The AstroStat Slog

The Χ² distribution plays an incredibly important role in astronomical data analysis, but it is pretty much a black box to most astronomers. How many people know, for instance, that its form is exactly the same as the γ distribution? A Χ² distribution with ν degrees of freedom is

p(z|ν) = (1/Γ(ν/2)) (1/2)^ν/2 z^ν/2-1 e^-z/2 ≡ γ(z;ν/2,1/2) , where z=Χ².

Continue reading ‘chi-square distribution [Eqn]’ »

Mmm.. chi-square!

The withering criticisms Hyunsook has been directing towards the faulty use of chisquare by astronomers brings to mind this classic comment by [astronomer] Jeremy Drake during the 2005 Chandra Calibration Workshop: Continue reading ‘Mmm.. donuts’ »

Leccardi & Molendi (2007) have a paper in A&A (astro-ph/0705.4199) discussing the biases in parameter estimation when spectral fitting is confronted with low counts data. Not surprisingly, they find that the bias is higher for lower counts, for standard chisq compared to C-stat, for grouped data compared to ungrouped. Peter Freeman talked about something like this at the 2003 X-ray Astronomy School at Wallops Island (pdf1, pdf2), and no doubt part of the problem also has to do with the (un)reliability of the fitting process when the chisq surface gets complicated.

Anyway, they propose an empirical method to reduce the bias by computing the probability distribution functions (pdfs) for various simulations, and then averaging the pdfs in groups of 3. Seems to work, for reasons that escape me completely.

[Update: links to Peter's slides corrected]

Despite some recent significant advances in Statistics and its applications to Astronomy (Cash 1976, Cash 1979, Gehrels 1984, Schmitt 1985, Isobe et al. 1986, van Dyk et al. 2001, Protassov et al. 2002, etc.), there still exist numerous problems and limitations in the standard statistical methodologies that are routinely applied to astrophysical data. For instance, the basic algorithms used in non-linear curve-fitting in spectra and images have remained unchanged since the 1960′s: the downhill simplex method of Nelder & Mead (1965) modified by Powell, and methods of steepest descent exemplified by Levenberg-Marquardt (Marquardt 1963). All non-linear curve-fitting programs currently in general use (Sherpa, XSPEC, MPFIT, PINTofALE, etc.) with the exception of Monte Carlo and MCMC methods are implementations based on these algorithms and thus share their limitations.
Continue reading ‘On the unreliability of fitting’ »