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Derivation of an uncertainty propagation factor for half-life determinations
Applied Radiation and Isotopes 158 (2020) 109046
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Applied Radiation and Isotopes journal homepage: http://www.elsevier.com/locate/apradiso
Derivation of an uncertainty propagation factor for half-life determinations S. Pomm�e a, *, T. De Hauwere a, b a b
European Commission, Joint Research Centre (JRC), Geel, Belgium TU Delft, Delft, the Netherlands
A R T I C L E I N F O
A B S T R A C T
Keywords: Half-life Uncertainty Propagation Activity Nuclear data
An analytical equation is derived for the uncertainty propagation factor for a half-life determination from a leastsquares fit to equidistant activity measurements performed with identical relative uncertainties. The obtained formula applies to a purely random statistical uncertainty component. It is equivalent to the solution published by Parker in Nucl. Instr. Meth. A 286, 502. A more general equation for weighted least-squares fitting is derived and presented in a compact manner. It is used as a benchmark to verify the applicability of Parker’s solution to non-equidistant data with unequal uncertainties.
1. Introduction The exponential-decay law (Rutherford, 1900; Semkow, 2007) is the foundation of the common measurement system for radioactivity (Judge et al., 2014). It relies on the invariability of the nuclear decay constants – or, inversely, the corresponding half-lives – which appears to apply rigorously within the boundaries of metrological accuracy (Emery, 1972; Hahn et al., 1976; Norman et al., 1995; Pomm� e et al., 2016a, 2017a,b,c,d, 2018a,b; Pomm�e, 2019; Pomm�e and De Hauwere, 2020). As a result, nuclear decay can be used as a clock (Dalrymple, 1991; Currie, 2004; Nir-El, 2004, Pomm� e, 2015; Pomm�e et al., 2014, 2016b). Half-life values of radionuclides are measured by various techniques, depending on their order of magnitude (Pomm�e, 2015). Half-lives in an intermediate range are measured by repeated activity measurements over periods of time, thus establishing a decay curve to which an exponential decay function is fitted. Generally, least-squares fitting routines are applied to determine the slope of the decay curve and the ensuing uncertainty is obtained from the covariance matrix (Bevington and Robinson, 2003). The latter procedure often gives rise to significant underestimation of the half-life uncertainty, since the model doesn’t account for systematic deviations due to metrological imperfections (Pomm� e, 2007, 2016). Medium- and long-term instability and non-linearity in the experiment propagate with a larger factor into the half-life uncertainty than the purely random variations (Pomm�e et al., 2008; Pomm� e, 2015). In state-of-the-art half-life measurements, the short-term random varia tions – such as counting statistics – may represent only a small fraction of the total uncertainty budget. For example, in a99mTc half-life
measurement (Pomm� e et al., 2018c), the uncertainty component due to random variability in more than 2000 measurements added very little to the total, since the attainable accuracy is limited by the linearity of the ionisation chamber, and possibly to an even larger extent by the chemical composition of the source influencing the probability for in ternal conversion of the 2.17 keV transition. On the other hand, the short-term reproducibility issues (including counting statistics for source and background) were the major uncertainty component for a55Fe half-life measurement, in spite of the presence of long-term, sys tematic effects due to pulse pileup, background subtraction and gas pressure variations (Pomm�e et al., 2019). Correct propagation of uncertainty components is an important issue. An approximate formula was proposed by Pomm� e et al. (2008) to propagate random uncertainty components from n equally spaced ac tivity measurements over a time period T to the fitted half-life T1/2: rffiffiffiffiffiffiffiffiffiffiffi � σ T1=2 2 2 σ ðAÞ (1) � λT n þ 1 A T1=2 The equation can also be used to evaluate medium and lowfrequency instabilities, if the parameter n receives a new meaning as the number of periods covered by a cyclic effect on the experiment. For low-frequency effects, the value n ¼ 1 is selected by default. A more rigorous equation for the propagation factor of random un certainties was published by Parker (1990), as part of a procedure to measure the half-life of a source by gamma-ray spectrometry, relative to a reference radionuclide with a well-established, long half-life. For a series of n measurements of activity ratios of both nuclides, R, assuming
* Corresponding author. E-mail address: [email protected] (S. Pomm�e). https://doi.org/10.1016/j.apradiso.2020.109046 Received 30 July 2019; Received in revised form 7 December 2019; Accepted 15 January 2020 Available online 20 January 2020 0969-8043/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
S. Pomm�e and T. De Hauwere
Applied Radiation and Isotopes 158 (2020) 109046
a constant measurement uncertainty, σR, Parker derived the following uncertainty formula: pffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 T1=2 ðn 1Þ n σ T1=2 ¼ σR (2) ln 2 T n ðn2 1Þ
σ 2b ¼ n ¼
On the basis of this equation, the following alternative to Eq. (1) was recommended specifically for high-frequency random variations in a review paper (Pomm� e, 2015) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ T1=2 2 3ðn 1Þ σ A � (3) T1=2 λT nðn þ 1Þ A
¼
In principle, the best fit of an exponential function to a nuclear decay curve can be obtained by means of the maximum likelihood estimation method using the Poisson distribution to account for the probability for the observed number of decays. For sufficiently high amounts of counts, a Gaussian distribution is an adequate approximation and the maximum likelihood search is conveniently replaced by a least-squares fit (Kendall and Stuart, 1961). Whereas a non-linear fit can be easily performed to obtain the decay constant from the slope of the exponential, it is ad vantageous to study the transform to a linear problem to derive the uncertainty propagation factor in Eq. (3). A variable y is considered which relates linearly with a variable x, through the free parameters a and b representing the intercept and the slope, respectively:
�2 xi
i¼1
σ
n P
(6)
�2 xi x
σy 2
(7)
varðxÞ
in which n-2 is the number of degrees of freedom of the linear fit. The
‘variance’ of xi can be calculated directly from the sum of all ðxi terms divided by n.
