*2.1. Fuzzy Numbers*

*A* fuzzy number (FN) is a fuzzy set *A* defined over the set of real numbers, and it is a fundamental concept of FST for representing uncertain quantities. Let us symbolize as *μA*-(*x*) the membership function of a fuzzy set *A* - . Hence, *A* - is also a FN if it is normal, i.e., max *x*∈*X μA*-(*x*) = 1, and convex, that is, its *α*-cuts are closed and bounded intervals. Hence, it can be represented as confidence intervals *A* = *A*(*α*), *<sup>A</sup>*(*α*), where *<sup>A</sup>*(*α*) (*A*(*α*)) increases (decreases) monotonously respect the membership degree *α* ∈ [0, 1]. A FN *A* - is a fuzzy quantity that matches "more or less" the real number *A*, such that *μA*-*A*) = 1. This paper uses triangular fuzzy numbers (TFNs), that are symbolized as *A* - = (*<sup>A</sup>*, *lA*,*rA*). Hence, *A* is the core, and it is the most reliable value: *μA*-(*A*) = 1. Likewise, *lA*,*rA* ≥ 0 are the left and right radius and measure the variability of *A* - respect *A*. Membership function and *α*-cuts of a FN are:


$$\mu\_{\bar{A}}(\mathbf{x}) = \begin{cases} \frac{\mathbf{x} - A + l\_A}{l\_A} & A - l\_A < \mathbf{x} \le A \\\\ \frac{A + r\_A - \mathbf{x}}{r\_A} & A < \mathbf{x} \le A + r\_A \\\ 0 & \text{otherwise} \end{cases} \tag{1a}$$


$$A = \begin{bmatrix} \underline{A}(\mathfrak{a}) , \overline{A}(\mathfrak{a}) \end{bmatrix} = \begin{bmatrix} A - l\_A(1 - \mathfrak{a}) , A + r\_A(1 - \mathfrak{a}) \end{bmatrix} \tag{1b}$$

The hypothesis of a triangular shape for uncertain variables is commonplace in papers on practical applications of FNs. We are aware that this hypothesis may suppose simplifying the complexity of available information. However, we feel that this drawback is balanced by several benefits:


In some cases, it will be useful to transform a fuzzy number *A* into a crisp equivalent. For example, when we are ranking alternatives from their scores in a variable that are done by FNs. Fuzzy literature provides a grea<sup>t</sup> deal of ordering methods (see [25]). In this paper, we will use the concept of the expected value of an FN in [26]. Let be an FN *A* - and a

parameter *λ* ∈ [0, 1] that quantifies the evaluator's optimism grade. The expected value of *A* - for a given *λ* is:

$$EV(\tilde{A};\lambda) = (1-\lambda)\int\_0^1 \Delta(\alpha)d\alpha + \lambda \int\_0^1 \overline{A}(\alpha)d\alpha \tag{1c}$$

Hence, for a TFN:

$$EV(\tilde{A}; \lambda) = A - \frac{l\_A}{2}(1 - \lambda) + \frac{r\_A}{2} \tag{1d}$$

#### *2.2. Modeling the Value of the Pearson Correlation Coefficient as a Linguistic Variable*

Linguistic variables are variables whose values are sentences from natural or artificial languages named linguistic labels [27]. They are built up by segmenting a universal set in a set of FNs where each one represents a linguistic label. For example, the variable coefficient of correlation has a reference set [−1, 1] and may be partitioned in several linguistic labels as, e.g., "no correlation", "low(−)", "medium(−)", "high(−)" ... }. Hence, "no correlation" may be quantified with the TFN (0, 0.005, 0.005).

Let be a linguistic variable *V* with a reference set [*Vmin*, *Vmax*]. It is built up by granulating the reference set into *J* levels (i.e., *J* fuzzy numbers), *j* = 1, 2, ... , *J*, which in this paper are assumed to be TFNs. Then, by considering *Vmin* = *V*1 < *V*2 < *V*3 < ... < *VJ*−<sup>1</sup> < *VJ* = *Vmax* we obtain:

$$\begin{aligned} \{\vec{V}\_{k\_1} = (V\_1, 0, V\_2 - V\_1); \,\,\vec{V}\_j = (V\_j, V\_j - V\_{j-1}, V\_{j+1} - V\_j), \, j = 2, 3, \dots, J - 1;\\ \vec{V}\_l = (V\_l, V\_l - V\_{l-1}, 0). \end{aligned} \tag{2}$$

Notice that it is accomplished that ∑*j μVj*(*x*) = 1 for any crisp value *<sup>x</sup>*<sup>∈</sup>[*Vmin*, *Vmax*].

