Linguistic approximation of symmetrical triangular fuzzy numbers using a linguistic scale

At first, we will consider a uniform linguistic scale with five linguistic termsT1, . . . ,T5. The meanings of these terms are represented (in respective order) by triangular fuzzy numbersT1= (0,0,0.25), T2= (0,0.25,0.5), T3= (0.25,0.5,0.75), T4= (0.5,0.75,1), T5= (0.75,1,1) that form a uniform Ruspini fuzzy partition of interval [0,1].

Each of the distance and similarity measures from section 4.3 is applied to identify the linguistic approximation of each fuzzy number from the set Out1.

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sults describing the performance of the Bhattacharyya distance d3 are depicted in Figure 4.3. Each approximating linguistic term is assigned a different color, areas with same color represent fuzzy numbers that are linguistically approximated by the same linguistic term. White areas consist of the representations of such sym-metrical triangular fuzzy numbers, that are not defined on [0,1]. This graphical representation was designed to provide insights into the performance of a selected distance/similarity measure that would be easily understandable. From Figure 4.3 we can e.g. see, that the result of linguistic approximation is highly COG driven;

the length of support plays only minor role.

Figure 4.3: A graphical representation of the results of linguistic approximation of symmetrical triangular fuzzy numbers using the Bhattacharyya distance d3 and a linguistic scale. Each color represents one term of the five term linguistic scale:

T1 (blue), T2 (green), T3 (black), T4 (pink) and T5 (yellow). Red color (visible on the borders between the black area and its neighboring areas) represents ambiguous cases, i.e. cases when more than one linguistic term is assigned.

Using this graphical representation more distance and similarity measures can be compared. Figure 4.4 summarizes the performance for all the investigated dis-tance/similarity measures presented in section 4.3. Differences in their performance are clearly visible. To provide more details about the performance of linguistic ap-proximation under selected measures we add the Table 4.2, that summarizes the frequencies of assignment of each of the elementary linguistic terms by linguistic approximations under different distance/similarity measures. This information can be used not only to verify our findings based on the graphical summary provided by Figure 4.4 (see the following list), but also to highlight some unexpected/easily overlooked behavior of linguistic approximation under some distance/similarity mea-sures. This becomes more important when enhanced linguistic scales are used and

fuzzy numbers. Therefore if the possible outputs of the model are only fuzzy numbers with higher cardinality, the choice of a distance/similarity measure (out of the ones discussed in this thesis) is of no consequence. In such cases it is therefore reasonable to use measures that are e.g. easy to compute or readily available in the software we are using.

• Distance measured2does not seem to be appropriate for the linguistic approx-imation of triangular symmetrical fuzzy numbers. Firstly, linguistic terms T1

(blue) andT5(yellow) are not used at all. The set of obtainable linguistic terms is thus reduced, moreover the border terms (i.e. the terms with meanings clos-est to the endpoints of [0,1] interval) are eliminated. This could be undesirable, because the border terms can be the most important ones (e.g. excellent eval-uation; extremely dangerous...). Secondly, there are four “triangle-shaped”

areas (red) that represent fuzzy numbers, for which an unambiguous linguistic approximation can not be determined (distance measured2selects more than one linguistic term as a result of linguistic approximation).

• The remaining distance measures d1, d3 and d4 provide similar results. The results of linguistic approximation using Bhattacharyya distance d3 depend almost exclusively on the center of gravity of the approximated fuzzy number (we can see from Figure 4.3 that the border between linguistic termsT1(blue) andT2(green) is not completely vertical, nor is the border betweenT4(pink) andT5(yellow)). The results of linguistic approximation using distance mea-sures d1 and d4 exhibit the same pattern, the border between the blue and the green area is more dependent on the length of the support of the approx-imated fuzzy numbers. The same holds for the border between the pink and the yellow areas for these two distances.

• Differences between the outputs of linguistic approximation using the four selected similarity measures are clearly visible. Linguistic approximation using the similarity measure s1 provides results similar to Bhattacharyya distance d3 – i.e. it focuses mostly on the center of gravity of the approximated fuzzy numbers.

• Similarity measures2is more focused on the shape of the approximated fuzzy number (it uses perimeters of fuzzy numbers) thans1. Therefore fuzzy

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bers with smaller lengths of support and centers of gravity close to the borders of the interval [0,1] tend to be linguistically approximated by linguistic term T1instead of T2 orT5 instead of T4. This is due to the fact, that narrow tri-angles close to 0 or 1 are more similar toT₁ otT₅respectively (note, that the perimeter ofT1orT5 is smaller than the perimeter ofT2, T3orT4).

• The performance of similarity measuress3ands4is significantly different from the performance of all the other similarity and distance measures considered in this thesis. Thes3 ands4measures focus not only on the perimeters (as sim-ilarity measures2 does), but also on the areas of fuzzy numbers. This results in the amplification of the effect observed for similarity measures₂. Note that even some fuzzy numbers that were consistently linguistically approximated by the middle termT3(black) by all the previous measures (with the exception of distance measured2) are approximated by linguistic termsT1orT5unders3

ands4. This effect is even stronger using similarity measures4. In essence,T2

and T4 are never assigned as linguistic approximation of low uncertain fuzzy numbers.

Table 4.2: Frequencies of assignment of each of the elementary linguis-tic terms T1, . . . ,T5 as linguistic approximations of the symmetrical triangular fuzzy numbers from the set Out1 by the examined distance/similarity measures d1, d2, d3, d4, s1, s2, s3 and s4. The column ambiguous represents cases where more than one linguistic term was recommended, i.e. the shortest distance/maximum similarity of the approximated fuzzy number to the meanings of the linguistic terms was identical for two or more terms.

4.4.2 Linguistic approximation of symmetrical triangular fuzzy numbers