xÞ2
2.3. Equidistant measurements Applying an additional condition that the xi data are equidistant by an amount Δx, such that xi ¼ iΔx, one can simplify Eq. (6) further. For a set of n data, the total distance between the extremes, x1 and xn, is T ¼ Δxðn 1Þ and the mean value is x ¼ Δxðn þ 1Þ=2. The term in the denominator of Eq. (6) equals
(4)
In the context of this paper, y represents the logarithm of the activity, y ¼ lnA, and x corresponds with time, t. Consequently, the intercept refers to the initial activity, a ¼ lnA0, the slope is related to the decay constant, b ¼ –λ, and the focus of this work is on the uncertainty of the least-squares estimate of b. The uncertainty on the xi values is assumed to be negligible, σx ¼ 0, or otherwise absorbed into the uncertainties of the corresponding yi values, σi, propagated by the slope b. In this paper, the ‘spread’ of the nominal xi and yi values, will be quantified by their ‘variance’, e.g. P P varðxÞ ¼ ðxi xÞ2 =n and covðx; yÞ ¼ ðxi xÞðyi yÞ= n, which should not be confound with their measurement uncertainty (i.e. varðxÞ 6 ¼ σ2x ). Minimisation of the weighted sum of the squares of the deviations between measured and fitted values, �2 n � X yi a bxi χ2 ¼ (5)
2 n � X � xi x ¼ ðΔxÞ2 i
n X i¼1
i¼1
¼ ðΔxÞ2
n X
i2
i¼1
� nþ1 n 2
�2 nþ1 2 �! 2
nðn2 1Þ nðn þ 1Þ ¼ ðΔxÞ2 ¼ T2 12 12ðn 1Þ
(9)
in which use was made of the Faulhaber-Bernouilli formulas (Knuth, 1993): n X
i¼ i¼1
n X nðn þ 1Þ nðn þ 1Þð2n þ 1Þ and i2 ¼ 2 6 i¼1
Combining Eqs. (6) and (9), one obtains sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σy 3ðn 1Þ σb ¼ 2 nðn þ 1Þ T
σi
through ∂χ 2 =∂a ¼ 0 and ∂χ 2 =∂b ¼ 0 yields analytical equations for the estimated values of a and b (Kendall and Stuart, 1961; Bevington and _
n P
variables. Eq. (7) facilitates the interpretation: the variance of the slope decreases when the uncertainty on the y values is small and many data are taken over a wide range of x values, or alternatively when the un certainty of the mean of the y values is small compared to the spread of the x data. If the standard uncertainty of the y data is not known a priori, it can be estimated from the spread of the residuals to the linear fit, through � �2 n P yi a bxi σ 2y � s2 ¼ i¼1 (8) n 2
2.1. Linear least-squares fit
Robinson, 2003); They can be reduced to a ¼ y P P ðxi xÞðyi yÞ= ðxi xÞ2 ¼ covðx; yÞ=varðxÞ.
x2i
Eq. (6) is valid independently of the statistical distribution of the y data. If a normal distribution applies to y, than it also applies to the _ estimated intercept a and slope b b, since they are sums of normal random
2. Derivation of Equation 3
i¼1
i¼1 2 y
nσ2y �
i¼1
No mathematical derivation was published to justify its validity. In this paper, it will be demonstrated how Eq. (3) is found starting from the uncertainty equations for linear least-squares fitting and transforming them to the case of exponential decay.