The association of a given kind of social expense with the efficiency of PPPs is done by means of a correlation index. As far as decision-making is concerned, it is usual to interpret the value of the correlation coefficient qualitatively by means of linguistic labels as "high (+) correlation" or "weak (−) correlation" that may depend on the context. Table 1 shows three scales exposed in [27] that are used in psychology, political science and medicine.

**Table 1.** Interpretation of the Pearson's and Spearman correlation coefficients.


Source: [28], Akoglu, H. (2018). User's guide to correlation coefficients. *Turkish Journal of Emergency Medicine*, *18*(3), 91–93.

By applying (2) on the scale by the Department of Politics at Quinnipiac University in Figure 1, we built up the fuzzy linguistic variable "correlation coefficient" used in this paper to qualitatively interpret the correlation. It is shown in Table 2.

**Figure 1.** Linguistic variable "coefficient of correlation" built up from the scale by the Department of politics of Quinnipiac University. Source: own elaboration by using [28], Akoglu, H. (2018). User's guide to correlation coefficients. *Turkish Journal of Emergency Medicine*, *18*(3), 91–93.

**Table 2.** Fuzzy linguistic variable "coefficient of correlation" built up from de Quinnipiac University scale.


Source: own elaboration by using Table 1 in [28], Akoglu, H. (2018). User's guide to correlation coefficients. *Turkish Journal of Emergency Medicine*, *18*(3), 91–93.

#### *2.3. Aggregating Crisp Observations by Means of a Triangular Fuzzy Number*

In this paper, we capture the uncertainty in data by using FNs. We are aware that Soft Computing Science provides several tools apart from fuzzy sets to represent uncertain data: rough sets, gray sets, intuitionistic and neutrosophic sets ... Using FNs instead other alternatives presents pros and cons. In any case we feel that using TFN in our analysis is suitable for the following reasons:

•Tools as intuitionistic fuzzy sets (IFSs) or neutrosophic fuzzy sets (NFSs) provide an analytical framework to quantify uncertainty more precisely than FNs. Therefore, they are able to capture more nuances from data and its imprecision. For example, NFS state for any element not only a truth membership degree but also an indeterminacy and a falsity degree. However, their estimation implies a greater cost since the number of parameters to fit for each uncertain observation is three times that number than in the case of FNs. On the other hand, gray numbers (GNs) provide a simpler representation of uncertain quantities than FNs. To define a GN is enough to fit its kernel and a grayness measure. Hence, in several circumstances GNs may oversimplify information. For example, GNs suppose a symmetrical structure for a uncertain quantity when perhaps available information does not sugges<sup>t</sup> so. TFNs balances capturing much of the uncertainty in available information (less than, e.g.,

NFS, but more than GNs), but with a smooth shape (less than GNs, but more than NFSs).


Cheng in [29] proposes a method that allows transforming a set of crisp observations on a given variable in an FN. Let us symbolizing as {*<sup>a</sup>*1, *a*2, ... , *a*n} the set of crisp observations and *A*- = (*<sup>A</sup>*, *lA*,*rA*) the TFN than will embed these observations. To fit *A*- the following steps must be followed:

Step 1. Calculate the distance between *i*th and *j*th value as *dij* = *ai* − *aj* . Of course, *dii* = 0, *dij* = *dji*. Hence, we can build up a distance matrix *D* = *dij n*×*n*.

Step 2. Calculate the mean distance of *i*th opinion the other *n* − 1 as:

$$\overline{d}\_{i} = \frac{\sum\_{i=1}^{n} d\_{ij}}{n-1} \tag{3a}$$

Hence, *di* measures the distance of *i*th opinion to the center of gravity of the opinion pool. Of course, the weight of the value *ai* to determine *A* is decreasing respect to *di*.

