Abstract
In scientific metrology practise the application of Monte Carlo simulations with the aid of the GUM Supplement 2 (GS2) technique for performing multivariate uncertainty analyses is now more prevalent, however a key remaining challenge for metrologists in many laboratories is the implicit assumption of Gaussian characteristics for summarizing and analysing measurement model results. Whilst non-Gaussian probability density functions (PDFs) may result from Monte Carlo simulations when the GS2 is applied for more complex non-linear measurement models, in practice results are typically only reported in terms of multivariate expected and covariance values. Due to this limitation the measurement model PDF summary is implicitly restricted to a multivariate Gaussian PDF in the absence of additional higher order statistics (HOS) information. In this paper an earlier classical theoretical result by Rosenblatt that allows for an arbitrary multivariate joint distribution function to be transformed into an equivalent system of Gaussian distributions with mapped variables is revisited. Numerical simulations are performed in order to analyse and compare the accuracy of the equivalent Gaussian system of mapped random variables for approximating a measurement model’s PDF with that of an exact non-Gaussian PDF that is obtained with a GS2 Monte Carlo statistical simulation. Results obtained from the investigation indicate that a Rosenblatt transformation offers a convenient mechanism to utilize just the joint PDF obtained from the GS2 data in order to both sample points from a non-Gaussian distribution, and also in addition which allows for a simple two-dimensional approach to estimate coupled uncertainties of random variables residing in higher dimensions using conditional densities without the need for determining parametric based copulas.
Highlights
1.1 Research motivationThe original Guide to the Uncertainty of Measurement (GUM) [1] utilizing an algebraic sensitivity coefficient and mixed frequency and Bayesian statistics approach has found widespread utilization in metrology practise for uncertainty analysis, but is only strictly valid for linearised measurement models
As a result both the GUM Supplement 1 (GS1) and GUM Supplement 2 (GS2) techniques are considered suitable for application to generalized and nonlinear measurement models which are problematic for the original GUM in its earlier form, where a key limitation in the application of both the GS1 and the GS2 is in the reporting of the summaries of the Monte Carlo simulation (MCS) simulation results for a measurement model’s uncertainty if this deviates from a univariate or multivariate Gaussian distribution
This paper focuses on using numerical simulations in order to investigate and compare the accuracy of a mathematically equivalent sequential system of mapped Gaussian probability density functions (PDFs) with that of an exact non-Gaussian trivariate joint PDF, that is obtained with representative synthesized GS2 MCS statistical data of a pressure balance
Summary
The original Guide to the Uncertainty of Measurement (GUM) [1] utilizing an algebraic sensitivity coefficient and mixed frequency and Bayesian statistics approach has found widespread utilization in metrology practise for uncertainty analysis, but is only strictly valid for linearised measurement models. When the GS1 and GS2 are utilized a fully Bayesian statistics framework is utilized for modelling consistency, whilst newer mathematical refinements to modernise and update the original GUM are being further developed as discussed by Bich et al [4] As a result both the GS1 and GS2 techniques are considered suitable for application to generalized and nonlinear measurement models which are problematic for the original GUM in its earlier form, where a key limitation in the application of both the GS1 and the GS2 is in the reporting of the summaries of the MCS simulation results for a measurement model’s uncertainty if this deviates from a univariate or multivariate Gaussian distribution. As a result the formulation of a higher dimensional joint PDF into an equivalent readily reported system of mapped variables which follow a Gaussian distribution with expected values and variances using a Rosenblatt based transformation, presents a potentially simpler and more convenient approach for many practising metrologists for analysing GS2 data which may potentially exhibit non-Gaussian characteristics if a functional form of the joint PDF is available
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