Download characterisation of non-constant background in

Radiation Protection Dosimetry (2015), Vol. 164, No. 3, pp. 408 – 421
Advance Access publication 15 October 2014
doi:10.1093/rpd/ncu297
CHARACTERISATION OF NON-CONSTANT BACKGROUND
IN COUNTING MEASUREMENTS
John Klumpp*, Guthrie Miller and Alexander Brandl
Environmental and Radiological Health Sciences, Colorado State University, 1618 Campus Delivery,
Fort Collins, CO 80523, USA
*Corresponding author: [email protected]
Received 28 April 2014; revised 28 August 2014; accepted 29 August 2014
A ‘moving-target’ method for characterising background in a counting measurement in which the instantaneous background count
rate is a function of time, rather than being fixed, is proposed. This model treats the average Poisson mean in observation period
P as coming from a gamma distribution with parameters aP and bP. This model is applied to a large dataset of replicate observations, consisting of 242 234U method blank measurements collected over a 2-y period. Point estimates of the model parameters are
determined by comparing the mean and variance of the observed data and by maximising the likelihood function. Posterior distributions of the parameters are obtained by Markov Chain Monte Carlo. Assuming time-invariant fluctuations of the background
count rate, the variation of the instantaneous count rate is described by a correlation function, which can be interpreted as describing how rapidly the background changes with time, or how likely the background is to change between measurements. An ‘exponential-correlation’ model of background time dependence is proposed, with parameters a, b and correlation time t. Once determined,
these parameters fully describe the distribution of background, just as NB and TB in the fixed-target model.
INTRODUCTION
The ability to accurately characterise background in
counting measurements is of interest in a wide range
of fields. Radiation measurements, which are of
counts per unit time, are common in fields such as
health physics, radiation detection, medical and industrial radiography and astrophysics. Background
measurements are also important for counting measurements over an area or volume, such as counting
the number of cells in a petri dish, animals in a forest
or stars in a region of space.
Accurate characterisation of background is essential
in a measurement used to determine the true amount
of some material in a sample. It is also needed to determine the rate of false alarms for measurements with
decision thresholds. In addition, background measurements might be taken to understand the origins of
background itself.
It has long been understood that background which
does not come from any single fixed source varies
with time of day, location, meteorological conditions
and many other factors(1). In cases where background
is varying, this results in a broader inferred distribution of background count rate than a ‘fixed-target’
background. Left unaccounted for, this leads to inaccurate posterior distributions, higher than expected
false positive rates in measurements with decision
levels and faulty performance estimates of detectors.
The fixed-target assumption is that some fixed true
mean background count rate does indeed exist and
that uncertainty on that count rate can therefore be
reduced indefinitely by repeated observations. As a
result, it is common practice to attempt to obtain very
accurate estimates of true mean count rates by conducting long background measurements(2, 3).
The fixed-target background model compiles background data into sufficient statistics (e.g. total number
of counts NB and total measurement time TB), and
if, for example, the times for individual counts are
recorded, discards a great deal of the available information. In particular, the complete measurement data
contain information about the time dependence of
background, which is lost when it is compiled into sufficient statistics. This simplification was necessary when
routine statistical calculations were performed by hand,
but with the widespread adoption of computers there
has been an increase of interest in developing models,
which can make use of the great wealth of information
obtained in long background measurements(4).
Because of the nature of radioactive decay, the
number of counts produced by any fixed source in a
given time or spatial interval will follow a Poisson distribution which, for high enough count rates, can be
well approximated by a Gaussian distribution. Widely
used statistical techniques—classical and Bayesian—
operate on the assumption that the background count
rate is sampled from a Gaussian distribution (ISO 2010).
The ‘exact Poisson’ likelihood function for sample
measurements(5, 6) is important when the number of
detected counts is not large, but so far the treatment of
background has been based only on the fixed-target
model.
Published by Oxford University Press 2014. This work is written by (a) US Government employee(s) and is in the public domain in the US.
NON-CONSTANT BACKGROUND
This paper makes use of Markov Chain Monte
Carlo (MCMC) methods discussed by Miller(5). A
Fortran sub-routine for General MCMC is available
from www.netlib.org/misc/gmcmc.
MOVING-TARGET MODEL OF
BACKGROUND COUNT DATA
0
A ‘moving-target’ model is proposed in which the instantaneous background count rate is time varying
rather than being constant. An example of nonconstant count rate is shown in Figure 1. As in the
traditional model, the number of counts N observed
in a given observation period P is Poisson distributed:
PðNÞ ¼
mN
P
expðmP Þ
N!
ð1Þ
where the mean, mP, is the integral of the instantaneous count rate over P,
mP ¼
ðP
lðTÞdT
The mean of the gamma distribution is given by
aT =b0T , and the variance by aT =ðb0T Þ2. The probability distribution of observed counts, called the sampling distribution, is given by
ð1
PoissonðNjmT Þ
PðNjaT ; b0T Þ ¼
gammaðmT jaT ; b0T ÞdmT
This integral evaluates to a normalised parametric
probability distribution, which through a simple reparametrisation (discussed in Appendix 1) takes the
form of an NB distribution(7):
1
PðNjaT ; b0T Þ ¼ NB Njr ¼ aT ; p ¼
ð5Þ
0
1þbT
The mean of the NB distribution is given by
pr=ð1 pÞ ¼ aT =b0T . The variance of the NB distribution is given by
pr
ð2Þ
2
0
ð1 pÞ
However, as the Poisson mean mP is not constant
from period to period, it is modelled as a random
variable drawn from a probability distribution, P(mP).
It is convenient to use a gamma distribution with
parameters aP and b0P ; bP =P (bP has dimensions of
time, b0P is dimensionless). The gamma distribution is
a convenient choice because it allows the mean and
variance to vary independently and results in a closedform sampling distribution (the negative binomial, NB).
