- Email: [email protected]
Overdispersion in nuclear statistics
Nuclear Instruments and Methods in Physics Research A 422 (1999) 444—449
Overdispersion in nuclear statistics Thomas M. Semkow Wadsworth Center, New York State Department of Health, Empire State Plaza, Albany, NY 12201-0509, USA State University of New York, Albany, NY, USA
Abstract The modern statistical distribution theory is applied to the development of the overdispersion theory in ionizingradiation statistics for the first time. The physical nuclear system is treated as a sequence of binomial processes, each depending on a characteristic probability, such as probability of decay, detection, etc. The probabilities fluctuate in the course of a measurement, and the physical reasons for that are discussed. If the average values of the probabilities change from measurement to measurement, which originates from the random Lexis binomial sampling scheme, then the resulting distribution is overdispersed. The generating functions and probability distribution functions are derived, followed by a moment analysis. The Poisson and Gaussian limits are also given. The distribution functions belong to a family of generalized hypergeometric factorial moment distributions by Kemp and Kemp, and can serve as likelihood functions for the statistical estimations. An application to radioactive decay with detection is described and working formulae are given, including a procedure for testing the counting data for overdispersion. More complex experiments in nuclear physics (such as solar neutrino) can be handled by this model, as well as distinguishing between the source and background. 1999 Elsevier Science B.V. All rights reserved. Keywords: Stastistical distribution theory; Overdispersion
1. Introduction The validity of Poisson distribution [1] in ionizing-radiation statistics is well established [2]. The attempts to find deviations from the Poisson statistics, based on a fundamental phenomenon such as the flicker noise postulated by Handel [3], were unsuccessful (see Concas and Lissia [4] for a recent
Address for correspondence: Wadsworth Center, New York State Department of Health, Empire State Plaza, Albany, NY 12201-0509, USA.
work as well as a review). The coefficient of dispersion d "p/k (equal to 1 for the Poisson distribuV V tion) can be used as a measure of the deviations. There are, however, known cases of data overdispersion (d '1) discussed by Currie [5] and Tries V [6] in radioactivity counting, Aurela [7] in solar neutrino experiment, and To ke et al. [8] in nuclear multifragmentation. Fro¨hner [9] considered an overdispersion caused by time uncertainties. To the best of our knowledge, no formal overdispersion theory of nuclear statistics, based on mathematical distribution theory, has been developed and is therefore a subject of the present investigation.
0168-9002/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 1 1 4 - 0
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
To incorporate a rapidly decaying source, it is necessary to study the overdispersion in the binomial statistics first, and subsequently obtain the Poisson and Gaussian limits. The binomial statistics depends on the probability of decay in time t, p (t)"1!e\HR, (1) where j is a decay constant [10]. We learn about ionizing-radiation by detecting it. Thus, the overall statistics combines that from the original radiations with that of a measurement, described globally by the probability of detection, i.e., the detector efficiency e [10]. This picture can be generalized to a multistep experiment involving a production, extraction, survival, decay, and detection (or any combination thereof), each depending on a characteristic probability. The probability of production in time t is given below [11] j (e\HR!e\HR), p (t)" (2a) j !j where j is a production rate and j is a decay rate. A Poisson limit to Eq. (2a) can be obtained [12] when j t 1 and j j , j p (t)" (1!e\HR). (2b) j The probability of extraction e is time independent [13], whereas the probability of survival in time t is [11] p (t)"e\HR. (3) In a traditional approach to the statistical distributions, these probabilities are assumed constant in a fixed time interval. However, in Section 2 we present reasons why the probabilities are actually fluctuating in the course of an experiment and show under which circumstances these fluctuations lead to an overdispersion. In Section 3 we derive the formula for the generating functions and probability distribution functions of the overdispersed nuclear system, as well as perform moment analysis. Section 4 discusses the applications to counting and more complex experiments. For the purpose of this work we define a statistical event as a physical process occurring for a single atom, such as a decay
445
or a count. Several events combined together constitute a statistical trial, or a measurement.
