**4. Total Ignorance in Measurements**

Let us here consider a simple example to show how, in measurement procedures, the situation called total ignorance by Shafer is very often present.

A calibrator provides a reference voltage of 24 V, and some multimeters of the same typology (4 <sup>1</sup> <sup>2</sup> Leader 856) are employed to measure this voltage. The instrument data sheet provides the measuring accuracy as ± % of reading ± number of digits, and the value of each digit is given by the resolution in the considered range. According to the data sheet, Table 7 provides the resolution and the measuring accuracy in the different ranges. For the measurand *Vx* = 24 V, which is the reference voltage in the proposed example, the range is 30 V and therefore, according to the specifications, the measurement accuracy is ±(0.05% *Vx* + 2 mV) = ±0.014 V.


**Table 7.** The multimeter data sheet.

Two different measurement procedures are considered:

1. All multimeters are employed to measure the reference voltage.

2. Only one multimeter is employed to measure the reference voltage.

Figure 2 shows, with the orange line the reference voltage and with the pink crosses the value measured by 10 different multimeters. Since different instruments are employed, it is likely to happen that the measured values fall around the reference value. In this situation, it could be possible to apply a probabilistic approach by considering the following: the mean of the measured values; an uncertainty interval around the evaluated mean, and a pdf over this interval (but only if a high number of different instruments are employed).

**Figure 2.** Up: values measured by different multimeters. Down: values measured by a single multimeter.

This situation represents very well the calibration procedure that is performed by the instruments' manufacturer to provide the accuracy interval, which reflects the behavior of all instruments of the same typology. However, this situation is very seldom met in practice, because generally only one instrument is available and employed. Under this more common situation, when only one multimeter is employed, the value measured by the multimeter will be shifted with respect to the reference value. Moreover, if different measurements were taken, this would not help to better estimate the reference value since all measured values would be shifted more or less the same amount with respect to the reference value, as shown by the green circles in Figure 2. In fact, all measured values are taken, in this case, by the same instrument and, therefore, are affected by the same systematic error, even if a small variation can be observed, due to the presence of also random phenomena.

In this last case, even if the mean of the measured values is taken, no better estimate of the measurand can be obtained. Additionally, even if an interval is built, according to the dispersion of the measured values, this interval would not contain the value of the measurand. Therefore, to provide a good uncertainty interval, it is necessary to refer to the accuracy interval provided by the data sheet. The data sheet does not provide any pdf associated with this interval, and therefore, no pdf can be assigned to the obtained interval.

When we have a pdf over a given support, it is possible to assign a confidence interval (or degree of belief) to any subintervals of the support. However, when no pdf is assigned and no knowledge is available to assign a specific pdf, it is not possible to associate any confidence interval (or degree of belief) to any subintervals of the support. We are, therefore, perfectly in the case of Shafer's total ignorance, where a degree of belief can be assigned to the support (or universal set), but no degree of belief can be assigned to the subintervals (to the subsets of the universal set).

It clearly follows that total ignorance is present in the measurement field. Since belief functions better represent total ignorance, it is worth exploring these functions and the theory of evidence to find an alternative, more general way to handle measurement uncertainty and measurement results. It is not the aim of this paper to provide all definitions and mathematical details, for which the readers are referred to the published literature [10–14]. The next section will, therefore, give only some introduction to come to the possibility distributions (PD) and the random-fuzzy variables (RFV).

#### **5. The Random-Fuzzy Variables**

In the previous sections, belief functions are introduced and it is shown how they can suitably represent the available knowledge, including total ignorance. It is interesting to observe that belief functions are a generalization of the probability functions and the necessity functions. In this respect, it is first necessary to know what a focal element is.

Let us first define the basic probability assignment function:

$$\begin{aligned} m: P(X) &\to [0, 1] \\ m(\mathcal{Q}) &= 0 \\ \sum\_{A \in P(X)} m(A) &= 1 \end{aligned} \tag{3}$$

where *X* is the universal set, *P*(*X*) is the power set of *X* and ∅ is the empty set. According to (3), *m*(*A*) represents the degree of belief that an element *x* belongs to set *A* (only to set *A* and not to its subsets).

