The Group Abundance Fraction: A statistically robust measure of particle composition and of spatial structure in images

[摘要] When two processes (A and B) are measured by counting instruments, we wish to present the results as an "abundance" along with an associated uncertainty. When the number of individual counts of either component is low, the usual ratio (A/B) is statistically ill-defined because there is a finite probability of zero counts in the denominator. However, the "Group Abundance Fraction" (F), e.g., A/(A+B), is a statistically robust parameter (under the condition that there must be at least one total count between A and B in order to constitute a measurement). We rigorously derive expectation values (over a conditional Poisson statistical ensemble) for 〈F〉 and its standard deviation σF that are valid for all values of total counts greater than one. We then derive useful estimators for these ensemble expectation values in terms of simple algebraic functions of the actual observed accumulated counts. We also incorporate the practical effects of instrument response functions and background rates. The results are not restricted to particle species composition (or spectra), but are also applicable when A and B represent counts in adjacent "pixels" of spatial images formed by the particles.

[发布机构] Johns Hopkins University, Applied Physics Laboratory, Laurel; MD; 20723-6099, United States^1

[效力级别] 天文学 [学科分类] 天文学(综合)

[关键词] Algebraic functions;Expectation values;Finite probabilities;Instrument response functions;Particle composition;Spatial structure;Species composition;Statistical ensembles