y ¼ a þ bx
n P
(10)
(11)
_
bx and b ¼
2.4. Transform to half-life When transforming a non-linear fit of an exponential curve into a linear fit through y ¼ lnA, the uncertainties of the measured quantities must by modified by
2.2. Uncertainty of the slope Ignoring systematic errors which would introduce correlations be tween uncertainties, the uncertainty of the slope can be derived from P 2 σ 2b ¼ σ2i ð∂b=∂yi Þ . In the special case of constant uncertainties, σi ¼ σy , this results in (Bevington and Robinson, 2003)
σy ¼
dlnA σ σA ¼ A dA A
(12)
which relates the standard uncertainty of y to the relative uncertainty on 2
S. Pomm�e and T. De Hauwere
Applied Radiation and Isotopes 158 (2020) 109046
the activity values A. The relative uncertainty of the slope equals the relative uncertainty on the decay constant (or the half-life), i.e. σ b = b ¼ σλ =λ. Combining this with Eqs. (11) and (12), one obtains sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ λ 2 3ðn 1Þ σA ¼ (13) λ λT nðn þ 1Þ A
�
� 1=2 ðσAi =Ai Þ 2 σλ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii¼1 λ λ P n n P ðσ Ai =Ai Þ 2 ðti tw Þ2 ðσ Ai =Ai Þ 2 i¼1
1 1 σA ¼ pffiffiffiffiffiffiffiffiffiffiffi w λ varðtÞ Aw
which is, according to the Gauss-Markov theorem (Wooldridge, 2013), the best linear unbiased estimator (BLUE) under the conditions of equidistant time between measurements and the same relative uncer tainty in each activity value. This result is equivalent to Eq. (2) proposed by Parker (1990).
Since the case of a constant uncertainty in y and fixed spacing in x is quite particular, it is of interest to consider the general case in which σi and Δxi vary for each data point. The uncertainty on the xi values is assumed to be negligible, σx ¼ 0, or otherwise absorbed into the un certainties of the corresponding yi values, σi. Ignoring systematic errors which would introduce correlations between uncertainties, the uncer tainty of the slope obtained from an inverse-variance weighted leastsquares fit – i.e. minimising χ2 in Eq. (5) – results in (Bevington and Robinson, 2003)
σ ¼
i¼1 n n 2 P P xi 1 i¼1
σ 2i
i¼1
4. Case studies 4.1. Few measurements
1
σ2i
�
σ2i
n P xi i¼1
(14)
�2
The uncertainty propagation formula for the half-life obtained from the ratio of two activity measurements with the same relative uncer tainty is (Pomm�e, 2015) pffiffi σ T1=2 2 σA ¼ (17) T1=2 λT A
σ 2i
The interpretation suggested by Eq. (7) for constant σi can also be looked for in Eq. (14) for weighted least-squares fitting. A weighted mean of the xi and yi values and a weighted ‘variance’ of the xi data – P P P xw ¼ pi xi , yw ¼ pi yi and varðxÞ ¼ pi ðxi xw Þ2 , respectively – can be calculated using normalised inverse-variance weights pi ¼ P 1=σ 2i = 1=σ 2j . The variance of the weighted mean, σ 2yw , equals 1= P 1=σ2j . Eq. (14) can be rearranged to obtain a similar result as in Eq. (7): � n P1 1 σ2 i¼1 i 2 σb ¼ � �2 �� n �2 � n 2 n n P P1 P P xi xi 1 i¼1
¼
n P i¼1
σ 2i
i¼1
σ2i
i¼1
σ2yw
�
�2 �
xi xw
σ 2i
n P i¼1
¼ 1
σ 2i
σ2yw varðxÞ
i¼1
(16)
resents the spread of the reference times and σ Aw =Aw ¼ P ð ðσ Ai =Ai Þ 2 Þ 1=2 is the relative uncertainty of the weighted mean ac tivity. In Fig. 1, the meaning of the variables involved is depicted for a hypothetical example. Explicit calculations via Eq. (16) involve the following steps: (1) calculate the sum of the inverse square of the relative uncertainties on the activity values, (2) normalise these values individually by their sum to obtain the weighting factors pi, (3) calculate the uncertainty of the weighted mean activity as the inverse square root of the normalisation factor, (4) calculate the weighted mean time tw, and the ‘variance’ of t, and (5) apply Eq. (16).
3.1. Uncertainty of the slope
2 b
i¼1
in which ti is the reference time of the measurement i expressed in the P same time unit as 1/λ, tw ¼ pi ti is the weighted mean of the reference times of the measurements, using the weighting factors pi based on the P relative uncertainties of the activity values, varðtÞ ¼ pi ðti tw Þ2 rep
3. Weighted least-squares fit
n P
n P
The same result is obtained from the approximation in Eq. (1) for n ¼ 3 and from the BLUE in Eq. (3) for n ¼ 2 and n ¼ 3. The third point in the middle of the decay curve does not influence the slope, only the pffiffiffiffiffiffiffiffi amplitude. For large n, the BLUE is proportional to 3=n, whereas the pffiffiffiffiffiffiffiffi approximation in Eq. (1) is somewhat lower, ~ 2=n.
