Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis
The application of the Guide to the Expression of Uncertainty in Measurement (GUM) for multivariate measurand equations requires an expected vector value and a corresponding covariance matrix in order to accurately calculate measurement uncertainties for models that involve correlation effects. Typi...
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| Format: | Article |
| Language: | English |
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EDP Sciences
2022-01-01
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| Series: | International Journal of Metrology and Quality Engineering |
| Subjects: | |
| Online Access: | https://www.metrology-journal.org/articles/ijmqe/full_html/2022/01/ijmqe220013/ijmqe220013.html |
| _version_ | 1811189045088223232 |
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| author | Ramnath Vishal |
| author_facet | Ramnath Vishal |
| author_sort | Ramnath Vishal |
| collection | DOAJ |
| description | The application of the Guide to the Expression of Uncertainty in Measurement (GUM) for multivariate measurand equations requires an expected vector value and a corresponding covariance matrix in order to accurately calculate measurement uncertainties for models that involve correlation effects. Typically in scientific metrology applications the covariance matrix is estimated from Monte Carlo numerical simulations with the assumption of a Gaussian joint probability density function, however this procedure is often times considered too complex or cumbersome for many practicing metrologists in industrial metrology calibration laboratories, and as a result a problem which occurs is that correlation effects are frequently omitted so that uncertainties are approximated through a simple root-sum-square of uncertainties which leads to inaccuracies of measurement uncertainties. In this paper, a general purpose deterministic approach is developed using a computer algebra system (CAS) approach that avoids the need for Monte Carlo simulations in order to analytically construct the covariance matrix for arbitrary nonlinear implicit multivariate measurement models. An illustrative example for a multivariate Sakuma-Hattori pyrometer equation with the proposed method is demonstrated with explanations of underlying Python code. |
| first_indexed | 2024-04-11T14:29:11Z |
| format | Article |
| id | doaj.art-dcfb9dd65a0648db9f8d39bb7f585000 |
| institution | Directory Open Access Journal |
| issn | 2107-6847 |
| language | English |
| last_indexed | 2024-04-11T14:29:11Z |
| publishDate | 2022-01-01 |
| publisher | EDP Sciences |
| record_format | Article |
| series | International Journal of Metrology and Quality Engineering |
| spelling | doaj.art-dcfb9dd65a0648db9f8d39bb7f5850002022-12-22T04:18:43ZengEDP SciencesInternational Journal of Metrology and Quality Engineering2107-68472022-01-0113910.1051/ijmqe/2022008ijmqe220013Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysisRamnath Vishal0https://orcid.org/0000-0002-7411-8058Department of Mechanical Engineering, University of South Africa, Private Bag X6The application of the Guide to the Expression of Uncertainty in Measurement (GUM) for multivariate measurand equations requires an expected vector value and a corresponding covariance matrix in order to accurately calculate measurement uncertainties for models that involve correlation effects. Typically in scientific metrology applications the covariance matrix is estimated from Monte Carlo numerical simulations with the assumption of a Gaussian joint probability density function, however this procedure is often times considered too complex or cumbersome for many practicing metrologists in industrial metrology calibration laboratories, and as a result a problem which occurs is that correlation effects are frequently omitted so that uncertainties are approximated through a simple root-sum-square of uncertainties which leads to inaccuracies of measurement uncertainties. In this paper, a general purpose deterministic approach is developed using a computer algebra system (CAS) approach that avoids the need for Monte Carlo simulations in order to analytically construct the covariance matrix for arbitrary nonlinear implicit multivariate measurement models. An illustrative example for a multivariate Sakuma-Hattori pyrometer equation with the proposed method is demonstrated with explanations of underlying Python code.https://www.metrology-journal.org/articles/ijmqe/full_html/2022/01/ijmqe220013/ijmqe220013.htmlcovariance matrixguide to the expression of uncertainty in measurement (gum)montecarlo simulation (mcs)multivariate measurement uncertaintypyrometry |
| spellingShingle | Ramnath Vishal Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis International Journal of Metrology and Quality Engineering covariance matrix guide to the expression of uncertainty in measurement (gum) montecarlo simulation (mcs) multivariate measurement uncertainty pyrometry |
| title | Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis |
| title_full | Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis |
| title_fullStr | Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis |
| title_full_unstemmed | Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis |
| title_short | Determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis |
| title_sort | determining the covariance matrix for a nonlinear implicit multivariate measurement equation uncertainty analysis |
| topic | covariance matrix guide to the expression of uncertainty in measurement (gum) montecarlo simulation (mcs) multivariate measurement uncertainty pyrometry |
| url | https://www.metrology-journal.org/articles/ijmqe/full_html/2022/01/ijmqe220013/ijmqe220013.html |
| work_keys_str_mv | AT ramnathvishal determiningthecovariancematrixforanonlinearimplicitmultivariatemeasurementequationuncertaintyanalysis |