Материал: Крючков Фундаменталс оф Нуцлеар Материалс Пхысицал Протецтион 2011

Covariance and correlation ratio

If the random quantities Х and Y are dependent (or we know nothing about the dependence of these), the variance of the sum or of the difference should be written as follows:

D(X±Y) = D(X)+D(Y)±2M((X– M(X))(Y–M (Y))) = = D(X)+D(Y)±2M(XY)– M(X)M(Y).

It is obvious that for independent quantities: M(X– M(X))(Y–M (Y)) = = M(XY)– M(X)M(Y) = 0.

For dependent quantities, we introduce the concept of covariance (joint variance) for the random quantities Х and Y:

cov(X,Y) = M((X– M(X))(Y– M(Y))) = M(XY)– M(X)M(Y). (4.34)

And the variance in this case is determined as

Note that, essentially, covariance is the joint first-order moment about mean.

It follows from (4.34) that covariance has the following property:

cov(X,X) = M((X– M(X))(X– M(X))) =D(X).

In fact, covariance (covariance ratio) is to a certain degree a measure of the relationship (dependence) between some random quantities since it possesses the following properties:

∙for independent random quantities, the covariance is equal to zero;

∙it is positive for the random quantities Х and Y tending to vary unidirectionally, and it is negative for those showing the opposite trend.

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However, this ratio is capable of assuming values across the number axis so it is not quite good and convenient for measuring the dependence degree. More convenience in this sense is provided by standardized covariance ratio (the so-called correlation ratio) which is normally introduced as

In fact:

∙for the independent random quantities x and y, ρ(x,y) = 0, because the covariance is equal to zero;

∙for linearly dependent random quantities | ρ | = 1;

∙ for the rest, the correlation ratio varies in a range of –1 to +1 (–1< ρ <+1).

Let us note that the closer | ρ | is to 1, the more reasons one has to treat X and Y as linearly dependent.

Strictly speaking, if the correlation ratio is zero, this does not always mean independence of random quantities. In this case, they say the quantities are uncorrelated (or do not correlate). Independence implies strictly lack of correlation and it is not always the case vice versa. However, it has been strictly shown that the properties of lack of correlation and independence are equivalent for normally distributed random quantities.

Therefore, one can characterize the function of random quantities or a multidimensional random quantity only by a covariance matrix, say, for the case X1 and X2:

Law of large numbers

It has long been observed that the arithmetic-mean value for numerical characteristics of random events (relative frequency) in a great number of

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such similar events tends to vary to a certain extent. On the average, a dependence shows itself which is inherent in the essence of phenomena, this having the influence of some factors that randomized single observation results cancelled therein.

Chebyshev inequality

Chebyshev inequality is a mathematical from of the law of large numbers: the probability that a deviation of random quantities from its mathematical expectation will exceed ε > 0 in the absolute magnitude, and will be no

greater than the relation of its variance to ε 2 .

Let Х be a random quantity, а its mathematical expectation and D(х) the variance, then

Chebyshev inequality gives a nontrivial estimate to the probability of the event Х - а £ e only where D( X ) < ε 2 , the estimate being trivial and not informative in other cases.

Conclusion. Where independent random quantities have identical mathematical expectations equaling а, the variance is limited to one and the same constant с, and the number is great, then for ε > 0 , the probability that the arithmetic mean of these random quantities will deviate from а is arbitrarily close to unity:

This conclusion makes the simplest law of large numbers. Practically, this means that the arithmetic mean of the measurement results is taken for the approximated value of an unknown random quantity. Given that the number of measurements is rather great, this will prove the practical reasoning which is important in terms of (4.39). The equality of the mathematical expectation for the same quantity а is required to show the absence of the so-called systematic component of measurement error.

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Limit theorems

If random quantities are many and satisfy some quite general conditions, then, whatever distribution these have, it is practically certain that the arithmetic mean thereof has an arbitrarily small deviation from the constant or the arithmetic mean of the mathematical expectations thereof, i.e. is practically a constant.

Thus, a common sampling technique making it possible to conclude on the universe based on the limited sample test findings, gets a rationale from the law of large numbers:

the number of grains in a sample (measure), п, is enough to infer the size of the whole grain batch, N, from; the number of the tested items (e.g. fuel assemblies), п, is enough to infer the size of the whole FA batch, N, from; i.e. enough for the law of great numbers to manifest itself with a satisfactory accuracy, though n < N, but n is rather great and the law works.

It can be shown that the sum of any finite number of normally distributed random quantities is normally distributed. Then, if independent random quantities are not distributed normally or are actually distributed in an unknown fashion, then it turns out that they may have some rather soft restrictions imposed on them such that the sum thereof will be distributed normally. This makes the essence of the so-called limit theorems. A form of central limit theorems is Lyapunov theorem [3].

4.2. Basic statistical requirements to NM A&C systems

Upgrades to and evolutions of NM accounting and control systems suggests a greater role of measurements in determining and verifying NM inventories. One can conclude on NM inventories against the background of uncertainties, that are inherent in measurements, only through using statistical methods and employing statistical criteria [4].

We shall consider the major uncertainties inherent in accounting and control of NM:

∙uncertainty inherent in applicable measurement methods;

∙uncertainty inherent in items to be measured;

∙uncertainty inherent in calculation techniques;

∙uncertainty due to the sampling procedure used;

∙uncertainty inherent in “documented” NM data.

Let us underline again that measurements are decisive to all operations on checking the presence and integrity of NM. Of crucial importance in this is to achieve the needed level of measurement accuracy to ensure the level

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of nuclear material control one desires. One of the most important tasks of MS is to analyze uncertainties, estimate measurement errors and search for ways to reduce these. Test measurement results normally deviate from the value declared for the measured characteristic of the NM under control. The cause of a deviation may be a statistically estimated uncertainty (measurement error) or the actually observed difference between the value measured and the one declared. The final conclusion with respect to the cause of the deviation can be made only using a statistical analysis based on accepted statistical criteria.

The need for statistical criteria to be used in inspections also arises due to limited time and funds. Production processes have to be stopped for the inspection, this leading to an economic loss. So, in such cases, sampling checks are done with only some of the items measured. MS is also employed to have the sample size and sequence rationally planned.

The prime objective of NM accounting and control (confident traceability of NM) may be achieved only through an extensive use of statistical and probabilistic methods.

The framework for both determining the inventory of NM in an MBA and detecting anomalies in using NM is formed by NM physical inventory takings. NM physical inventory takings involve audits of accounting data and measurements of the NM quantities in the material balance areas, material balance closing, determination of inventory difference (ID) and estimation of the ID error for each NM.

Absence of anomalies in using NM and of deficiencies in the NM accounting and control system is concluded on based on statistical criteria, that is, statistical guidelines for making decisions from the ID quantity obtained, the ID error and the quantities of the probabilities as set by the guidelines, and detection of a shortage/surplus in NM threshold quantities. We shall highlight again the principles of NM accounting and control that define the statistical requirements:

∙NM shall be subject to state accounting and control starting with the smallest quantities;

∙NM classification depends on the quantity, type and form of the NMcontaining product;

∙onsite MBAs are organized;

∙key measurement points (KMP) are identified in each MBA;

∙NM has access controls (AC) applied thereto to extend the confidence of earlier measurements;

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