2. Fluctuation of probabilities The detection probability for charged particles depends on the geometric arrangement and properties of materials involved in the construction of the detector as well as surroundings [14]. Let us consider a detection of charged particles emitted from an extended source and a detector model consisting of an air space, window, and gas. Let us begin with the decays from a fixed point in the source leading to the emission of charged particles always in one direction. In the limit of a large number of such experiments in direction 1, the efficiency e is the ratio of the number of counts to the total number of decays, and it is a constant. The equivalent considerations apply to another point of decay in the source and emission in direction 2, leading to a different and constant e . Since there is an infinite number of possibilities for positioning the radionuclide within the source as well as the directions of charged particles, the efficiency e is a continuous stochastic variable which fluctuates with a frequency equal to the counting rate. In general, the fluctuation of the efficiency can have a whole frequency spectrum. We subdivide the fluctuations into two categories. The fluctuations which are centered at the counting rate we call an event-level fluctuations (ELF). Other processes contribute to the ELF such as gas/air pressure and electronic fluctuations, as well as light transmission characteristics and magnetic field effect on a photomultiplier tube in the case of scintillation counting. The efficiency for c-ray detection is also subjected to ELF [15]. A second category of efficiency fluctuations we call trial-level fluctuations (TLF) which are centered at the measurement rate (an inverse of the measurement time). The TLF include a change of sample positioning in multiple measurements and the effect of temperature on a photomultiplier tube. They may also include pressure, magnetic field or electronics fluctuations. The fluctuations of the extraction probability e can be described in the same manner as e, however, with appropriately different physical processes.
IV. INFORMATION PROCESSING
446
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
Since the radioactive decay rate j is a quantummechanical constant [16], the fluctuations of probabilities p , p and p from Eqs. (2a), (3) and (1), N Q B respectively, depend on the fluctuation of time. A typical example is the fluctuation of the measurement time due to imprecise timing devices. The time fluctuations belong to TLF. Under certain circumstances j may also fluctuate, for instance, the fluctuation of cosmic-background rate or an inbeam formation rate. Such fluctuations can be both ELF and TLF. It can be shown [13] that the fluctuations of probabilities p can be described by a beta distribu tion [17] p?G\(1!p )@G\?G\ G b(p "a ,b )" G , G G G B(a ,b !a ) G G G 0)p )1, a '0, b 'a , G G G G
(4)
where B is the beta function (an integral over the numerator in Eq. (4)), whereas the parameters a and b are functions of the mean probability pN and the coefficient of variation v "p /pN , G G G G 1!pN a G!pN , b " G. a" G G G pN v G G
recognized when, in addition to p fluctuating withG in a trial, pN fluctuates from a trial to trial. The TLF GI could be named as the random Lexis binomial sampling scheme, leading to the overdispersion in statistics as outlined in the next section.
3. Overdispersion in nuclear statistics Let us consider a stable or radioactive atom subjected to sequential processes such as a transmutation, extraction, survival, decay, detection, etc., with the probabilities p , i"1, 2, n, where n is G a total number of processes. It can be shown [13], using the laws for probabilities [18], that the joint probability L P" p . G G
(6)
If a trial consists of N independent atoms undergoing the n-step process, the probability generating function pgf [11,18] for the trial is given by G(z)"[1#P(z!1)],,
(7)
(5)
The fluctuations of probabilities (or parameters) in statistical distributions belong to a category of binomial sampling schemes in mathematical statistics (see Ref. [18] for a review), with the difference that the fluctuations considered here are random, while the binomial sampling schemes employ parameters with nonrandom variations (i.e., taken from a fixed set). Let us consider the fluctuations of a probability p in a trial k, giving raise to an average value of G such fluctuations, pN . In the following, we give GI a quantitative definitions of ELF and TLF. If the fluctuations are such that this average remains constant for all trials k, we call them ELF, regardless of their exact frequencies. The ELF could be termed as the random Poisson binomial sampling scheme. It has been shown by Breitenberger [19] that the fluctuations from event to event do not disturb ionizing-radiation statistics. This applies more generally to ELF as well. In this case only pN matter, G while the dispersions of p are irrelevant. The situG ation is entirely different for the TLF, which are
which is a generating function of the binomial distribution. Furthermore, it can be shown [13] that pgf for the n-step process with the TLF present can be obtained by mixing Eq. (7) with the beta distributions from Eq. (4), using a procedure developed by Tripathi and Gurland [20] which yields G(z)" F [!N,a , 2, a ; b , 2, b ; 1!z], (8) L> L L L where F is the generalized hypergeometric L> L function defined as [21] (a ) 2(a ) xH I H , (9) F [a , , a ; b , , b ; x]" H I J 2 I 2 J (b ) 2(b ) j! JH HV H where (a) "a(a#1)2(a#j!1) abbreviates an H ascending factorial, (a) "1. The distributions with pgf given by the generalized hypergeometric function with an argument proportional to 1-z were discovered by Kemp and Kemp [22] and are named the generalized hypergeometric factorial moment distributions (GHFD) (see also Ref. [18] for a review).