Step 3. Find the matrix *P* = *pij n*×*n* that indicates the importance of *i*th opinion over the *j*th to fix *A* by doing:

$$p\_{ij} = \frac{d\_j}{\overline{d}\_i} \tag{3b}$$

Moreover, so, *pii* = 1 and *pij* = 1 *pji* . Notice that *P* is obtained from a comparison of distances and so, it ensures its consistency, i.e., that there is a coherent judgment in specifying the pairwise comparison of score importance.

Step 4. Fit coefficients *wi*, *i* = 1, 2, ... , *n*, which measure the degree of importance of *i*th observation to fit *A*-, in such a way 0 ≤ *wi* ≤ 1, *i* = 1, 2, ... , *n*. These weights are adjusted by taking into account the relative degree of importance of *i*th observation respect *j*th, *j* = 1, 2, ... , *n* (3b). Following [29], if we symbolize as *w* the vector of weights *n*x1, *Pw* = *nw*, where *n* is an eigenvalue of *P* and *w* an eigenvector. Likewise, given that it must accomplished that ∑*n i*=1 *wi* = 1, the weights are solved from (3b) by doing:

$$w\_i = \frac{1}{\sum\_{j=1}^{n} p\_{ji}}\tag{3c}$$

Hence, ref. [29] indicates that the consistency of *P* lead to:

$$p\_{ij} = \frac{w\_i}{w\_j} \tag{3d}$$

Step 5. Calculate the center of *A*- as:

$$A = \sum\_{i=1}^{n} w\_i a\_i \tag{3e}$$

Step 6. Estimate so-called mean deviation (*σ*) of the FN *A*- as a first step to adjusting their spreads. Hence, ref. [29] defines the mean deviation of a FN as *σ* = *<sup>A</sup>*+*rA A*−*l A* |*<sup>x</sup>*−*<sup>A</sup>*|*μAx*)*dx <sup>A</sup>*+*rA A*−*l A μAx*)*dx*

and then for a TFN *σ* = *<sup>l</sup>*2*A*+*r*2*A* <sup>3</sup>(*lA*+*rA*) By using the rate of the left spread respect to the right *η* = *lArA*:

$$d\_A = \frac{3\left(1+\eta\right)\eta\sigma}{1+\eta^2} \text{ and } r\_A = \frac{3\left(1+\eta\right)\sigma}{1+\eta^2} \tag{3f}$$

Notice that *σ* and *η* are, in fact, unknown parameters because they are built up from *lA* and *rA*. Hence, ref. [29] proposes the following approximation for *σ*, *σ*ˆ:

$$
\hat{\sigma} = \sum\_{i=1}^{n} w\_i |A - a\_i| \tag{3g}
$$

Step 7. Find the estimate of *η*, *η*ˆ. By defining as *al* = ∑*ni*=1 *ai*<*<sup>A</sup> wiai* ∑*n i*=1 *ai*<*<sup>A</sup>wi* and *a<sup>r</sup>* =

$$\frac{\sum\_{\substack{i=1 \\ a\_i > A}}^n w\_i a\_i}{\sum\_{\substack{i=1 \\ a\_i > A}}^n w\_i}, \text{ } \text{ $\mathfrak{H}$  is:} $$

$$\hat{\eta} = \frac{A - a^l}{a^r - A} \tag{3h}$$

Step 8. Find *lA* and *rA* by doing

$$l\_A = \frac{3\left(1+\hat{\eta}\right)\hat{\eta}\hat{\sigma}}{1+\hat{\eta}^2} \text{ and } r\_A = \frac{3\left(1+\hat{\eta}\right)\hat{\sigma}}{1+\hat{\eta}^2} \tag{3i}$$

Numerical Application 1

Table 3 shows the values provided by Eurostat within 2014–2018 of the social expenses over GPD, SER = SE/GPD for Belgium. Let us fitting that variable for the quinquennium as a TFN *SER* = (*SER*, *lSER*,*rSER*) by using (3a)–(3i).

**Table 3.** Annual social expenses (over GPD) by Belgium in the period 2014–2018.


Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017).

The matrix of distances between observations is exposed in Table 4, and the relative importance of each annual value of SER in the final TFN is provided in Table 5.

**Table 4.** Matrix of distances to build up the fuzzy number "Belgian SER within 2014–2018".


Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017).


**Table 5.** Relative importance of social expenses over GPD (SER) in each year in the triangular fuzzy numbers (TFN) "Belgian SER in 2014–2018".

Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017).

Then, the vector of weights is: *w* = (0.178, 0.217, 0.138, 0.233, 0.233) and, therefore SER = 29.286. To fit the spreads, *lSER* and *rSER* we find that *σ*ˆ = 0.478, *SER<sup>l</sup>* = 28.89 and *SERr* = 29.89. Hence, *η*ˆ = 29.286−28.89 29.89−29.286 = 0.654, *lSER* = 3 (<sup>1</sup>+0.654)·0.654·0.478 1+0.654<sup>2</sup> = 1.085 and *rSER* = 3 (<sup>1</sup>+0.654)·0.478 1+0.654<sup>2</sup> = 1.660. Hence, annual SER by Belgium for 2014–2018 is fitted as *SER* = (29.286%, 1.085%, 1.660%).

Notice that quantifying Belgian SER as the TFN *SER* = (29.286%, 1.085%, 1.660%) is very suited to the intuition that comes after a visual inspection of Table 3 "Belgium SER has been around 29% within 2014–2018". Of course, more sophisticated representations of uncertainty as IFSs or NFSs can capture a greater amount of information. However, the cost of fitting these kind of sets is much greater and not very reliable with the information available for our analysis (identical to that in Table 3 for Belgium SER). On the other hand, if information came from an extended and structured questionnaire submitted to experts, surely NFSs will provide a better representation of that information than FNs.

Belgium SER in Table 3 admits a gray number representation. Following the exposition in [30] and taking into account that SER in Table 3 is within the interval [28.8, 30] since is the discrete set {28.8, 28.8, 29.2, 29.8, 30}, the kernel of SER is *SER*<sup>ˆ</sup> = (28.8 + 28.8 + . . . +30)/5 = 29.32. Grayness degree can be estimated by taking into account that SER for any country must be within [0, 100] and so its value is (30 − 28.8)/100 = 0.012. By using notation in [30], SER is 29.32(0.012). Notice that this parameterization is simpler than *SER* = (29.286%, 1.085%, 1.660%), but on the other hand, the TFN captures the asymmetric distribution of values around the gravity center of the data that GN does not.

Due to the kind of data that we will use in our analysis, we feel that using TFN parameterization from [29] provides an adequate compromise between applying the principle of parsimony in vagueness modeling and avoiding unnecessary loss of information.

#### *2.4. Correlation Coefficients for Fuzzy Data*

Pearson's correlation coefficient (PCC) is a real-valued function in <sup>2</sup>*n* of the pairwise observations over the variables *X* and *Y*: {(*<sup>x</sup>*1, *y*1);(*<sup>x</sup>*2, *y*2);...;(*xn*, *yn*)}. Hence, PCC between *X* and *Y* is estimated as:

$$corr\_{\mathbf{X},Y} = f(\mathbf{x}\_1, \dots, \mathbf{x}\_n; y\_1, \dots, y\_n) = \frac{\sum\_{i=1}^n \left(\mathbf{x}\_i - \frac{\sum\_{i=1}^n \mathbf{x}\_i}{n}\right) \left(y\_i - \frac{\sum\_{i=1}^n y\_i}{n}\right)}{\sqrt{\sum\_{i=1}^n \left(\mathbf{x}\_i - \frac{\sum\_{i=1}^n \mathbf{x}\_i}{n}\right)^2 \sum\_{i=1}^n \left(y\_i - \frac{\sum\_{i=1}^n y\_i}{n}\right)^2}} \tag{4a}$$

i.e., *corrX*.*Y* is a function *f*(*<sup>x</sup>*1,..., *xn*; *y*1,..., *yn*). Hence, if pairwise observations are given by FNs {(*X* - 1,*Y* - 1); (*X* - 2,*Y* - 2);...;(*<sup>X</sup>* - *<sup>n</sup>*,*Y* - *<sup>n</sup>*)}, *corrX*.*Y* induce a FN:

$$\overrightarrow{corr}\_{X,Y} = f(\check{X}\_1, \dots, \check{X}\_n; \check{Y}\_1, \dots, \check{Y}\_n) = \frac{\sum\_{i=1}^n \left(\check{X}\_i - \frac{\sum\_{i=1}^n \bar{X}\_i}{n}\right) \left(\check{Y}\_i - \frac{\sum\_{i=1}^n \bar{Y}\_i}{n}\right)}{\sqrt{\sum\_{i=1}^n \left(\check{X}\_i - \frac{\sum\_{i=1}^n \bar{X}\_i}{n}\right)^2} \sqrt{\sum\_{i=1}^n \left(\check{Y}\_i - \frac{\sum\_{i=1}^n \bar{Y}\_i}{n}\right)^2}}\tag{4b}$$

