El concepto de nueva gobernanza

Originated from Karr’s work (Karr, 1991), the GA approach in fuzzy systems was initially

utilised to adjust the parameters of membership functions, which leads to no significant difference when compared to other learning paradigms. The real significance of employing EAs for optimising FRBSs comes from EAs’ flexibility in terms of being able to encode and

evolve almost every component of the FRBS (Herrera, 2008). Such a flexibility offers a

solution so that one can take into account the interpretability (structure) and the performance of the FRBS in a more coherent way. Broadly speaking, there currently exist two different EA-based streams to tackle the interpretability issues: the first stream is mainly concerned

with the linguistic modelling, in which a set of pre-specified fuzzy partitions are given a

priori by experts or users (grid partition); the task is then to find an optimal FRBS in terms of

its compactness and performance (Ishibuchi et al., 1995; Ishibuchi et al., 1997; Ishibuchi et

al., 2001; Ishibuchi et al., 2004; Alcal ́ et al., 2007; Cococcioni et al., 2007); the second

stream generally uses the approximate fuzzy model as the starting point; hence, the task is to improve the model’s explanatory ability, which may have been lost during the automatic learning process, through a set of similarity-driven simplification and parameter adjusting

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Jim ́nez et al., 2002; Jin et al., 1999; Jin et al., 2000; Wang et al., 2005; Gonz ́lez et al.,

2007, Chen et al., 2004).

In the first stream, the earliest noticeable attempt was made by Ishibuchi et al. (1995), in

which a fuzzy classifier is built using the pre-specified linguistic terms (fuzzy sets). These linguistic terms are fixed during the course of the evolution so that their physical meanings are retained. Only the fuzzy rules are subject to the selection via GA so that a compact rule- base can be evolved from a large number of candidate rules, which should lead to a more interpretable FRBS. Since the selection process removes irrelevant and inconsistent rules, the

accuracy is also improved. In the works of Ishibushi et al. (1997), Ishibushi et al. (2001) and

Ishibushi et al. (2004), extensions to the above ‘rule selection’ idea were made in both single

objective and multi-objective configurations. It is worth mentioning that, in Ishibushi et al.’s

work (2004), the GA is not only used to select the optimal combination of rules but also to learn the granularity of different fuzzy partitions for each input, which leads to a more accurate fuzzy model while the linguistic feature is not compromised. Further relevant

researches include those which were proposed by Alcal ́ et al. (2007) and Cococcioni et al.

(2007). In Alcal ́ et al.’s work (2007), apart from the rule selection, the authors also tuned

the linguistic terms by a modified GA. However, such tuning is only operated in a local sense in order to maintain their original semantics. One interesting paper in the second stream is

attributed to Setnes et al. (1998), in which the TSK model is elicited via a fuzzy clustering

algorithm for its premises and a parameter estimation method for its consequents. A similarity measure is taken so that similar fuzzy sets can be merged. Consequently, similar rules are merged as well. Hence, the distinguishability of membership functions and the compactness of the rule-base are improved. Although this rule-base simplification method does not relate to the EA directly, it has since inspired many EA-based fuzzy modeling

algorithms within this trend (Setnes et al., 2000; Roubos et al., 2001; Jim ́nez et al., 2001;

Jim ́nez et al., 2002; Jin et al., 1999; Jin et al., 2000; Wang et al., 2005). In Gonz ́lez et al.’s

work (2007), the idea of rule pruning is used to delete less relevant rules within a multi- objective optimisation framework. The similarity measure is not explicitly used in this work. Comparing the two streams leads to the following: in the linguistic modelling stream, the target problems are normally associated with classifications and low-dimensional function approximations; hence, the effect of the ‘curse of dimensionality’ due to the grid partition and the need for the parameter tuning due to the performance requirement are not serious issues. In the latter case, high-dimensional approximations are often the case; as a result, an

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approximate FRBS is a better choice to start with due to the accuracy and compactness requirements. Within the second stream, EA-based multi-objective fuzzy modelling has become a recent hotspot for function approximations due to its ability of producing a set of

compromised FRBSs (Jim ́nez et al., 2001; Jim ́nez et al., 2002) and (Wang et al., 2005;

Gonz ́lez et al., 2007). However, this is a rather new developing area with several other

issues to be addressed. Among which, it is believed that the following considerations are the most important:

— most well-known multi-objective optimisation algorithms used in fuzzy modeling, e.g.

