the parameter is known as the Rao-Cramer bound. In the case of the standard error, the sample of observations is fixed and one wonders by how much one can change the parameters without changing the resulting normalized chi square too much. That is the role of s (eq. 32 on Wolfram). It should be noted that nowhere on that page there is the notion of error on the observations: the data are what they are and no alternative sampling should be considered. Please, have a look at http://en.wikipedia.org/wiki/Standard_deviation http://en.wikipedia.org/wiki/Standard_error for further details, especially the last section of the Standard_error page as it compares std. error and deviation. Regards, Dim. ---------------------------------------------------------------------------- Dimitri Pourbaix * Don't worry, be happy Institut d'Astronomie et d'Astrophysique * and CARPE DIEM. CP 226, office 2.N4.211, building NO * Universite Libre de Bruxelles * Tel : +32-2-650.35.71 Boulevard du Triomphe * Fax : +32-2-650.42.26 B-1050 Bruxelles * NAC: HBZSC RG2Z6 http://sb9.astro.ulb.ac.be/~pourbaix * mailto:pourb...@astro.ulb.ac.be --------------------------------------------------------------------- To unsubscribe, e-mail: dev-unsubscr...@commons.apache.org For additional commands, e-mail: dev-h...@commons.apache.org