447
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
Let us define L V>I\ 1#j/a G, 1#j/b C " G G H V>I 1,
if x#k'1, if x#k)1,
(10)
where a and b are given by Eq. (5). The probability G G distribution function for an overdispersed nuclear system can be obtained by expanding G(z) from Eq. (8)
N!x ,\V Pr[X"x]" PM V C (!PM )I, V>I x k I N
(11)
where x is the number of counts and PM is the average of the joint probability from Eq. (6). If there are no TLF fluctuations, a , b PR, so G G C "1 (Eq. (10)). Then the sum in Eq. (11) yields V>I (1!PM ),\V, so it becomes a binomial. Note that, in this formulation, it is not necessary that all the ith processes in the n-step process have TLF fluctuations. For those processes that do not exhibit TLF fluctuations, the factors in the product in Eq. (10) are set to 1.We study the dispersion by performing a moment analysis according to the established procedures [13,18]. The mean k and the coefficient of dispersion d are given below V k"NPM , (12a)
L (1#v)!1 , (12b) G G where v is the index of variation of TLF fluctuG ations of the probability p . The last term in G Eq. (12b) is the overdispersion term. Since it is proportional to k, the larger the mean, the larger the overdispersion. Overdispersion disappears when all the ith processes have no TLF fluctuations (v "0) and, interestingly, when N"1. G The Poisson limit can be obtained when N 1 and any of pN 1 such that PM 1. Then pgf is given G by [13] 1 d "1!PM #k 1! V N
G(z)" F [a , 2, a ; b , 2, b ; N(z!1)], (13) L L L L which also belongs to a GHFD family. The probability distribution function is given by kV (!k)I Pr[X"x]" C . V>I k! x! I
(14)
If there are no TLF, C "1 and the sum in V>I Eq. (14) yields e\I, so it becomes a Poisson distribution. If k 1 and "k!x" k, one obtains a Gaussian limit 1 Pr[X"x]" e\V\I I (2nk
k kI I ; C . (15) (!1)J V>J k! l I J For the Poisson and Gaussian limits the mean continues to be given by Eq. (12a), while d can be V obtained directly from Eq. (12b).
L d "1#k (1#v)!1 . V G G
(16)
4. Applications As a first application we consider the statistics of counts from radioactive decay and detection. We describe only the Poisson limit, which corresponds to a measurement of a long-lived radionuclide and is of the most practical use. In this case n"2 and the average joint probability PM "pN (t)eN (from Eqs. (6) and (1)). We can have at most the TLF of the measurement time and efficiency. The parameters a , b , a , and b are calculated from Eq. (5), C C using pN (tM )"jtM , v "v , eN , and v . Then the distribu R C tion of counts x is described by Eq. (14), in which C is calculated from Eq. (10). The coefficients of V>I dispersion d as well as variation of counts v are V V obtained from Eq. (16) d "1#k[v#(1#v)v], V C C R
(17a)
(17b) v "(1/k#v#(1#v)v. V C C R One way of obtaining the five independent parameters: k, tM , v , eN and v by fitting Eq. (14), by either R C the maximum likelihood or higher-moment analysis [18], requires a precise experimental determination of the distribution function. The situation is simplified by the fact that k and tM are determined in the same series of measurements, whereas eN is usually known from the detector calibration. Therefore, only v and v need to be determined from the C R distribution function. In this model, it is not
IV. INFORMATION PROCESSING
448
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
necessary that both efficiency and time TLF are present at the same time, the former being more common. Neglecting the time fluctuations leaves only one parameter v to be determined from C the fit. Another way to fit the data is by using Eq. (17a). A typical detector performance sequence is to measure a standard radioactive source at prescribed intervals (e.g., daily), calculate empirical d (which is equal to s/l for Poisson data), and V perform s test to justify the results at a given confidence level. However, Eq. (17a) shows that if overdispersion is present, d is proportional to k. V One can thus measure a standard several times to determine good values of d and k. The sequence is V then repeated with increasingly higher radioactivity levels to have higher k. If the plot of d vs. k has V a slope, it is an unmistakable signature of overdispersion without the need of relying on the s test. Then, the correction factor K"v#(1#v)v is C C R calculated from the slope. The standard deviation of a single measurement x is then calculated as (x(1#Kx), rather than (x. The index of variation in Eq. (17b) is essentially the relative error of radioactivity measurement. It has been an established practice, based on the first term under the square root, that the more counts detected, the smaller is the relative error. However, as this term diminishes