σ2i
(15)
σ2i
The variance on the slope of a weighted fit is proportional to the variance of the weighted mean of the yi values and inversely to the weighted spread of the xi data. 3.2. Transform to half-life As derived from Eq. (12), when transforming a non-linear fit of an exponential curve into a linear fit through y ¼ lnA, the standard un certainties of the y values equal the relative uncertainties on the activity values, σy ¼ σA =A. The weighting factors to be applied are therefore P pi ¼ ðσAi =Ai Þ 2 = ðσAj =Aj Þ 2 . Combining this with Eq. (15), one obtains
Fig. 1. Example of a linearization by y ¼ lnA of a decay curve of n ¼ 5 activity measurements with variable relative uncertainties σy ¼ σi ¼ σ(Ai)/Ai obtained over a time interval T. The uncertainty on the slope of the fitted line is pro portional to the uncertainty of the weighted mean of y, σyw ¼ σAw =Aw , and inversely proportional to the spread of the measurement reference times pffiffiffiffiffiffiffiffiffiffiffiffi t, varðtÞ. 3
S. Pomm�e and T. De Hauwere
Applied Radiation and Isotopes 158 (2020) 109046
4.2. Systematic error
Table 1 Uncertainty calculation of a decay constant (λ ¼ 1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the measurement uncertainty, the approximate uncer tainty from Eq. (3) deviates only by 6% from the exact uncertainty from Eq. (16), if in Eq. (3) use is made of the average value of the relative uncertainty at the beginning and end of the measurement campaign, <σА/A>i¼1,5.
The reduction of the propagation factor by n 1=2 (Eqs. (1)–(3)) ap plies to stochastic effects which randomly vary with each measurement, not to systematic errors which affect subsequent data in a correlated manner. Non-linearity in detector output or changes in source integrity, geometrical and environmental conditions may occur at any phase of the project. Live-time corrections for pileup and dead time may introduce a systematic error at the start of the measurement campaign, not so much at the end. On the other hand, systematic errors in background sub traction may affect the measurements at the end of the campaign, while being negligible at the start. It is suggested to apply n ¼ 1 in Eq. (1) to propagate an average value of each systematic uncertainty component at the start and end of the campaign, which effectively leads to a correct
propagation: with Eq. (3).
σλ λ
¼
2 σ A =Aþ0 λT 2
¼
1 σA λT A .
This effect cannot be reproduced
4.3. Variable uncertainties The condition of constant relative uncertainties on the activity measurements is difficult to fulfil throughout the measurement campaign. Whereas statistical accuracy can be maintained by measuring the source at increasingly longer time intervals during the decay, the statistical influence of background subtraction gains importance as the activity of the source diminishes. When using Eqs. (1)–(3) in practice, it has been suggested to take an average σA =A value of these uncertainty components at the beginning and end of the campaign, since the most extreme measurement data carry the largest weight. The adequacy of this approach can be tested by comparing the results of Eq. (3) with that of the general equation Eq. (16). In Table 1, a hy pothetical set of n ¼ 5 data is shown, for which the uncertainty increases progressively with time. When applying the above procedure, the approximating uncertainty from Eq. (3) still agrees quite well with the exact value from Eq. (16). The situation is more complex when the un certainties vary more randomly from one measurement to another. Large differences in relative uncertainties σA =A can diminish the ‘effective’ number of measurements n that define the slope. The uncer tainty estimations through Eqs. (1)–(3) should then be focussed on data with comparably small uncertainties.
Table 2 Uncertainty calculation of a decay constant (λ ¼ 1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the time gap between measurements, the approximate uncertainty from Eq. (3) deviates only by 3% from the exact uncertainty from Eq. (16). Larger differences are found if measurements are extremely focussed to the centre or to the beginning and end of the campaign.
4.4. Non-equidistant measurements under the conditions that the time intervals between the measurements and the relative uncertainties of the activity values are constant. The validity of Eq. (3) can be extended to cases where relative uncertainties in the activity and/or time differences between measurements show mild variations. Large variations in uncertainty as well as extreme grouping of data towards the centre, start or end of the measurement campaign affect the number of data ‘effectively’ contributing to the uncertainty of the fit. Medium and long-term systematic errors can be propagated with Eq. (1).
Measurements may also deviate from the prerequisite that they are performed at regular time intervals. In Table 2, a hypothetical set of n ¼ 5 data is shown, for which the time spacing increases progressively with each measurement. In spite of the large variations in Δt, the approxi mate uncertainty calculated from Eq. (3) remains surprisingly close to the correct value from Eq. (16). Larger variations may occur when a majority of data would be taken in the middle of the measurement campaign, which would lead to an underestimation of the half-life un certainty by Eq. (3). This could be solved by decreasing the ‘effective’ number n of measurement data. Conversely, if the measurements would be mainly taken at the beginning and end of the campaign, Eq. (3) would overestimate the real uncertainty. The data at the extremes have the highest impact on the overall uncertainty.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
5. Conclusions
CRediT authorship contribution statement
Eq. (16) presents a rigorous formula for calculating the random un certainty component for a half-life determination by means of a weighted least-squares fit of an exponential function to a decay curve obtained by consecutive measurements of activity with well-established relative uncertainties on the activity values. It does not account for systematic errors, which introduce correlations in the uncertainties. Eq. (3) is a mathematical reduction of Eq. (16), which rigorously applies
�: Conceptualization, Investigation, Writing - original draft. S. Pomme T. De Hauwere: Investigation.