NSGA II (Deb, 2001), are originally designed to solve real-valued problems; in order to use such type of algorithms to simultaneously optimise the rule-base structure and the membership function parameters, similarity-driven simplifications are normally

selected as the mutation operators for the former (Jim ́nez et al., 2001; Jim ́nez et al.,

2002; Wang et al., 2005), and the heuristic variations (crossover) are proposed for the

latter (Jim ́nez et al., 2001; Jim ́nez et al., 2002; Wang et al., 2005; Gonz ́lez et al.,

2007); however, the search power of these optimisation algorithms relies heavily on their original variation (search) operators; other components of the algorithms are mainly used to advocate diversity and elitism; without using the original variation operators, even if the general framework is kept fixed it is likely that the search capability, in terms of the real-valued optimisation part, may be compromised, and this is the partial reason to explain the necessity to include a gradient-based

optimisation for the enhancement of the parameter optimisation in Gonz ́lez et al.’s

work (2007);

— The reason behind the use of the heuristic variation operators for the parameter

optimisation is that the structure optimisation leads to individuals with different sizes, e.g. rule base length, which makes the conventional variation operators invalid. Hence, new techniques that can cope with the variable length coding and can facilitate the use of the original variation operators are needed.

With the aim of solving high-dimensional approximation problems, the proposed modelling framework-IMOFM falls into the second stream. To address the above two issues, the research work in Chen & Mahfouf’s works (2006, 2008a) (refer to Chapter 3) is extended, which has been shown to be effective for real-valued multi-objective optimisation, to a fuzzy modeling scenario. A new distance index (Chen & Mahfouf, 2008b; 2009) that is able to cope with the variable-length individuals and unconstraint optimisation is also proposed. The main

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focus points are two types of FRBS, viz. Singleton FRBS and Mamdani FRBS, due to their simplicity and their ability to express semantics in both premises and consequents. In the next section, IMOFM is introduced, which is in essence a three-stage modelling procedure which mimics the proposed multi-stage immune optimisation procedure already discussed in Section 3.5.

5.2 The Framework of the Proposed Modelling Method

IMOFM is a systematic multi-objective fuzzy modelling framework, which can be regarded as a three-stage modelling procedure. The first two stages are equivalent to the vaccination process in the first stage of the immune optimisation procedure (see Section 3.5). By doing so, an initial ‘vaccine model’ (prior knowledge, in some sense) can efficiently be elicited. Another reason of including the first two modelling stages, especially the second one, is that by doing so the most complex-rule base can survive under the pressure of ‘Pareto’ selection. Without including the refining step (the second stage), the rule-base with a complex structure may be regarded inferior to the less complex-rule base in a ‘Pareto’ sense. Even if both the most complex and less complex rule-bases are inaccurate in the early evolutionary stages, the ‘Pareto’ selection favours the one with a simpler structure. Hence, one may lose the chance of evolving the most accurate FRBS, which normally comes with a complex structure (refer to Section 5.6.1). The ‘vaccine model’ is then used in the third stage to seed the initial population of PAIA2 in order to obtain a set of Pareto fuzzy models with improved interpretability.

To tackle the problem of simultaneously optimising the rule-base structure and parameters, a variable length coding scheme is adopted, and a new distance index is proposed to cope with the variable-length individuals, which should improve the efficiency of the search (see Section 5.5.3 for more details).

Figure 5.3 represents a schematic diagram of such a framework and each stage depicted in this figure is expanded in depth in the following sections.

- 116 - First Stage: Extracting The Initial FRBS Second Stage: Refining The Initial FRBS Third Stage: Multi-objective Fuzzy Modelling A Set of Fuzzy Models

An Immune Algorithm Based Fuzzy Predictive Modeling Mechanism

Activation

Clone

Initial Population Pool

The Initial FRBS

Variable Length Coding

Clone Affinity Maturation Reselection Network Suppression Stop No Model Simplification Ye s

Figure 5.3 The proposed IMOFM framework.

5.3 First Stage: Elicitation of Initial FRBSs

Section 4.4.2 gives detailed steps on how to elicit an initial Singleton FRBS from data using the G3Kmeans algorithm, which serves as the first modelling stage in IMOFM_S (IMOFM_S stands for the Singleton version of IMOFM). Hence, in the following space, special attentions have been given to the Mamdani version of IMOFM, viz. IMOFM_M. IMOFM_M differs from the original Mamdani FRBS (Mamdani, 1974) in that IMOFM_M adopts a different T- norm, S-norm and defuzzification mechanism.