Workshop on Detection Limits without Noise … · Loveland et al. 2017. Poisson distribution 7 1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 0369 12 15 Probability mass function ... Seaborg

Thomas M. Semkow, Xin Li, Liang T. Chu

Wadsworth Center, New York State Department of Health, Empire State Plaza, Albany, NY

12201, USA

Department of Environmental Health Sciences, School of Public Health, University at Albany,

SUNY, Rensselaer, NY 12144, USA

[email protected]

Workshop on Detection Limits without NoiseAmerican Society for Testing and Materials

InternationalWebinar April 2, 2020

Statistics of noiseless detectors

1

Where can we expect zero counts?

2

Rare-event physics: neutrino Astrophysics: dark matter Radioactivity counting Mass spectrometry Asbestos counting under microscope Bacteria counting on Petri dish Zero accident

Example: alpha spectrometry

3

239Pu 236Pu

300 400 500 600 700 800

Rel

ativ

e in

tens

ity, s

hifte

d

Pulse height (channel)

LCS

MB = 0

MB > 0

BKG = 0

BKG > 0

Pu-239

Pu-236

Signal and noise trials

Signal trial (gross)

Noise trial

(bkg)Net Comment

General case

Any Regular case

Special case

Ideal detector

Special case

How can you do statistics when signal is zero?

4

The following approaches are described to study theory of zero counts

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Simple approaches: One-count upper limit Probabilistic approach

Bayesian statistics using distributions: Poisson Negative binomial Zero-inflated Poisson

Simple solution to zero counts

One-count upper limit:• Assume 1 count• Calculate upper limit for propagated quantities

such as rate or concentration of analyte

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Loveland et al. 2017

Poisson distribution

7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

0 3 6 9 12 15

Prob

abili

ty m

ass

func

tion

Counts/10 ms

DataPoisson

137Cs source 3109922 events

collected mean counts𝜇 2.8738/10 ms dispersion

coefficient𝛿 1.0017

Parameter Symbol RequirementMean 𝜇 𝑁𝑝

𝑁 ≫ 1𝑝 ≪ 1

Variance 𝜎 𝜇

Dispersion coefficient 𝛿

𝜎𝜇 1

In radioactivity:

𝜇 𝑁𝑝 𝑁 𝜆𝑡𝜀𝑁𝜆𝜀 𝑡 𝜌𝑡

Johnson et al. 2005

Beach et al. 2017

Probabilistic approach to zero counts

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𝑃 𝑥 𝜇𝜇 𝑒𝑥! , 𝑥 ∈ 0,1, …

𝑃 0 𝜇 𝑒

Poisson distribution

Probability of 0 counts

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

exp(

-μ)

μ

ConfidenceSignificanceExponentialMean valueUpper limit

Significance Confidence Level

Upper limit

𝛼 𝐶𝐿 1 𝛼 ln 1/𝛼0.01 0.99 4.60.05 0.95 3.00.10 0.90 2.30.37 0.63 1.0

Loveland et al. 2017

Likelihood function, sufficiency, bias

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𝑥 , 𝑖 1, … ,𝑛 individual 𝑛 measurements of counts, each in time 𝑡

𝑆 𝑥

𝐿 𝜌 𝑆 ~𝜌 𝑒 likelihood function

𝑆 is sufficient statistics

𝜌 rate of Poisson process

Maximum Likelihood Estimators (MLE), unbiased

𝜌𝑆𝑛𝑡

𝜎𝑆𝑛𝑡

sum of counts

Upper limit for zero counts

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How can we estimate an upper limit on rate ?

Unbiased MLE estimators for zero counts are:

Bayesian statistics

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likelihood function prior

posteriorBox et al. 1994

Priors for rate

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Prior class Prior Parameters Formulaconjugate gamma 𝑎 𝑏 𝜌 𝑒

noninformativeuniform 1 0 𝑐𝑜𝑛𝑠𝑡Jeffreys 1/2 0 1/ 𝜌Jaynes 0 0 1/𝜌

Box et al. 1994, Jeffreys 2003, Jaynes 1968, Semkow 2007

Bayesian estimators

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Bayesian point estimators for rate

Bayesian upper limit for rate:

solve for

Gamma function

Incomplete Gamma function

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Prior

uniform Jeffreys Jaynes𝑎 1 0.5 0 Formula

Mean 1 0.5 0 𝑎 𝑛𝑡 𝑏⁄Variance 1 0.5 0 𝑎 𝑛𝑡 𝑏⁄

Bias biased biased unbiased𝛼 𝐶𝐿 1 𝛼 𝑢

0.01 0.99 4.6 3.3 na0.05 0.95 3.0 1.4 na0.10 0.90 2.3 1.2 na0.37 0.63 1.0 One-count