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S. Pomm�e and T. De Hauwere
Applied Radiation and Isotopes 158 (2020) 109046
References
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Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: http://www.elsevier.com/locate/apradiso
Derivation of an uncertainty propagation factor for half-life determinations S. Pomm�e a, *, T. De Hauwere a, b a b
European Commission, Joint Research Centre (JRC), Geel, Belgium TU Delft, Delft, the Netherlands
A R T I C L E I N F O
A B S T R A C T
Keywords: Half-life Uncertainty Propagation Activity Nuclear data
An analytical equation is derived for the uncertainty propagation factor for a half-life determination from a leastsquares fit to equidistant activity measurements performed with identical relative uncertainties. The obtained formula applies to a purely random statistical uncertainty component. It is equivalent to the solution published by Parker in Nucl. Instr. Meth. A 286, 502. A more general equation for weighted least-squares fitting is derived and presented in a compact manner. It is used as a benchmark to verify the applicability of Parker’s solution to non-equidistant data with unequal uncertainties.
1. Introduction The exponential-decay law (Rutherford, 1900; Semkow, 2007) is the foundation of the common measurement system for radioactivity (Judge et al., 2014). It relies on the invariability of the nuclear decay constants – or, inversely, the corresponding half-lives – which appears to apply rigorously within the boundaries of metrological accuracy (Emery, 1972; Hahn et al., 1976; Norman et al., 1995; Pomm� e et al., 2016a, 2017a,b,c,d, 2018a,b; Pomm�e, 2019; Pomm�e and De Hauwere, 2020). As a result, nuclear decay can be used as a clock (Dalrymple, 1991; Currie, 2004; Nir-El, 2004, Pomm� e, 2015; Pomm�e et al., 2014, 2016b). Half-life values of radionuclides are measured by various techniques, depending on their order of magnitude (Pomm�e, 2015). Half-lives in an intermediate range are measured by repeated activity measurements over periods of time, thus establishing a decay curve to which an exponential decay function is fitted. Generally, least-squares fitting routines are applied to determine the slope of the decay curve and the ensuing uncertainty is obtained from the covariance matrix (Bevington and Robinson, 2003). The latter procedure often gives rise to significant underestimation of the half-life uncertainty, since the model doesn’t account for systematic deviations due to metrological imperfections (Pomm� e, 2007, 2016). Medium- and long-term instability and non-linearity in the experiment propagate with a larger factor into the half-life uncertainty than the purely random variations (Pomm�e et al., 2008; Pomm� e, 2015). In state-of-the-art half-life measurements, the short-term random varia tions – such as counting statistics – may represent only a small fraction of the total uncertainty budget. For example, in a99mTc half-life
measurement (Pomm� e et al., 2018c), the uncertainty component due to random variability in more than 2000 measurements added very little to the total, since the attainable accuracy is limited by the linearity of the ionisation chamber, and possibly to an even larger extent by the chemical composition of the source influencing the probability for in ternal conversion of the 2.17 keV transition. On the other hand, the short-term reproducibility issues (including counting statistics for source and background) were the major uncertainty component for a55Fe half-life measurement, in spite of the presence of long-term, sys tematic effects due to pulse pileup, background subtraction and gas pressure variations (Pomm�e et al., 2019). Correct propagation of uncertainty components is an important issue. An approximate formula was proposed by Pomm� e et al. (2008) to propagate random uncertainty components from n equally spaced ac tivity measurements over a time period T to the fitted half-life T1/2: rffiffiffiffiffiffiffiffiffiffiffi � σ T1=2 2 2 σ ðAÞ (1) � λT n þ 1 A T1=2 The equation can also be used to evaluate medium and lowfrequency instabilities, if the parameter n receives a new meaning as the number of periods covered by a cyclic effect on the experiment. For low-frequency effects, the value n ¼ 1 is selected by default. A more rigorous equation for the propagation factor of random un certainties was published by Parker (1990), as part of a procedure to measure the half-life of a source by gamma-ray spectrometry, relative to a reference radionuclide with a well-established, long half-life. For a series of n measurements of activity ratios of both nuclides, R, assuming
* Corresponding author. E-mail address: [email protected] (S. Pomm�e). https://doi.org/10.1016/j.apradiso.2020.109046 Received 30 July 2019; Received in revised form 7 December 2019; Accepted 15 January 2020 Available online 20 January 2020 0969-8043/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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Applied Radiation and Isotopes 158 (2020) 109046
a constant measurement uncertainty, σR, Parker derived the following uncertainty formula: pffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 T1=2 ðn 1Þ n σ T1=2 ¼ σR (2) ln 2 T n ðn2 1Þ
σ 2b ¼ n ¼
On the basis of this equation, the following alternative to Eq. (1) was recommended specifically for high-frequency random variations in a review paper (Pomm� e, 2015) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ T1=2 2 3ðn 1Þ σ A � (3) T1=2 λT nðn þ 1Þ A
¼
In principle, the best fit of an exponential function to a nuclear decay curve can be obtained by means of the maximum likelihood estimation method using the Poisson distribution to account for the probability for the observed number of decays. For sufficiently high amounts of counts, a Gaussian distribution is an adequate approximation and the maximum likelihood search is conveniently replaced by a least-squares fit (Kendall and Stuart, 1961). Whereas a non-linear fit can be easily performed to obtain the decay constant from the slope of the exponential, it is ad vantageous to study the transform to a linear problem to derive the uncertainty propagation factor in Eq. (3). A variable y is considered which relates linearly with a variable x, through the free parameters a and b representing the intercept and the slope, respectively:
�2 xi
i¼1
σ
n P
(6)
�2 xi x
σy 2
(7)
varðxÞ
in which n-2 is the number of degrees of freedom of the linear fit. The
‘variance’ of xi can be calculated directly from the sum of all ðxi terms divided by n.
