MAXIMUM-LIKELIHOOD FITTING OF DATA DOMINATED BY POISSON STATISTICAL UNCERTAINTIES

The fitting of data by chi(2) minimization is valid only when the unce rtainties in the data are normally distributed. When analyzing spectro scopic or particle counting data at very low signal level (e.g., a Tho mson scattering diagnostic), the uncertainties are distributed with a Poisson distribution. We have developed a maximum-likelihood method fo r fitting data that correctly treats the Poisson statistical character of the uncertainties. This method maximizes the total probability tha t the observed data are drawn from the assumed fit function using the Poisson probability function to determine the probability for each dat a point. The algorithm also returns uncertainty estimates for the fit parameters. We compare this method with a chi(2)-minimization routine applied to both simulated and real Thomson scattering data. Difference s in the returned fits are greater at low signal level (less than simi lar to 10 counts per measurement). The maximum-likelihood method is fo und to be more accurate and robust, returning a narrower distribution of values for the fit parameters with fewer outliers. (C) 1997 America n Institute of Physics.