Science and Technology of Nuclear Installations

Volume 2019, Article ID 1769149, 10 pages

https://doi.org/10.1155/2019/1769149

## Analysis of Variance for Item Differences in Verification Data with Unknown Groups

SGIM/Nuclear Fuel Cycle Information Analysis, International Atomic Energy Agency, Vienna, Austria

Correspondence should be addressed to T. Burr; vog.lnal@rrubt

Received 10 July 2018; Accepted 22 November 2018; Published 1 January 2019

Academic Editor: Arkady Serikov

Copyright © 2019 T. Burr et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For sequentially collected data, this paper introduces a lag-one differencing method to estimate the random error standard deviation and then uses the estimate to calculate a change detection threshold in a moving window method to detect shifts in the short-term systematic error. Performance results on simulated and real data are presented. Fortunately, the impact of having to perform change detection on the estimated short-term systematic and random error variances is anticipated to be modest or small. The motivating example arises from facilities under nuclear safeguards agreements, where inspector data collected during International Atomic Energy Agency (IAEA) verifications are compared to corresponding operator data. The differences between the operator and inspector values are evaluated using an application of analysis of variance (ANOVA). Typically, it is assumed that short-term systematic errors change across inspection periods, so inspection periods form the groups used in the ANOVA. In some data sets, it appears that the short-term errors have changed at other times, so change detection methods could be used to detect the actual change times.

#### 1. Introduction and Background

For facilities under nuclear safeguards agreements, inspector data collected during International Atomic Energy Agency (IAEA) verifications are compared to corresponding operator data. The differences between the operator and inspector values are evaluated using an application of analysis of variance (ANOVA). It is typically assumed that short-term systematic errors change across inspection periods (groups); however, in some data sets, it appears that the short-term errors have changed at other times, so change detection methods could be used to detect the actual change times. This paper introduces a lag-one differencing method to estimate the within-group random error standard deviation and then uses the estimate to calculate a change detection threshold in a moving window method to detect shifts in the short-term systematic error. Performance results on simulated and real data are presented.

An effective measurement error model for inspector measurement data must account for variation within and between groups, where in this context a group is usually (but not always) an inspection period. A model for multiplicative errors for the inspector () (and similarly for the operator ) iswhere for item (from 1 to ) from group (from 1 to ), is the inspector’s measured value, is the true unknown value, is the short-term systematic error in group , and is a random error. The notation means that has a normal distribution with mean 0 and variance , and similarly for [1]. For simplicity, it is assumed here that there are measurements in each inspection period; allowing for a differing number of measurements in each inspection period is straightforward.

The measurement error model in (1) sets the stage for applying ANOVA with random effects [1–4]. Neither nor are observable. However, for various types of observed data, the variances and can be estimated. The variance of is , or for a fixed value of , , where the item variance is the variance of the , defined as , and is the overall mean of all the true values. The values , by linear error variance propagation, have approximate variance (an accurate approximation for large , up to approximately = 0.60). Regarding notation, and . This paper focuses on estimating and .

#### 2. ANOVA

The usual ANOVA decomposition (which is a standard analysis technique) iswhere , , and In (2), SSW is the within-group sum of squares, and SSB is the between-group sum of squares. In random effects ANOVA (the group means are random), the expected value and , so reasonable estimators are and . It follows that . The vector is a complete sufficient statistic for so by the Lehmann-Sheffé theorem, the estimators and have the minimum variance among all possible unbiased estimators (MVUE) [1]. However, biased estimators can have smaller mean-squared error (MSE), where . Also, the MVUE property relies on the normality assumption. Therefore, other estimators are sometimes considered [1]; for example, setting to 0 if it is negative leads to a positive bias, but also to a lower MSE.

Figure 1(a) plots 10 simulated values in each of 3 groups. The alarm limit on the right side of the plot in Figure 1(a) is estimated on the basis of the = 3 sets of = 10 values of The groups in data such as that in Figure 1(a) are typically inspection periods; however, in some data sets, it appears that the short-term errors have changed at other times, so change detection methods can be used to detect the actual change times. This paper introduces a lag-one differencing method to estimate the random error standard deviation and then uses the estimate to calculate a change detection threshold in a moving window method to detect shifts in the short-term error . Performance results on simulated and real data are presented.