Thus, the complete model for the counts observed
in some time interval T is
mT gammaðmT jaT ; b0 T Þ
N PoissonðNjmT Þ
¼
aT
aT
þ
b0T ðb0T Þ2
ð6Þ
The variance of the moving-target sampling distribution is therefore equal to the sum of the mean
ðaT =b0T Þ; as for a fixed target), plus the variance of
the gamma distribution from which the Poisson mean
is sampled. It follows that the variance of the sampling distribution will be greater than that of the
Poisson distribution in proportion to the variability of
the Poisson mean, and defining the relative excess
variance Vx(T ) as the excess (over Poisson) variance
of the counts observed in time T,
Vx ðTÞ ¼
ð3Þ
Figure 1. Example of non-constant instantaneous count
rate l(T).
ð4Þ
ðNÞ
1
T
1¼ 0 ¼
MeanðNÞ
bT bT
ð7Þ
Figure 1 shows that there is some degree of correlation of the instantaneous count rate. There is some
timescale over which the count rate can be expected to
be constant. This timescale is determined by the correlation time, defined more precisely later on in terms
of a correlation function C(T ) that mathematically
describes how the instantaneous count rates at two
time points separated by T are related. In a time interval T very small compared with the correlation time,
the instantaneous count rate will be constant and there
will be no excess variance. For small time T, a linear
dependence as in Equation (7) with bT constant is
expected. In the other extreme of a time T very much
longer than the correlation time, the relative excess
variance will be independent of time. This is because
the total time can be broken up into a large number of
subintervals having uncorrelated instantaneous count
rates, and the total variance from the independent subintervals will be additive just like the total mean from
409
J. KLUMPP ET AL.
the independent subintervals. The relative excess variance Vx (T ) can be expressed as an integral over the
correlation function C(T ), and thus C(T ) can be
obtained from the time dependence of Vx (T ).
It is useful to define a particular model of time
correlation of instantaneous count rate. An ‘exponential-correlation model’ is defined, which is used to
generate the instantaneous count rate versus time
curve shown in Figure 1. In this model, the instantaneous count rate, l(t), is constant for some time, tc,
which is drawn from an exponential distribution so
that Pðtc Þ ¼ expð2tc Þ=tÞ2dtc =t, where t is the correlation time. At the end of time tc, a new l is drawn
from the gamma distribution, gamma (lja, b), and a
new tc is generated. Numerical studies show that
within this model, the distribution of the average count
rate in some time interval T, /mT , is approximately
gamma (mTjaT, bT). The quantities a and b are the
same as the measured parameters aT and bT in the limit
of small time T. This model is discussed in greater detail
in Appendix 1. The exponential-correlation model was
found adequate to represent the dataexamined.
over a 3-y period. These are measurements of samples
known to not contain 234U, carried through the entire
chemical processing and alpha-spectrometry counting
procedure, using the exact analysis procedures routinely employed for actual samples. The counting period
was 42 h for one measurement. The total number of
detected counts for all 242 replicate measurements was
2200 (only 0.2 counts per hour). These data are
illustrated in Figure 2.
DETERMINATION OF aT AND bT
Methods
In the moving-target model, the observed number of
counts in time T is a random number drawn from a
Poisson distribution with mean mT. Rather than being
constant, a new mT is drawn from a gamma distribution
with parameters aT and a0T ¼ b=T after each observation. Point estimates of the model parameters aT
and b0T were determined by maximising the likelihood
function, and posterior distributions of the parameters
were obtained by MCMC.
The likelihood of the observed data is given by
EXPERIMENTAL DATA
The data for this study consist of 242 replicate 234U-inurine ‘method blank’ background measurements taken
LðaP ; b0P Þ /
m
Y
PðNi jaP ; b0P Þ
ð8Þ
i¼1
Figure 2. The experimental data plotting 234U ‘method blank’ background measurements taken over a 3-y period. The error
bars represent the square root of the number of observed counts.
410
NON-CONSTANT BACKGROUND
where Ni represents the number of counts observed in
the ith observation, and subscript P denotes the duration of each observation. Maximising the likelihood
is equivalent to minimising the negative logarithm of
this function. Therefore, it is defined that an error
function to be minimized, for which each data point
is defined as follows:
!
PðNi j aP ; b0P Þ
2
0
xi ðaP ; bP Þ ; 2 ln
maxaP ;b0P PðNi j aP ; b0P Þ
NBðNi j aP ; b0P Þ
¼ 2 ln
PoissonðNi j mP ¼ Ni Þ
ð9Þ
where NB denotes the negative binomial distribution
discussed in Appendix 1.
The denominator is the maximum value of
PðNi jaP ; b0P Þ, so the ratio is ,1. Therefore, x2i is positive, and a square root can be taken to obtain xi. When
defined in this way, the average value of x 2/m for a
large number of repetitions is somewhat greater than,
but approximately equal to, 1. Therefore, x 2 becomes a
useful metric for interpreting the quality of the fit.
Setting the partial derivative with respect to b0P
at the minimum,
equal to zeroP
implies that b0P ¼ aP N
¼ð
where N
N
Þ=m.
Given
that
the mean of the
i
i
gamma distribution is equal to aP =b0P ; this leads to
the intuitive result that the mean of the Maximum
Likelihood (ML) distribution of the Poisson mean
will be equal to the observed mean count rate.
Incorporating this result, the error function becomes
x2i ðaP Þ ¼
"
2 ln
aP
aP þ N
aP Ni
eN
Ni ðaP þ NÞ
ðNi þ aP 1Þ!
ðaP 1Þ!
#
ð10Þ
For a single data point, the minimum value is zero,
which occurs for aP !1. For multiple observations
that are not identical, the minimum of the x 2 function
occurs for some positive value of aP, which may be
finite and can easily be determined numerically.
