Methods applied to achieve the semantic-based constraints

Much of the works related to achieve the semantic-based constraints are attributed to developing methods that aim to maintain the distinguishability property in fuzzy rule- based systems. This property can be characterized using mathematical expressions that can be included in the optimization function of the fuzzy rule-based system. Basically, distinguishability can be formally defined as a relation between fuzzy sets defined on the same universe of discourse. In the following, the main approaches that have been proposed to mathematically formalize this interpretability constraint are presented.

a) - Similarity measure

Similarity measures approach is the most adopted way to characterize distinguishability constraint (Mencar & Fanelli, 2008). Similarity is usually defined as the degree to which the concepts represented by fuzzy sets belonging to the same universe of discourse are equal (Setnes et al., 1998). More specifically, similarity between fuzzy sets 𝐴 and 𝐵 is quantified by a function called similarity measure which assigns a similarity value 𝑆 to the pair of fuzzy sets (𝐴, 𝐵). Similarity measure function is given by the following expression:

𝑆(𝐴, 𝐵) = |𝐴 ∩ 𝐵|

|𝐴| + |𝐵| − |𝐴 ∩ 𝐵| (3.5)

Where |. | and ∩ represent the cardinality of a set and intersection, respectively.

This approach was applied not only to ensure the distinguishability requirement in fuzzy models but also for rule base simplification procedure (Setnes et al., 1998). The main objective of the simplification procedure, which is applied after producing the initial fuzzy rule-based system via a clustering method, is to remove redundancy that may be present in the form of similar fuzzy sets (or non-distinguishable fuzzy sets) that represent compatible concept and replace them by a new fuzzy set representative of the merged fuzzy sets. By replacing the merged fuzzy sets with the new one, the number of fuzzy sets in the rule base decreases and rule base simplification is achieved. Since

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similar fuzzy sets are merged, the distinguishability in this way is also achieved. A threshold 𝜆 controls the value of similarity at which the two fuzzy sets 𝐴 and 𝐵 are considered similar and thus should be merged. The choice of a suitable threshold value depends on the application and the user-preference of accuracy versus interpretability. Lower threshold value will enforce the distinguishability constraint and results in less fuzzy sets and generally lower accuracy.

In fact, similarity measures have been adopted by many studies to enhance the interpretability in fuzzy models using different learning methods especially evolutionary-based methods such as Genetic Algorithms (Jimenez, Gomez-Skarmeta, Roubos, & Babuska, 2001; Meesad & Yen, 2002; Roubos & Setnes, 2001; Yaochu, 2000), Symbiotic Evolution (Jamei, Mahfouf, & Linkens, 2001), Coevolution (Peña- Reyes & Sipper, 2003). Although similarity measures are well-suited for capturing the characteristics of the distinguishability requirement, this approach was criticized for the calculation needed which is usually computationally intensive (Mencar et al., 2007).

b) - Possibility measure

Another approach to quantify distinguishability was based on a possibility measure. It is defined as the degree of applicability of the soft constraint “x is B” for x = A (Mencar et al., 2007). The possibility measure can be evaluated using the following expression:

Π(𝐴, 𝐵) = sup

𝑥∈𝑈min {𝜇𝐴(𝑥), 𝜇𝐵(𝑥)} (3.6)

A useful interpretation of the expression (3.6) is the extent to which 𝐴 and 𝐵 overlap (W. Pedrycz & Gomide, 1998). Possibility measure has some advantages such as the computational efficiency of the calculation procedure comparing with similarity. In addition, it can be used for both ensuring the high distinguishability and high coverage of fuzzy sets (Mencar et al., 2007). Furthermore, and despite similarity measure is different from possibility measure but under mild conditions (such as continuity, normality, and convexity) similarity and possibility are related monotonically, so that

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small values of possibility implies small values of maximal similarity. In fact, the possibility measure has some good features but it needs more investigation (Mencar et al., 2007).

c)- Pointwise property approach

De Oliveira (1999) has proposed a pointwise property to characterize the distinguishability constraint which states that any element 𝑥 of 𝑋 will not have simultaneously high membership value in different fuzzy sets defined on 𝑋. In other word, if an element in 𝑋 has a high membership value for a fuzzy set, then it must have a low membership value for all the other fuzzy sets. This reasoning was formulated by the following constraint:

∀ 𝑥 ∈ 𝑋: √∑𝐴∈𝐹(𝜇𝐴(𝑥))𝑃 𝑃

≤ 1 (𝑃 ≥ 1) (3.7)

Where 𝑃 is a user-defined parameter that control the strength of the constraint imposed on the fuzzy sets. For the case 𝑃 = 1, the distinguishability constraint is strong whereas the strength of the constraint is reduced for higher values of 𝑃 and vanished as 𝑃 → ∞. This constraint is especially effective for controlling the distinguishability in an online training session where it has been used in (De Oliveira, 1999) as the inequality can be checked when just an input is given. In the case where the inequality in (3.7) is violated, a modification of the fuzzy sets will be made through an appropriate learning algorithm (Mencar et al., 2007).

d)- Pre-defined fuzzy partition

Actually, there is a special type of partition called “Strong Fuzzy Partition (SFP)” proposed by Ruspini (1969) that satisfies all the semantic constraints to the highest level particularly when the membership functions are also uniform (Gacto et al., 2011). Using this kind of partitions, however, should consider the accuracy of the system. For example, Fazendeiro and de Oliveira (2005) show how it is possible to get more accurate and less interpretable fuzzy rule-based system by breaking the SFP property.

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This issue, namely, interpretability-accuracy trade-off, is an important topic and has been especially addressed in the framework of Multi-Objective Evolutionary Algorithms (MOEA). Another interesting approach for maintaining distinguishability property is proposed by Ishibuchi and his co-researchers (see for example (Ishibuchi & Nojima, 2007; Ishibuchi, Yamamoto, et al., 2005) where they suggested pre-defined fuzzy partitions with different level of granularity ranging from 2 to 5 for each input variable. The partitions compose of well-defined and distinguishable fuzzy sets with clear semantic meaning. In the first stage, all the possible rules are generated using these linguistic terms and then a rule and fuzzy set selection procedure are performed using MOEA to choose the most relevant rules and fuzzy sets. In this way, distinguishability constraint is satisfied to the highest level.