Материал: Крючков Фундаменталс оф Нуцлеар Материалс Пхысицал Протецтион 2011

·the input to estimate the quantity Y is the values уi obtained by measurements, as well as random errors S(yi) and the systematic error θ (Y) calculated in one of the experiments;

·the result of determining the quantity Y shall be estimated as follows:

Calculation of ID errors

A system to account for NM as any other material and articles of value is based on balance closing, the major tasks in which are inventory taking, handling and statistical analysis of the data obtained (determination of the statistical significance). The basic assumptions one needs to analyze the ID significance are as follows:

·it can be assumed fairly credibly that the ID obeys to normal distribution law with a zero mathematical expectation and the variance of

σID2 ;

·the confidence interval is to be selected subject to regulations and

requirements [4]. Therefore, estimating the ID value and the variance σ ID2 is enough to conclude on potential anomalies.

Since this is essential, let us discuss in more details the procedures and methodology to estimate the ID statistical significance for detecting anomalies during the material balance closing for an MBA.

The ID is determined and its statistical significance is estimated based on the physical inventory taking data so, as to anomalies, in the given case

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we will deal only with an anomaly showing itself in the ID modulus exceeding the threshold values set in the General Rules [4].

To calculate the ID variance, the equation is expressed as the function

of:

∙ the input quantities – direct and indirect measurem ents (calculation or another determination) data for nuclear material parameters (concentrations, volumes, contents, weights, masses and so on) x1,..., xn ;

∙the random deviations of the given parameters r1,..., rn as conditioned by the random errors of parameter measurements;

∙the systematic deviations of the parameters s1,..., sn as conditioned by

the residual components of the systematic errors of parameter measurements:

ID = f (x1,..., xn ; r1 ,..., rn ; s1,..., sn ) .

The ID variance will be found by the following expression:

The ID variance or the total root-mean-square error is obtained by expanding the ID function into a Taylor series about the mathematical expectations of input quantities with the terms of the given expansion being limited only to linear terms. Individual derivatives are taken at the point x1 ,..., xn while no allowance is made for the deviations and variances of

parameter measurements for the NM inventory as of the inventory taking time in the products not changed since the preceding inventory taking and not measured for the accounting purposes in the given material balance period. Estimation of the ID statistical significance consists in finding out if the following two conditions are fulfilled [4]:

∙the ID modulus does not exceed the threefold root-mean-square ID error value;

∙the ID value does not exceed the found threshold quantities (G) with the confidential probability of 0.95.

The ID modulus exceeding the threefold ID root-mean-square error:

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ID 3σ ID ,

testifies that there are anomalies present in the accounting and control system. The failure to fulfill the second condition:

ID G + 2σ ID ,

evidences of an anomaly present in the accounting and control system, this manifesting itself in the threshold quantities of nuclear material being exceeded.

4.4. Tests of hypotheses and sampling studies

A measurement process occasionally requires decisions to be made on the significance of the measurement data obtained, say, to find out if data from several series of measurements are consistent, if measurement data from several laboratories agree, or if the ID value has significance.

The final decision with respect to the significance can be made following a hypothesis test. For example, if the ID is significantly above

zero in a given period at a given facility.

Let Н0 be a zero hypothesis that consists in that ID≤0; and Н1 is an alternative hypothesis consisting in that ID>0.

As such, testing a hypothesis about a set of experimental data statistically fails to give any proofs as to whether this hypothesis is true or false. A detailed test simply indicates to what degree the hypothesis agrees with the experiment result.

A hypothesis test consists in calculating a certain criterion and comparing the value of this to the tabulated one. Therefore, a test of the zero hypothesis Н0 may suggest the following cases:

1)the hypothesis Н0 is just and the criterion permits Н0;

2)the hypothesis Н0 is just but the criterion discards Н0;

3)the hypothesis Н1 is just and the criterion discards Н0;

4)the hypothesis Н1 is just but the criterion permits Н0.

It is only in cases 1 and 3 that the hypothesis test leads to a correct result. The probability of a type I error (case 2) is numerically equal to the significance level α set for the hypothesis test. If the probability of a type II error (case 4) is equal to β, then the quantity 1– β is called the strength of the test. One often succeeds in giving a test more strength just by increasing α. In other words, a compromise is possible between the level of

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significance and the test strength, a high strength occasionally turning out to be greater than small α.

Several basic hypothesis types are normally tested.

1.Is the measurement data described by the given distribution?

2.Is the difference between two average values significant?

3.Do the variances of several measurement series differ significantly?

Criterion χ 2

It happens sometimes that for the preset random sample хi (i = 1, 2,…, n) of the random quantity Х one needs to test the hypothesis that the function f(x) is the probability density for Х. To get the measure of the available data deviation from the anticipated data as per a hypothetical distribution, the quantity χ 2 is used. The criterion for testing the hypothesis is the comparison of χ 2 against the tabulated value χ 2р, which meets the preset р–percent significance level.

First, the whole range of the values Х in the given sample is broken down into m nonoverlapping intervals; m is found by one of the following formulas:

m = log2 n +1; m = 5 lg n;

m = n;

m = 1.9 n0.4 ,

or from Table 4.3.

Table 4.3

Recommended number of breakdown intervals depending on the number of events

Interval length: d = 1.02 X max

m

The boundaries of the intervals are determined as follows:

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X min − D; X min + d − D;

............................................................

where D = 0.01d.

The number of the values falling into the i interval is equal to νi. If νi <5, the intervals merge. Let pi be the hypothetical probability that Х takes a value that falls into the i interval. Then

i =1

If the assumed number of values in each interval is more than 10 (i.e. npi

> 10), then the quantity vi − npi is distributed asymptotically normally.

npi

Let there be some χ 2p , for which χ 2 > χ 2p with the probability p in the

event of n– 1 degrees of freedom, then it should be taken that the given sample exhibits a significant difference from the hypothetical distribution,

and then the hypothesis Н0 is rejected. Otherwise (χ 2 < χ 2p ) , the

hypothesis is accepted. According to this rule, the probability of a correct hypothesis to be rejected is equal to p.

t–criterion (Student’s)

This criterion is normally used to check if a calculated average value differs from the quantity preset (X = μ ).

To this end, the following criterion is used:

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