The research carried out in this thesis aims to improve the quality of FRBCSs. To do so, we have developed both a theoretical and a practical part.
The generalization of the FRM of the Chi et al. algorithm by the Choquet integral led to an increase of the system’s performance. In this thesis, we intended to apply this methodology
in the FRM of the FARC-HD fuzzy classifier. This classifier is considered as one of the most interpretable and accurate fuzzy classifiers nowadays and we tried to enhance its performance.
We have followed an incremental research methodology. Which means that we started from generalizations with averaging characteristics (delimited by the maximum of the elements to be aggregated) and went to generalizations with non-averaging characteristics (not limited by their maximum). The mentioned incremental line, as will be seen in the next paragraphs.
To start, a first generalization was constructed by the replacement of the product of the standard Choquet integral by different t-norms. This generalization was supported by an important theoretical concept that we introduced: the pre-aggregation functions. Differently of a standard aggregation function, pre-aggregation functions are monotone along some di- rection, being an important contribution in the field of aggregation operators. We noticed that this first generalization, produced averaging functions and, when used to cope with clas- sification problems, enhanced the performance of the classifier. Thus, we continued this line of work.
At this point, we were aiming at producing a generalization of the Choquet integral that resulted in aggregation functions. To do so, we used the distributivity property of the product used in the Choquet integral, which was called, the Choquet integral in its expanded form. Then, we generalized this expanded form by copulas, introducing the concept of Choquet- like Copula-Based aggregation functions (CC-integral for short). These CC-integrals are averaging generalizations of the expanded Choquet integral. However, we demonstrated that they could produce results that could compete with the pre-aggregations and could be even more accurate than the standard Choquet integral or than the classical FRM of the Winning Rule.
In the previous step, we introduced the CC-integrals. One of them was based on a copula that uses anα parameter, CαC-integral. Then, at this point, we also introduced a methodology to tune this parameter by adapting the evolutionary part of the FARC-HD algorithm. The
CαC-integral is a CC-integral, consequently, is also an averaging operator. We showed in the experimental, that this approach is also able to increase the performance of the classifier.
Up to this point, we have only presented generalizations of the Choquet integral with av- eraging characteristics. However, the state-of-the-art fuzzy classifiers use a non-averaging approach. Thus, in order to produce even more competitive generalizations, we introduced
the family of left 0-absorbing fusion functions F. Additionally, the generalization of the standard Choquet integral by a function F introduced the concept of CF-integrals. These functions are averaging or non-averaging, it depends on the considered functionF that gener- alizes the Choquet integral. We demonstrated that the averagingCF-integrals present good results when compared with another averaging operators. Furthermore, the non-averaging
CF-integrals were comparable with classical non-averaging operators. Finally, we showed that the non-averagingCF-integrals statistically overcame the averaging ones, reinforcing the idea that not being limited by the maximum is a good option to tackle classification problems.
The summit of our generalizations was reached when we generalized the extended Choquet integral by two functions F1 and F2. The result of this generalization was named CF1F2- integrals. These functions are Ordered Directional increasing functions (OD increasing) and, therefore, represent a different level of aggregation operators. We showed a methodology to select different functions as F1 and F2, based on the concept of dominance and strength
degrees. Then, for the considered CF1F2-integrals we demonstrated that in five different combinations ofF1 andF2 we produced generalizations that are equivalent, or even superior,
than classical fuzzy classifiers like FARC-HD, IVTURS and FURIA.
Finally, we draw the main conclusions of the thesis related to our initial objectives:
◦ We have applied the developed generalizations of the Choquet integral in the FRM of the FARC-HD fuzzy classifier, which is one of the most accurate and interpretable fuzzy classifier nowadays, and we enhanced its performance. We must point out, that this was the main objective of this thesis.
◦ The generalizations were constructed by replacing the product operator of the original Choquet integral and its extended form by different aggregation functions. This allowed us to define important concepts in the field of aggregation operators, like:
CT-integrals: Generalizations of the original Choquet integral by t-norms.
CC-integrals: Generalizations of the extended Choquet integral by copulas.
CF-integrals: Generalizations of the standard Choquet integral by functionsF. Where
F, is a family of fusion functions that we have introduced.
CF1F2-integrals: Generalizations of the extended Choquet integral by two functions,
F1 and F2.
◦ We have introduced a methodology to adapt the evolutionary part of the FARC-HD to learn a fuzzy measure which is adapted for each class of the problem. In this way we
increased the performance of the system.
◦ The developed generalizations presented both averaging and non-averaging behavior. In this way, we compared them against classical FRMs presented in the literature and state-of-the-art fuzzy classifiers.
– When comparing our averaging methods against the classical averaging FRM of WR we have that most of the developed generalizations are superior in terms of performance.
– Comparing the generalizations having non-averaging characteristics versus state- of-the-art fuzzy classifiers, we have that the generalizations are equivalent, or even superior, than the considered methods.