In the Bayesian, i.e. probabilistic approach, prior
probability distributions need to be assumed for
aP and bP. These two parameters are highly correlated, with the most likely values being those whose
ratio is equal to the observed mean count rate. For
that reason, the analysis is done using parameters
mP ¼ aP =b0P , which is the mean of the distribution,
and 1=b0P , which is the variance divided by the mean
minus 1 (excess relative variance). Both parameters
are assigned log-uniform priors, which are broad
enough to avoid truncating the posterior distribution.
The parameter space is then explored using MCMC.
RESULTS
By maximising the likelihood function given in
Equation (8) with respect to aP and b0P , the point
estimates aP ¼ 9.3 and b0P ¼ 1:1 for the 234U dataset
are obtained, which implies that the Poisson mean
mP is drawn from
mP gammaðaP ¼ 9:3; b0P ¼ 1:1Þ
ð11Þ
where mP has units of counts per observation, and
each observation is of time P. The sampling distribution for the expected counts observed in time interval
P, NP, is given by
PðNP jaP ; b0P Þ ¼ NBðaP ¼ 9:3; b0P ¼ 1:1Þ
ð12Þ
The mean and variance of the sampling distribution
are given by
Mean½NBðaP ; b0P Þ ¼ aP =b0P
¼ 8:5
0
Variance½NBðaP ; bP Þ ¼ aP =b0P þ aP =b0P 2
¼ 16:1
ð13Þ
These parameters are the ML estimates of the
mean and variance of the observed counts over the
242 multiple measurements of time P, whereas, for
comparison, the actual mean and variance of the 242
observations are 8.5 and 17. The ML estimate of the
mean is equal to the mean of the observed counts.
This sampling distribution is analogous to the
Poisson distribution, which would be computed using
the traditional fixed-target model. For the traditional
model, the expected variance of the observed counts
is equal to the mean, aP =b0P , while for the movingtarget model there is excess variance aP =ðb0P Þ2. The
excess variance divided by the mean (the excess relative variance, Vx, defined by Equation (7) and discussed in Appendix 1) is given by
Vx ðPÞ ¼
aP =ðb0P Þ2
1
¼ 0 ¼ 0:9
aP =b0P
bP
ð14Þ
The probabilistic analysis provides probability distributions rather than point estimates for the parameters. Probability distributions from analysis of the
234
U data obtained using MCMC are shown in
Figure 3. These are well aligned with the ML point
estimates, as they must be given the use of a logarithmic prior, which is effectively flat with respect to the
binning. The average value of x 2/m is 1.7 for m ¼
242 data. For simplicity of the figures, the unit of time
is chosen to equal the observation period P, so that
b0P is the same as bP.
411
J. KLUMPP ET AL.
Figure 3. Probability distributions of the parameters of the Poisson mean. These figures illustrate the uncertainty on the
values of the parameters of the gamma distribution from which the Poisson mean is drawn. The central vertical lines indicate
ML point.
Figure 4. Scatter plot of the parameters of the Poisson mean
distribution. If the Poisson mean were in fact constant, the
fixed-target point would be close to the moving-target
distribution. From inspection, it is clear that the posterior
distribution of the model parameters effectively rules out the
fixed-target model.
Another way of visualising the output of the parameter estimates is through scatter plots, shown in
Figure 4. This plot shows interpretations of the data
in terms of the model parameters aP/bP and 1=b0P ,
produced by the fixed-target and moving-target
models. The ML and fixed-target methods result in
point estimates of the parameters, while the probabilistic method produces a sample of interpretations
from the posterior distribution. In the probabilistic
method, two MCMC runs (MCMC1 and MCMC2)
with different seeds and different extreme starting
points are shown to demonstrate convergence. The
y-axis shows PaP/bP , which is the mean number of
counts in time period P and corresponds to the
Poisson mean during that time period, mP. The x-axis
shows P=bP ¼ 1=b0P, which is the excess relative
variance of the distribution in time period P, Vx(P).
Small values of 1=b0P correspond to a narrow countrate distribution.
In the moving-target model, the gamma distribution represents variation and uncertainty of the
Poisson mean. In contrast, in the fixed-target model
the gamma distribution represents uncertainty on the
Poisson mean (implicitly assuming that it has some
fixed value), and therefore becomes arbitrarily narrow
as more data are collected. This fact is represented by
the small value of 1=b0P in the plot of the fixed-target
estimate, which corresponds to a narrow count-rate
distribution.
The probabilistic analysis gives a sample of the
model parameters conditioned on the data (i.e. a
sample from the posterior distribution). From this,
one can calculate a sample of the probability distribution of any model quantity. One such model quantity
is the cumulative distribution of the number of
observed counts, which is shown in Figure 5. This
figure shows the mean and standard deviation of the
samples (from two Monte Carlo runs with different
extreme initial values), along with cumulative distributions from the ML and fixed-target estimates for
comparison.
DETERMINATION OF THE CORRELATION
TIME
Methods
The correlation time, t, was determined by comparing the theoretical and experimental values of the
412
NON-CONSTANT BACKGROUND
Figure 5. The posterior cumulative distribution function of
the number of observed counts per time interval P. The data
are shown together with calculations using the probabilistic
method, the ML method and the fixed-target method. The
ML estimate comes from the NB (a ¼ 9.3, b 0 ¼ 1.1) while
the fixed-target result is indistinguishable from the Poisson
distribution, Poisson (m ¼ a/b 0 ).
determinations of aT and b0T , one for T ¼ P as shown
before, and another with T ¼ 150P.
In determining the correlation time, experimental
values of the excess relative variance were obtained directly from the observed variance and mean of the replicate samples, rather than with MCMC. Referring to
the total time T in terms of the number of sequential
samples that are summed, j, for j ¼ 1, the 242 original
observations are used. For j ¼ 2, samples 1 and 2, 3
and 4, 5 and 6 etc. are summed forming 121 replicate
observations of counts in twice the observation period.