xÞ2
2.3. Equidistant measurements Applying an additional condition that the xi data are equidistant by an amount Δx, such that xi ¼ iΔx, one can simplify Eq. (6) further. For a set of n data, the total distance between the extremes, x1 and xn, is T ¼ Δxðn 1Þ and the mean value is x ¼ Δxðn þ 1Þ=2. The term in the denominator of Eq. (6) equals
(4)
In the context of this paper, y represents the logarithm of the activity, y ¼ lnA, and x corresponds with time, t. Consequently, the intercept refers to the initial activity, a ¼ lnA0, the slope is related to the decay constant, b ¼ –λ, and the focus of this work is on the uncertainty of the least-squares estimate of b. The uncertainty on the xi values is assumed to be negligible, σx ¼ 0, or otherwise absorbed into the uncertainties of the corresponding yi values, σi, propagated by the slope b. In this paper, the ‘spread’ of the nominal xi and yi values, will be quantified by their ‘variance’, e.g. P P varðxÞ ¼ ðxi xÞ2 =n and covðx; yÞ ¼ ðxi xÞðyi yÞ= n, which should not be confound with their measurement uncertainty (i.e. varðxÞ 6 ¼ σ2x ). Minimisation of the weighted sum of the squares of the deviations between measured and fitted values, �2 n � X yi a bxi χ2 ¼ (5)
2 n � X � xi x ¼ ðΔxÞ2 i
n X i¼1
i¼1
¼ ðΔxÞ2
n X
i2
i¼1
� nþ1 n 2
�2 nþ1 2 �! 2
nðn2 1Þ nðn þ 1Þ ¼ ðΔxÞ2 ¼ T2 12 12ðn 1Þ
(9)
in which use was made of the Faulhaber-Bernouilli formulas (Knuth, 1993): n X
i¼ i¼1
n X nðn þ 1Þ nðn þ 1Þð2n þ 1Þ and i2 ¼ 2 6 i¼1
Combining Eqs. (6) and (9), one obtains sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σy 3ðn 1Þ σb ¼ 2 nðn þ 1Þ T
σi
through ∂χ 2 =∂a ¼ 0 and ∂χ 2 =∂b ¼ 0 yields analytical equations for the estimated values of a and b (Kendall and Stuart, 1961; Bevington and _
n P
variables. Eq. (7) facilitates the interpretation: the variance of the slope decreases when the uncertainty on the y values is small and many data are taken over a wide range of x values, or alternatively when the un certainty of the mean of the y values is small compared to the spread of the x data. If the standard uncertainty of the y data is not known a priori, it can be estimated from the spread of the residuals to the linear fit, through � �2 n P yi a bxi σ 2y � s2 ¼ i¼1 (8) n 2
2.1. Linear least-squares fit
Robinson, 2003); They can be reduced to a ¼ y P P ðxi xÞðyi yÞ= ðxi xÞ2 ¼ covðx; yÞ=varðxÞ.
x2i
Eq. (6) is valid independently of the statistical distribution of the y data. If a normal distribution applies to y, than it also applies to the _ estimated intercept a and slope b b, since they are sums of normal random
2. Derivation of Equation 3
i¼1
i¼1 2 y
nσ2y �
i¼1
No mathematical derivation was published to justify its validity. In this paper, it will be demonstrated how Eq. (3) is found starting from the uncertainty equations for linear least-squares fitting and transforming them to the case of exponential decay.
