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. 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.