Multinomial, Poisson and Gaussian statistics in count data analysis
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Multinomial, Poisson and Gaussian statistics in count data analysis. / Lass, Jakob; Boggild, Magnus Egede; Hedegard, Per; Lefmann, Kim.
In: Journal of Neutron Research, Vol. 23, No. 1, 2021, p. 69-92.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 269910456