Linguistic summaries rely on the theory of fuzzy sets and fuzzy logic, where belonging to a set is a matter of degree. A fuzzy set F is defined by a membership function μ F , a mapping from the universe of discourse X to a completely distributive lattice (in this case unit interval) that matches each element of X with its degree of membership to the set F (Zadeh, 1965) (1) μ F ( x ) : X → [ 0 , 1 ] , where μ F ( x ) = 0 means that an element x clearly does not belong to F, while μ F ( x ) = 1 means that x is a full member of F. A value of μ F ( x ) between 0 and 1 indicates the intensity by which the element x ∈ X belongs to F.

The next concept required for this work is Linguistic Variable (LV). It is a variable, whose values (often called labels) are words of natural language determined by a quintuple ( L , G ( L ) , X , M , H ) (Zadeh, 1975), where: •

L is the name of the variable,

The syntactic rule explains the required number of linguistic labels and their names (in this case three, but a finer or coarser granularity can be created), whereas the semantic rule assigns the context-dependent meaning to each label by fuzzy sets. Generally, the fuzzy set long (or other terms expressing larger amounts like high or big) is expressed as an increasing function. In this work, we adopted the linear functions due to their simplicity, but non-linear ones can be straightforwardly applied (Holzinger et al., 2017; Hudec et al., 2018). In this context, the fuzzy set long is formalized as (see Fig. 1): (2) μ long ( x ) = 1 for x ⩾ x 4 , x − x 4 x 4 − x 3 for x 3 < x < x 4 , 0 for x ⩽ x 3 .

Value x c in Fig. 1 is the maximal uncertainty point. In a smooth transition from sets medium to long, x c belongs to both with 0.5 degree.

The next key element in LSs is the fuzzy relative quantifier. In this work, we adopted the sigma-counts approach (Zadeh, 1983) for its simplicity. In this way, all building blocks of LSs are modelled by the same approach, which makes the whole process effective, especially for visualization on maps. Within that approach, the proportional non-decreasing quantifier is formalized by a function where μ Q ( 0 ) = 0 and μ Q ( 1 ) = 1 (Kacprzyk and Zadrożny, 2005) as (3) μ Q ( y ) = 1 for y ⩾ 0.8 , y − 0.3 0.5 for 0.3 < y < 0.8 , 0 for y ⩽ 0.3.

When formalizing quantifier most of, the non-decreasing function starts to increase in 0.5, to cover the natural meaning of majority, i.e. (4) μ Q ( y ) = 1 for y ⩾ 0.8 , y − 0.5 0.3 for 0.5 < y < 0.8 , 0 for y ⩽ 0.5.

By the above defined fuzzy sets and the quantifier, we can calculate the truth value of LS of the form Q entities in X are P (we call it the basic structure) as Yager (1982) (5) v L S b ( X ) = μ Q ( 1 n ∑ i = 1 n μ P ( x i ) ) , where n is the number of entities in a considered data set X, and membership function μ formalizes quantifier Q and predicate P. The truth value of a summary is usually denoted as t or T. To avoid any confusion with t-norms, we adopt the letter v (considered also as validity).

The truth value of LS of the form Q R entities in X are P (we call it the structure with restriction) is computed as (Yager, 1982; Rasmussen and Yager, 1997) (6) v L S r ( X ) = μ Q ( ∑ i = 1 n μ P ( x i ) ∧ ∑ i = 1 n μ R ( x i ) ∑ i = 1 n μ R ( x i ) ) , where the membership function μ formalizes quantifier Q, restriction R, and predicate P of a considered data set X. The restriction R focuses on a particular subdomain of attribute when evaluating relation among subdomains like most of elderly patients have long stay, where set elderly expresses R and long stay explains S. The convention 0 / 0 = 0 is used in order to avoid undefined proportions (Rasmussen and Yager, 1997); this situation occurs when not a single record does meet R (and as a logical consequence, not a single record does simultaneously meet R and P) (Hudec et al., 2018), where ∧ stands for the conjunction. Theoretically, any t-norm function for conjunction can be applied (Klement et al., 2000), but t-norms having downward reinforcement property unnaturally reduce the value of proportion. In the extreme case (drastic product t-norm), we get T DP ( 0.8 , 0.9 ) = 0. Considering this fact, the minimum t-norm can be the solution.