This process is continued up to using the sum of 60
separate observations (total time ¼ 60` TD ¼ 150P),
which gives only 4 replicates to calculate the variance
and mean.
Theoretical values of Vx(T ) were derived from the
theoretical excess relative variance given by Equation
A.30 of Appendix 1:
Vx ð jÞ
2
¼1þ
Vx ð1Þ
TD
excess relative variance of counts observed in time
T, Vx(T ), where T is the total time for j observations,
T ¼ jTD. As already discussed, Vx(T ) is equal to
1=b0T . This is illustrated in Figure 6, which shows two
Cðtd Þð1 td =TÞdtd
ð15Þ
TD
Thus, Vx( j ) is dependent on the correlation function, C(td), whose form will depend on the mechanism causing the count rate to change. One sees from
Equation (15) that the excess relative variance
increases from its initial value to a new plateau as
T!1 over a time that depends on the correlation
function C(td). Any parametrisation of C(td) must
approach 1 as td !0, and 0 as td !1.
The 234U data were analysed using the exponentialcorrelation model defined above, which has a correlation function, CðTÞ ¼ expð2td =tÞ, where t is the
correlation time given by Equation (A.20). Note that
this parametrisation meets the constraints regarding
T!0 and T!1.
The exponential-correlation model has three parameters: a, b and t. Numerical studies showed that
within this model, the distribution of the average count
rate in some time interval T, mT, is approximately
gamma mT j aT, bT. In terms of the measured quantities
aP and b0P , using Equations A.27 and A.28 and equating mean values so that a/b ¼ aP/bP,
t
t
2P
1
1 exp P
2P
t
aP
a¼b
bP
b ¼ bP
b0T
Figure 6. Two determinations of aT and
for T ¼ P and
T ¼ 150P. For T ¼ P, 242 observations were used. For T ¼
150P, 60 of these observations were summed, giving only 4
replicates of the sums. The points corresponding to the
actual means and variances of the replicate observations are
also shown.
ð T¼jTD
ð16Þ
In this way, the exponential-correlation model is
reduced to a single unknown parameter t.
Likelihood functions and prior probabilities are
required in order to interpret the data. The prior
probability distribution for t was assumed to be loguniform from t/P ¼ 1 to t/P ¼ 1000. Assuming a
Gaussian likelihood function, and using bold face to
413
J. KLUMPP ET AL.
denote matrix quantities, the following is obtained:
!
ðy cÞT H1 ðy cÞ
LðtÞ / exp ð17Þ
2
where y is the observed data for the excess relative
variance, c is the theoretical excess variance as a
function of t and H is the covariance matrix, with elements
Hi;j ; kðyi ci Þðyj cj Þl:
ð18Þ
Because yj ¼ Vx( j ) at different times j involves the
same measured count quantities Ni, the observed data
are necessarily correlated. As a result, the covariance
matrix is not diagonal.
RESULTS
Given the above likelihood and prior distributions,
the posterior distribution of t was calculated by
MCMC. The covariance matrix was recalculated at
each iteration by simulating a large number of replicates of the data using the exponential-correlation
model. Figure 7 plots the theoretical curves as the
posterior means, with error bars corresponding to
one standard deviation of the posterior. The two
curves using chains with different extreme starting
points line up almost perfectly, demonstrating convergence of the chains. The data uncertainty bars are the
diagonal elements of the covariance Hj,j. The large
uncertainty bars for the correlation function reflect
the large uncertainty of t, which seems to be caused
by the correlation of the data.
The parameter values used for these calculations were
aP ¼ 9.3 and b0P ¼ 1:1. The posterior average value of
x 2/m was 1.3, with m ¼ 60 and x2 ¼ 22log(L(t)).
The cumulative probability of the correlation time is
shown in Figure 8.
Using a simple likelihood function that neglected
data correlation produced a much narrower
t distribution centred at t/P 30. Studies using simulated data showed that this likelihood function was too
narrow to reliably include the correct value of t.
DISCUSSION
Characterising background is of importance primarily
with respect to measurement of something else, ‘the
sample’, but it also might be done to understand the
origins of the background itself. Inference on a sample
Figure 7. Excess relative variance from 234U data. The data were interpreted using the exponential-correlation model for the
instantaneous count rate. The data uncertainty bars are the diagonal terms of the covariance matrix. The curves show
the posterior means of the correlation function and the excess relative variance, while the uncertainty bars on the curves show
the posterior standard deviations.
414
NON-CONSTANT BACKGROUND
Figure 8. Cumulative probability of the correlation time
from the 234U data, computed using the exponentialcorrelation model for the instantaneous count rate.
is made by constructing a likelihood function. For a
counting measurement with background, the exact
Poisson likelihood function(5, 6) involves the integral of
the posterior probability of background count rate l,
given the observed background counts. In the ‘fixedtarget’ model of background, the number of observed
background counts, NB, in time TB, is presumed to
come from a Poisson distribution with a constant but
unknown mean lTB, which, as a function of l, is proportional to ðlTB ÞNB expðlTB Þ. When normalised,
this is a gamma distribution, gamma (lj a, b), with
a ¼ NB þ 1 and b ¼ TB. For a non-informative prior
on background count rate, the posterior distribution of
l is this gamma distribution. Analytic formulas for
the exact Poisson likelihood function using the fixedtarget treatment of background and depending on the
number of background counts NB in time TB are given
in Chapter 7 of the book by Miller(5).
One sees that in the limit of a very long background
observation period, the fixed-target model results in a
perfectly well-known background, because in that
case a and b are very large and the standard deviation
divided
pffiffiffi by the mean of the gamma distribution is
1= a.
