Accounting for the kurtoses of input quantities in the procedure of evaluating measurement uncertainty using the example of weight calibration

Example 9.3.1.1 of JCGM-S1 “Mass calibration” is analyzed, which describes the comparison in air of reference and calibrated weights having the same nominal mass. JCGM-S1 compares uncertainty evaluation procedures based on the GUM uncertainty framework and the Monte Carlo method. The article uses the procedure developed by the authors and consists in decomposing the measurement model into a Taylor series of the second order taking into account the kurtoses of the distributions of input quantities. To facilitate the calculations, the finite increment method is used. To find expanded uncertainty, the kurtosis method is used. Good agreement between the results obtained by the proposed method and the result obtained by the Monte Carlo method is shown.


Introduction
Weight calibration is carried out by comparing its mass with a reference weight having the same mass using balance working in the air. Measurement uncertainty evaluating for this example is discussed in regulatory document JCGM-S1, 9.3.1.1. [1].
In [1], the measurement uncertainty evaluation using the methods of GUM [2] and Monte Carlo [3] is performed. At this, a significant bias in the measurement uncertainty evaluation obtained using GUM is discovered.
To implement the Monte Carlo method, one can apply, for example, the "Uncertainty machine" program developed by NIST [4]. However, one of the drawbacks of such programs is the lack of a total measurement uncertainty budget. Elimination of the above disadvantages is possible using the methods proposed by the authors in [5][6][7].
The purpose of this article is to implement the methods [5][6][7] for measurement uncertainty evaluation during calibration of weights with verification of their adequacy.

Development of a measurement model
Mass of calibrated weight m W with mass density ρ W is compared with a reference weight with mass density ρ R , having nominally the same mass using balance working in air with mass density ρ a . Since ρ W and ρ R are generally different, buoyancy of air should be taken into account. Therefore, based on the Archimedes law, we can write the following formula [1]: where m R -actual mass of the reference weight; δm R -actual mass of small weight with density ρ R , added to equalize balance. Usually the conventional mass of the calibrated weight m W , c -mass of hypothetical weight with density ρ 0 = 8000 kg/m 3 , which balances the calibrated weight in the air with density ρ a0 = 1.2 kg/m 3 , is determined. Therefore, taking into account the notation of the conventional masses of reference weights m R , c , δm R , model (1) Insofar as the equation (4) is nonlinear, in numerical value evaluation of the measurand and its uncertainty it is necessary to consider higher terms in the expansion (4) in the Taylor series of the second order considering kurtoses of input quantity distributions.

Calculation of the numerical value of the measurand
In the first approximation, the calculation of the numerical value of the measurand is carried out according to the formula: ,  nom (5) in which the values of input quantities are replaced by their numerical values marked with hats.
For given in [1]  Formula (2) gives an unbiased estimate of the numerical value of the measurand only in the absence of uncertainties of input quantities.
The bias ∆ y of the numerical value of the measurand can be estimated taking into account the partial derivatives of the second order of the measurand with respect to the corresponding input quantities c(x i ) 2 [5]: where u(x i ), i =1,2,...,N -standard uncertainties of input quantities X 1 , X 2 ,...,X N . For equation (4) the calculation of the bias for  δm is performed by the formula: The expressions for c(x i ) 2 are given in Table 1.
Since the second-order partial derivatives of the measurand with respect to all input quantities turned out to be equal to zero, the bias of the measurand numerical value will also be zero.

Calculation of the standard uncertainty of the measurand
In the first approximation, the calculation of the standard uncertainty of the measurand is carried out based on the equation: Second-order partial differential expressions where u(x i ) and c(x i ) -the standard uncertainty of the input quantity X i and the partial derivative of the measurand with respect to this input quantity (sensitivity coefficient), respectively. Expressions for c(x i ) and their values are given in Table 2. Table 2 Expressions for sensitivity coefficients For the above evaluations of input quantities, as well as for the standard uncertainty of the input quantities specified in [1]: u(m R,c ) = 0.05 mg; u(δm R,c ) = = 0.02 mg; u(ρ a ) = 0.05774 kg/m 3 ; u(ρ W ) = 577.35 kg/m 3 ; u(ρ R ) = 28.87 kg/m 3 , we obtain u(δm) = 0.05385 mg.
Formula (6) provides an unbiased estimate of the standard uncertainty of the measurand only in the absence of uncertainties of the input quantities.
The bias of the variance of the measurand is calculated by the formula [6]: where η(x i ) -kurtosis of the distribution of the i-th input quantity, which is taken from Table 3 [5], c(x i , x j )a mixed second-order partial derivative of the measurand with respect to the i-th and j-th input quantities, which is estimated for known values of the input quantities.
where c(x i , x j ) -second-order mixed partial derivatives of the measurand with respect to the corresponding input quantities; η(x i ) -kurtoses of input quantities. Expressions for c(x i , x j ) and their values are given in Table 4.
For the above input quantities estimates and their standard uncertainties, as well as for kurtoses corresponding to the distribution laws of input quantities taken in [ For the above estimates of input quantities, standard uncertainties, and their kurtoses we have η(δm) = 0.
The value of the coverage factor for the probability of 0.95 is calculated by the formula [7]: . , Since η(δm) = 0 we take k 0.95 = 1.96. In this case U(δm) = 1.47 mg, which is very close to the value obtained in [1] by the Monte Carlo method (1.4955 mg).

Uncertainty budget
The results are summarized in the uncertainty budget (Table 5).
In contrast to the usual uncertainty budget for the kurtosis method [7], two columns are added to Table 5 that take into account the nonlinearity of the measurement model.
The unbiased estimate of the combined standard uncertainty calculated by the formula: For the above values u(δm) and ∆(u 2 ), we obtain u 0 (δm) = 0.075 mg, which is very close to the value calculated in [1] by the Monte Carlo method (0.0754 mg).

Calculation of expanded uncertainty
Since there is no bias of the measurand, this indicates that the nonlinearity of the model does not introduce additional asymmetry into the distribution law of the measurand, therefore, to calculate the expanded uncertainty, we can use the kurtosis method proposed by the authors [7].

Conclusions
An example of measurement uncertainty evaluation using the methods developed by the authors, at weight calibration as an example, is considered. Unbiased estimates of the measurand and its uncertainty are obtained.
It is shown that when calculating the standard and expanded uncertainty of the measurand, it is necessary to take into account the kurtoses of the input quantities. The obtained estimates of the standard and expanded uncertainties of the measurand showed good agreement with the estimates obtained by the Monte Carlo method [1], which proves the advantage of the proposed approaches in comparison with the GUM method [2]. чення вимірюваної величини також дорівнюватиме нулю. Здійснюється обчислення стандартної невизначеності вимірюваної величини з урахуванням часткових похідних другого порядку та ексцесів вхідних величин. Показано, що отримане значення стандартної невизначеності суттєво відрізняється від аналогічного значення, отриманого за процедурою GUM.