Poisson Cost Function

Description

The Poisson cost function is designed to be applied to data, which has a low number of counts/events. In this scenario a fit is being performed using a model (function) to some low stats data.

The number of counts/events in bin i is given by yi and the corresponding value for the model is μi. Where the values of yi and μi are strictly positive.

The quality of the fit can be represented numerically by the deviance (D)

2iN{yilog(yi)yilog(μi)(yiμi)},

where the first two terms provide a measure for small variations between yi and μi, whereas the third term provides a measure of large variations. Therefore, by minimizing the deviance the best fit can be identified. This method is also known as the Poisson deviance or Poisson log-linear model.

When performing a fit on Poisson distributed data care needs to be taken when selecting the cost function. For a large number of events/counts the least squares cost function can be used to minimize Poisson distributed data. This is because in the limit of large numbers of events/counts the Poisson and Gaussian distributions are approximately equal. In the limits of low events/counts using a least squares cost function can lead to biased results.

Example

Given a workspace with low counts, a fit can be done in a script as follows: Fit(Function=f,InputWorkspace=workspace,Output="outputName",CreateOutput=True,CostFunction="Poisson",Minimizer="Levenberg-MarquardtMD")

The plot below demonstrates the difference between using the Poisson and Least Squares cost functions for a given parameter over a series of fits to data with different number of counts. The Poisson model gives a much more consistent result and means no special treatment is required for zero counts.

Poisson cost function compared with least squares cost function

References

[1] Rodríguez, G. (2007). Lecture Notes on Generalized Linear Models.