Fuzzy rough set:
This post explains the definition of fuzzy rough set and the operations of fuzzy rough set. We cover fuzzy lower and upper approximations and provide a fuzzy rough set with example for hybrid data mining.
Fuzzy Rough Set (FRS)
Introduction
A Fuzzy Rough Set (FRS) is a mathematical framework that combines two types of uncertainty. While Fuzzy Sets handle the “gradualness” of belonging to a category, Rough Sets handle the “indiscernibility” between objects. By merging them, FRS allows us to define boundaries of a set using fuzzy membership functions, making it highly effective for feature selection, machine learning, and data mining.
Definition of Fuzzy Rough Set
Given a universe X and a fuzzy similarity relation R, a Fuzzy Rough Set is defined by two fuzzy sets: the Lower Approximation and the Upper Approximation.
For any x ∈ X, the membership degrees are defined as:
Represents elements that “certainly” belong to the fuzzy set.
Represents elements that “possibly” belong to the fuzzy set.
Fuzzy Rough Number (FRN)
An element in a fuzzy rough set is called a Fuzzy Rough Number. It is represented as an interval-like pair of its lower and upper membership degrees: α = [ μα , μ̄α ], where 0 ≤ μα ≤ μ̄α ≤ 1.
Detailed Mathematical Operations
Let α = [ μα , μ̄α ] and β = [ μβ , μ̄β ] be two Fuzzy Rough Numbers. The operations are defined as:
Let α = [ 0.4, 0.7 ] and β = [ 0.5, 0.8 ].
Union (α ∪ β):
• Lower: max(0.4, 0.5) = 0.5
• Upper: max(0.7, 0.8) = 0.8
Result: [ 0.5, 0.8 ]
Intersection (α ∩ β):
• Lower: min(0.4, 0.5) = 0.4
• Upper: min(0.7, 0.8) = 0.7
Result: [ 0.4, 0.7 ]
Complement (αc):
• Lower: 1 – 0.7 = 0.3
• Upper: 1 – 0.4 = 0.6
Result: [ 0.3, 0.6 ]
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator