Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances that are characterized by vagueness and uncertainty. Rough set theory uses a table called an information system, and knowledge is defined as classifications of an information system. In this paper, we introduce the concepts of information entropy, rough entropy, knowledge granulation and granularity measure in incomplete information systems, their important properties are given, and the relationships among these concepts are established. The relationship between the information entropy E(A) and the knowledge granulation GK(A) of knowledge A can be expressed as E(A)+GK(A) = 1, the relationship between the granularity measure G(A) and the rough entropy E _r(A) of knowledge A can be expressed as G(A)+E _r(A) = log₂|U|. The conclusions in Liang and Shi ( 2004 Liang, J.Y. and Shi, Z.Z. 2004. The information entropy, rough entropy and knowledge granulation in rough set theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12(1): 37–46. [Crossref], [Web of Science ®] [Google Scholar] ) are special instances in this paper. Furthermore, two inequalities − log₂ GK(A) ≤ G(A) and E _r(A) ≤ log₂(|U|(1 − E(A))) about the measures GK, G, E and E _r are obtained. These results will be very helpful for understanding the essence of uncertainty measurement, the significance of an attribute, constructing the heuristic function in a heuristic reduct algorithm and measuring the quality of a decision rule in incomplete information systems.

Attribute subset selection is an important issue in data mining and information processing. However, most automatic methodologies consider only the relevance factor between samples while ignoring the diversity factor. This may not allow the utilization value of hidden information to be exploited. For this reason, we propose a hybrid model named intuitionistic fuzzy (IF) rough set to overcome this limitation. The model combines the technical advantages of rough set and IF set and can effectively consider the above-mentioned statistical factors. First, fuzzy information granules based on IF relations are defined and used to characterize the hierarchical structures of the lower and upper approximations of IF rough set within the framework of granular computing. Then, the computation of IF rough approximations and knowledge reduction in IF information systems are investigated. Third, based on the approximations of IF rough set, significance measures are developed to evaluate the approximation quality and classification ability of IF relations. Furthermore, a forward heuristic algorithm for finding one optimal reduct of IF information systems is developed using these measures. Finally, numerical experiments are conducted on public datasets to examine the effectiveness and efficiency of the proposed algorithm in terms of the number of selected attributes, computational time, and classification accuracy.

Multi-modal classification (MMC) aims to integrate the complementary information from different modalities to improve classification performance. Existing MMC methods can be grouped into two categories: traditional methods and deep learning-based methods. The traditional methods often implement fusion in a low-level original space. Besides, they mostly focus on the inter-modal fusion and neglect the intra-modal fusion. Thus, the representation capacity of fused features induced by them is insufficient. The deep learning-based methods implement the fusion in a high-level feature space where the associations among features are considered, while the whole process is implicit and the fused space lacks interpretability. Based on these observations, we propose a novel interpretative association-based fusion method for MMC, named AF. In AF, both the association information and the high-order information extracted from feature space are simultaneously encoded into a new feature space to help to train an MMC model in an explicit manner. Moreover, AF is a general fusion framework, and most existing MMC methods can be embedded into it to improve their performance. Finally, the effectiveness and the generality of AF are validated on 22 datasets, four typically traditional MMC methods adopting best modality, early, late and model fusion strategies and a deep learning-based MMC method.

Ordinal classification is an important classification task, in which there exists a monotonic constraint between features and the decision class. In this paper, we aim at developing a method of fusing ordinal decision trees with fuzzy rough set based attribute reduction. Most of the existing attribute reduction methods for ordinal decision tables are based on the dominance rough set theory or significance measures. However, the crisp dominance relation is difficult in making full use of the information of attribute values; and the reducts based on significance measures are poor in interpretability and may contain unnecessary attributes. In this paper, we firstly define a discernibility matrix with fuzzy dominance rough set. With this discernibility matrix, multiple reducts can be found, which provide multiple complementary feature subspaces with original information. Then diverse ordinal trees can be established from these feature subspaces, and finally, the trees are fused by majority voting. The experimental results show that the proposed fusion method performs significantly better than other fusion methods using dominance rough set or significance measures.