for a large number of counts, the relative error is essentially determined by the combined TLF of efficiency and time (if present) and not by the statistics of radioactive decay. It is not useful to continue the measurement beyond this point. We describe a possible application of this model to complex experiments. In modern solar neutrino experiments GALLEX [23] and SAGE [24], one measures the decay of Ge (¹ "11.4 d) induced in a Ga target by solar neutrinos according to reaction Ga (l , e\) Ge. The quantity sought is a number of solar neutrino units (1 SNU"10\ solar neutrino captures per target atom/s) and its dispersion. These experiments are analyzed by the maximum likelihood method by Cleveland [25]. Opendak and Wildenhain [26] have shown that the complete statistical process from production to detection of Ge is Poisson. However, in the analysis of an earlier solar neutrino experiment using
Bayes technique, Aurela [7] has pointed out the need for some probabilities to be stochastic variables and expressed this by using some ad hoc chosen three- and five-point discrete distributions. In the present formulation, let N abbreviate the (known) number of Ga atoms (in excess of the Avogadro number). The probability for the combined process is described by the average Eq. (6), PM "pN (t )pN (t )eM pN (t )pN (t )eN . The times are as signed as follows: t production of Ge, t extrac tion, t survival, and t decay during detection. The time fluctuations can be neglected. In Eq. (2b), j is the number of SNU sought (v is its coefficient of variation), j is the decay constant of Ge; so is j in Eqs. (1) and (3). Other parameters are eM , v (extraction of Ge atoms) and eN , v (detection of C C Ge decays), all of which can be determined in separate experiments. From the known parameters one calculates a , b according to Eq. (5). Then the G G distribution of counts x from Ge decay is described by Eq. (14), where k is the mean counts and a , b parameters enter C in Eq. (10). Although G G V>I n"6 in this case, only j , e, and e can possibly contribute to overdispersion. The only unknown parameters to be determined are jM (the average number of SNU) and its v . Distinguishing between the source and the background is done as follows within the framework of overdispersion theory. Let C be a random variable of the (measured) total counts from the source plus background, B a random variable of the (independently measured) counts from the background, and S a random variable of the (unknown) counts from the source itself. Then the likelihood function is given by [13] Pr[C"c,B"b"k ,k ]"Pr[B"b"k ] A ; Pr[S"i"k ]Pr[B"c!i"k ], G
(18)
where k , k are the (unknown) true means of the source and background, respectively. Pr[S] and Pr[B] are taken from Eq. (14). To estimate the means and their confidence intervals, one has to maximize Eq. (18) using maximum likelihood principle, or invert it with the Bayes method.
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
Acknowledgements The author is indebted to P.P. Parekh for valuable comments.
References [1] H. Bateman, Phil. Mag. 20 (Suppl. 6) (1910) 704. [2] F. Cannizzaro, G. Greco, S. Rizzo, E. Sinagra, Int. J. Appl. Rad. Isot. 29 (1978) 649. [3] P.H. Handel, Phys. Rev. A 22 (1980) 745. [4] G. Concas, M. Lissia, Phys. Rev. E 55 (1997) 2546. [5] L.A. Currie, Nucl. Instr. and Meth. 100 (1972) 387. [6] M.A. Tries, Health Phys. 72 (1997) 458. [7] A.M. Aurela, Comp. Phys. Comm. 13 (1977) 281. [8] J. To ke, D.K. Agnihotri, B. Djerroud, W. Skulski, W.U. Schro¨der, Phys. Rev. C 56 (1997) R1683. [9] F.H. Fro¨hner, Nucl Sci. Eng. 126 (1997) 1. [10] A. Ruark, L. Devol, Phys. Rev. 49 (1936) 355. [11] T. Jorgensen, Am. J. Phys. 16 (1948) 285. [12] P.G. Groer, Y. Lo, Rad. Prot. Dos. 69 (1997) 281.
449
[13] T.M. Semkow, Unpublished. [14] H. Yamamoto, T. Norimura, A. Katase, Nucl. Instr. and Meth. A 396 (1997) 418. [15] M. Korun, A. Likar, T. Vidmar, Nucl. Instr. and Meth. A 390 (1997) 203. [16] H. Jakobovits, Y. Rothschild, J. Levitan, Am. J. Phys. 63 (1995) 439. [17] N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, vol. 1, 1994, vol. 2, 1995, Wiley, New York. [18] N.L. Johnson, S. Kotz, A.W. Kemp, Univariate discrete distributions, Wiley, New York, 1993. [19] E. Breitenberger, Prog. Nucl. Phys. 4 (1955) 56. [20] R.C. Tripathi, J. Gurland, Comm. Statist.—Theor. Meth. A 8 (1979) 855. [21] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966. [22] A.W. Kemp, C.D. Kemp, Comm. Statist. 3 (1974) 1187. [23] P. Anselmann et al., Phys. Lett. B 357 (1995) 237; Erratum, 361 (1995) 235. [24] J.N. Abdurashitov et al., Phys. Lett. B 328 (1994) 234. [25] B.T. Cleveland, Nucl. Instr. and Meth. 214 (1983) 451. [26] M. Opendak, P. Wildenhain, Nucl. Instr. and Meth. A 345 (1994) 570.
