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Radiation Protection Dosimetry
, Volume Advance Article – Mar 15, 2018

/lp/ou_press/using-the-hermite-polynomials-in-radiological-monitoring-networks-gHpXgvAGw3

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com
- ISSN
- 0144-8420
- eISSN
- 1742-3406
- DOI
- 10.1093/rpd/ncy040
- Publisher site
- See Article on Publisher Site

Abstract The most interesting events in Radiological Monitoring Network correspond to higher values of H*(10). The higher doses cause skewness in the probability density function (PDF) of the records, which there are not Gaussian anymore. Within this work the probability of having a dose >2 standard deviations is proposed as surveillance of higher doses. Such probability is estimated by using the Hermite polynomials for reconstructing the PDF. The result is that the probability is ~6 ± 1%, much >2.5% corresponding to Gaussian PDFs, which may be of interest in the design of alarm level for higher doses. INTRODUCTION Currently, radiological surveillance networks have an essential role in the protection and safety of the population. It is possible to differentiate diverse types of networks: those with a global extension(1); with state range(2, 3); and, a smaller size, those of local extension(4, 5). However, all the networks have in common the traditional method of surveillance based on: the average, standard deviation, maximum value and a confidence interval of 2σ(6). The confidence interval of 2σ presupposes a Gaussian behavior on the recorded data distribution by the surveillance network, which gives a confidence of 95%. On the contrary, empirically, it has been observed that the data have skewness and a kurtosis, which does not correspond to the normal distribution. The observed skewness is positive, which revealed that the probability of finding extreme events above values at 2σ increases, and the presumption of coverage at 95% of confidence is not correct. Due to the few extreme events it is not easy to estimate the probability of finding events beyond 2σ; P(>2σ). Therefore, in this work, a possible probability density function (PDF) of the empirical data is reconstructed from the first four central moments, using the orthogonal property of the Hermite polynomials(7–9); it is required for the PDF to be quasi-Gaussian. Under these conditions, the calculation of P(>2σ) is quite simple, and can be used as a monitoring parameter of extreme events, which entail more risk for the population from the point of view of radiological safety. THEORETICAL BASIS There are numerous widely used probability distributions, such as the Gaussian, the Beta, the Exponential, the Gamma distributions, etc. However, they do not always fit properly to the empirical distributions obtained in radiological protection measurements. In this sense, PDFs of radiological records can be calculated empirically. It is observed that the proper characterization for the PDF needs at least four parameters, because the standard description is insufficient using the mean value and the variance; typical of Gaussian distributions. Hence, let us reconstruct PDF from the first four central moments: the mean value, variance, skewness and kurtosis. In addition, due to the PDF should be quasi-Gaussian; a polynomial correction can be used with respect to the normalized Gaussian function. The mathematical definitions are shown below. f(x)≡PDF W(x)=12πe−x22≡Normaldistribution Hn(x)=(−1)nex22dndxne−x22≡Hermitepolynomials From the definition of the Hermite polynomials the first four terms can be calculated: H0(x)=1 H1(x)=x H2(x)=x2−1 H3(x)=x3−3x H4(x)=x4−6x2+3 The variable x stands here for the H*(10) values in normalized non-dimensional units: x=d−d¯σd where d stands for the recorded dose rate in μSv/h; d¯ is the mean value of the recorded data; and σd is the standard deviation. On the other hand, Hermite polynomials orthogonal property means that: <HiHj>=∫−∞∞W(x)Hi(x)Hj(x)dx=δijj!; where δij is the Kronecker delta. In this case, the expression ∫−∞∞W(x)Hi2(x)dx=i! is fulfilled. Therefore, the calculation for xn can be performed with the Hermite polynomials: 1=H0 x=H1 x2=H2+H0 x3=H3+3H1 x4=H4+6(H2+H0)−3H0=H4+6H2+3H0 The PDF is now expressed as a linear combination of the Hermite polynomials: f(x)=W(x)∑k≥0akHk(x) where the coefficients ak, in this case, can be found by using the first four central moments: μn=∫−∞∞xnf(x)dx,n=0,1,2,3,4 The coefficients for the first four central moments can be calculated as follows: Zero order, n = 0 (normalization) μ0=1=∑k=04ak<H0Hk>=a0 First order, n = 1 (average) μ1=∑k=04ak<xHk>=∑k=04ak<H1Hk>=a1 Second order, n = 2 (variance) μ2=∑k=04ak<x2Hk>=∑k=04ak<(H2+H0)Hk>=2a2+a0 Third order, n = 3 (skewness) μ3=∑k=04ak<x3Hk>=∑k=04ak<(H3+3H1)Hk>=6a3+3a1 Fourth order, n = 4 (kurtosis) μ4=∑k=04ak<x4Hk>=∑k=04ak<(H4+6H2+3H0)Hk>=∑k=04ak<H4Hk>+6ak<H2Hk>+3ak<H0Hk>=24a4+12a2+3a0 Finally, four equations with four unknowns are defined, which are solved and yields the values for the coefficients: a0=1;a2=μ2−12;a3=μ3−3μ16;a4=124[μ4−3−12(μ2−12)] Finally the PDF equation would be: f(x)=12πe−x22[1+μ1x+μ2−12(x2−1)+μ3−3μ16(x3−3x)+⋯⋯+124(μ4−3−12(μ2−12))(x4−6x2+3)] It’s highlighted that in normalized units, μ1=0 and μ2=1. Consequently, Eq. 1 is used for obtaining the final PDF results: f(x)=12πe−x22[1+μ36(x3−3x)+…..