y ¼ a þ bx
n P
(10)
(11)
_
bx and b ¼
2.4. Transform to half-life When transforming a non-linear fit of an exponential curve into a linear fit through y ¼ lnA, the uncertainties of the measured quantities must by modified by
2.2. Uncertainty of the slope Ignoring systematic errors which would introduce correlations be tween uncertainties, the uncertainty of the slope can be derived from P 2 σ 2b ¼ σ2i ð∂b=∂yi Þ . In the special case of constant uncertainties, σi ¼ σy , this results in (Bevington and Robinson, 2003)
σy ¼
dlnA σ σA ¼ A dA A
(12)
which relates the standard uncertainty of y to the relative uncertainty on 2
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Applied Radiation and Isotopes 158 (2020) 109046
the activity values A. The relative uncertainty of the slope equals the relative uncertainty on the decay constant (or the half-life), i.e. σ b = b ¼ σλ =λ. Combining this with Eqs. (11) and (12), one obtains sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ λ 2 3ðn 1Þ σA ¼ (13) λ λT nðn þ 1Þ A
�
� 1=2 ðσAi =Ai Þ 2 σλ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii¼1 λ λ P n n P ðσ Ai =Ai Þ 2 ðti tw Þ2 ðσ Ai =Ai Þ 2 i¼1
1 1 σA ¼ pffiffiffiffiffiffiffiffiffiffiffi w λ varðtÞ Aw
which is, according to the Gauss-Markov theorem (Wooldridge, 2013), the best linear unbiased estimator (BLUE) under the conditions of equidistant time between measurements and the same relative uncer tainty in each activity value. This result is equivalent to Eq. (2) proposed by Parker (1990).
Since the case of a constant uncertainty in y and fixed spacing in x is quite particular, it is of interest to consider the general case in which σi and Δxi vary for each data point. The uncertainty on the xi values is assumed to be negligible, σx ¼ 0, or otherwise absorbed into the un certainties of the corresponding yi values, σi. Ignoring systematic errors which would introduce correlations between uncertainties, the uncer tainty of the slope obtained from an inverse-variance weighted leastsquares fit – i.e. minimising χ2 in Eq. (5) – results in (Bevington and Robinson, 2003)
σ ¼
i¼1 n n 2 P P xi 1 i¼1
σ 2i
i¼1
4. Case studies 4.1. Few measurements
1
σ2i
�
σ2i
n P xi i¼1
(14)
�2
The uncertainty propagation formula for the half-life obtained from the ratio of two activity measurements with the same relative uncer tainty is (Pomm�e, 2015) pffiffi σ T1=2 2 σA ¼ (17) T1=2 λT A
σ 2i
The interpretation suggested by Eq. (7) for constant σi can also be looked for in Eq. (14) for weighted least-squares fitting. A weighted mean of the xi and yi values and a weighted ‘variance’ of the xi data – P P P xw ¼ pi xi , yw ¼ pi yi and varðxÞ ¼ pi ðxi xw Þ2 , respectively – can be calculated using normalised inverse-variance weights pi ¼ P 1=σ 2i = 1=σ 2j . The variance of the weighted mean, σ 2yw , equals 1= P 1=σ2j . Eq. (14) can be rearranged to obtain a similar result as in Eq. (7): � n P1 1 σ2 i¼1 i 2 σb ¼ � �2 �� n �2 � n 2 n n P P1 P P xi xi 1 i¼1
¼
n P i¼1
σ 2i
i¼1
σ2i
i¼1
σ2yw
�
�2 �
xi xw
σ 2i
n P i¼1
¼ 1
σ 2i
σ2yw varðxÞ
i¼1
(16)
resents the spread of the reference times and σ Aw =Aw ¼ P ð ðσ Ai =Ai Þ 2 Þ 1=2 is the relative uncertainty of the weighted mean ac tivity. In Fig. 1, the meaning of the variables involved is depicted for a hypothetical example. Explicit calculations via Eq. (16) involve the following steps: (1) calculate the sum of the inverse square of the relative uncertainties on the activity values, (2) normalise these values individually by their sum to obtain the weighting factors pi, (3) calculate the uncertainty of the weighted mean activity as the inverse square root of the normalisation factor, (4) calculate the weighted mean time tw, and the ‘variance’ of t, and (5) apply Eq. (16).
3.1. Uncertainty of the slope
2 b
i¼1
in which ti is the reference time of the measurement i expressed in the P same time unit as 1/λ, tw ¼ pi ti is the weighted mean of the reference times of the measurements, using the weighting factors pi based on the P relative uncertainties of the activity values, varðtÞ ¼ pi ðti tw Þ2 rep
3. Weighted least-squares fit
n P
n P
The same result is obtained from the approximation in Eq. (1) for n ¼ 3 and from the BLUE in Eq. (3) for n ¼ 2 and n ¼ 3. The third point in the middle of the decay curve does not influence the slope, only the pffiffiffiffiffiffiffiffi amplitude. For large n, the BLUE is proportional to 3=n, whereas the pffiffiffiffiffiffiffiffi approximation in Eq. (1) is somewhat lower, ~ 2=n.