IV. INFORMATION PROCESSING
Overdispersion in nuclear statistics Thomas M. Semkow Wadsworth Center, New York State Department of Health, Empire State Plaza, Albany, NY 12201-0509, USA State University of New York, Albany, NY, USA
Abstract The modern statistical distribution theory is applied to the development of the overdispersion theory in ionizingradiation statistics for the first time. The physical nuclear system is treated as a sequence of binomial processes, each depending on a characteristic probability, such as probability of decay, detection, etc. The probabilities fluctuate in the course of a measurement, and the physical reasons for that are discussed. If the average values of the probabilities change from measurement to measurement, which originates from the random Lexis binomial sampling scheme, then the resulting distribution is overdispersed. The generating functions and probability distribution functions are derived, followed by a moment analysis. The Poisson and Gaussian limits are also given. The distribution functions belong to a family of generalized hypergeometric factorial moment distributions by Kemp and Kemp, and can serve as likelihood functions for the statistical estimations. An application to radioactive decay with detection is described and working formulae are given, including a procedure for testing the counting data for overdispersion. More complex experiments in nuclear physics (such as solar neutrino) can be handled by this model, as well as distinguishing between the source and background. 1999 Elsevier Science B.V. All rights reserved. Keywords: Stastistical distribution theory; Overdispersion
1. Introduction The validity of Poisson distribution [1] in ionizing-radiation statistics is well established [2]. The attempts to find deviations from the Poisson statistics, based on a fundamental phenomenon such as the flicker noise postulated by Handel [3], were unsuccessful (see Concas and Lissia [4] for a recent
Address for correspondence: Wadsworth Center, New York State Department of Health, Empire State Plaza, Albany, NY 12201-0509, USA.
work as well as a review). The coefficient of dispersion d "p/k (equal to 1 for the Poisson distribuV V tion) can be used as a measure of the deviations. There are, however, known cases of data overdispersion (d '1) discussed by Currie [5] and Tries V [6] in radioactivity counting, Aurela [7] in solar neutrino experiment, and To ke et al. [8] in nuclear multifragmentation. Fro¨hner [9] considered an overdispersion caused by time uncertainties. To the best of our knowledge, no formal overdispersion theory of nuclear statistics, based on mathematical distribution theory, has been developed and is therefore a subject of the present investigation.
0168-9002/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 1 1 4 - 0
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
To incorporate a rapidly decaying source, it is necessary to study the overdispersion in the binomial statistics first, and subsequently obtain the Poisson and Gaussian limits. The binomial statistics depends on the probability of decay in time t, p (t)"1!e\HR, (1) where j is a decay constant [10]. We learn about ionizing-radiation by detecting it. Thus, the overall statistics combines that from the original radiations with that of a measurement, described globally by the probability of detection, i.e., the detector efficiency e [10]. This picture can be generalized to a multistep experiment involving a production, extraction, survival, decay, and detection (or any combination thereof), each depending on a characteristic probability. The probability of production in time t is given below [11] j (e\HR!e\HR), p (t)" (2a) j !j where j is a production rate and j is a decay rate. A Poisson limit to Eq. (2a) can be obtained [12] when j t 1 and j j , j p (t)" (1!e\HR). (2b) j The probability of extraction e is time independent [13], whereas the probability of survival in time t is [11] p (t)"e\HR. (3) In a traditional approach to the statistical distributions, these probabilities are assumed constant in a fixed time interval. However, in Section 2 we present reasons why the probabilities are actually fluctuating in the course of an experiment and show under which circumstances these fluctuations lead to an overdispersion. In Section 3 we derive the formula for the generating functions and probability distribution functions of the overdispersed nuclear system, as well as perform moment analysis. Section 4 discusses the applications to counting and more complex experiments. For the purpose of this work we define a statistical event as a physical process occurring for a single atom, such as a decay
445
or a count. Several events combined together constitute a statistical trial, or a measurement.
