In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets.
The following section contains an overview of the basic framework of rough set theory, as originally proposed by Zdzisław I. Pawlak, along with some of the key definitions. More formal properties and boundaries of rough sets can be found in Pawlak (1991) and cited references. The initial and basic theory of rough sets is sometimes referred to as "Pawlak Rough Sets" or "classical rough sets", as a means to distinguish from more recent extensions and generalizations.
Let be an information system (attribute–value system), where is a non-empty, finite set of objects (the universe) and is a non-empty, finite set of attributes such that for every . is the set of values that attribute may take. The information table assigns a value from to each attribute and object in the universe .
With any there is an associated equivalence relation :
The relation is called a -indiscernibility relation. The partition of is a family of all equivalence classes of and is denoted by (or ).
If , then and are indiscernible (or indistinguishable) by attributes from .
The equivalence classes of the -indiscernibility relation are denoted .
For example, consider the following information table:
{| class="wikitable" style="text-align:center; width:30%" border="1"
|+ Sample Information System
! Object !! !! !! !! !!
|-
!
| 1 || 2 || 0 || 1 || 1
|-
!
| 1 || 2 || 0 || 1 || 1
|-
!
| 2 || 0 || 0 || 1 || 0
|-
!
| 0 || 0 || 1 || 2 || 1
|-
!
| 2 || 1 || 0 || 2 || 1
|-
!
| 0 || 0 || 1 || 2 || 2
|-
!
| 2 || 0 || 0 || 1 || 0
|-
!
| 0 || 1 || 2 || 2 || 1
|-
!
| 2 || 1 || 0 || 2 || 2
|-
!
| 2 || 0 || 0 || 1 || 0
|}
When the full set of attributes is considered, we see that we have the following seven equivalence classes:
Thus, the two objects within the first equivalence class, , cannot be distinguished from each other based on the available attributes, and the three objects within the second equivalence class, , cannot be distinguished from one another based on the available attributes.
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Learn to optimize on smooth, nonlinear spaces: Join us to build your foundations (starting at "what is a manifold?") and confidently implement your first algorithm (Riemannian gradient descent).
Dans une première partie, nous étudierons d’abord comment résoudre de manière très concrète un problème au moyen d’un algorithme, ce qui nous amènera dans un second temps à une des grandes questions d
Dans une première partie, nous étudierons d’abord comment résoudre de manière très concrète un problème au moyen d’un algorithme, ce qui nous amènera dans un second temps à une des grandes questions d
This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous
Granular computing is an emerging computing paradigm of information processing that concerns the processing of complex information entities called "information granules", which arise in the process of data abstraction and derivation of knowledge from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like.
Type-2 fuzzy sets and systems generalize standard Type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Lotfi A.
La théorie des sous-ensembles flous est une théorie mathématique du domaine de l’algèbre abstraite. Elle a été développée par Lotfi Zadeh en 1965 afin de représenter mathématiquement l'imprécision relative à certaines classes d'objets et sert de fondement à la logique floue. Les sous-ensembles flous (ou parties floues) ont été introduits afin de modéliser la représentation humaine des connaissances, et ainsi améliorer les performances des systèmes de décision qui utilisent cette modélisation.
Explore les limites et les limites dans les catégories de functeurs, en mettant l'accent sur les égaliseurs, les retraits et leur importance dans la théorie des catégories.
Déplacez-vous dans le calcul et la réalisation géométrique de petites catégories, explorant la relation entre les nerfs et les structures géométriques.
We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive (d-6)\documentclass[12pt]{minimal} \usepackage{ ...
In this note, we study certain sufficient conditions for a set of minimal klt pairs ( X, triangle) with kappa ( X, triangle) = dim( X ) - 1 to be bounded. ...
In this article, we propose a dynamical system to avoid obstacles which are star shaped and simultaneously converge to a goal. The convergence is almost-global in a domain and the stationary points are identified explicitly. Our approach is based on the id ...