+124(μ4−3)(x4−6x2+3)] (1) Observe that f(x) is reduced to a normal Gaussian distribution, when μ3=0 and μ4=3; so that f(x)=W(x). Also for the case that it was a symmetric distribution, where μ3=0; it is got: f(x)=12πe−x22[1+124(μ4−3)(x4−6x2+3)] Because W(x) is used as a weight for Hermite polynomials, and a good representation of the empirical PDF is not expected when it is far from Gaussian. EXPERIMENTAL RESULTS The data correspond to the environmental equivalent dose rate, H*(10), recorded every hour throughout the year 2003 by a Geiger Müller detector (Gamma Tracer); which is part of the Gamma Network owned by CIEMAT. The data was fragmented monthly in order to evaluate the resulting PDF and establish a statistically more advanced surveillance system. Traditionally, the Gaussian distribution is used, where P(>2σ) = 0.025; however, for the measured distributions it is showed that this probability is noticeably greater. In the next graphs, the recorded environmental dose rate and the correspondent normalized PDFs for January and February is showed. Figures 1 and 2 show that are clearly not Gaussian, because they have positive skewness; but they are quasi-Gaussian and therefore they fit into the proposed method. The corrections on the PDF in normalized units are made using the Hermite polynomials. Figure 1. View largeDownload slide (A) H*(10) for January 2003. (B) Fit of the experimental distribution with Hermite polynomials. Figure 1. View largeDownload slide (A) H*(10) for January 2003. (B) Fit of the experimental distribution with Hermite polynomials. Figure 2. View largeDownload slide (A) H*(10) for February 2003. (B) Fit of the experimental distribution with Hermite polynomials. Figure 2. View largeDownload slide (A) H*(10) for February 2003. (B) Fit of the experimental distribution with Hermite polynomials. On the other hand, Figure 3 shows an anomalous graph. Figure 3. View largeDownload slide (A) H*(10) for April 2003. (B) Fit of the experimental distribution with Hermite polynomials. Figure 3. View largeDownload slide (A) H*(10) for April 2003. (B) Fit of the experimental distribution with Hermite polynomials. A bimodal distribution was observed in April, which is rather far from the quasi-Gaussian presumption. The Hermite polynomials do not fit properly, because appears negative significant values on the resulting PDF. The correction to this problem could be: To add more polynomial terms. However, it is discarded because the usual PDF should be a quasi-Gaussian distribution. Moving average smoothing around the maximum values of H*(10) of the time series, which are responsible for the bimodal distribution. Observe on the Figure 4, how the bimodal distribution is corrected as a consequence of the moving average smoothing. Figure 4. View largeDownload slide (A) Moving average smoothing around the maximum values of H*(10) for April 2003. (B) Fit of the experimental distribution smoothing the maximum values with Hermite polynomials. Figure 4. View largeDownload slide (A) Moving average smoothing around the maximum values of H*(10) for April 2003. (B) Fit of the experimental distribution smoothing the maximum values with Hermite polynomials. However, this solution, although accurate, is discarded as a surveillance method, also due to the loss of information from the time series that involves eliminating the maximum values, which are the most relevant events for the time series. November also showed the same anomaly. It shown an increase of H*(10) and the proposed method did not fit properly. It is observed in Figure 5. Figure 5. View largeDownload slide (A) H*(10) for November 2003. (B) Fit of the experimental distribution with Hermite polynomials. Figure 5. View largeDownload slide (A) H*(10) for November 2003. (B) Fit of the experimental distribution with Hermite polynomials. The existence of the bimodal distribution in both months was investigated, a correlation with rainfall was found precisely in those months. They were the 2003 rainiest months and, due to this, the Rn-222 and its progeny produced an increase of H*(10)(10). It is shown in Figure 6. Figure 6. View largeDownload slide (A) Rainfall April 2003. (B) Rainfall November 2003. Figure 6. View largeDownload slide (A) Rainfall April 2003. (B) Rainfall November 