A moving-target method has been developed for
background, which allows the background count rate
and the Poisson mean to vary randomly with time. In
contrast to the fixed-target method, which requires
only one measurement, the moving-target method
requires a large dataset of replicate measurements.
This method was tested on a dataset consisting of 242
replicate 234U method blank measurements.
In the moving-target method, the variation and
uncertainty of the average count rate in time period P
are modelled as coming from a gamma distribution
with parameters aP and bP. The simplest use of these
parameters for a sample measurement is to replace the
probability distribution of the background count rate
gamma (ljNB þ 1, TB) in the integral for the exact
Poisson likelihood function with gamma (ljaP, bP).
This is done by replacing in the formulas the number
of background counts measured, NB, by aP –1 and
the background observation period, TB, by bP as if
aP –1 background counts had been detected in time
bP. For the 234U dataset, assuming a constant background count rate and combining all the measurements gives NB 2000 in TB ¼ 242P in contrast to
aP 1 ffi 8:1 in bP ffi 1:1P with the moving-target
method.
Although a Classical approach would not be
recommended, as an example these numbers can be
used to determine the Classical decision threshold
under fixed-target and moving-target assumptions. In
terms of net counts, y ¼ N NB =R, where R is the
ratio of background to sample count time, and N is
the number of gross sample counts in sample count
time P, the a ¼ 0.05 decision threshold y* is 1.645
times the uncertainty standard deviation of net
countspfor
zero true amount (8), which works out to be
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1:645 ðNB þ 1Þð1 þ RÞ=R, where NB þ 1 needs to
appear rather than NB (note the case NB ¼ 0). Using
NB ¼ 2000 or 8.1 in time period TB ¼ 242P or 1.1P,
the decision threshold is either 4.7 or 6.5 counts per
time period P, that is, 50 % higher using the
moving-target method. This simple calculation does
not take into account the time correlation of the background, effectively assuming it to be short compared
with the sample measurement time. The effect of time
correlation is discussed in Appendix 2.
For the 234U dataset, the moving-target method
implies a broader background distribution than the
fixed-target method. In the limit of an infinitely long
background observation period, the fixed-target model
predicts a perfectly well-known background, while
when this long observation period is broken up into a
large number of replicate background measurements, a
background distribution with finite width may be
obtained. This would be indicated by finite values of
aP and bP.
Under the assumption of time-invariant fluctuation
of the background count rate, the variation of the instantaneous count rate is governed by a correlation
function, which can be interpreted as describing how
rapidly the background changes, or how likely the
background is to change between measurements.
Depending on the presumed mechanism of background variation, the correlation function might refer
to the time during which the sample was prepared in
a ‘contaminated reagents’ view of the background, or
it might refer to the natural time dependence of the
background (e.g. cosmic ray background).
A specific exponential-correlation model of time dependence of the instantaneous count rate was derived,
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J. KLUMPP ET AL.
which has three parameters, a, b and correlation time
t. The quantities a and b are the parameters aT and
taT in the limit of small time T. The 234U data, rebinned to obtain replicate measurements of the sum of
counts in different numbers j of counting periods, was
also analysed using this model to determine t.
The variability of the Poisson mean can be characterised in a number of different ways depending on the
complexity of the analysis. The simplest approach is to
consider only the mean and relative excess variance
Vx(T) (variance/mean 2 1) for replicate measurements
of summed total counts in total time T ¼ jTD for
j counting periods. In terms of gamma distribution
parameters aT and bT, the mean number of counts in
time period P is given by PaT/bT and the relative
excess variance by T/bT, which allows one to solve for
aT and bT given the mean and relative excess variance.
The dependence of Vx(T ) on T is related to the correlation function, which depends only on the correlation
time t, and by a trial-and-error comparison process a
‘best’ value of t can be determined.
It is more complex to use the probabilistic approach, which in general is to determine the likelihood function of the observed data as a function of
the parameters, assign a prior probability distribution
for the parameters and then calculate the probability
distribution of the parameters using MCMC or numerical integration.
For the determination of aT and bT, the data are
the replicate observations in time period T, and the
likelihood function is the NB distribution. The mean
and fractional excess variance were calculated by
maximising the likelihood function or probabilistically using log-scale uniform priors. The results are in
good agreement with mean and fractional excess variance from the replicate measurements and also yield
the uncertainties of these quantities.
To determine the correlation time t, given aP and
t, given aP and bP, a Gaussian likelihood function
taking into account data correlation was constructed.
With a log-scale uniform prior, the probabilistic analysis showed a very broad distribution of t for the
234
U dataset, which shows that the 242 measurements
were barely enough for the determination of t.
If, as in the exponential-correlation model, the instantaneous background count rate is assumed to be
time dependent, this has implications for the optimal
way to take data. If the background is changing
rapidly enough so that background measurement
would be uncorrelated with the sample measurement,
there would be no benefit from performing an individual background measurement. One would instead use
aT and bT derived from a large dataset of background
measurements, where T is the sample measurement
time. In contrast, if the background varies slowly
enough that there is an appreciable correlation
between the background and sample measurements,
it is beneficial to perform individual background measurements. In this case, increasing the duration of the
individual background measurement beyond a
certain point is counter-productive, because it reduces
the correlation between the two measurements. This
is discussed in detail in Appendix 2.
Variation of the background count rate can occur
for a number of reasons. Failure to account for this
variation may produce a dramatic misrepresentation of
the true background distribution. If the Poisson mean
is varying, the moving-target model is needed to accurately describe the background. Our Vx(T ) curves seem
an important tool that might be routinely used to determine if the background is time varying, and if so to
characterise the background correlation time.
ACKNOWLEDGEMENTS
The authors would like to thank Dawn Lewis, then at
the Los Alamos National Laboratory, for providing
the 234U method blank dataset.
REFERENCES
1. Hemic, G. Environmental radiation monitoring in the
context of regulations on dose limits to the public. J. Off.
