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The original Mamdani FRBS is based on the so-called ‘sup-star compositional rule of inference’ (see Section 2.6.2 and Eqs. 5.1~5.3) and the overall implied fuzzy set (see Section 2.6.2 and Eq. 5.3) (Passino et al., 1998, p. 63), which are defined as follows:

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    (5.1)

· · … · (5.2)

,    1 … (5.3) where, and are the inputs and output of the data point; indicates the input of the data point; and is the number of fuzzy rules in the rule-base. The ‘sup’ corresponds to the operation, and the ‘star’ corresponds to *. A special instance of the ‘sup-star’, which uses maximum for and minimum for *, was adopted in the original Mamdani implementation, and the centre of average defuzzification was applied on the overall implied fuzzy set in order to derive a crisp output, which leads to two problems as mentioned by Passino (1998, p. 64):

(1) The overall implied fuzzy set is itself difficult to compute;

(2) The defuzzification techniques based on the overall implied fuzzy set are also difficult to compute.

More importantly, if an analytical solution cannot be deducted from the defuzzification step the gradient based optimisation method, such as the BEP technique, cannot be utilised. Hence, in this work, the centre of gravity defuzzfication is applied on the implied fuzzy set (Eq. 5.1). Instead of using minimum and maximum, ‘product’ is used for * and ‘plus’ is used for . Unlike traditional Mamdani FRBS which may use the same type of membership functions for premises and consequents, IMOFM_M uses Gaussian membership functions for the premises (refer to Section 4.4.2) and the bell-shape membership functions for the consequents (Eq. 5.4).

(5.4)

Where, and are the centre and the spread of the membership function of the output. Hence, a Mamdani FRBS can be formulated as follows:

∑ ·

∑

∑ · ·

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where, bi is the centre of area of the membership function and is the peak ( ) if is symmetric; is the final defuzzified output of the FRBS. , , , is the parameter vector in which each individual parameter is linked directly to the identified cluster centres and spreads. This vector is subject to further fine-tuning in a bid to improve the model’s predictive performance. denotes the area under over the output interval : , and is calculated using Eq. 5.6.

, (5.6)

Hence, after the first stage, a Singleton/Mamdani FRBS with the pre-specified number of rules is extracted from the numerical data, which is analytical and can be refined further using gradient based techniques, as will be introduced in Section 5.4.

5.3.2 An Example of Application

The benchmark example tested in Section 4.4.3 is employed again to demonstrate the results of the first modelling stage using IMOFM_M. The number of rules is again set to 5. Figure 5.4 shows individual rules of the initial FRBS and the membership functions on each dimension (including the output dimension).

Comparing Figure 5.4 with Figure 4.19, one can find that the premises of Mamdani FRBS and Singleton FRBS for this particular problem are the same since they are all extracted by G3Kmeans. The only difference lies in their consequents. Instead of singleton values, Mamdani FRBS uses fuzzy sets for its consequents as well, which makes Mamdani FRBS more interpretable when compared to the Singleton one. Fuzzy outputs convey vagueness information that is inherent in the model’s knowledge-base and may be well designated by linguistic terms (Mencar et al., 2005).

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Figure 5.4 (a) individual rules in a Mamdani FRBS; (b) membership functions of each

dimension.

Table 5.1 summarised the predictive performance of IMOFM_M and IMOFM_S, which are the average values of 20 independent runs. The results of IMOFM_S are adapted from Table 4.5. The detailed comparison of IMOFM_S and IMOFM_M can be found in Section 6.4.

TABLE5.1

THE PREDICTIVE PERFORMANCES OF THE FIRST MODELING STAGE OF IMOFM_S AND IMOFM_M ON A

NONLINEAR STATIC SYSTEM WITH FIVE RULES

Modeling Methods  The Predictive Performance of Initial FRBS 

RMSE (average)  Std. 

IMOFM_S  0.5954  0 

IMOFM_M  0.6078  0 

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5.4 Second Stage: Refinement of Initial FRBSs

The initial fuzzy model extracted from the first modeling stage is not optimal from two perspectives:

(1) The structure of FRBS is not optimal as far as the interpretability is concerned. As one can see from Figures 4.21 and 5.4, the FRBS elicited from the first modeling stage contains redundant fuzzy sets and rules.

(2) The membership function parameters need to be tuned further as far as the accuracy is concerned.

A constrained BEP algorithm is thus utilised to first improve the accuracy of the initial FRBS so that a ‘vaccine model’ can be obtained for the next operation in the multi-objective optimisation stage. As mentioned by Gonz ́lez et al. (2007), if the initial population can be constructed using some heuristics, e.g. an optimised FRBS in terms of its predictive performance, then many generations of evolutionary search can be saved. The ‘vaccine model’ constructed by the first two stages acts similarly to these heuristics. In the subsequent Sections, the BEP updating formulas for IMOFM_S and IMOFM_M are given. Interested readers are referred to Passino’s book (Passino, 1998, p. 246-252) for the detailed BEP deduction for Singleton FRBS, and to Appendix A for the Mamdani FRBS.