*2.2. Multivariate Real Quantity*

	- Identification of the measurand **Y** = (*Y*1,...,*Ym*) and the input quantities **X** = (*X*1,..., *XN*) on which the measurand depends.
	- Assignment of the mathematical relationship between the measurand and the input quantities, e.g.,

$$\mathbf{Y} = \mathbf{f}(X\_1, \dots, X\_N) \equiv (f\_1(\mathbf{X}), \dots, f\_m(\mathbf{X})) \; ^\top \dots$$

	- Assign a number *M* of trials.
	- For *k* = 1, ... , *M*, sample values *x*1,*k*, ... , *xN*,*<sup>k</sup>* from the probability distributions for the input quantities and evaluate

$$\mathbf{y}\_k \equiv (y\_{1,k}, \dots, y\_{m,k})^\top = \mathbf{f}(\boldsymbol{\pi}\_{1,k}, \dots, \boldsymbol{\pi}\_{N,k}).$$

• From the values **y***k*, *k* = 1, ... , *M*, calculate an estimate **y** = (*y*1,..., *ym*) of the measurand and its associated covariance matrix

$$V\_{\mathbf{Y}} = \begin{bmatrix} u^2(y\_1) & u(y\_1, y\_2) & \dots & u(y\_1, y\_{m-1}) & u(y\_1, y\_m) \\ u(y\_2, y\_1) & u^2(y\_2) & \dots & u(y\_2, y\_{m-1}) & u(y\_2, y\_m) \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ u(y\_{m-1}, y\_1) & u(y\_{m-1}, y\_2) & \dots & u^2(y\_{m-1}) & u(y\_{m-1}, y\_m) \\ u(y\_m, y\_1) & u(y\_m, y\_2) & \dots & u(y\_m, y\_{m-1}) & u^2(y\_m) \end{bmatrix},$$

where *u*(*yi*) is the standard uncertainty associated with *yi* and *u*(*yi*, *yj*) ≡ *u*(*yj*, *yi*) is the covariance associated with *yi* and *yj*.

• Use the approximation to the distribution function for the measurand to determine a coverage region corresponding to a specified coverage probability.