The truth value of a summary can be calculated by other approaches. The conjunction of Q entities in X are P ∧ Q ˆ entities in X are ¬ P, where Q ˆ is an antonym of Q and ¬ P is a complement of P, is calculated by the method based on the Sugeno integral (Jain and Keller, 2015; Wilbik et al., 2020). The same observation, as in the previous paragraph for the most suitable conjunctive function, holds here.

The truth value is a significant measure, but it is not sufficient (Kacprzyk and Yager, 2001). Hirota and Pedrycz (1999) have introduced five features for measuring the quality of mined and aggregated information (not necessarily linguistically): validity (corresponds to truth value), novelty, usefulness, simplicity, and generality. Based on this observation, Wu et al. (2010) have proposed equations for calculating quality measures for LSs with restriction (see Eq. (6)) for transforming them into the IF-THEN rules. For instance, the degree of usefulness is computed as a minimum of truth value and coverage. Usually, quality measures focus on a single summary or a set of summaries to recognize the most representative ones (Bugarín et al., 2015). Aggregation of several quality measures is examined in Hudec (2017). Kacprzyk and Strykowski (1999) have introduced quality measures: truth value, degree of fuzziness, degree of coverage, degree of appropriateness, and length of summary mainly related to the basic structure of LSs (see Eq. (5)). Recently, the quality measure conjunctively aggregating a summary and the summary consisting of antonym Q and negation of predicate P by the Sugeno integral for the summaries with restriction has been considered in Wilbik et al. (2020).

Another problem occurs when evaluating several subsets of data (in our case districts) by a single summarized sentence. The data distribution among regions varies and therefore it might cause skewed summaries, even though quality measures like data coverage, degree of fuzziness, or degree of focus do not report problems. However, not a single quality measure, to our best knowledge, is related to the quality of a summary evaluated over a hierarchical data, e.g. the most of entities in a district have the low value of attribute A calculated for each district. We need a quality measure for evaluating summaries on subsets of different sizes and compare the computed results. In the literature, theoretical works often illustrate achievements with smaller data sets for diverse summaries. The problem and proposed solution are discussed in the next section.

To illustrate the problem of a summary Q entities in X are P, let us evaluate the truth value of the sentence the most of patients are old for each district. The fuzzy set old is a linguistic term on a domain of attribute age (see e.g. Fig. 5). The quantifier most of is expressed by Eq. (4). In Table 1, there is an example of the numbers of patients in three districts, belonging to sets young, middle-aged and old. For the simplicity reason (which do not affect generality), all the patients belong to one of these sets with the degree equal to 1. In this case, ∑ i = 1 n μ P ( x ) (expressing the cardinality of a fuzzy set P) assigns a natural number.

District D1 has a higher truth value than district D2, which means a slightly darker hue on a map. But, a higher concern should be focused on D2, instead of on D1. Thus, we should include the data distribution among districts to emphasize D2 on a map. It has a significantly higher number of patients which should be reflected in the summary, while a lower number of patients should reduce the relevance (alarm) of a summary. Theoretically, not a single record in a district might be recorded, which leads to undefined operation 0 / 0 in Eq. (5), i.e. n = 0. For this case, we adopt 0 / 0 = 0.

The first (and simple) option is considering the proportion index p as a weight of the summary, i.e. (7) v i ‾ ( X ) = p i v i ( X ) for i = 1 , … , n , where n is the number of districts. The proportion can be expressed as (8) p i = N i max { N 1 , … , N n } , where N i is the number of patients in district i, N i ⩽ max { N 1 , … , N n } causing p i ∈ [ 0 , 1 ], i = 1 , … , n. If max { N 1 , … , N n } = 0, not a single case is recorded and therefore interpreting summaries on a map is irrelevant. For simplicity, we denoted v L S b i = v i . Instead of the number of patients, the ratio of patients to the number of inhabitants in districts can be used. Firstly, it does not solve this problem. Secondly, many inhabitants might have a temporal address somewhere else. Assigning weights is a widely applied approach, but we should be careful. Discussions related to weights can be found in e.g. (Dujmović, 2018; Zadrożny et al., 2008).

Next, v i ‾ ( X ) = v i ( X ) holds only for a district where N i = max { N 1 , … , N n }. However, a high truth value for a district of a bit lower number of cases is attenuated, which is problematic.

