*4.3. Unique Identifiers and Digital Records*

Details about the digital storage format used by GTC are outlined in [13]. The role of unique identifiers associated with influence quantities is of interest.

Uncertain-number algorithms must keep track of the identity of all the influence factors, which is analogous to the need for adequate notation in mathematical expressions. In the GUM, a general measurement function is represented as:

$$\mathcal{Y} = f(X\_1, X\_2, \dots) \; , \tag{6}$$

where *Y* is the quantity intended to be measured and *X*1, *X*2, ... are the quantities that influence the measurement. In the GUM notation for a component of uncertainty, *ui*(*y*) is understood to be the component of uncertainty in *y*, as an estimate of *Y*, due to the uncertainty in an estimate, *xi*, of the influence quantity *Xi*. Therefore, the subscript *i* may take the value of any of the *X*'s subscripts in Equation (6). Uncertain-number software and digital records must somehow keep track of all the *i*'s as well. This is more complicated

than it first appears, because measurements are carried out in stages that occur in different locations and at different times.

GTC uses a standard algorithm to produce universally unique 128 bit integers, which it uses to form unique digital identifiers [13]. The format is simple, and the identifier reveals nothing about influence. Would a more sophisticated type of identifier be appropriate [20]? For the purposes of data processing alone, there is no need to complicate matters: the only requirement is uniqueness. However, GTC does allow text labels to be associated with identifiers (used as influence quantity labels in uncertainty budgets), and a planned enhancement to GTC will allow unique identifiers to index a manifest of information about each influence quantity. A manifest could accompany the digital record of uncertainnumber data, which would address needs for metadata about influence quantities without the burden of minting and configuring digital objects that give access to information on the Internet.

## *4.4. Comparison Analysis by Generalized Least Squares*

The analysis equations used in this work take a fairly straightforward mathematical form. However, we have alluded to the complexity, due to the large number of terms, in handling the associated uncertainty calculations, and in [6], we suggested that a more practical analysis tool is generalized least squares (GLS). GLS is a more opaque "black box" method, but software packages are available to perform the linear algebra once the required matrices have been prepared (see [6], §4).

It is interesting to note that, in order to link an RMO comparison, GLS has been used to simultaneously process CIPM comparison data and RMO comparison data [19]. This is another example of comparison analysis compensating for the lack of transferability in standard reporting formats.

A GLS algorithm could also be applied to uncertain-number data, in which case results such as those described here would be obtained. In the formulation of the GLS calculation (Equation (69) of [6], §4, repeated here; note that bold Roman type is used to represent matrices):

$$
\begin{bmatrix} \mathbf{m} \\ \mathbf{d} \end{bmatrix} = (\mathbf{x}^\mathsf{T}\mathbf{u}^{-1}\mathbf{x})^{-1}\mathbf{x}^\mathsf{T}\mathbf{u}^{-1} \begin{bmatrix} \mathbf{y} \\ \mathbf{d}\_l \end{bmatrix} \,\prime\tag{7}
$$

the elements of the design matrix, **x**, are pure numbers, as are the elements of the covariance matrix, **u**, so conventional numerical routines can be used to evaluate the matrix:

$$\mathbf{g} = (\mathbf{x}^{\mathsf{T}} \mathbf{u}^{-1} \mathbf{x})^{-1} \mathbf{x}^{\mathsf{T}} \mathbf{u}^{-1} \,. \tag{8}$$

Then, with uncertain-number elements in the vector of participant results, **y**, and in the vector of linking participant DoEs, **d***l*, the final calculation of degrees of equivalence:

$$
\begin{bmatrix} \mathbf{m} \\ \mathbf{d} \end{bmatrix} = \mathbf{g} \begin{bmatrix} \mathbf{y} \\ \mathbf{d}\_I \end{bmatrix} \tag{9}
$$

obtains the vector of linking participant DoEs, **d**, as a linear combination of uncertain numbers. The results would, as in this work, reflect the influence of all terms contributing to participants' measurements. This would be much more informative than the information available from the covariance matrix usually obtained as an additional calculation in GLS analysis.

#### **5. Conclusions**

This case study of comparison analysis and linking has identified benefits in a particular approach to digitalization using a digital format called an uncertain number. Because comparison participants must provide more information than is available in standard calibration certificates, the context of the study highlights deficiencies in current reporting formats. These deficiencies can be summarized as a lack of support for transferability and internal consistency in the expression of uncertainty. However, if the uncertain-number

format were widely adopted, as was assumed in this case study, transferability and internal consistency would be achieved.

The study shows that more rigorous uncertainty calculations are enabled by uncertain numbers. Algorithms for data processing can be expressed in a more intuitive and streamlined manner, and it is no longer necessary to formulate separate calculations for measurement uncertainty. Because the approach keeps track of all influences, it can deliver more accurate uncertainty statements. Uncertain numbers would be advantageous to a wider range of measurement problems than just international comparisons. Adopting the format for DCCs could therefore enhance the quality of new digital infrastructures.

**Author Contributions:** Conceptualization, B.D.H. and A.K.; methodology, B.D.H.; software, B.D.H.; formal analysis, A.K.; data curation, A.K.; writing–original draft preparation, B.D.H.; writing–review and editing, A.K.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the New Zealand government.

**Data Availability Statement:** A dataset for the case study in this article is available [14]. It consists of digital records (JSON files) of the participant results, as well as JSON files containing the uncertainnumber DoE results obtained from the data processing. Python modules that display the contents of these files are provided, as are modules to carry out the data processing. The Python software package GUM Tree Calculator (GTC) is required (Version 1.3.6 or above) [11].

**Acknowledgments:** The authors are grateful to Peter Saunders for carefully reviewing this work.B.D.H. is a member of the team supporting the CIPM-TG-DSI, and A.K. is a member of the CCPR. The opinions expressed by the authors are not necessarily shared by other members of those groups.

**Conflicts of Interest:** The authors declare no conflict of interest.