2. Fluctuation of probabilities The detection probability for charged particles depends on the geometric arrangement and properties of materials involved in the construction of the detector as well as surroundings [14]. Let us consider a detection of charged particles emitted from an extended source and a detector model consisting of an air space, window, and gas. Let us begin with the decays from a fixed point in the source leading to the emission of charged particles always in one direction. In the limit of a large number of such experiments in direction 1, the efficiency e is the ratio of the number of counts to the total number of decays, and it is a constant. The equivalent considerations apply to another point of decay in the source and emission in direction 2, leading to a different and constant e . Since there is an infinite number of possibilities for positioning the radionuclide within the source as well as the directions of charged particles, the efficiency e is a continuous stochastic variable which fluctuates with a frequency equal to the counting rate. In general, the fluctuation of the efficiency can have a whole frequency spectrum. We subdivide the fluctuations into two categories. The fluctuations which are centered at the counting rate we call an event-level fluctuations (ELF). Other processes contribute to the ELF such as gas/air pressure and electronic fluctuations, as well as light transmission characteristics and magnetic field effect on a photomultiplier tube in the case of scintillation counting. The efficiency for c-ray detection is also subjected to ELF [15]. A second category of efficiency fluctuations we call trial-level fluctuations (TLF) which are centered at the measurement rate (an inverse of the measurement time). The TLF include a change of sample positioning in multiple measurements and the effect of temperature on a photomultiplier tube. They may also include pressure, magnetic field or electronics fluctuations. The fluctuations of the extraction probability e can be described in the same manner as e, however, with appropriately different physical processes.
IV. INFORMATION PROCESSING
446
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
Since the radioactive decay rate j is a quantummechanical constant [16], the fluctuations of probabilities p , p and p from Eqs. (2a), (3) and (1), N Q B respectively, depend on the fluctuation of time. A typical example is the fluctuation of the measurement time due to imprecise timing devices. The time fluctuations belong to TLF. Under certain circumstances j may also fluctuate, for instance, the fluctuation of cosmic-background rate or an inbeam formation rate. Such fluctuations can be both ELF and TLF. It can be shown [13] that the fluctuations of probabilities p can be described by a beta distribu tion [17] p?G\(1!p )@G\?G\ G b(p "a ,b )" G , G G G B(a ,b !a ) G G G 0)p )1, a '0, b 'a , G G G G
(4)
where B is the beta function (an integral over the numerator in Eq. (4)), whereas the parameters a and b are functions of the mean probability pN and the coefficient of variation v "p /pN , G G G G 1!pN a G!pN , b " G. a" G G G pN v G G
recognized when, in addition to p fluctuating withG in a trial, pN fluctuates from a trial to trial. The TLF GI could be named as the random Lexis binomial sampling scheme, leading to the overdispersion in statistics as outlined in the next section.
3. Overdispersion in nuclear statistics Let us consider a stable or radioactive atom subjected to sequential processes such as a transmutation, extraction, survival, decay, detection, etc., with the probabilities p , i"1, 2, n, where n is G a total number of processes. It can be shown [13], using the laws for probabilities [18], that the joint probability L P" p . G G
(6)
If a trial consists of N independent atoms undergoing the n-step process, the probability generating function pgf [11,18] for the trial is given by G(z)"[1#P(z!1)],,
(7)
(5)
The fluctuations of probabilities (or parameters) in statistical distributions belong to a category of binomial sampling schemes in mathematical statistics (see Ref. [18] for a review), with the difference that the fluctuations considered here are random, while the binomial sampling schemes employ parameters with nonrandom variations (i.e., taken from a fixed set). Let us consider the fluctuations of a probability p in a trial k, giving raise to an average value of G such fluctuations, pN . In the following, we give GI a quantitative definitions of ELF and TLF. If the fluctuations are such that this average remains constant for all trials k, we call them ELF, regardless of their exact frequencies. The ELF could be termed as the random Poisson binomial sampling scheme. It has been shown by Breitenberger [19] that the fluctuations from event to event do not disturb ionizing-radiation statistics. This applies more generally to ELF as well. In this case only pN matter, G while the dispersions of p are irrelevant. The situG ation is entirely different for the TLF, which are
which is a generating function of the binomial distribution. Furthermore, it can be shown [13] that pgf for the n-step process with the TLF present can be obtained by mixing Eq. (7) with the beta distributions from Eq. (4), using a procedure developed by Tripathi and Gurland [20] which yields G(z)" F [!N,a , 2, a ; b , 2, b ; 1!z], (8) L> L L L where F is the generalized hypergeometric L> L function defined as [21] (a ) 2(a ) xH I H , (9) F [a , , a ; b , , b ; x]" H I J 2 I 2 J (b ) 2(b ) j! JH HV H where (a) "a(a#1)2(a#j!1) abbreviates an H ascending factorial, (a) "1. The distributions with pgf given by the generalized hypergeometric function with an argument proportional to 1-z were discovered by Kemp and Kemp [22] and are named the generalized hypergeometric factorial moment distributions (GHFD) (see also Ref. [18] for a review).
