Rough Sets Theory

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Rough Sets Theory

• Rough Set Theory. Introduction
• Rough Sets and Information Systems
• Approximation of sets
• Undefinable sets
• A Simple Informal Example
• Reducts in Information Systems
• Example
• Summary

Kersti Antoi
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Rough Sets Theory. Introduction
The rough sets theory is a mathematical tool to
deal with vagueness and uncertainty.

• Knowledge has granular structure.
• Some objects of interest cannot be discerned and
  appear as the same (or similar).
• Objects characterised by the same information are
  indiscernible
• Any set of all indiscernible objects is called
  elementary set.
• Any union of some elementary sets is referred to
  as precise set - otherwise a set is rough
  (imprecise, vague).

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Rough Sets Theory. Introduction
In the proposed approach is any vague concept
replaced by a pair of precise concepts - called
the lower and the upper approximation
The lower approximation consists of all objects
that surely belong to the concept
The upper approximation contains all objects that
possibly belong to the concept
The boundary region (doubtful region) of the
vague concept is the difference between the upper
and the lower approximation constitute
Approximations are two basic operations in the
rough set theory
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Rough Sets Theory. Introduction
The basic operations of the rough set theory are
used to discover fundamental patterns in data.

• The main specific problems addressed by this
  theory are
• representation of uncertain or imprecise
  knowledge
• empirical learning and knowledge acquisition from
  experience
• knowledge analysis
• analysis of conflicts
• identification and evaluation of data
  dependencies
• approximate pattern classification
• reasoning with uncertainty
• information-preserving data reduction

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Rough Sets Theory. Introduction

• The important application areas for rough sets
  are
• medical diagnosis
• pharmacology
• stock market prediction and financial data
  analysis
• banking
• market research
• information storage and retrieval systems
• pattern recognition, including speech and
  handwriting recognition
• control system design
• image processing
• digital logic design
• and many others

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Rough Sets and Information Systems
Information system S ?U, A, V, f? U universe
of S-elements (nonempty, finite set of objects) A
(finite) set of attributes V domain of the
attribute a (set of values of attributes) f
description/information function (f U?A ? V,
f(x, a) ? Va, ?x ? U, ?a ? A)
Indiscernible IND(P) P-elementary sets
equivalence classes of IND(P) DESp(X)?(a,
v)f(x, a)v, ?x?X, ?a?P?
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Rough Sets and Information Systems
Let P?A and Y?U. The P-lower approximation of Y,
denoted by PY and the P-upper approximation of Y,
denoted by?PY, are defined as
PY ??X?U P X?Y? ?PY ??X?U P X?Y???
The P-boundary (doubtful region) of set Y is
defined as
BnP(Y) ?PY - PY
Set PY is the set of all elements of U which can
be certainly classified as elements of Y,
employing the set of attributes P. Set ?PY is the
set of elements of U which can be possibly
classified as elements of Y, using the set of
attributes P. The set BnP(Y) is the set of
elements which cannot be certainly classified to
Y using the set of attributes P.
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Rough Sets and Information Systems
Undefinable sets

• Set Y is definable in P iff PY ?PY, otherwise
  set Y is undefinable in P.

• If PY ? ? and ?PY ?U, Y will be called roughly
  definable in P.

• If PY ? ? and ?PY U, Y will be called
  externally undefinable in P.

• If PY ? and ?PY ? U, Y will be called
  internally undefinable in P.

• If PY ? and ?PY U, Y will be called totally
  undefinable in P.