Repub. Fr. 4, 88–521 (1988).
2. Eckelman, W. A handbook of radioactivity measurements
procedures: National Council on Radiation Protection and
Measurements. NCRP Report No. 58, 2nd edn. NCRP
(1985).
3. Turner, J. Atoms, Radiation, and Radiation Protection.
John Wiley & Sons (2008).
4. Brandl, A. Statistical considerations for improved signal
identification from repeated measurements at low signalto-background ratios. Health Phys. 3, 256 –263 (2013).
5. Miller, G. Probabilistic Interpretation of Data—A
Physicist’s Approach. Lulu Publications (2013) downloadable from Research Gate.
6. Miller, G., Martz, H., Little, T. and Guilmette, R. Using
exact Poisson likelihood functions in Bayesian interpretation of counting measurements. Health Phy. 4, 512– 518
(2002).
7. Ntzoufras, I. Bayesian Modeling using WinBUGS. Wiley
(2011).
8. International Standards Organization. Determination
of the characteristic limits (decision threshold, detection
limit and limits of the confidence interval) for measurements of ionizing radiation “Fundamentals and application. ISO 11929– 1, ISO (2010).
APPENDIX 1. MATHEMATICAL MODEL
OF NON-CONSTANT BACKGROUND
Correlated poisson random variables
Consider two Poisson random variables, N1 and N2.
Our mathematical model is that the joint probability
416
NON-CONSTANT BACKGROUND
P(N1, N2) is of the form
ð1
ð1
PðN1 ; N2 Þ ¼
0
The variance of N is given by
dm 1
0
dm2 PðN1 jm1 Þ
VarðNÞ ¼ k ðN kNlÞ2 l
ðA:1Þ
¼ kN 2 l kNl2
PðN2 jm2 ÞPðm1 ; m2 Þ
¼ a=b0 þ a= b0
in terms of some joint probability P(m1, m2) for the
Poisson means. The first and second moments of the
Poisson distribution are given by
kNjml ¼
1
X
kN 2 jml ¼
kPðkjmÞ ¼ m
¼
k2 PðkjmÞ ¼ m2 þ m
so that for the correlated distribution given by
Equation A.1,
kNi Nj l ¼ kmi ldi;j þ kmi mj l
ðA:3Þ
where di,j ¼ 0 unless i ¼ j in which case it is 1. The
averaging denoted by k l is defined by context so that
on the left-hand side in Equation A.3 it is defined as
in Equation A.2 with
f ðk1 ; k2 ÞPðk1 ; k2 Þ
ðA:4Þ
k1 ¼0 k2 ¼0
with P(k1, k2) from Equation A.1, while on the righthand side
kf l ¼
ð1
ð1
0
dm1
0
The integral is evaluated by rewriting the integrand
in terms of a normalised gamma distribution using
Equation A.9.
The distribution given by Equation A.9 is referred
to in the literature as the ‘NB.
General form of the excess variance
The sum of counts Ni detected in m observation
periods i ¼ 1. . .m is expressed in the following
equation:
N¼
dm2 f ðm1 ; m2 ÞPðm1 ; m2 Þ
0
b0 a
ma1 eb m
ða 1Þ!
ðA:6Þ
In this case the distributions of N1 and N2 are
independent, and, because the first and second moments
2
of the gamma distribution are a/b0 and ða þ 1Þa= b0 ,
m
X
where Ni is the number of counts detected in one observation period of length Pi.
The average value of the total number of detected
counts is given by
kNl ¼
m
X
kmi l
where mi ¼ l (ti)Pi in terms of the average count rate
l in the interval. The average of the square of the
number of counts is given by
kNl ¼ a=b0
m X
m
X
i¼1
2
ðA:11Þ
i¼1
kN 2 l ¼
kNi Nj l ¼ ½a=b0 þ ða þ 1Þa= b0 di;j
ðA:10Þ
Ni
i¼1
ðA:5Þ
As an example, let P(m1, m2) be the product of
gamma distributions with parameters a and b 0. For a
gamma distribution of m,
PðmÞ ¼ gammaðm j a; b0 Þ ¼
PðNjmÞPðmja; b0 Þdm
ð1
0
b0 a
dmmNþa1 emðb þ1Þ
N!ða 1Þ! 0
0 a N
b
1
ðN þ a 1Þ!
¼
ða 1Þ!N!
b0 þ 1
b0 þ 1
ðA:9Þ
ðA:2Þ
1 X
1
X
ð1
0
k¼0
kf l ¼
2
The probability distribution of counts N is given
by
PðNja; b0 Þ ¼
k¼0
1
X
ðA:8Þ
kNi Nj l
j¼1
ðA:7Þ
¼ kNl þ
m X
m
X
i¼1
417
j¼1
ðA:12Þ
km i m j l
J. KLUMPP ET AL.
using Equation A.3. Algebraically,
m X
m
X
i¼1
kmi mj l ¼
j¼1
¼
m X
m
X
i¼1
j¼1
m
X
m
X
i¼1
j¼1
2
kðmi kmi lÞðmj kmj lÞl þ kNl
Vx ðmÞ ¼
Ci;j si sj þ kNl2
ðA:13Þ
where si is the standard deviation of mi and the
autocorrelation matrix, Ci,j, is defined as
kðmi kmi lÞðmj kmj lÞl
Ci;j ;
si sj
i¼1
Ci;j si sj
ðT
ðT
dt
0
dt0 Cðjt t0 jÞ;
ðA:18Þ
0
where VarðmP Þ ¼ kðmP ðtÞ kmP ðtÞlÞ2 l, which is
assumed to be independent of time t.