447
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
Let us define L V>I\ 1#j/a G, 1#j/b C " G G H V>I 1,
if x#k'1, if x#k)1,
(10)
where a and b are given by Eq. (5). The probability G G distribution function for an overdispersed nuclear system can be obtained by expanding G(z) from Eq. (8)
N!x ,\V Pr[X"x]" PM V C (!PM )I, V>I x k I N
(11)
where x is the number of counts and PM is the average of the joint probability from Eq. (6). If there are no TLF fluctuations, a , b PR, so G G C "1 (Eq. (10)). Then the sum in Eq. (11) yields V>I (1!PM ),\V, so it becomes a binomial. Note that, in this formulation, it is not necessary that all the ith processes in the n-step process have TLF fluctuations. For those processes that do not exhibit TLF fluctuations, the factors in the product in Eq. (10) are set to 1.We study the dispersion by performing a moment analysis according to the established procedures [13,18]. The mean k and the coefficient of dispersion d are given below V k"NPM , (12a)
L (1#v)!1 , (12b) G G where v is the index of variation of TLF fluctuG ations of the probability p . The last term in G Eq. (12b) is the overdispersion term. Since it is proportional to k, the larger the mean, the larger the overdispersion. Overdispersion disappears when all the ith processes have no TLF fluctuations (v "0) and, interestingly, when N"1. G The Poisson limit can be obtained when N 1 and any of pN 1 such that PM 1. Then pgf is given G by [13] 1 d "1!PM #k 1! V N
G(z)" F [a , 2, a ; b , 2, b ; N(z!1)], (13) L L L L which also belongs to a GHFD family. The probability distribution function is given by kV (!k)I Pr[X"x]" C . V>I k! x! I
(14)
If there are no TLF, C "1 and the sum in V>I Eq. (14) yields e\I, so it becomes a Poisson distribution. If k 1 and "k!x" k, one obtains a Gaussian limit 1 Pr[X"x]" e\V\I I (2nk
k kI I ; C . (15) (!1)J V>J k! l I J For the Poisson and Gaussian limits the mean continues to be given by Eq. (12a), while d can be V obtained directly from Eq. (12b).
L d "1#k (1#v)!1 . V G G
(16)
4. Applications As a first application we consider the statistics of counts from radioactive decay and detection. We describe only the Poisson limit, which corresponds to a measurement of a long-lived radionuclide and is of the most practical use. In this case n"2 and the average joint probability PM "pN (t)eN (from Eqs. (6) and (1)). We can have at most the TLF of the measurement time and efficiency. The parameters a , b , a , and b are calculated from Eq. (5), C C using pN (tM )"jtM , v "v , eN , and v . Then the distribu R C tion of counts x is described by Eq. (14), in which C is calculated from Eq. (10). The coefficients of V>I dispersion d as well as variation of counts v are V V obtained from Eq. (16) d "1#k[v#(1#v)v], V C C R
(17a)
(17b) v "(1/k#v#(1#v)v. V C C R One way of obtaining the five independent parameters: k, tM , v , eN and v by fitting Eq. (14), by either R C the maximum likelihood or higher-moment analysis [18], requires a precise experimental determination of the distribution function. The situation is simplified by the fact that k and tM are determined in the same series of measurements, whereas eN is usually known from the detector calibration. Therefore, only v and v need to be determined from the C R distribution function. In this model, it is not
IV. INFORMATION PROCESSING
448
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
necessary that both efficiency and time TLF are present at the same time, the former being more common. Neglecting the time fluctuations leaves only one parameter v to be determined from C the fit. Another way to fit the data is by using Eq. (17a). A typical detector performance sequence is to measure a standard radioactive source at prescribed intervals (e.g., daily), calculate empirical d (which is equal to s/l for Poisson data), and V perform s test to justify the results at a given confidence level. However, Eq. (17a) shows that if overdispersion is present, d is proportional to k. V One can thus measure a standard several times to determine good values of d and k. The sequence is V then repeated with increasingly higher radioactivity levels to have higher k. If the plot of d vs. k has V a slope, it is an unmistakable signature of overdispersion without the need of relying on the s test. Then, the correction factor K"v#(1#v)v is C C R calculated from the slope. The standard deviation of a single measurement x is then calculated as (x(1#Kx), rather than (x. The index of variation in Eq. (17b) is essentially the relative error of radioactivity measurement. It has been an established practice, based on the first term under the square root, that the more counts detected, the smaller is the relative error. However, as this term diminishes for a large number of