limit0.16 0.84 1.0

Formula ln 1 𝛼⁄ 𝑛𝑡⁄ 𝐶𝐿 erf 𝑛𝑡𝑢 divergent

𝑆 0 𝑏 0 𝑛 1 𝑡 1

Bayesian numerical examples for zero counts

Overdispersion and negative binomial

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0.01

0.02

0.03

0.04

0.05

0.06

0.07

30 35 40 45 50

Pmf

Counts/0.1 s

DataPoissonNB

1E-04

1E-03

1E-02

15 25 35 45 55 65

Pmf

Counts/0.1 s

DataPoissonNB

gamma source 385375 events collected mean counts

𝜇 41.07/0.1 s dispersion coefficient

𝛿 1.041

𝑁𝐵 𝑥 𝑐,𝑑𝑐 𝑑

𝑥! 1 𝑑 , 𝑥 ∈ 0,1, …

𝜇 𝑐 𝑑⁄𝛿 1 1 𝑑⁄

Johnson et al. 2005

Müller 1978

How to set an upper limit for zero counts using Bayesian solution for negative binomial distribution?

We have 3 parameters: evaluate experimentally or theoretically

dispersion coefficient make assumptions about prior

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Zero-inflated Poisson

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1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

0 3 6 9 12 15

Pmf

Counts

Poissonz-PoissonPoisson meanz-Poisson mean

Poisson z-Poisson ExampleParameters 𝜇 𝜇, 0 𝜔 1

𝜇 4𝜔 0.25

Mean 𝜇 1 𝜔 𝜇Dispersion coefficient 𝛿 1 𝛿 1 𝜔𝜇

Bayesian solution 3 parameters: 𝜇, 𝛿 , 𝑎

Johnson et al. 2005

Summary and conclusions

The key solution to zero counts is to calculate an upper-limit for the mean.

A 1-count upper limit results in poor confidence.

A simple probabilistic approach to zero counts is effectively a Bayesian method with uniform prior, and it is biased.

All Bayesian methods based on Poisson distribution used for zero counts are biased, except for the Jaynes prior, the latter is however improper and does not have the upper-limit solution.

The best choice for the upper limit appears to be using Jeffreys prior.

Methods based on negative binomial or z-Poisson require an additional parameter from theory or experiment to estimate the upper limit.

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Selected references

• Beach S.E., Semkow T.M., Remling D.J., Bradt C.J. (2017) Demonstration of fundamental statistics by studying timing of electronics signals in a physics-based laboratory. Am. J. Phys. 85, 515-521.

• Box G.E.P., Tiao G.C. (1994) Bayesian Inference in Statistical Analysis. J. Wiley & Sons, New York, NY.

• Jaynes E.T. (1968) Prior probabilities. IEEE Trans. Sys. Sci. Cyb., SSC-4, 227-241.• Jeffreys H. (2003) Theory of Probability. Clarendon Press, Oxford.• Johnson N.L., Kotz S., Kemp A.W. (2005) Univariate Discrete Distributions. Wiley-Interscience,

Hoboken, NJ.• Loveland W.D., Morrissey D.J., Seaborg G.T. (2017) Modern Nuclear Chemistry. Wiley,

Hoboken, NJ.• Müller J.W. (1978) A test for judging the presence of additional scatter in a Poisson process.

BIPM Report, 78/2, Bureau International des Poids et Mesures, Sèvres, France.• Semkow T.M. (2007) Bayesian inference from the binomial and Poisson processes for multiple

sampling. In Applied Modeling and Computations in Nuclear Science, ACS Symposium Series 945, ACS/OUP, Washington, DC, 335-356.

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AbstractIn physical and analytical measurements, we have a noise (or background) trial and a signal trial(i.e., signal plus noise). One can have three experimental outcomes: i) both signal and noisetrials are positive, ii) signal trial is positive whereas noise trial is zero (an ideal detector), as wellas iii) where the signal and noise trials are both zero, which is the subject of this work. The noisetrial could be even greater than zero in an excessively long measurement but may be zero in ameasurement at hand. Can we infer anything about the signal in the latter case iii)? We assumethat the detector is properly calibrated and tested using strong-signal sources. This topic isrelated to the science of rare events, which are often encountered in physics, astrophysics, aswell as in analytical measurements utilizing ionizing radiation, mass spectrometry, asbestosdetection, among others. In our laboratory, the noiseless detection is often encountered in alphaspectrometry performed for radiological health protection. Poisson distribution is central to thescience of rare events and we present a brief introduction to it. The statistics of noiselessdetectors can be handled using Bayesian statistics applied to the Poisson likelihood. We discussthe use of priors: conjugate as well as noninformative (uniform, Jeffreys and Jaynes). Wecalculate the posteriors for the noiseless detector using several priors. From the posteriors, wecalculate statistical measures for the signal such as upper limits, mean values, as well asvariances, when the measured signal is zero. Then, we discuss the effect of different priors onsuch calculated statistical measures as well as their interpretation. The Poisson distribution is asingle-parameter distribution which cannot accommodate overdispersion. A possible presence ofoverdispersion in the detection systems can be determined using strong signals. An extension ofthis work is discussed to include a negative binomial distribution, which can handleoverdispersion. We also describe a zero-inflated Poisson distribution which can handle anenhancement at zero signal as well as overdispersion. 20