A comparison of methods for Poisson regression in the presence of background

Abstract

This paper provides a statistical analysis of three common methods of regression for Poisson data in the presence of Poisson background, namely the joint fit with two parametric models for the source and the background, the use of a non-parametric model for the background known as the wstat method, and the regression with a fixed background. The non-parametric background method, which is a popular method for spectral data, is found to be significantly biased, especially in the low-count and background-dominated regimes. Similar conclusions apply to the fixed-background regression. The joint-fit method, on the other hand, simultaneously affords reliable hypothesis testing by means of the usual Cash statistic and unbiased reconstruction of source parameters. We also investigate the effect of non-parametric regression on the number of effective degrees of freedom by means of the Efron degree of freedom function. We find that the wstat method adds a significantly larger number of degrees of freedom, compared to the number of free parameters in the source model. The other two methods have a number of degrees of freedom consistent with the number of adjustable parameters, at least for the simple models investigated in this paper.

Figures (4)

• Figure 1: Sample data set and best-fit models for $\theta=\beta=1$ and $t_S=t_B=1$, with $N=100$. Data points for the source region (red) are drawn from a Poisson distribution with mean $\theta+\beta$, and for the background region (blue) from a Poisson distribution with mean $\beta$.
• Figure 2: Experimental Cumulative Distribution Function (eCDF) for the statistics and best-fit parameters, based on 1,000 simulations with intensity $\theta=\beta=1$ and $N=100$ data points.
• Figure 3: (Top): eCDFs for the simulations with $N=100$, $\theta=\beta=0.1$. (Bottom): eCDFs for the simulations with $N=100$, $\theta=0.1, \beta=10$
• Figure 4: eCDFs for the most extreme case of $\theta=0.1$, $\beta=100$, $N=100$.

Theorems & Definitions (10)

• Conjecture 1
• Conjecture 2
• Remark 1: Accuracy of $C_{\mathrm{min}}$ for joint fit
• Remark 2: $W_{\mathrm{min}} \leq C_{\mathrm{min}}(\mathrm{FB})$
• Remark 3: Model mis-specification for $C_{\mathrm{min}}$(FB)
• Remark 4: Unbiasedness of joint-fit
• Remark 5: Biases with $W_{\mathrm{min}}$ and $C_{\mathrm{min}}$(FB) in low-mean data
• Remark 6: Empirical validation of conjectures \ref{['eq:CminDf']} and \ref{['eq:cminMomentsDf']}
• Remark 7: Overfitting and $\mathrm{df}(\mu) \gg p$ with $W_{\mathrm{min}}$
• Remark 8: Hypothesis testing with $W_{\mathrm{min}}$