counts, the relative error is essentially determined by the combined TLF of efficiency and time (if present) and not by the statistics of radioactive decay. It is not useful to continue the measurement beyond this point. We describe a possible application of this model to complex experiments. In modern solar neutrino experiments GALLEX [23] and SAGE [24], one measures the decay of Ge (¹ "11.4 d) induced in a Ga target by solar neutrinos according to reaction Ga (l , e\) Ge. The quantity sought is a number of solar neutrino units (1 SNU"10\ solar neutrino captures per target atom/s) and its dispersion. These experiments are analyzed by the maximum likelihood method by Cleveland [25]. Opendak and Wildenhain [26] have shown that the complete statistical process from production to detection of Ge is Poisson. However, in the analysis of an earlier solar neutrino experiment using
Bayes technique, Aurela [7] has pointed out the need for some probabilities to be stochastic variables and expressed this by using some ad hoc chosen three- and five-point discrete distributions. In the present formulation, let N abbreviate the (known) number of Ga atoms (in excess of the Avogadro number). The probability for the combined process is described by the average Eq. (6), PM "pN (t )pN (t )eM pN (t )pN (t )eN . The times are as signed as follows: t production of Ge, t extrac tion, t survival, and t decay during detection. The time fluctuations can be neglected. In Eq. (2b), j is the number of SNU sought (v is its coefficient of variation), j is the decay constant of Ge; so is j in Eqs. (1) and (3). Other parameters are eM , v (extraction of Ge atoms) and eN , v (detection of C C Ge decays), all of which can be determined in separate experiments. From the known parameters one calculates a , b according to Eq. (5). Then the G G distribution of counts x from Ge decay is described by Eq. (14), where k is the mean counts and a , b parameters enter C in Eq. (10). Although G G V>I n"6 in this case, only j , e, and e can possibly contribute to overdispersion. The only unknown parameters to be determined are jM (the average number of SNU) and its v . Distinguishing between the source and the background is done as follows within the framework of overdispersion theory. Let C be a random variable of the (measured) total counts from the source plus background, B a random variable of the (independently measured) counts from the background, and S a random variable of the (unknown) counts from the source itself. Then the likelihood function is given by [13] Pr[C"c,B"b"k ,k ]"Pr[B"b"k ] A ; Pr[S"i"k ]Pr[B"c!i"k ], G
(18)
where k , k are the (unknown) true means of the source and background, respectively. Pr[S] and Pr[B] are taken from Eq. (14). To estimate the means and their confidence intervals, one has to maximize Eq. (18) using maximum likelihood principle, or invert it with the Bayes method.
T.M. Semkow/Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 444—449
Acknowledgements The author is indebted to P.P. Parekh for valuable comments.
References [1] H. Bateman, Phil. Mag. 20 (Suppl. 6) (1910) 704. [2] F. Cannizzaro, G. Greco, S. Rizzo, E. Sinagra, Int. J. Appl. Rad. Isot. 29 (1978) 649. [3] P.H. Handel, Phys. Rev. A 22 (1980) 745. [4] G. Concas, M. Lissia, Phys. Rev. E 55 (1997) 2546. [5] L.A. Currie, Nucl. Instr. and Meth. 100 (1972) 387. [6] M.A. Tries, Health Phys. 72 (1997) 458. [7] A.M. Aurela, Comp. Phys. Comm. 13 (1977) 281. [8] J. To ke, D.K. Agnihotri, B. Djerroud, W. Skulski, W.U. Schro¨der, Phys. Rev. C 56 (1997) R1683. [9] F.H. Fro¨hner, Nucl Sci. Eng. 126 (1997) 1. [10] A. Ruark, L. Devol, Phys. Rev. 49 (1936) 355. [11] T. Jorgensen, Am. J. Phys. 16 (1948) 285. [12] P.G. Groer, Y. Lo, Rad. Prot. Dos. 69 (1997) 281.
449
[13] T.M. Semkow, Unpublished. [14] H. Yamamoto, T. Norimura, A. Katase, Nucl. Instr. and Meth. A 396 (1997) 418. [15] M. Korun, A. Likar, T. Vidmar, Nucl. Instr. and Meth. A 390 (1997) 203. [16] H. Jakobovits, Y. Rothschild, J. Levitan, Am. J. Phys. 63 (1995) 439. [17] N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, vol. 1, 1994, vol. 2, 1995, Wiley, New York. [18] N.L. Johnson, S. Kotz, A.W. Kemp, Univariate discrete distributions, Wiley, New York, 1993. [19] E. Breitenberger, Prog. Nucl. Phys. 4 (1955) 56. [20] R.C. Tripathi, J. Gurland, Comm. Statist.—Theor. Meth. A 8 (1979) 855. [21] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966. [22] A.W. Kemp, C.D. Kemp, Comm. Statist. 3 (1974) 1187. [23] P. Anselmann et al., Phys. Lett. B 357 (1995) 237; Erratum, 361 (1995) 235. [24] J.N. Abdurashitov et al., Phys. Lett. B 328 (1994) 234. [25] B.T. Cleveland, Nucl. Instr. and Meth. 214 (1983) 451. [26] M. Opendak, P. Wildenhain, Nucl. Instr. and Meth. A 345 (1994) 570.
IV. INFORMATION PROCESSING