Object-First Incremental Algorithms for Updating Approximations in Multi-Granulation Fuzzy Probabilistic Rough Sets Over Two Universes

Multi-granulation fuzzy probabilistic rough set (MFPRS) model over two universes has various applications in recent years. However, data are typically changing with time in real-life applications. When data changes, people have to spend a lot of unnecessary time to recalculate the approximations in MFPRS over two universes. To address this problem, we design incremental algorithms based on the object-first approach for updating approximations in MFPRS over two universes while the objects are added into or removed from the two universes. Firstly, considering a case where the equivalence class of the object with respect to the fuzzy relation could be empty, we introduce a model of type-empty MFPRS over two universes. Then, based on the matrix, a traditional algorithm is introduced to calculate approximations in MFPRS over two universes. In addition, by optimizing the traditional algorithm, we design the object-first algorithm which could calculate approximations more efficient in MFPRS over two universes. After that, based on the object-first algorithm, four object-first incremental algorithms are designed for updating approximations in MFPRS over two universes while the objects are added into or removed from two universes. Finally, experimental results show that the proposed object-first incremental algorithms for updating approximations in MFPRS over two universes are more efficient in comparison with the non-incremental algorithms and the traditional incremental algorithms.