Dynamic numerical recipes to improve detection schemes for the interdiction of radioactive shipment

Abstract

Radiation Portal Monitors (RPM) acquire sequential measurements of gamma activity that can be modeled as a sequence of Poisson distributed random numbers. These series can be analyzed with Statistical Process Control (SPC) methods to trigger alarms when the statistical distribution of the ambient dose during a transit changes. Some usual SPC methods have been compared and modified to narrow the time required to bound an alarm. Both the probabilities of detection and the Average Run Length (ARL) are used and fitted to curves to benchmark the SPC methods, which are ranked as: SR, EWMA, CUSUM, SPRT, FLR, and Shewhart.

Appendix

${{\upalpha '}}$	Error type I (false positive) in a Bayesian scheme
${\text{a}}_{\text{n}}$	Bottom limit of the interval defined in ROIn
A, B, C	Auxiliary variables to fit the response of $\frac{{\mathop \sum \nolimits_{{\updelta_{\rm i} \ge \updelta^{*} }} \widetilde{\text{ARL}}_{{{{\updelta }}^{*} }} /{\text{ARL}}_{0} }}{{\mathop \sum \nolimits_{{\updelta_{\rm i} \ge \updelta^{*} }} i}}$ versus UCL, where ${\text{A, B, C}} \ge 0$
a, b, c	Auxiliary variables to fit some curves, where ${\text{a, b, c}} \ge 0$
ARL	Average Run Length. It is defined as the average number of samples from a Poisson distribution before the SPC method signals
${\text{ARL}}_{ 0}$	ARL of the in-control state in a quasi-infinite sequence being monitored (background case)
$\widetilde{\text{ARL}}_{ 0}$	ARL of the in-control sub-sequence representing the time through the RPM being monitored (background case)
${\text{ARL}}_{{{\updelta }}}$	ARL of a shift of size ${{\updelta }}$ in the state in a quasi-infinite sequence being monitored (source case)
$\widetilde{\text{ARL}}_{{{\updelta }}}$	ARL a shift of size ${{\updelta }}$ in the sub-sequence representing the time through the RPM being monitored (source case)
$\widetilde{\text{ARL}}_{{{{\updelta }}^{*} }} / {\text{ARL}}_{ 0}$	Ratio between the ARL of a change that triggers a true alarm in a transit over the time needed to trigger a false alarm in an infinite sequence. Then, values can be compared between methods as the FAR is the same
${{\upbeta '}}$	Error type II (false negative) in a Bayesian scheme
${\text{b}}_{\text{n}}$	Top limit of the interval defined in ROIn
${\text{C}}_{\text{in}}$	Sequence of gross counts in the channel i and integrated between the channels defined in ROIn
${\text{C}}_{\text{n}}^{{{\# }}}$	Detection limit of the gross counts ${\text{C}}_{\text{in}}$ being monitored in ROIn, where ${{\updelta }} = {{\updelta }}^{ *}$ (minimum shift to trigger an alarm)
CUSUM	Cumulated Sums
d	Perpendicular distance between the radioactive source and the detector
${{\updelta }}$	Size of the shift introduced in the mean above the background level. The size ${{\updelta }}$ of the shift is sampled every 0.25 units between 0 and 10
${{\updelta }}^{ *}$	Period of time in the profile when the alarm is triggered (coverage factor). It is obtained as twice the normal inverse cumulative distribution of the detection probabilities: 0.505, 0.525, 0.6, 0.7, 0.775, 0.8, 0.85, 0.9, 0.9332, 0.95, 0.965, 0.975, 0.98, 0.9825, 0.985, 0.9875, 0.99, 0.995, 0.999, 0.9999
EWMA	Exponentially Weighted Moving Average
${\text{f}}_{\text{in}}$	Background measured at ith position in the window ROIn