2003. The summary of the measurements is shown in Figure 7, where the P(>2σ) is plotted with respect to the months of the year. Figure 7. View largeDownload slide Probabilities for high events monthly in 2003. Figure 7. View largeDownload slide Probabilities for high events monthly in 2003. Due to the abundant precipitations in the month of November the fit of the polynomials of Hermite is not adequate. Therefore, the calculated value of P(>2σ) for this month is anomalous. However, as it shows Figure 7, the other months show a probabilities range between 5 and 7%. It is significantly greater than probabilities in Gaussians distributions, 2.5%. DISCUSSION It has been observed that the fitting of the PDF with four polynomial terms of Hermite is very good for quasi-Gaussian distributions; but not so much when the distributions are bimodal. In this case, more central moments are needed for the correct fitting; and, consequently, more polynomial terms. The use of this methodology has shown that it is possible the appearance of negative values in the lower end on the PDF fitting, which indicates an anomaly. That is, it indicates the suspicion of some extreme radiological event, which needs further investigation. When events of greater amplitude are sought, a possible solution is to consider the PDF between −2σ and 3σ, suppressing the negative values found for the lower amplitudes, which lack of interest. In this sense, the values of large amplitudes, responsible for the sudden increases of the environmental equivalent dose rate, would be fairy represented. Sometimes the statistic is not enough, <30 data (1 data per day), and the empirical construction of PDF is not reliable; but it is possible to calculate the four central moments and obtain, with a greater confidence, the probabilities for radiological extremes events. The P(>2σ) monthly for 2003 are of the order of 6 ± 1%, while for a Gaussian distribution they should be 2.5%. When the distribution is quasi-Gaussian, the estimate of extreme events with the Hermite polynomials concludes that the confidence of 99.5% is obtained with a coverage factor between 3.6σ and 4σ. Therefore, it is recommended to use an upper limit of surveillance for amplitudes >4σ. The proposed methodology is applicable to other fields, for example Metrology. The GUM recommends investigating the most appropriate PDF for calculating the 95% coverage factor, even for distribution that are not Gaussian. CONCLUSIONS An appropriate methodology for quasi-Gaussian PDFs has been constructed with the Hermite polynomials using the first four central moments of the distribution. Its usefulness has been tested, from the point of view of Radiological Safety, with the Gamma Network during the months of the year 2003. The formation of bimodal distributions, far from the Gaussian hypothesis, denotes anomalous events that must be investigated. It is proposed to use P(>2σ) as a monthly monitoring parameter; in this way, the probability of the occurrence of risky radiological events can be evaluated. It has been shown that P(>2σ) is of the order of 6 ± 1%, much greater than that corresponding to a Gaussian distribution. Due to the PDF have a positive skewness, it is recommended the use an upper limit of 4σ in radiological surveillance for evaluating the extreme events, which involves a greater risk for the population. ACKNOWLEDGEMENT Thanks to the human effort of the Radiological Protection Service at CIEMAT and to the Sub-Direction of Security and Improvement of the Facilities. REFERENCES 1 Commission for the Comprehensive Nuclear-Test-Ban Organization . http://www.ctbto.org. 2 De Felice , P. The quality assurance programme for the national radioactivity surveillance network in Italy . Radiat. Prot. Dosim. 97 ( 4 ), 313 – 316 ( 2001 ). https://doi.org/10.1093/oxfordjournals.rpd.a006678. Google Scholar CrossRef Search ADS 3 Pereira , M. F. , Pereira , J. , Rangel , S. , Saraiva , M. , Santos , L. M. , Cardoso , J. V. and Alves , J. G. 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Google Scholar CrossRef Search ADS 8 Abramowitz , M. and Stegun , I. S. Handbook of Mathematical Functions ( New York, NY : Dover Publications, Inc ) ( 1965 ). 9 Andrews , L. C. Special Functions for Engineers and Applied Mathematicians ( New York, NY : MacMillan Publishing Co ) ( 1985 ). 10 Márquez , J. L. , Benito , G. , Saez , J. C. , Navarro , N. , Alvarez , A. and Quiñones , J. The influence of radon (gas and progeny) and weather conditions on ambient dose equivalent rate . Radiat. Prot. Dosim. 174 ( 3 ), 423 – 430 . ( 2017 ) doi:10.1093/rpd/ncw219 . © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

Radiation Protection Dosimetry – Oxford University Press

**Published: ** Mar 15, 2018

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