A Cox model for radioactive counting measure: Inference on the intensity process
Introduction
The practical problem that arises in the laboratory of assessing the proper functioning of a spectrometer or other detection device led us to derive a solution. These detectors give as an output the counts of particles emitted by a source such as a radioactive isotope at several time points over an interval of time. First of all, it is necessary to establish a model for the counting process when the detector is working properly. Then, this model could be used as a control model for testing new observed data. The present paper proposes to model the counting process as a Cox process (CP) and provides a stochastic estimation of its intensity. Thus, on the one hand it is possible to reconstruct the functional sample paths of the intensity in an interval of time and on the other hand, the required model is obtained. This estimation is applied to the radioactive counting of the isotopes 226Ra and 137Cs. The first theoretical novelty of this paper is that by means of the specific stochastic representation of the two isotopes intensities, a confidence band for the intensity process in an interval of time is obtained. Then, this representation of the intensity enables us to build a hypothesis test to decide whether the new observed data follow the same model which is the second novelty. If this is the case, the detector is assumed to be working properly but if not it is understood to be out of control.
In the framework of chemical counting processes, the homogeneous or nonhomogeneous Poisson processes (by which the intensity is a constant or a function of time, respectively) have traditionally been used to model data. The intensity estimation of these processes is a very useful tool in many fields such as instrument quality control, signal processing, nuclear medicine or image processing in photon counting (Bityukov et al. [1]; Jansen [2], among many others). Even so, experimental data suggest that these models are insufficient. In this respect, Bayne et al. [3] stated “a more complex decay function may be required to approximate decreasing ion intensity”. See also the conclusions section in Nádai and Várlaki [4] as another example. A generalization of those models, the CP, began to be used time ago (see for example Snyder [5]; Teich and Saleh [6]). In recent years, the use of doubly stochastic processes has expanded (see Nádai and Várlaki [4], Molski [7]), even though, most of the studies in this field still uses homogeneous or nonhomogeneous Poisson process for modeling these counting phenomena. The reason for this may be the intractability of explicit expressions of the moments of doubly stochastic processes. That is why in this paper we use the CP but from the point of view of functional data analysis (FDA) which allows us to obtain the results mentioned above.
The CP or doubly stochastic Poisson process is a Poisson process whose intensity is also a stochastic process. Due to the stochastic nature of its intensity, the CP is quite flexible and realistic for the purposes of modeling real phenomena. CP was first defined by Cox [8] and it has been studied at length for example by Daley and Vere-Jones [9], Snyder and Miller [10], Andersen et al. [11], Last and Brand [12] or Grigoriu [13]. From these references it can be observed that this process-counting model has been used in various fields including risk analysis, economics, population theory, biology, catastrophe analysis, medicine, signal processing and optics.
Estimating the intensity process of a CP is a problem that has been widely considered. For instance, Boel and Beneš [14], Snyder and Miller [10], Manton et al. [15], Nádai and Várlaki [4], among many others, have formulated approaches using filtering methodology but with these models it has always been necessary to impose a fixed model on the intensity moments. When stochastic processes are observed at discrete time points, FDA models reconstruct the functional form of their sample paths (see Ramsay and Silverman [16] and Valderrama et al. [17]). The advantage of using FDA is that it does not require to impose a distribution on the process or to have known moments. Bouzas et al. [18] proposed an estimation of the intensity process of a CP from the FDA point of view, just from observed sample paths of the CP in a finite set of points. In a subsequent paper, Bouzas et al. [19] proposed applying FDA in order to estimate the mean process of a CP and with the novelty of preserving the monotonicity of its sample paths, thus providing an ad hoc stochastic estimation of this mean process. Finally, Bouzas et al. [20] extended the estimation of the intensity process by providing a new stochastic estimation by means of its relation with the mean process.
The above-mentioned ad hoc estimation of the intensity process is applied in this paper to both 226Ra and 137Cs radioactive counting processes. The counts are observed in a discrete set of time points and using Functional Principal Components Analysis (FPCA), the intensity is expressed in terms of an expansion of random variables (r.v.'s) multiplied by functions of time. From this stochastic estimation, the intensity can be estimated for any instant of time of the whole observation interval. Furthermore, knowledge of the joint distribution of those r.v.'s allows us to estimate the marginal intensity distribution and then to obtain a confidence band of the intensity for an interval of time, and as a final contribution, to perform a hypothesis test. The latter is used as a tool for calibrating the counting device, and so it fulfils the first aim of this paper.
The rest of the paper is structured as follows: in the first subsection of Section 2, we review the theoretical background for estimating the intensity process while the second subsection describes how to build a confidence band for the intensity and to design a hypothesis test to determine whether the new data are in accordance with the estimated model. In Section 3, the procedure is applied to the data counts of radioactive 226Ra and 137Cs obtaining explicit expressions for these real examples. Finally, some conclusions are drawn in Section 4.
Section snippets
The stochastic Cox model
A CP {N(t) : t ≥ t0} with intensity {λ(t) : t ≥ t0} is defined as a Poisson process with an intensity described as the stochastic process {λ(t) : t ≥ t0}. Then, its mean process is given by Λ(t) = ∫ t0t λ(σ) dσ. Besides the mean, all the characteristics of the CP depend on its intensity process. That is why it is so important to estimate the intensity in general and in our application. Our aim is to estimate λ(t) in [t0, t0 + T) having no previous knowledge about its structure, from several sample paths of N(t)
Application to real counting data
In order to minimize the influence of radioactive decay in the data analysis, we used the long life isotopes 226Ra (half life: 1600 years) and 137Cs (30.07 years), so in these conditions decay effect is negligible. Therefore, the counts can be considered independent in disjoint intervals of time and the proposal to model them by CP is consistent. Measurements were made with an IMPO MC24E event counter, attached to a Geiger–Mullar probe supplied by Fredeiksen (Denmark). The polarization potential
Results and discussion
The radioactivity of 226Ra and 137Cs is studied modeling both of them as a CP or doubly stochastic Poisson process. This fact implies that their intensities are considered as stochastic processes and not just a simple constant parameter or a function of time.
For both isotopes and from the observed counts of the emitted particles, the intensity process is estimated by means of FPCA. Thereby, a stochastic estimation in terms of an expansion of uncorrelated random variables is derived without
Acknowledgments
This work was partially supported by projects MTM2007-63793 and MTM 2007-66791 of Plan Nacional I + D + I, Ministerio de Ciencia e Innovación, P06-FQM-01470 from Consejería de Innovación, Ciencia y Empresa de la Junta de Andalucía and grant FQM-307 of Conserjería de Innovación de la Junta de Andalucía, all of them being Spanish institutions. The authors wish to thank the anonymous reviewers for their helpful suggestions.
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