Fuzzy literature has proposed two ways to estimate PCC when the observations are done by FNs (FPCC). The first approach to FPCC, ref. [31], applies Zadeh's extension principle to (4b). So:

$$\mu\_{\overrightarrow{\text{cov}}\boldsymbol{\chi},\boldsymbol{\chi}}(\boldsymbol{z}) = \max\_{\substack{\boldsymbol{z} = \boldsymbol{f}(\boldsymbol{x}\_{1}, \ldots, \boldsymbol{x}\_{n} \boldsymbol{y}\_{1}, \ldots, \boldsymbol{y}\_{n})}} \min \{ \mu\_{\overrightarrow{\boldsymbol{X}}\_{1}} \boldsymbol{x}\_{1} \boldsymbol{), \ldots, \boldsymbol{\iota}\_{1} \boldsymbol{\mu}\_{\overrightarrow{\boldsymbol{X}}\_{n}} \boldsymbol{x}\_{n} \}; \mu\_{\overrightarrow{\boldsymbol{Y}}\_{1}} \boldsymbol{y}\_{1} \rangle, \ldots, \mu\_{\overrightarrow{\boldsymbol{Y}}\_{n}} \boldsymbol{y}\_{n} \rangle ] \tag{5a}$$

Notice that it is often difficult computing the membership function of *corr X*.*Y*. Following [32], it may be easier computing *corrX*,*Y<sup>α</sup>* such as:

$$\begin{aligned} corr\_{X,Ya} &= \begin{bmatrix} \overbrace{corr\_{X,Y}}(a), \overbrace{corr\_{X,Y}}(a) \end{bmatrix} \\ &= \begin{cases} z = f(\mathbf{x}\_1, \dots, \mathbf{x}\_n; y\_1, \dots, y\_n) \Big| \mathbf{x}\_j \in \left[ \underline{X}\_j(a), \overline{X}\_j(a) \right], y\_j \\ &\in \left[ \underline{Y}\_j(a), \overline{Y}\_j(a) \right], j = 1, 2, \dots, n \end{cases} \end{aligned} \tag{5b}$$

Hence, in (5b) *corrX*,*<sup>Y</sup>*(*α*) (*corrX*,*<sup>Y</sup>*(*α*)) are the global minimum (maximum) of *f*(·) within the rectangular domain in (5b).

$$
\underline{\operatorname{corr}\_{X,Y}(a)} = \min\_{j,k} \{ f(V\_j), f(E\_k) \} \text{ and } \overline{\operatorname{corr}\_{X,Y}(a)} = \max\_{j,k} \{ f(V\_j), f(E\_k) \} \tag{5c}
$$

Being the vector in 2n *Vj*, *j* = 1, 2, ... , 22*n* a vertex of (5b), *f*(*Ek*) *k* = 1, 2, ... , *K* an extreme point of the function and *Ek* an interior point of (5b). Hence, to find the lower (upper) extreme of α-cuts, a nonlinear minimizing (maximizing) mathematical program must be solved.

The second approach to fit fuzzy correlation uses the weakest T-norm (Tw -norm) in [33] instead of the min operator. So:

$$T\_W(a,b) = \begin{cases} \quad a \text{ if } b = 1\\ \quad b \text{ if } a = 1\\ \quad 0 \text{ otherwise} \end{cases} \tag{6a}$$

where *TW*(*<sup>a</sup>*, *b*) ≤ *min*(*<sup>a</sup>*, *b*).