Apparently, we should emphasize a summary for districts where both the truth value and proportion of data are high, and reduce the relevance of summaries when these two values are low. This observation leads to the aggregation by functions known as uninorms.

Uninorms generalize t-norms and t-conorms using the fact that these two classes of aggregation functions are defined by the same axioms of associativity, commutativity, monotonicity, and the presence of a neutral element. It means that uninorms consider neutral element e inside the unit interval (Beliakov et al., 2007).

A uninorm is a bi-variate aggregation function U : [ 0 , 1 ] 2 → [ 0 , 1 ] which is associative, commutative, and has a neutral element e ∈ ] 0 , 1 [. For e ∈ { 0 , 1 } we have the limiting cases of t-conorm and t-norm. Next, •

∀ ( x , y ) ∈ [ 0 , e ] 2 U ( x , y ) = e T u ( x e , y e ) has a conjunctive behaviour;

Representative uninorms are continuous everywhere except for the corners ( 0 , 1 ) and ( 1 , 0 ). For conjunctive uninorm holds U ( 0 , 1 ) = 0 (annihilator a = 0), whereas for disjunctive uninorm holds U ( 0 , 1 ) = 1 (annihilator a = 1). The former case might solve the problem with the quality of summaries. Due to commutativity, the same observation holds for U ( 1 , 0 ). The presence of annihilator prevents uninorms from being strict on the whole unit square, i.e. they are strict on ] 0 , 1 [ 2 .

An important family of parametrized representative uninorms is (Fodor et al., 1997; Klement et al., 1996): (9) U λ ( x , y ) = λ x y λ x y + ( 1 − x ) ( 1 − y ) , ( x , y ) ∈ [ 0 , 1 ] 2 / { ( 0 , 1 ) , ( 1 , 0 ) } , where λ ∈ ] 0 , ∞ [ and either U λ ( 0 , 1 ) = 0 or U λ ( 0 , 1 ) = 1, the neutral element is e λ = 1 1 + λ . Taking λ = 1, we get the well-known 3 − ∏ function (Yager and Rybalov, 1996) (10) U λ ( x , y ) = x y x y + ( 1 − x ) ( 1 − y ) , where e = 0.5 and convention 0 / 0 = 0 holds for the conjunctive uninorm.

This function can meet the needs for aggregating the truth value (consider v i = x) and data proportion (consider p i = y). When p i = 0, the validity is also 0 (the validity of a summary on empty set is 0, when adopting 0 / 0 = 0 in Eq. (5)). When v i = 0, the solution should be 0, regardless of value p i . These observations hold for the 3 − ∏ function.

Moving back to the example in Table 1, we get p 1 = 0.0273, p 2 = 1, p 3 = 0.5844 by Eq. (8) and therefore we get by Eq. (7) v 1 ‾ = 0.019, v 2 ‾ = 0.7013 and v 3 ‾ = 0. The resulting relevance by Eq. (10) of summary for district D1 is 0.0649 (averaging function), for district D2 is 1 (disjunctive function), and for district D3 is 0. District D2 is emphasized because a truth value and the proportion of entities influencing a truth value are higher than 0.5. District D1 is attenuated by the averaging behaviour of validity higher than 0.5 and the proportion lower than 0.5. District D3 gets the value of 0 as the validity of summary is 0 (0 is an annihilator for the conjunctive behaviour).

On the other hand, due to discontinuity in the proximity of (0, 1) this function is unstable, especially for the imprecision of input data, i.e. U ( 1 , ϵ ) = 1 for any ϵ > 00$]]> like 0.001. A simple solution is replacing 1 (for truth value and proportion) with 0.999 to get the average of 0.999 and 0.001. Anyway, we have searched for a better solution.

The next option is aggregating a truth value and data proportion by the ordinal sums, which are an extension for semigroups (Clifford, 1954) or for posets (Birkhoff, 1967). In the framework of fuzzy sets theory, they were considered to build new t-norms/t-conorms from the scaled versions of existing ones (Klement et al., 2000). The ordinal sum of conjunctive and disjunctive functions has been proposed by De Baets and Mesiar (2002) as follows.

