3.2.2. Multivariate Real Quantity

Frequently in metrology, it is necessary to consider a multivariate real quantity, i.e., a vector of real quantities. The D-SI allows multivariate quantities to be treated by employing a 'list' structure. In its most general form, the list structure allows a multivariate real quantity to be represented as shown in Table 4, where each element is of a type specified in Tables 1–3.

**Table 4.** Component structure for a multivariate quantity comprising a series of real quantities.


GUMS2 [11] describes how both hyper-ellipsoidal and hyper-rectangular coverage regions can be defined for multivariate quantities. Tables 5 and 6 list the components of the D-SI that can be used to represent a multivariate real quantity with uncertainty information provided in the form of a hyper-ellipsoidal coverage region and hyper-rectangular coverage region, respectively. For each case, uncertainty and covariance information is provided in the form of an additional component which itself comprises a number of components.

**Table 5.** Component structure for a multivariate real quantity with a hyper-ellipsoidal coverage region.


**Table 6.** Component structure for a multivariate real quantity with hyper-rectangular coverage region.


Consider the covariance matrix

$$V\_{\mathbf{y}} = \begin{bmatrix} u^2(y\_1) & u(y\_1, y\_2) \\ u(y\_2, y\_1) & u^2(y\_2) \end{bmatrix}$$

of size 2 × 2. Table 7 lists the components of the D-SI that can be used to represent *V***y**. Information is provided one column at a time, starting at column one, and within each column information is presented one row at a time, starting at row one. Therefore, in Table 7, information is presented in the order *<sup>u</sup>*(*y*1, *<sup>y</sup>*1) ≡ *<sup>u</sup>*2(*y*1), *<sup>u</sup>*(*y*2, *<sup>y</sup>*1), *<sup>u</sup>*(*y*1, *<sup>y</sup>*2) and *<sup>u</sup>*(*y*2, *<sup>y</sup>*2) ≡ *<sup>u</sup>*2(*y*2). The approach generalises straightforwardly for covariance matrices of larger size.

**Table 7.** Component structure for a covariance matrix of size 2 × 2.


A multivariate quantity may consist of multiple measurements of quantities of the same type, e.g., measurements of temperature at a particular location taken at regular time intervals, or quantities of different types, e.g., measurements of different environmental factors within a laboratory. In the former case, if all quantities have the same unit of measurement, the list structure as presented leads to unnecessary repetition of information. The D-SI has been adapted to allow for more efficient representation in such cases. Table 8 shows how the same unit of measurement may be assigned to all individual quantities in a vector of real quantities. (For ease of reading, the optional label and dateTime components have been omitted.) When the **listUnit** component is used, there is no longer the mandatory requirement to provide a unit of measurement for each quantity (c.f. Table 1).

**Table 8.** Component structure for a multivariate quantity comprising a series of real quantities with the same unit of measurement.


The D-SI also allows for the same expanded uncertainty or coverage interval to be associated with all the real quantities of a multivariate quantity but this functionality is not discussed further in this paper. A complex quantity is treated as a special case of a multivariate quantity and is considered in [16].
