Thomas M. Semkow
Wadsworth Center, New York State Department of
Health, Empire State Plaza,
Albany, NY 12201-0509, Telephone 518-474-6071,
Fax 518-474-8590, E-mail
semkow@wadsworth.org
School of Public Health, University at Albany,
State University of New York
According to an
idealized model, the Poisson distribution describes fluctuations in nuclear
statistics. Nearly exact Poisson statistics has been reached under very
stringent conditions (Concas and Lissia, 1997). It is known from practice, however, that the Poisson model is
often inadequate. A dispersion coefficient d, defined as a ratio of a variance
to mean, is used as a test for Poisson statistics (where it is equal to
1). One is often faced with a so called
overdispersion in nuclear data caused by either excess fluctuations (Semkow,
1999a,b) or sequential decay (Inkret et al.,1990), which lead to the dispersion
coefficient d>1. The opposite
situation is an underdispersion (d<1), which occurs due to dead-time effects
and is not considered in this work. There are two approaches to the
overdispersed statistics. One is with the aid of the dispersion coefficient (or
chi-square), as studied recently by Tries (1997,2000). Another is a
distributional view represented in this work.
We have proposed a
general stochastic model of sequential processes, to describe overdispersion in
nuclear statistics (Semkow, 1999a,b).
The processes typically encountered in nuclear science are production,
extraction, survival, decay and detection of radioactive atoms. The probability
of a count is thus a product of the probabilities of individual processes. The
probabilities of processes may fluctuate in the course of experiments. We have
shown that these fluctuations can be well described by a beta distribution. The
statistics becomes overdispersed if the average probability changes from a
measurement to measurement. In this case, the fluctuations are called random
Lexis fluctuations. If the fluctuations do not change the average probability
(random Poisson fluctuations), the statistics remain Poisson. We have shown
that this model leads to a generalized hypergeometric factorial moment
distribution (GHFD) by Kemp and Kemp (1974).
In this presentation, we
show how the overdispersed counting data is recognized in practice as well as
how to apply our model to it. Therefore, we concentrate on a limited version of
the model. To register a count we have to have a minimum of 2 processes:
radioactive decay and detection. They
are represented by a probability of decay and probability of detection (i.e.,
detector efficiency). The probability of decay can fluctuate due to time
fluctuations of the timer. The efficiency can fluctuate due to temperature,
electronic setup, sample positioning, etc. While both processes are needed to
register a count, we assume further that only one of them undergoes
fluctuations in a particular experiment leading to overdispersion. This considerably simplifies the mathematical
framework and leads to known statistical distributions, discussed below, that
can describe the overdispersed nuclear statistics.
If 0<p<1, where p
is the probability of the excess fluctuating process, then the overdispersed
statistics is described by a beta-Poisson distribution. However, since the Poisson process requires
N>>1 and p<<1 in general (N is the number of radioactive atoms),
the Poisson character has been ensured by another, nonfluctuating p. For
instance, if p<<1 is the nonfluctuating probability of decay and 0<e<1
is the fluctuating detection efficiency.
If p<<1 is fluctuating then the statistics is described by a
negative binomial distribution. In either case, if mean m>>1, we obtain
an overdispersed Gaussian distribution, which is a Gaussian distribution having
additionally d>1. In this work, we present experimental data consistent with
the beta-Poisson, negative binomial and overdispersed Gaussian
distributions. We also show how to use
the formula for dispersion coefficient, d=1+m*v**2, where v is a variation
coefficient (a standard deviation divided by the mean of the fluctuating
probability).
The first data we
discuss is a measurement of 226-Ra in equilibrium with the daughters in the
Lucas cell. That data exhibit an unusual d_.2 due to temperature effect on the
efficiency. In fact, one could measure a temperature with this radiation
detector, if properly calibrated. Next, we show the measurements of the alpha
background on a gas proportional detector showing d=1.22. These data are well
described by the beta-Poisson distribution. The 239-Pu standard measurements on
the same counter are consistent with the overdispersed Gaussian distribution
(d=1.27). We also present the scalar data by Mueller (1978) which are
overdispersed due to time fluctuations (d=1.04) and are fitted with the
negative binomial distribution. Another
overdispersed background data is from plastic/NaI beta/gamma coincidence system
(d=1.05), which is a borderline between the beta-Poisson and overdispersed
Gaussian distributions. In all the fits described above, a moment analysis was
performed. We also discuss a possible application of the beta-Poisson
distribution to sequential decay. Finally, we present an application of the
dispersion formula to the NaI data acquired through a multiplexer, which
suffers from overdispersion due to a low quality timer. The application of the overdispersed
statistics requires that the processes are fluctuations. An example of
overdispersion caused by systematic outliers due to electronics noise is shown
(d=1.45), which is in the background data on a beta proportional counter. In
this case, one cannot use the methodology described here.
The main effect of the
overdispersed statistics is that it decreases the precision of radioassay. In
addition, the detection limits increase, as shown by Tries (1997).
References
Concas G. and Lissia M.
(1997) Search for non-Poissonian behavior in nuclear beta decay. Phys. Rev.
E55:2546-2550.
Inkret W.C., Borak T.B.
and Boes D.C. (1990) Estimating the mean and variance of measurements from serial
radioactive decay schemes with emphasis on 222-Rn and its short-lived
progeny. Rad. Prot. Dos. 32:45-55.
Kemp A.W. and Kemp C.D.
(1974) A family of discrete distributions defined via their factorial moments.
Comm. Statist. 3:1187-1196.
Mueller J.W. (1978) A
test for judging the presence of additional scatter in a Poisson process.
Bureau International des Poids et Mesures Report BIPM--78/2, Sevres, France.
Semkow T.M. (1999a)
Theory of overdispersion in counting statistics caused by fluctuating
probabilities. Appl. Rad. Isot. 51:565-579.
Semkow T.M. (1999b)
Overdispersion in nuclear statistics. Proceedings of the Ninth Symposium on
Radiation Measurements and Applications, Ann Arbor, MI, May 1998, Nucl. Instr.
Meth. Phys. Res. A422:444-449.
Tries M.A. (1997) Detection
limits for samples that give rise to counting data with extra-Poisson variance.
Health Phys. 72:458-464.
Tries M.A. (2000)
Applications of a quadratic variance model for counting data. Health Phys.
78:322-328.