This work focuses on fuzzy orthopartitions and credal partitions, which are distinct mathematical models representing partitions where the membership of elements to classes is only partially known. Firstly, we show that fuzzy orthopartitions and credal partitions are special cases of generalized fuzzy orthopartitions, which we introduce in this article as a new structure for modelling partitions with uncertainty. Next, we examine the connections between credal partitions and fuzzy orthopartitions, considering that both can be seen as types of fuzzy partitions (in particular, we deal with fuzzy probabilistic and Ruspini partitions). Moreover, we find that each generalized fuzzy orthopartition corresponds to a collection of zero, one, or infinitely many credal partitions; conversely, a credal partition maps to at most one generalized fuzzy orthopartition. Finally, we identify the class of all credal partitions that coincide with fuzzy orthopartitions.

The aim of this article is to study the relationship between two popular Cautious Learning approaches, namely: Three-way decision (TWD) and conformal prediction (CP). Based on the novel proposal of a technique to transform three-way decision classifiers into conformal predictors, and vice versa, we provide conditions for the equivalence between TWD and CP. These theoretical results provide error-bound guarantees for TWD, together with a formal construction to define cost-sensitive cautious classifiers based on CP. The proposed techniques are then applied and evaluated on a collection of benchmark and real-world datasets. The results of the experiments show that the proposed techniques can be used to obtain cautious learning classifiers that are competitive with, and often out-perform, state-of-the-art approaches. Further, through a qualitative medical case study we discuss the usefulness of cautious learning in the development of robust Machine Learning.

We propose a method to approximate Intuitionistic Fuzzy Sets (IFSs) with Shadowed Sets that could be used, in decision making or similar tasks, when the full information about membership values is not necessary, is difficult to process or to interpret. Our approach is based on an information-theoretic perspective and aims at preserving the uncertainty, represented through an entropy measure, in the original IFS by minimizing the difference between the entropy in the input IFS and the output Shadowed Set. We propose three different efficient optimization algorithms that retain Fuzziness, Lack of Knowledge, or both, and illustrate their computation through an illustrative example. We also evaluate the application of the proposed approximation methods in the Machine Learning setting by showing that the approximation, through the proposed methods, of IFS k-Nearest Neighbors is able to outperform, in terms of running time, the standard algorithm.