For an n-ary aggregation function B : [ 0 , 1 ] n → [ 0 , 1 ] and [ a , b ] ⊂ R, denote B [ a , b ] ( x ) = a + ( b − a ) · B ( x − a b − a ). Then B [ a , b ] is an n-ary aggregation function on [ a , b ]. For B 1 , … , B k : [ 0 , 1 ] n → [ 0 , 1 ], k ⩾ 2, and 0 ⩽ a 0 < a 1 < ⋯ < a k = 1. Let A i : [ a i − 1 , a i ] n → [ a i − 1 , a i ] be given by A i = ( B i ) [ a i − 1 , a i ] . Then the ordinal sum A : [ 0 , 1 ] n → [ 0 , 1 ], A = ( ⟨ a i − 1 , a i , A i ⟩ ) | i = 1 , … , k given by De Baets and Mesiar (2002): (11) A ( x ) = ∑ i = 1 k ( A i ( a i ∧ ( a i − 1 ∨ x ) ) − a i − 1 ) is an aggregation function on [ 0 , 1 ]. If all B 1 , … , B k are t-norms (t-conorms, copulas, means) then also A is a t-norm (t-conorm, copula, mean) (De Baets and Mesiar, 2002).

Analogously, A ( x ) = ∑ i = 1 k ( a i − a i − 1 ) · B i ( 1 ∧ ( 0 ∨ x − a i − 1 a i − a i − 1 ) ). For our purposes, n = k = 2 is considered. Denoting a 1 = a ( a 0 = 0 , a 2 = 1 ), we have two next forms of ordinal sums (Hudec et al., 2021):

(i) B 1 , B 2 : [ 0 , 1 ] 2 → [ 0 , 1 ], (12) A ( x , y ) = a · B 1 ( 1 ∧ x a , 1 ∧ y a ) + ( 1 − a ) · B 2 ( 0 ∨ x − a 1 − a , 0 ∨ y − a 1 − a ) , (ii) A 1 : [ 0 , a ] 2 → [ 0 , a ], A 2 : [ a , 1 ] 2 → [ a , 1 ], (13) A ( x , y ) = A 1 ( a ∧ x , a ∧ y ) + A 2 ( a ∨ x , a ∨ y ) − a .

Functions B i cover subsquares of the unit square. In order to be inside the respective subsquares, conjunction between the attribute values (considering the separation point a) and 1, and disjunction between the values and 0 is applied. Observe that the function A covers the conjunctive part A 1 ( x , y ) when x and y are lower or equal to a, i.e. a ∧ x = x, and a ∧ y = y. Then, for A 2 we get a ∨ x = a and a ∨ y = a. As a consequence ( a ∨ a ) − a = 0. More details are in De Baets and Mesiar (2002), Hudec et al. (2021).

Then:

The next task is the suitable variation of conjunctive, disjunctive, and averaging functions in ordinal sums. In this work, we need upward reinforcement when both values are high, downward reinforcement when both are low and averaging behaviour when one measure is high and another is low. In addition, we need the stability in the ϵ neighbour of ( 0 , 1 ) and ( 1 , 0 ).

The option is a strict t-norm for the conjunctive part, strict t-conorm for the disjunctive part and a logically neutral averaging function (arithmetic mean), because we do not consider inclinations towards conjunctive or disjunctive areas provided by, e.g. geometric and quadratic mean, respectively.

The representative function of strict behaviour is product t-norm (expressed as C P ( x , y ) = x y), whereas its dual t-conorm is a probabilistic sum (expressed as D P ( x , y ) = x + y − x y).

The result for A ( 0.5 , 0.5 ), when a = 0.5 should be 0.5 for conjunctive, averaging and disjunctive function (see Fig. 3). In order to keep the expected value on this point, the product t-norm on [ 0 , 0.5 ] 2 is expressed as (Hudec et al., 2021): (14) C P ( x , y ) = A 1 ( x , y ) = 2 x y .

Analogously, strict t-conorm on [ 0.5 , 1 ] 2 is expressed as: (15) D P ( x , y ) = A 2 ( x , y ) = − 1 + 2 x + 2 y − 2 x y .

Finally, the aggregation on the averaging part is expressed as: (16) A M ( x , y ) = A 1 ( x , 1 / 2 ) + A 2 ( 1 / 2 , y ) − 1 / 2 = x + y − 1 / 2 .

From the logic perspective, the arithmetic mean and its variants thereof (weighted arithmetic mean and the like) are the logically neutral averaging functions, with the ORNESS measure equal to 0.5. The other functions either incline towards conjunction ( ORNESS < 0.5 ) or disjunction ( ORNESS > 0.5 )0.5)$]]> (Dujmović, 2018). Applying the other averaging functions will increase the complexity. This is the reason, why we applied arithmetic mean. However, the other averaging functions could be examined in the future work.