Since the max-operator is still the T-conorm to apply the use of the norm (6a), suppose we reformulate the membership function of the correlation between *X* and *Y* as:

$$\mu\_{\overrightarrow{\text{cov}}\_{\mathbf{X},\mathbf{Y}}}(z) = \max\_{z = f(\mathbf{x}\_1, \dots, \mathbf{x}\_n; y\_1, \dots, y\_n)} T\_{\mathcal{W}}[\mu\_{\mathbf{X}\_1} \mathbf{x}\_1), \dots, \mu\_{\mathbf{X}\_n} \mathbf{x}\_n); \mu\_{\mathbf{Y}\_1} y\_1), \dots, \mu\_{\mathbf{Y}\_n} y\_n) \tag{6b}$$

Tw-norm lets obtaining less uncertain results than min-norm. Likewise, Tw-norm allows an easier computation of (4b) when the observations are LR fuzzy numbers [34] since *corr X*,*Y* will conserve L-R shape. In the particular case of TFNs, the calculation of *corr X*.*Y*. is developed in [35]. In [33] following arithmetical rules to handle arithmetically two TFNs *A* - = (*<sup>A</sup>*, *lA*,*rA*) and *B* - = (*<sup>B</sup>*, *lB*,*rB*) are stated:

$$
\widetilde{A} + \widetilde{B} = (A, l\_A, r\_A) + (B, l\_B, r\_B) = (A + B, \max(l\_A, l\_B), \max(|r\_A, r\_B))\tag{6c}
$$

$$
\dot{A} - \dddot{B} = (A, l\_{A\prime} r\_A) - (B, l\_{B\prime} r\_B) = \left(A - B, \max(l\_{A\prime} r\_B), (r\_{A\prime} l\_B)\right) \tag{6d}
$$

$$
\lambda \tilde{A} = \lambda (A, l\_{A\prime} r\_A) = \begin{cases}
& (\lambda A, \lambda l\_{A\prime} \lambda r\_A), \,\lambda > 0 \\
& (\lambda A\_\prime - \lambda r\_{A\prime} \lambda l\_A), \,\lambda \le 0
\end{cases} \tag{6e}
$$

$$
\sqrt{A} \approx \left(\sqrt{A}, \frac{l\_A}{\sqrt{A}}, \frac{r\_A}{\sqrt{A}}\right), \ A - l\_A > 0 \tag{6f}
$$

$$-\frac{1}{\tilde{A}} \approx \left(\frac{1}{A}, \frac{r\_A}{A^2}, \frac{l\_A}{A^2}\right), \ A - l\_A > 0\tag{6g}$$

*A* - ·*B* - = (*<sup>A</sup>*, *lA*,*rA*)·(*<sup>B</sup>*, *lB*,*rB*) = = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ (*<sup>A</sup>*·*B*, *max*(*<sup>A</sup>*·*lB*, *<sup>B</sup>*·*lA*), *max*(*<sup>A</sup>*·*rB*, *<sup>B</sup>*·*rA*)) *if A*, *B* ≥ 0 (*<sup>A</sup>*·*B*, <sup>−</sup>*max*(*<sup>A</sup>*·*lB*, *<sup>B</sup>*·*lA*), <sup>−</sup>*max*(*<sup>A</sup>*·*rB*, *<sup>B</sup>*·*rA*)) *if A*, *B* ≤ 0 (*<sup>A</sup>*·*B*, *max*(−*A*·*rB*, *<sup>B</sup>*·*lA*), *max*(−*A*·*lB*, *<sup>B</sup>*·*rA*)) *if A* ≤ 0, *B* ≥ 0 (*<sup>A</sup>*·*B*, <sup>−</sup>*max*(*<sup>A</sup>*·*lB*, <sup>−</sup>*B*·*rA*), <sup>−</sup>*max*(*<sup>A</sup>*·*rB*, −*B*·*lA*)) *if A*, *B* ≤ 0 (6h) *A* - *B* - = (*<sup>A</sup>*, *lA*,*rA*) 1*B* , *rBB*2 , *lBB*2 = ⎧⎪⎨⎪⎩ *<sup>A</sup>*·*B*, *max A*·*rB B*<sup>2</sup> , *<sup>B</sup>*·*lA*, *max A*·*lB B*<sup>2</sup> , *<sup>B</sup>*·*rA if A*, *B* ≥ 0 *<sup>A</sup>*·*B*, *max*−*<sup>A</sup> A*·*lB B*<sup>2</sup> , *<sup>B</sup>*·*lA*, *max*− *A*·*rB B*<sup>2</sup> , *<sup>B</sup>*·*rA if A* ≤ 0, *B* ≥ 0 being *B* − *lB*> 0 (6i)

