scope:

This Supplement to the \Guide to the expression of uncertainty in measurement" (GUM) is concerned with measurement models having any number of input quantities (as in the GUM and GUM Supplement 1) and any number of output quantities. The model and quantities involved might be real or complex. Two approaches are considered for treating such models. The first approach is a generalization of the GUM uncertainty framework. The second is a Monte Carlo method as an implementation of the propagation of distributions. In cases where the applicability of the GUM uncertainty framework is questionable, appropriate use of the Monte Carlo method would be expected to provide valid results. The approach based on the GUM uncertainty framework is applicable when the input quantities are summa- rized (as in the GUM) in terms of estimates and standard uncertainties associated with these estimates and, when appropriate, covariances associated with pairs of these estimates. Formulae and procedures are provided for obtaining estimates of the output quantities and for evaluating the associated standard uncertainties and covariances. Variants of the formulae and procedures relate to models for which the output quantities (a) can be expressed directly in terms of the input quantities as measurement functions, and (b) are obtained through solving a measurement model (which links indirectly the input and output quantities). The counterparts of the formulae in the GUM for the standard uncertainty associated with an estimate of the output quantity would be algebraically cumbersome. Such formulae are provided in a more compact form in terms of matrices and vectors, the elements of which contain variances (squared standard uncertainties), covariances and sensitivity coeficients. An advantage of this form of presentation is that these formulae can readily be implemented in the many computer languages and systems that support matrix algebra. The Monte Carlo method is based on (i) the assignment of probability distributions to the input quantities in the model [JCGM 101:2008 6], (ii) the determination of a discrete representation of the (joint) probability distribution for the output quantities, and (iii) the determination from this discrete representation of estimates of the output quantities and the evaluation of the associated standard uncertainties and covariances. This approach constitutes a generalization of the Monte Carlo method in Supplement 1 to the GUM, which applies to a single scalar output quantity. For a prescribed coverage probability, this Supplement can be used to provide a coverage region for the output quantities of a multivariate model, the counterpart of a coverage interval for a single scalar output quantity. The provision of coverage regions is limited to those taking the form of a hyper-ellipse or a hyper-rectangle. These coverage regions are produced from the results of the two approaches described here. This Supplement contains detailed examples to illustrate the guidance provided. This document is a Supplement to the GUM and is to be used in conjunction with it and GUM Supplement 1. The audience of this Supplement is that of the GUM and GUM Supplement 1.