Poisson distribution
Assume that each data point has an error that is independently random and distributed as a Poisson
distribution. The log likelihood function, L(p)
, as a function of the fit parameters,
p
, is minimized using a Gauss-Newton method. Since logarithms are involved, a good
first approximation is required before starting the Poisson fit, so try a normal fit first, and
use the resultant parameter values to start off the Poisson fit.
Weights do not have meaning, and so are not used, in a Poisson fit.
Assume that each data point, yk
, has an error that is
independently random and distributed as a Poisson distribution, that is,
We want to minimize:
but ∑ln(yk!)
is a constant. So, the goal is to minimize
Consider the Taylor expansion of :
Define:
Then:
Linearize, and the problem reduces to solving the matrix equation