Considering again the example in Table 1 (consider x = v – truth value and y = p – proportion), the resulting quality of the summary ( v ‾) for district D1 is 0.2413 (an example of averaging behaviour), for district D2 is 1 (an example of disjunctive behaviour), and for district D3 is 0 (due to restriction put on summary). We get the expected results. Moreover, we get A ( 1 , 0.001 ) = 0.501, which is an averaging behaviour. Next, when the proportion is 0, the truth value gets the same value. It is worth noting that, when a truth value is 0, the solution should be zero. This is not a problem, because quality measures should be activated when a truth value is greater than zero.

The downward and dual upward reinforcement behaviour is also a property of nilpotent t-norms and t-conorms, respectively. The representative functions are the Łukasiewicz t-norm and its dual Łukasiewicz t-conorm. In order to keep the expected value on edges of subinterval [ a , 1 ] 2 when a = 0.5, the Łukasiewicz t-conorm on [ 0.5 , 1 ] 2 is expressed as (Hudec et al., 2021) (17) D L ( x , y ) = A 2 ( x , y ) = min ( 1 , x + y − 0.5 ) .

When the truth value and proportion are higher than 0.75, the solution is equal to 1. Thus, we cannot distinguish between two summaries, for which the truth value and proportion are ( 0.75 , 0.76 ) and ( 0.95 , 1 ), respectively. Considering this fact and the need for providing continued hues on maps, the option is a strict t-norm and its dual t-conorm.

In summary Q R entities in X are P, when, for instance, two of 10 6 entities satisfy R and only the same two entities satisfy P (Eq. (6)), the truth value gets value 1, but it is based on outliers (Hudec et al., 2018). To overcome this problem, Wu et al. (2010) proposed a coverage measure that expresses the proportion of entities included in both R and P to avoid summaries on outliers. We refer this measure here. The ratio of included entities in a summary is (18) i C = 1 n ∑ i = 1 n m i , where n is the number of records and m i = 1 for μ P ( x i ) > 0 ∧ μ R ( x i ) > 0 , 0 otherwise . 0\wedge {\mu _{R}}({x_{i}})>0,\\ {} 0\hspace{1em}& \text{otherwise}.\end{array}\right.$]]>

Since a summary of the structure (Eq. (6)) covers a subset of the entire database, i c is considerably smaller than 1. Thus, the following function converts this ratio into the degree of sufficient coverage (Wu et al., 2010) (19) C = f ( i c ) = 0 for i c ⩽ r 1 , 2 ( i c − r 1 ) 2 / ( r 2 − r 1 ) 2 for r 1 < i c < ( r 1 + r 2 ) / 2 , 1 − 2 ( r 2 − i c ) 2 / ( r 2 − r 1 ) 2 for ( r 1 + r 2 ) / 2 ⩽ i c < r 2 , 1 for i c ⩾ r 2 , where the suggested values for parameters r 1 and r 2 depend on the length of the summary (the number of attributes in R and P).

To illustrate these calculations, let us have 2000 records, r 1 = 0.02 and r 2 = 0.15 (i.e. when 15% of data are included in the R and S parts, it is considered as a fully relevant coverage). Next, we have three summarized sentences S 1 , S 2 and S 3 covering 175, 320 and 13 records, respectively. The ratio of included records and coverage are shown in Table 2. Summary S 2 fully covers the relevant subset of data, whereas summary S 3 should be excluded, even when its validity is equal to 1.

The method for calculating a truth value of conjunction of summary and its antonym based on the Sugeno integral (Jain and Keller, 2015) also solves this problem (Wilbik et al., 2020).

The same problem as for a basic structure of a summary holds here. Even though quality measures filter summaries on outliers, the distinction among subsets of different sizes should be reflected on the map. Districts, where the number of patients is higher, should be emphasized when the truth value of a summary and data coverage are high. In addition, we have a truth value of a summary, data coverage, and proportion. Hence, we should aggregate these three measures.

The truth value and coverage are measures for evaluating different summaries on the same data set. As both should be satisfied, we aggregate them by t-norm function (Hudec, 2017). In the next step, we apply ordinal sums (see Eqs. (14), (15), (16)).