σ2i
(15)
σ2i
The variance on the slope of a weighted fit is proportional to the variance of the weighted mean of the yi values and inversely to the weighted spread of the xi data. 3.2. Transform to half-life As derived from Eq. (12), when transforming a non-linear fit of an exponential curve into a linear fit through y ¼ lnA, the standard un certainties of the y values equal the relative uncertainties on the activity values, σy ¼ σA =A. The weighting factors to be applied are therefore P pi ¼ ðσAi =Ai Þ 2 = ðσAj =Aj Þ 2 . Combining this with Eq. (15), one obtains
Fig. 1. Example of a linearization by y ¼ lnA of a decay curve of n ¼ 5 activity measurements with variable relative uncertainties σy ¼ σi ¼ σ(Ai)/Ai obtained over a time interval T. The uncertainty on the slope of the fitted line is pro portional to the uncertainty of the weighted mean of y, σyw ¼ σAw =Aw , and inversely proportional to the spread of the measurement reference times pffiffiffiffiffiffiffiffiffiffiffiffi t, varðtÞ. 3
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Applied Radiation and Isotopes 158 (2020) 109046
4.2. Systematic error
Table 1 Uncertainty calculation of a decay constant (λ ¼ 1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the measurement uncertainty, the approximate uncer tainty from Eq. (3) deviates only by 6% from the exact uncertainty from Eq. (16), if in Eq. (3) use is made of the average value of the relative uncertainty at the beginning and end of the measurement campaign, <σА/A>i¼1,5.
The reduction of the propagation factor by n 1=2 (Eqs. (1)–(3)) ap plies to stochastic effects which randomly vary with each measurement, not to systematic errors which affect subsequent data in a correlated manner. Non-linearity in detector output or changes in source integrity, geometrical and environmental conditions may occur at any phase of the project. Live-time corrections for pileup and dead time may introduce a systematic error at the start of the measurement campaign, not so much at the end. On the other hand, systematic errors in background sub traction may affect the measurements at the end of the campaign, while being negligible at the start. It is suggested to apply n ¼ 1 in Eq. (1) to propagate an average value of each systematic uncertainty component at the start and end of the campaign, which effectively leads to a correct
propagation: with Eq. (3).
σλ λ
¼
2 σ A =Aþ0 λT 2
¼
1 σA λT A .
This effect cannot be reproduced
4.3. Variable uncertainties The condition of constant relative uncertainties on the activity measurements is difficult to fulfil throughout the measurement campaign. Whereas statistical accuracy can be maintained by measuring the source at increasingly longer time intervals during the decay, the statistical influence of background subtraction gains importance as the activity of the source diminishes. When using Eqs. (1)–(3) in practice, it has been suggested to take an average σA =A value of these uncertainty components at the beginning and end of the campaign, since the most extreme measurement data carry the largest weight. The adequacy of this approach can be tested by comparing the results of Eq. (3) with that of the general equation Eq. (16). In Table 1, a hy pothetical set of n ¼ 5 data is shown, for which the uncertainty increases progressively with time. When applying the above procedure, the approximating uncertainty from Eq. (3) still agrees quite well with the exact value from Eq. (16). The situation is more complex when the un certainties vary more randomly from one measurement to another. Large differences in relative uncertainties σA =A can diminish the ‘effective’ number of measurements n that define the slope. The uncer tainty estimations through Eqs. (1)–(3) should then be focussed on data with comparably small uncertainties.
Table 2 Uncertainty calculation of a decay constant (λ ¼ 1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the time gap between measurements, the approximate uncertainty from Eq. (3) deviates only by 3% from the exact uncertainty from Eq. (16). Larger differences are found if measurements are extremely focussed to the centre or to the beginning and end of the campaign.
4.4. Non-equidistant measurements under the conditions that the time intervals between the measurements and the relative uncertainties of the activity values are constant. The validity of Eq. (3) can be extended to cases where relative uncertainties in the activity and/or time differences between measurements show mild variations. Large variations in uncertainty as well as extreme grouping of data towards the centre, start or end of the measurement campaign affect the number of data ‘effectively’ contributing to the uncertainty of the fit. Medium and long-term systematic errors can be propagated with Eq. (1).
Measurements may also deviate from the prerequisite that they are performed at regular time intervals. In Table 2, a hypothetical set of n ¼ 5 data is shown, for which the time spacing increases progressively with each measurement. In spite of the large variations in Δt, the approxi mate uncertainty calculated from Eq. (3) remains surprisingly close to the correct value from Eq. (16). Larger variations may occur when a majority of data would be taken in the middle of the measurement campaign, which would lead to an underestimation of the half-life un certainty by Eq. (3). This could be solved by decreasing the ‘effective’ number n of measurement data. Conversely, if the measurements would be mainly taken at the beginning and end of the campaign, Eq. (3) would overestimate the real uncertainty. The data at the extremes have the highest impact on the overall uncertainty.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
5. Conclusions
CRediT authorship contribution statement
Eq. (16) presents a rigorous formula for calculating the random un certainty component for a half-life determination by means of a weighted least-squares fit of an exponential function to a decay curve obtained by consecutive measurements of activity with well-established relative uncertainties on the activity values. It does not account for systematic errors, which introduce correlations in the uncertainties. Eq. (3) is a mathematical reduction of Eq. (16), which rigorously applies
�: Conceptualization, Investigation, Writing - original draft. S. Pomme T. De Hauwere: Investigation.
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Applied Radiation and Isotopes 158 (2020) 109046
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