In the integration on the right-hand side of
Equation A.18, variables are changed to
ðA:14Þ
VarðNÞ ¼ kN 2 l kNl2
m X
m
X
VarðmP Þ 1
kmP l TTD
td ¼ t t0
It can be shown algebraically that the autocorrelation
matrix satisfies the inequality jCi;j j 1. In terms of
the autocorrelation matrix the variance of the number
of detected counts N is given by
¼ kNl þ
Combined with Equations A.16 and using kNl ¼
mkmP l ¼ T=TD kmP l, this results in
ðA:15Þ
ta ¼
ðA:19Þ
t þ t0 T
2
The determinate of the Jacobean matrix of this
transformation is 1, and the limits of integration are
td from 2T to T and ta from (td –T )/2 to (T–td)/2 as
shown in Figure A1.
This allows the integral to be evaluated as follows:
ð
ð
dtdt0 Cðjt t0 jÞ ¼ dtd dta Cðjtd jÞ
j¼1
¼
ðT
dtd Cðjtd jÞ
T
Time-invariant fluctuations
At this point, the index i is associated with the time ti,
assuming that the individual counts have equal observation periods P and are spaced equally an amount
TD ¼ T/m in time. Approximating the sums by time
integrals, the following expression for the excess variance is obtained:
VarðNÞ
1
kNl
ð
ðT
1
dt T dt0
kðmP ðtÞ kmP ðtÞlÞ
¼ kNl t¼0 TD t0 ¼0 TD
Vx ðmÞ ;
¼ 2T
!
ð ðTtd Þ=2
dta
ðtd TÞ=2
ðT
0
td dtd Cðtd Þ 1 T
; TgðTÞ
ðA:20Þ
where a new function g(T ) involving an integration of
the correlation function has been defined. This integral is easily solved in the limits T!0 and T!1. By
taking C(0) ¼ 1 outside the integral, it is found that
for very small values of T,
lim gðTÞ ¼ T
T!0
ðmP ðt0 Þ kmP ðt0 ÞlÞl
ðA:21Þ
ðA:16Þ
where mP(t) is the Poisson mean in time period
P centred around time t (time integral of the instantaneous count rate over period P), and T ¼ mTD.
In order to have a more tractable form of the
model, time-invariant fluctuations are assumed, and
the correlation is a time-invariant function of the time
difference given by
kðmP ðtÞ kmP ðtÞlÞðmP ðt0 Þ kmP ðt0 ÞlÞl
Cðt; t0 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kðmP ðtÞ kmP ðtÞl2 ÞðmP ðt0 Þ kmP ðt0 Þl2 Þl
¼ Cðjt t0 jÞ
ðA:17Þ
Figure A1. Integration area in terms of the new variables
td and ta.
418
NON-CONSTANT BACKGROUND
Similarly, for very large values of T,
lim gðTÞ ¼ t
T!1
ðA:22Þ
where the correlation time t is defined as such:
ð1
t;2
Cðtd Þdtd
ðA:23Þ
0
In terms of g, Equation A.18 yields
Vx ðmÞ ¼
VarðmP Þ gðmTD Þ
kmP l
TD
ðA:24Þ
More accurate than this is to separate out the diagonal terms in A.15 and write
Vx ðmÞ ¼
VarðmP Þ
ðgðmTD Þ gðTD ÞÞ
1þ
kmP l
TD
VarðmT Þ
1
T
¼ 0 ¼
kmT l
b T bT
ðA:27Þ
which shows its dependence on time T.
A limiting case of this theory is to let the individual
observation period P equal TD, which equals T/m so
that the entire time T consists of observations, and
consider the limit m ! 1. In this case, the approximation of the sums in Equation A.15 by integrals is
exact, and
Vx ðTÞ ¼
gðTÞ
b
For
an
exponential-correlation
Cðtd Þ ¼ expð2td =tÞ, and
t
2T
1 exp gðTÞ ¼ t 1 2T
t
ðA:26Þ
which can be seen from Equation A.25. If mT is
then
sampled
from
gammaðaT ; b0T Þ,
0
kmT l ¼ aT =bT , kmT l ¼ aT =b0T , and
Vx ðTÞ ¼
Vx ðmÞ
ðgðmTD Þ gðTD ÞÞ
¼1þ
Vx ð1Þ
TD
ðA:25Þ
If mT Tis defined to be the Poisson mean in time
period, then by the definition of relative excess
variance,
VarðmT Þ
Vx ðTÞ ¼
kmT l
At the end of time tc, a new l is drawn from the gamma
distribution gamma (lja, b), and a new tc is generated.
The process is initialised with random values of l and
tc from their respective distributions. During a single
counting period of whatever length, l is assumed to be
constant. The parameter values used in the simulation
were a ¼ 8.5, b ¼ 1 and t ¼ 20.
The correlation function that corresponds to this
model is Cðtd Þ ¼ expð2td =tÞ, which can be seen by
returning to Equation A.17 and noticing that
Cðjt t0 jÞ is either 1 or 0 depending on whether tc is
greater than or less than jt–t0 j, so that after averaging
over
the
probability
distribution
of
tc,
Cðtd Þ ¼ pðtc . td Þ ¼ expð2td =tÞ.
Using Equation A.25,
ðA:30Þ
function
ðA:31Þ
To summarise, these are the basic steps needed to
apply this model. First, determine aT and bT, which is
most simply done without consideration of uncertainties just using the mean and variance of replicate
background counts in time period T.
These quantities can be used to replace the number
of background counts NB observed in a single
ðA:28Þ
where the quantity b with dimensions of time is
defined as
b ¼ lim bT
T!0
ðA:29Þ
Validation using simulated data: an exponentialcorrelation model
Figure A2 shows a validation of Equation A.25 using
10 000 replicates of simulated data. In each simulation, a
lambda-change time, tc, during which time the instantaneous count rate, l, a constant is generated from the
exponential distribution pðtc Þ ¼ expð2tc Þ=tÞ2dtc =t.
Figure A2. Comparison of simulated data using 10 000
replicates with theory as discussed in the text, assuming a
counting period spacing of TD ¼ 2.5, a correlation time t ¼ 20
and an exponential-correlation function Cðtd Þ ¼ expð2td =tÞ.
419
J. KLUMPP ET AL.
observation of length TB using the substitutions
NB ¼ aT 1
TB ¼ bT
ðA:32Þ
The time dependence of relative excess variance
Vx(m)/Vx(1) using re-binned data can be fitted to determine the correlation time t. This requires only a
one-parameter fit using Equation A.30 except for the
matter of the uncertainties. Then to determine a and
b the following equations are used:
gðTÞ
T
aT
a¼b
bT
Figure A3. Three methods of measuring a blank sample
for time TB to determine background under a sample
measurement for time T.
b ¼ bT
ðA:33Þ
For example, assuming as before from the 234U data
aT ¼ 9.1, bT/T ¼ 1.1 , the following are obtained: b
¼ 1.076T and a ¼ 8.9 for t/T ¼ 30.
APPENDIX 2. THREE IDEALISED
BACKGROUND MEASUREMENT
METHODS
Three different methods of measuring background
are considered, using the exponential-correlation
model to describe the actual time dependence of the
background count rate (unlike in a contaminated
reagent view of background where the background,
once drawn, remains constant in time). The background count in time TB is used to infer the effective
background under the sample measurement, which
has counting time T. It is assumed that a large background study has already been done so that a, b and
t are known. Three different methods, distinguished
by the degree of correlation of the instantaneous
background count rates in the two measurement
periods, are defined as follows:
(I)
(II)
(III)
The two measurements are widely spaced in
time. There is no correlation.
The two measurements are as close together
sequentially in time as possible.
III. The two measurements overlap in time as
much as possible.
These methods are illustrated in Figure A3.
Method I has the smallest possible correlation, and
Method III has the largest possible correlation. In
order to have maximal correlation in Method III, the
difference in counting periods is split, and half of that
difference is before and half after the shared counting
period. With Method II, the background counting
period is split with half before and half after the sample
counting period. Of course, other arrangements could
be used, for example, background count immediately
following the sample count, but these would give somewhat less correlation.
The calculations in this section will be limited to
variance and covariance of numbers of counts. This
will allow calculation of the Classical decision threshold for the three background methods.
The Classical decision threshold for net counts
y ¼ N– NB/R is given by
y ¼ 1:645
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
VarðNÞ þ VarðNB Þ=R2 þ CoVarðN; NB Þ=R
ðA:34Þ
where N is the number of background counts in
sample observation period T, NB is the number of
background counts in background observation period
TB and R ¼ TB/T.
The covariance term is absent in the conventional
calculation and exists because of the correlation of
the fluctuations of the instantaneous count rate.
To evaluate the variances appearing in Equation
A.34 taking into account correlations, Equations
A.16 and A.17 are used, which give the variance of
the number of counts in a time interval T as
VarðNÞ ¼
Ta
gðTÞ
1þ
b
b
ðA:35Þ
and for the exponential-correlation model, g(T ) is
given by Equation A.31.
The covariance of observed counts between two separate intervals, the larger Dt centred on time t and the
smaller Dt0 centred on time t0 is obtained from the
generalisation of Equation A.20. Using the time-invariant fluctuation model of background, one can
420
NON-CONSTANT BACKGROUND
show that
dtdt0 Cðjt t0 jÞ ¼
ð ðDtDt0 Þ=2
dtd Cðjtd þ t t0 jÞ
ðDtþDt0 Þ=2
Dt0
Dt
1
ðDt þ Dt0 Þ jtd j
2
ð ðDtDt0 Þ=2
þ
dtd Cðjtd þ t t0 jÞDt0
ðDtDt0 Þ=2
þ
ð ðDtþDt0 Þ=2
dtd Cðjtd þ t t0 jÞ
ðDtDt0 Þ=2
1
ðDt þ Dt0 Þ jtd j
2
ðA:36Þ
The covariance is given by
CoVarðN; N 0 Þ ; kðN kNlÞðN 0 kN 0 lÞl
a
¼ 2 dtdt0 Cðjt t0 jÞ
b
ðA:37Þ
Equation A.36 is the same as Equation A.20 when
Dt ¼ Dt0 ¼ T and t ¼ t0 . Although these formulas are
complicated, they involve integrals of exponentials and
can be evaluated in terms of elementary functions.
Because of the complexity of these formulas, it is
simpler to simulate many replicates of y and compute
the variance directly.
An example of such a calculation is shown in
Figure A4. The graph shows the Classical decision
threshold as a function of R for three different values
of the correlation time t/T. Equations A.33 are used
to obtain a and b from the measured values aT and
bT by using the assumed value of t.
One can see from Figure A4 that the curves for the
three methods come together at small and large
values of R. Using different methods makes a difference for the intermediate values of R, and for longer
correlation time the difference is larger. When the correlation time is short compared with the sample count
time, there is no difference between Methods I and II,
because for both of these methods there is essentially
no correlation. When the correlation time is long
compared with the sample count time, there is no
difference between Methods II and III, because there is
almost the same correlation for both of these methods.
No difference between the methods means the covariance term in Equation A.37 is negligible. It also
means that there is no point in making an individual,
paired background measurement—the background
variance is obtained from Equation A.35 just using a,
b and t obtained from the large background study.
It is interesting to note that for the long correlation
time, t/T ¼ 30, with Methods II and III, at some
Figure A4. Comparison of the three background methods.
The simulated data parameter values were aT ¼ 9.1,
bT/T ¼ 1.1 , for three values of t/T: 0.03, 1 and 30.
point longer background count time actually increases
the decision threshold. This happens because the
covariance of background between sample and background measurement decreases as the background
count time increases beyond a certain point.
421