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Rough Sets and Information Systems
With every set Y?U, we can associate an accuracy
of approximation defined as
Subsets Yi , i1,...,n are categories of
partition Y. By P-lower (P-upper) approximation
of Y in S we mean sets PY?PY1, PY2, ..., PYn?
and PY??PY1,?PY2, ...,?PYn?, respectively.
The coefficient
is called the quality of approximation of
partition Y by a set of attributes P (quality of
sorting).
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A Simple Example
Name Education Descision Joe High
School No Mary High School Yes Peter Elementary
No Paul University Yes Cathy Doctorate Yes
O Mary, Paul, Cathy A Education R(A)
Joe, Mary, Peter, Paul,
Cathy POS(O) LOWER(O) Paul,
Cathy NEG(O) Peter BND(O) Joe,
Mary UPPER(O) POS(O) BND(O) Paul, Cathy,
Joe, Mary
(Education, University) or (Education, Doctorate)
--gt Good prospects (Education, Elementary) --gt
No good prospects (Education, High School) gt
Good prospects (i.e. possibly)
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Reducts in Information Systems
Set of attributes P is independent if for every
proper subset Q of P, that is Q ? P,
IND(P) ? IND(Q), otherwise P is dependent (in S).
A subset P ? Q ? A is a reduct of Q (in S) if P
is independent subset of Q and IND(P)  IND(Q).
For every P ? A, RED(P) ? ?, and if P is
independent then RED(P)  ?P?.
An element p ? P is said to be dispensable for P
if IND(P)  IND(P-?p?), otherwise an element p
is indispensable.
The set of all indispensable elements for P is
said to be a core of P and denoted by CORE(P).
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Example
Let U consist of five elements denoted by t1...t5
and let A  ?a, b, c, d, e?, Va  ?0, 1?,
Vb  ?0, 2?, Vc  ?1, 2, 3?, Vd  ?1, 3?,
Ve  ?0, 1, 2, 3?.
All partitions in this system are ?  ?t1, t2, t
3, t4, t5? a  ??t1, t5?, ?t2, t3, t4?? b   ??t
1, t3?, ?t2, t4, t5?? c   ??t1, t3?, ?t2, t4?, ?
t5?? d  ??t1, t3, t5?, ?t2, t4?? e  ??t1, t5?,
 ?t2?, ?t3?, ?t4?? (ab)  ??t2, t4?, ?t1?, ?t3?, 
?t5?? (ad)  ??t2, t4?, ?t1, t5?, ?t3?? (be)  ?
t1?, ?t2?, ?t3?, ?t4?, ?t5??
The table gives an information function
U a b c d e t1 0 2 1 3 1 t2 1 0 2 1 2 t3 1 2 1 3 3
t4 1 0 2 1 0 t5 0 0 3 3 1
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Example
This information system has nine different
indiscernibility relations IND(?), IND(a), IND(b)
, IND(c) ( IND(bc)  IND(bd)  IND(cd) 
IND(bcd)), IND(d), IND(e) ( IND(ae)  IND(de)  I
ND(ade)), IND(ab) ( IND(ac)  IND(abc)  IND(abd)
  IND(acd)  IND(abdc)), IND(ad), IND(A) IND(be)
  IND(ce)  IND(abc)  IND(ace)  IND(bce)  
IND(cde) IND(abce) IND(bcde)
IND(acde) IND(abde))
RED(A)  ?be, ce? and CORE(A)  e
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Summary
To remember

• A rough set is a set defined only by its lower
  and upper approximation. A set, O, whose boundary
  is empty is exactly definable.

• If a subset of attributes, A, is sufficient to
  create a partition R(A) which exactly defines
  set of objects, then we say that A is a reduct.

• The intersection of all reducts is known as the
  core.

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References

• Articles
• Rough Classification. Z. Pawlak, Polish Academy
  of Sciences (1983)
• Reducts in Information Systems. C. M. Rauszer,
  Uof Warsaw (1991)
• Variable Precision Extension of Rough Sets. J. D.
  Katzberg, W. Ziarko, University of Regina,
  Saskatchewan, Canada (1996)
• Rough Concept Analysis a Synthesis of Rough Sets
  and Formal Concept Analysis. R. E. Kent,
  University of Arkansas at Little Rock, USA (1996)

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