TY - JOUR

T1 - Multinomial, Poisson and Gaussian statistics in count data analysis

AU - Lass, Jakob

AU - Boggild, Magnus Egede

AU - Hedegard, Per

AU - Lefmann, Kim

PY - 2021

Y1 - 2021

N2 - It is generally known that counting statistics is not correctly described by a Gaussian approximation. Nevertheless, in neutron scattering, it is common practice to apply this approximation to the counting statistics; also at low counting numbers. We show that the application of this approximation leads to skewed results not only for low-count features, such as background level estimation, but also for its estimation at double-digit count numbers. In effect, this approximation is shown to be imprecise on all levels of count. Instead, a Multinomial approach is introduced as well as a more standard Poisson method, which we compare with the Gaussian case. These two methods originate from a proper analysis of a multi-detector setup and a standard triple axis instrument.We devise a simple mathematical procedure to produce unbiased fits using the Multinomial distribution and demonstrate this method on synthetic and actual inelastic scattering data. We find that the Multinomial method provide almost unbiased results, and in some cases outperforms the Poisson statistics. Although significantly biased, the Gaussian approach is in general more robust in cases where the fitted model is not a true representation of reality. For this reason, a proper data analysis toolbox for low-count neutron scattering should therefore contain more than one model for counting statistics.

AB - It is generally known that counting statistics is not correctly described by a Gaussian approximation. Nevertheless, in neutron scattering, it is common practice to apply this approximation to the counting statistics; also at low counting numbers. We show that the application of this approximation leads to skewed results not only for low-count features, such as background level estimation, but also for its estimation at double-digit count numbers. In effect, this approximation is shown to be imprecise on all levels of count. Instead, a Multinomial approach is introduced as well as a more standard Poisson method, which we compare with the Gaussian case. These two methods originate from a proper analysis of a multi-detector setup and a standard triple axis instrument.We devise a simple mathematical procedure to produce unbiased fits using the Multinomial distribution and demonstrate this method on synthetic and actual inelastic scattering data. We find that the Multinomial method provide almost unbiased results, and in some cases outperforms the Poisson statistics. Although significantly biased, the Gaussian approach is in general more robust in cases where the fitted model is not a true representation of reality. For this reason, a proper data analysis toolbox for low-count neutron scattering should therefore contain more than one model for counting statistics.

KW - Poisson statistics

KW - Multinomial statistics

KW - data analysis

KW - neutron scattering

KW - NEUTRON-SCATTERING

KW - CONFIDENCE-INTERVALS

U2 - 10.3233/JNR-190145

DO - 10.3233/JNR-190145

M3 - Journal article

VL - 23

SP - 69

EP - 92

JO - Journal of Neutron Research

JF - Journal of Neutron Research

SN - 1023-8166

IS - 1

ER -