${\text{f}}_{\text{Rn}}$	Background of reference integrated over the range defined in ROIn
${{\mathcal{F}}} ( {\text{x)}}$	Normal cumulative distribution
FAR	False alarm rate as the ratio provided by the total amount of times that $\widetilde{\text{ARL}}_{ 0}$ triggers an alarm with respect to the total number of assays
FLR	Full Likelihood Ratio as the ratio between the probability of measuring the detection limit ${\text{C}}_{\text{n}}^{{{\# }}}$ and the probability of measuring the background
FNR	False Negative Rate as the complement probability of the TAR
${{\uplambda}}$	EWMA parameter to balance how much distant measurements are considered in the average ($0\le {{\uplambda}} \le 1$)
${{\Lambda }}_{\text{in}}$	Full likelihood ratio in ROIn to detect a source as a change in the gross counts sequence ${\text{C}}_{\text{in}}$
${{\mathcal{G}}} ( {\text{x}}|\upmu ,\upsigma )$	Gaussian distribution of a process of mean ${{\upmu}}$ and standard deviation ${{\upsigma}}$
${\text{m}}_{\text{in}}$	Net counts after background subtraction
N	Total number of channel from a spectrum ($1\le {\text{a}}_{\text{n}} {\text{ < b}}_{\text{n}} \le {\text{N}}$)
${{\mathcal{N}}} ( 0 , 1 )$	Standardized normal distribution
NORM	Naturally Occurring Radioactive Material
${\text{N}}_{{{\upsigma}}}$	Sigma level significance of the Shewhart SPC method
$p_{1} , p_{2}$	Parameters to generalize the UCL according to the variable ${\text{x}}$ being monitored. When ${\rm x} = {\text{N}}_{{{{\upsigma}}_{\text{in}} }}$, then p1 = 0, and p2 = 1. Conversely, when ${\rm x} = {\text{C}}_{\text{in}}$, then $p_{1} = {\text{f}}_{\text{Rn}}$, and $p_{2} = \sqrt {{\text{f}}_{\text{Rn}} }$
${{\mathcal{P}}} ( {\text{x}}|\upmu)$	Poisson distribution of a process of mean ${{\upmu}}$
PVT	Polyvinyl-toluene
r	Distance sample-detector
ROI	Region of interest as a range of channels $[ {\text{a}}_{\text{n}} , {\text{b}}_{\text{n}} ]$ from the spectra where the integral is obtained
RPM	Radiation Portal Monitor
RSP	Radiation Sensor Panel
${{\upsigma}}_{{{\text{f}}_{\text{Rn}} }}$	Standard deviation of the background of reference in ROIn
${\text{s}}_{\text{ij}}$	Original spectra acquisition from the detector’s channel i at time j
${\text{S}}_{{ 0 {\text{n}}}}$	Initial condition of a recursive statistic from a SPC method. It can be set to a value different than zero to more steadily detect a shift at the beginning of the monitoring
${\text{S}}_{\text{in}}$	Recursive value of the statistic used in a SPC method. Its value is compared with the UCL to trigger an alarm when ${\text{S}}_{\text{in}} \ge {\text{UCL}}$
SPC	Statistical Process Control
SPRT	Wald’s Sequential Probability Ratio Test
SR	Shiryaev–Roberts procedure
t	Time spent during the sequence being monitored
T	Time (fixed) of the length of the subsequence corresponding to a radioactive profile/subsequence (transit)
TAR	True Alarm Rate as the ratio provided by the total amount of times that $\widetilde{\text{ARL}}_{{{\updelta }}}$ triggers an alarm with respect to the total amount of essays
TAR*	TAR probabilities when the FAR ${ = 0} . 0 1$%
TNR	True Negative Rate as the complement probability of the FAR
UCL	Upper Condition Limit of the SPC to trigger an alarm in the corresponding method
v	Velocity of the radioactive source during the profile/subsequence. Its value is sampled from 0.5 to 6 m/s by 0.5 m/s steps to represent most of the monitoring scenarios