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Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
A relation on X is called a tolerance relation if it is reflexive and symmetric. A tolerance space (X,τ) (or simply X ) is a set X with a tolerance relation τ on X. If x,x'∈τ, we say x is within tolerance of x' and write xτx';iX=X×X is called the big tolerance and δX={(x,x)∣x∈X} is called the little tolerance on X.
Fuzzy Logic
Published in Paresh Chra Deka, A Primer on Machine Learning Applications in Civil Engineering, 2019
Tolerance relation has only the properties of reflexivity and symmetry. A tolerance relation, R, can be reformed into an equivalence relation by at most (n − 1) compositions with itself, where n is the number of rows or columns of R.
Attribute reduction for set-valued data based on prediction label
Published in International Journal of General Systems, 2023
Taoli Yang, Zhaowen Li, Jinjin Li
An SVDIS refers to an IS with decision attributes. Its attribute values use the possible values of various evaluation indicators in information collection, so its attribute values are sets. At present, many researchers are studying an SVIS. For example, the methods for set-based Granular computing based on an SVIS are proposed (Yao and Liu 1999; Yao 2001). Kryszkiewicz (1998) researched a rough set model based on the tolerance relation. Qian et al. (2009) studied set-valued order IS and a rough set model based on dominance relation. Dai and Tian (2013) advanced the concepts of information entropy and granularity measure in an SVIS. Zhang and Chen (2021) considered the entropy measurement method of an incomplete SVIS considered. Huang et al. (2017) introduced the concept of a probability SVIS and proposed an extended variable precision rough set model based on λ-tolerance relation.
Missing data imputation for traffic flow based on combination of fuzzy neural network and rough set theory
Published in Journal of Intelligent Transportation Systems, 2021
Jinjun Tang, Xinshao Zhang, Weiqi Yin, Yajie Zou, Yinhai Wang
In set X, any member of may be a member of set A and any member of must be a member of set A. If R is a fuzzy tolerance relation, is the decision system and X is a subset of A, then the fuzzy rough upper approximation set and low approximation set of A are defined as: where, I is the implication operator, T is a t-modulus and B is a subset of set A. Attribute set A can be a ordinary set or a fuzzy set.
Feature selection for a set-valued decision information system based on fuzzy rough iterative computation model
Published in International Journal of General Systems, 2022
Feature selection is an effective way to eliminate negative effects caused by redundant features. In the framework of RST, feature selection is also called attribute reduction, which is to find a minimal feature subset that provides the same discriminating information as the whole feature set. Specifically, some features in an IS are redundant. We want to find a reduct that has the fewest features. Feature selection in an IS mean deleting redundant features under keeping the classification ability. The core step of feature selection is to construct the feature evaluation function. This function can be used to select key representative features from high-dimensional data. Thus, feature selection can simplify data and reduce the computational complexity for machine learning. Up to now, there have been many outstanding results. Dai et al. (2013) presented attribute reduction based on conditional entropies for incomplete decision systems. Meng and Shi (2009) proposed a fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets. Yao and Zhang (2017) brought up class-specific attribute reduction in RST. Cornelis et al. (2010) obtained a generalized model of attribute reduction based on fuzzy tolerance relation within the context of fuzzy rough set theory. Wang et al. (2019) constructed fuzzy rough set-based attribute reduction using distance measures. Lang, Li, and Guo (2015) gave attribute reduction in dynamic fuzzy covering information systems based on homomorphism. Z. W. Li et al. (2021) investigated attribute selection for heterogeneous data based on information entropy. Trabelsi and Elouedi (2010) put forward a heuristic method for attribute selection from partially uncertain data using rough sets. You and Li (2011) studied feature selection for multi-class problems by using pairwise-class and all-class techniques.