Numerical analysis of the accuracy of bivariate quantile distributions utilizing copulas compared to the GUM supplement 2 for oil pressure balance uncertainties

Abstract

Abstract. In the field of pressure metrology the effective area is A e = A 0 (1 + lP) where A 0 is the zero-pressure area and l is the distortion coefficient and the conventional practise is to construct univariate probability density functions (PDFs) for A 0 and l. As a result analytical generalized non-Gaussian bivariate joint PDFs has not featured prominently in pressure metrology. Recently extended lambda distribution based quantile functions have been successfully utilized for summarizing univariate arbitrary PDF distributions of gas pressure balances. Motivated by this development we investigate the feasibility and utility of extending and applying quantile functions to systems which naturally exhibit bivariate PDFs. Our approach is to utilize the GUM Supplement 1 methodology to solve and generate Monte Carlo based multivariate uncertainty data for an oil based pressure balance laboratory standard that is used to generate known high pressures, and which are in turn cross-floated against another pressure balance transfer standard in order to deduce the transfer standard's respective area. We then numerically analyse the uncertainty data by formulating and constructing an approximate bivariate quantile distribution that directly couples A 0 and l in order to compare and contrast its accuracy to an exact GUM Supplement 2 based uncertainty quantification analysis.

Comparison of straight line curve fit approaches for determining parameter variances and covariances

“…The key differences in the newer approach of Wuethrich and Souiyam [14] and an earlier approach of Ramnath [16] is that in the latter work a multivariate GUM Supplement 2 approach [17] was utilized that modelled the various inputs x by treating the two parameters A 0 and l as a single vector y = [A 0 , l] T with an equation h(y, x) = 0 where a multivariate PDF g y (h y ) of the output was to be determined from a multivariate input g x (j x ) by numerically solving the equation h(h y , j x ) = 0 for a large number M of Monte Carlo simulation events.…”

“…Differences in the two approaches is that firstly the method of Wuethrich and Souiyam [14] requires estimates for the uncertainties of the areas u 2 (S 1 ), … , u 2 (S n ) in the case of a WLS approach and additionally that of the covariances cov(S i , S j ), 1i, jn for each Monte Carlo event j, and secondly makes an assumption of a Gaussian correlation between A 0 and l. On the other hand the method of Ramnath [16] firstly avoids the explicit need for calculating u 2 (S i ) and cov(S i , S j ) at each Monte Carlo event j as the data is considered as statistical samples, and secondly does not assume a Gaussian correlation between A 0 and l since bivariate copulas can model non-Gaussian correlation effects.…”

Analysis of approximations of GUM supplement 2 based non-Gaussian PDFs of measurement models with Rosenblatt Gaussian transformation mappings

“…The case for non-Gaussian multivariate models is more complex but is in principle directly amenable with the use of multivariate copulas as originally developed by Possolo [7] for metrology uncertainty analysis problems. This approach has subsequently been implemented using parametrized vine copulas for an oil pressure balance problem as discussed by Ramnath [8] in order to demonstrate the utility of this approach for summarizing a non-Gaussian joint PDF in mechanical metrology applications.…”

“…Earlier work by Ramnath [8] studied a bivariate pressure balance model of the form A = A 0 (1 + λP ) where A is the effective area of the pressure balance, λ is a distortion coefficient and P is an independently varied applied pressure in the range 50 P/[MPa] 500. In this model the parameters A 0 and λ are coupled to each other and a bivariate PDF g A0,λ (η A0 , η λ ) is used to model the characteristics of the pressure balance.…”

“…as discussed by Yadav et al [29] where the parameters A 0 , λ 1 and λ 2 are coupled in a trivariate joint PDF g A0,λ1,λ2 (η A0 , η λ1 , η λ2 ), however relatively few experimental data sets are available of pressure balances which exhibit strongly non-Gaussian characteristics. In the earlier work by Ramnath [8] the experimental data-set exhibited weakly non-Gaussian characteristics i.e. the non-Gaussian joint PDF could be roughly approximated with a Gaussian joint PDF for a reduced level of accuracy, and as a result there exists a general research gap in the area of metrology uncertainty analysis for summarizing and analysing non-Gaussian joint PDFs that cannot be adequately approximated with Gaussian joint PDFs.…”