Hence, to fit *corr X*.*Y* = (*corrX*.*Y*, *lcorrX*.*<sup>Y</sup>* ,*rcorrX*.*<sup>Y</sup>* ). We must evaluate (4b) with (6c)–(6i). Of course, FPCC may be interpreted qualitatively by using the linguistic variable defined in Table 2. If *min* T-norm is used, the compatibility grade of *corr X*.*Y* with the *j*th linguistic label *V* - *j <sup>C</sup>*(*corr X*.*Y*, *V* - *j*) can be found by using the *max*-*min* rule as:

$$\mathbb{C}(\overline{coir}\_{X.Y}, \tilde{V}\_{\tilde{\jmath}}) = \max\_{\mathbf{x}} \min \Big[ \mu\_{\overline{coir}\_{X.Y}} \mathbf{x} \big), \mu\_{\overline{V}\_{\tilde{\jmath}}} \mathbf{x} \big] \tag{7a}$$

On the other hand, if Tw-norm is used and so the correlation is calculated by following (6c)–(6h), the compatibility between *corr X*.*Y* and *V* - *j* is measured by using a *max*-Tw rule:

$$\mathbb{C}(\overline{\alpha}\overline{r}r\_{X,Y}, \tilde{V}\_k) = \max\left[\mu\_{\overline{\alpha}\overline{r}r\_{X,Y}}V\_{\tilde{\jmath}}, \mu\_{\overline{V}\_{\tilde{\jmath}}}corr\_{X,Y}\right) \tag{7b}$$


In both cases, we can find the closest linguistic label in Table 1 to *corr X*.*Y*, *V k*, by doing:

$$\check{V}\_k = \arg\max \left\{ \mathbb{C} \left( \overline{\boldsymbol{\alpha}} \boldsymbol{\bar{r}} \boldsymbol{r}\_{\boldsymbol{X}, \boldsymbol{Y}}, \check{V}\_j \right) \right\}\_{k=1 \le j \le m}$$

Numerical Application 2

We fit for 2014–2018 FPCC between social expenses over GPD (SER) and the percentual diminution of poverty risk index (RRP) in EU-28 countries. Fuzzy observations of these variables are shown in Table 6. Likewise, we also fit crisp PCC by considering as observations the core triangular shapes SER and RRP (3e).

**Table 6.** Fuzzy observations on social expenses over GPD, relative reduction of poverty-at risk index and Debreu–Farrell measure in the period 2014–2018.



**Table 6.** *Cont.*

Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017). Variables SER and RRP are expressed over 100 and DF over 1.

> The α-cut representation of two FPCCs is given in Table 7. Of course, crisp PCC is simply the 1-cut of both FPCCs, i.e., 0.4585. Hence, FPCC generalizes the results of crisp PCC since this last is the core of FPCCs. Likewise, *α*-cuts of FPCCs can be understood as an structured set of simulations that range from maximum fuzziness scenario (generated by the 0-cut of fuzzy estimates of SER and RRP) to maximum reliability situation (that comes from the cores of the observations on SER and RRP).

**Table 7.** α-cut representation of [31,34] FPPC between SER and relative reduction of poverty in EU-28 countries within the period 2014–2018.


Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017).

FPCC in [31] does not preserve the triangular shape of input data. On the other hand, by using FPCC [34], we obtain *corr X*.*Y* = (0.4585, 0.0389, 0.0413). Table 8 shows that the closest linguistic label for both correlations is "strong (+) relation". However, max-min correlation is extremely imprecise since embed values from −0.0643 (no correlation) to 0.7635 (very strong (+)), and so it is compatible with 4 linguistic levels in a truth level above 0.5. Those levels vary from "negligible (+) correlation" to "very strong (+) correlation". On the other hand, the correlation [34] is clearly less uncertain and allows a better balance between maintaining all the information in the sample, which is not made by conventional PCC and providing a useful value to obtain conclusions.


**Table 8.** Qualitative interpretation of max-min, max- Tw-conorm and crisp correlation between SER and relative reduction of poverty in EU 28 countries within the period 2014–2018 by using *<sup>C</sup>*(*corr X*.*Y*, *V* - *k*).

> Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017).