Set *A* for which *m*(*A*) > 0 is called the focal elements of *X*. When the focal elements are singletons, then it can be proved [9–12] that belief functions are probability functions, and the theory of evidence enters in the particular case of the probability theory. This shows that the belief functions are, as wanted by Shafer, a generalization of the probability functions. However, it is also interesting to consider another particular case of belief functions, which are called necessity functions and are obtained when the focal elements are all nested, as shown in Figure 3.

**Figure 3.** Example of nested focal elements, for sets and for intervals.

The upper plot in Figure 3 clearly shows that, when sets are considered, all sets can be ordered in such a way that *A*<sup>1</sup> ⊂ *A*<sup>2</sup> ⊂ ... ⊂ *An* ≡ *X*. When, instead of sets, intervals are considered, the lower plot can be drawn, which still satisfies *A*<sup>1</sup> ⊂ *A*<sup>2</sup> ⊂ ... ⊂ *An* ≡ *X*. This case is very interesting from the metrological point of view because there could be an analogy between these nested intervals and the confidence intervals of a given pdf at different, increasing levels of confidence.

The necessity function is defined as follows:

$$\operatorname{Vec}\left(A\_{\bar{j}}\right) = \sum\_{k=1}^{\bar{j}} m(A\_k)$$

and represents the degree of belief that an element *x* belongs to set *A* and to all its subsets. When the belief functions are necessity functions, then the theory of evidence enters the particular case of the possibility theory.

In the same way that probability density functions are defined in probability, possibility distribution functions (PD) are defined in possibility as follows:

$$r: X \to [0, 1]$$

where:

$$\max(r(x)) = 1$$

when *x* ∈ *X*.

It can be proved [10,11] that the nested intervals of Figure 3, together with their corresponding necessity functions *Nec*- *Aj* , represent confidence intervals at specific levels of confidence, coverage probability, or degree of belief *Nec*- *Aj* . Therefore, remembering the GUM words that "*the ideal method for evaluating and expressing measurement uncertainty should be capable of readily providing such an interval, in particular, one with a coverage probability or level of confidence that corresponds in a realistic way to that required*" [1], it can be stated that the possibility theory, which provides all confidence intervals at all confidence levels, is perfectly GUM compliant.

If the intervals of Figure 3 are not overlapped with each other but are positioned at different vertical levels <sup>α</sup>, such as *<sup>α</sup><sup>j</sup>* = <sup>1</sup> − *Nec*- *Aj* , then a fuzzy variable is obtained, as in the example in Figure 4.

**Figure 4.** Example of possibility distributions and confidence intervals.

The fuzzy variable is commonly defined by its membership function which is, from the strict mathematical point of view, a PD.

Since a fuzzy variable (a PD) represents confidence intervals at all levels of confidence, a fuzzy variable can be used to represent in a very immediate way the result of a measurement [10–12]. Moreover, since different kinds of uncertainty contributions may affect the measurement procedure, the best way to represent the result of a measurement is the use of a fuzzy variable of type 2 and, in particular, a random-fuzzy variable (RFV). An RFV provides two PDs and can, hence, represent separately the effects on the measurement result of the different contributions to uncertainty. An example of RFV is given in Figure 5, with the red and violet lines. In an RFV, the uncompensated systematic contributions are represented by the internal PD *rint*(*x*) (violet line), while the random contributions are represented by the random PD *rran*(*x*) (green line). The external PD *rext*(*x*) (red line) is obtained by the combination of the two PDs *rint*(*x*) and *rran*(*x*) [10–13].

**Figure 5.** Example of RFV.

Extending the considerations made for the fuzzy variables, it can be stated that the cuts *X<sup>α</sup>* at levels α of the RFV are the confidence intervals associated to the measurement result at the confidence levels *Nec*(*Xα*) = 1 − *α* (as shown in Figure 6). In particular, the internal interval of each confidence interval is due to the effect on the measured value of the systematic contributions to uncertainty, while the external intervals are due to the effect of the random contributions.

**Figure 6.** Example of RFV and some of the confidence intervals, which show the systematic effects and the random effects on the final measured result.

If RFVs can suitably represent measurement results, then it is important to understand how an RFV can be built and how two RFVs can be combined with each other, as will be briefly explained below; we refer the readers to the literature for more details [10–14].

## *5.1. RFV Construction*

To build an RFV, it is necessary to define the shape of the PDs *rint*(*x*) and *rran*(*x*), whose construction is different [10–14] since they represent different kinds of contributions.

As far as *rint*(*x*) is concerned, this PD represents the uncompensated systematic contributions to uncertainty. As shown in the example of the multimeter in the previous Section 4, generally, the only available knowledge is, in this case, the accuracy interval given by the manufacturer of the employed instrument in the data sheet. Therefore, the available knowledge can be represented by Shafer's total ignorance. As is also shown in Section 3, total ignorance is mathematically represented by the belief function [9–11]:

$$Bel(X) = 1$$

$$Bel(A) = 0 \quad \forall A \subset X$$

and by the rectangular PD, such as the one in violet line in Figure 5. It follows that *rint*(*x*) is rectangular in most situations, even if situations may exist that could lead to different shapes [10–14].

On the other hand, *rran*(*x*) must represent the random contributions to uncertainty and therefore, in most cases, a pdf is known or can be supposed. In this case, the corresponding PD can be easily obtained by applying the suitable probability–possibility transformation (different probability–possibility transformations are available in the literature to transform pdfs into PDs. The suitable transformation when PDs are used to represent measurement results is the maximally specific probability–possibility transformation, which preserves all confidence intervals and corresponding confidence levels) [10,15].

As an example, when the pdf is uniform, then the corresponding PD is triangular; when the pdf is triangular, then the corresponding PD is the orange one in Figure 7; when the pdf is Gaussian, then the corresponding PD is the blue one in Figure 7.

**Figure 7.** Example of PDs coming from given pdfs. Blue line: PD from a Gaussian pdf. Orange line: PD from a triangular pdf.

#### *5.2. RFV Combination*

When the measurement results are represented by RFVs and they must be combined, it is possible to take into account all the available metrological information about the nature of the contributions to be combined and the way these contributions combine in the specific measurement procedure. According to that, since PDs can be combined using many different mathematical operators, the most proper one can be chosen.

Without entering the details, for which the readers are referred to [10,16,17], it can be stated that the random contributions to uncertainty always compensate with each other during the combination, and therefore, an operator that simulates this typical probabilistic compensation should be chosen. On the other hand, the systematic contributions to uncertainty could compensate or not with each other during the combination, according to the specific contributions and the specific measurement procedure. Therefore, there should be the possibility to choose between a mathematical operator that simulates compensation and another one which does not compensate.

Let us first consider the evaluation of the joint PD, starting from two PDs. As an example, Figure 8 shows the results obtained by combining the same two PDs with the use of two different t-norms (for the definition of the mathematical t-norms, the readers are addressed to [15]): the min t-norm (on the left) and the Frank t-norm (on the right). In the upper plots, the two-dimensional joint PDs are shown, while in the lower plots, the corresponding α-cuts are shown. It can be easily seen how compensation applies when the Frank t-norm is employed, while no compensation applies when the min t-norm is employed.

**Figure 8.** Combination of uncorrelated contributions. The same initial PDs are considered on both the left and right figures. On the left, the min t-norm is applied; on the right the Frank t-norm is applied. In the upper plots, the joint possibility distributions are shown. In the lower plots, the corresponding α-cuts are shown.

Figure 8 refers to the combination of uncorrelated contribution. Without entering the details, the correlation can also be considered, as shown, as an example, in Figures 9 and 10.

**Figure 9.** Combination of correlated contributions when the min t-norm is applied. The same initial PDs as in Figure 8 are considered. **Right**: joint PD. **Left**: shape of its α-cuts.

**Figure 10.** Combination of correlated contributions when the Frank t-norm is applied. The same initial PDs as in Figure 8 are considered. **Left**: joint PD. **Right**: its α-cuts.

From Figures 9 and 10, it can be easily seen how correlation modifies the joint PDs and the corresponding α-cuts.

Once the joint PDs *rran*(*x*, *y*) and *rint*(*x*, *y*) are obtained, it is possible to evaluate the joint PD *rext*(*x*, *y*) and the final RFV (this is obtained by applying the famous Zadeh extension principle. The readers are referred to [9,10] for the details) [9,10,16,17].
