Granular and Rough Computing: Incremental Development

1
Granular and Rough ComputingIncremental
Development

• Tsau Young (T.Y.) Lin
• tylin_at_cs.sjsu.edu
• Computer Science Department, San Jose State
  University, San Jose, CA 95192,
• and
• Berkeley Initiative in Soft Computing,
  UC-Berkeley, Berkeley, CA 94720

2
Introduction

• The term granular computing is first used
• by this speaker in 1996-97 to label a
• subset of Zadehs
• granular mathematics
• as his research topic in BISC.
• (Zadeh, L.A. (1998) Some reflections on soft
  computing, granular computing and their roles in
  the conception, design and utilization of
  information/intelligent systems, Soft Computing,
  2, 23-25.)

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Granular computing

• IEEE GrC-conference
• http//www.cs.sjsu.edu/grc/.

4
Historical Notes

• 1. Zadeh (1979) Fuzzy Sets and Information
  granularity(about Dempster-Shaffer Theory(DST))

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Notes

• Dempster-Shaffer Theory(DST)
• Note In general, basic probability assignment
  (bpa) ? classical probability(cp)
• but . . .

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Notes

• Dempster-Shaffer Theory(DST)
• 2. Note, but if the given focal elements are
  mutually disjoints, then bpacp. In this case
• 2.1. Bel inner probability(lower bound)
• 2.2. PlOuter probability (upper bound)
• (This is NOT general casesCommon errors)

7
Historical Notes

• 2. Pawlak (1982 Dec)
• 3. Tony Lee (1983 Jan)
• Study of relations via partitions

8
Historical Notes

• Pawlak Rough Sets, Information systems,
  Approximations
• Lee Algebraic Theory of Relational Databases

9
Historical Notes

• 4a T. Y. Lin 1988-89 Neighborhood Systems(NS)
• ( ? a set of general binary
  relations)
• 4b T. Y. Lin (1989) Chinese Wall Security Model
• (A study of non-reflexive, symmetric,
  non-transitive binary relation)
• 5. Stefanowski (1989) about Fuzzified Partitions

10
Historical Notes

• 6. Lin, Qing Liu James Huang (1990)
  Neighborhood system RS)
• 7. Lin (1992)Topological and Fuzzy Rough Sets
• Lin said, Intuitively, closure is smallest
  closed setincorrect.
• This is true for topological spaces, not for
  NS-space

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Historical Notes

• 8. Lin Liu (1993) Operator View of RS and NS
• Important Results
• 8.1. Axiomatic View of RS
• 8.2. clopen space defines a partition

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Historical Notes

• 8.3. Simple Proof of item 2
• Consider connected components(CC) of clopen
  space. From general topology theory,
• 8.3.1. CC is closed.
• 8.3.2. CCs intersect each other only on
  boundary.
• Since CC is also open, and open set has no
  boundary.
• So CC is disjoint and hence is a partition.

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Historical Notes

• 9. Lin Hadjimichael (1996) Non-classificatory
  hierarchy
• This paper builds a tree for nested binary
  relations
• This result is obvious for equivalence
  relations
• but is very hard for general binary
  relations

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Granular computing

• Granulation seems to be a natural
  problem-solving methodology deeply rooted in
  human thinking.

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Granular computing

• Human body has been granulated into head, neck,
  and etc. (there are overlapping areas)
• The notion is intrinsically fuzzy, vague, and
  imprecise.

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Partition theory

• Mathematicians have idealized the granulation
  into
• Partition (at least as far back to Euclid)

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Partition theory

• Mathematicians have
• developed it into
• a fundamental problem solving methodology in
  mathematics.

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Partition theory

• Rough Set community has applied the idea into
  Computer Science with reasonable results, called
• Rough Set Theory(RST)

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Key Views in RST

• 1. GranulationPartition E
• 2. (V, E) Approximation Space
• 3. Representations(Information Systems/Tables)
• 4. Table Processing(Knowledge Processing)
• 4.1. Reducts and Core
• 4.2. Value Reducts(Data Mining)

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Key Views in RST

• 5. Granular/Rough Logic Theory.

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1. GranulationPartition

• A Partition of V

Class B
i, j, k

f, g, h
Class C
Class A
l, m, n
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RS Approximations

Upper approximation
Lower approximation
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Lower/Interior Approximations

• L(X) ?B(p) B(p) ? X (Pawlak)
• I(X) p B(p) ? X (Lin topology based)
• L I in RS theory

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Upper/Closure Approximations

• U(X) ?B(p) B(p) ? X ? ? (Pawlak)
• C(X) p B(p) ? X ? ? (Lin - topology based)
• (A Closure,C(X), may not be closederror in Lin
  1992)
• UC if B is a partition

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Research Issues

• In general case, the coverings((?B(p)) are not
  unique, so Pawlaks notion of Upper/Lower
  approximations need MAX or MIN
• U(X) MIN (?B(p) B(p) ? X ? ? )
• L(X) MAX (?B(p) B(p) ? X)
• Please do some research (see measure theory)

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Research Issues

• From topological space point of view,
• Pawlak Style approximations should be
  replaced by Lins style in Topological View.

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Other Closure approximations

• Cl(X) ?iCi(X) (Sierpenski defined)
• where Ci(X) C(C(C(X)))
• (transfinite steps) Cl(X) is closed.
• This is the closure in classical topology.

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Other Closure approximations

• May consider finite steps Cls
• Clk(X) ?i1k Ci(X)
• Chinese Wall Security Policy Model
• (Consider all Clk, k 1, 2, . . .)

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Representation Theory

• Given a partition(named COLOR)

Class B
i, j, k

f, g, h
Class C
Class A
l, m, n
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Representation Theory

• Review RS approaches
• V the universe, V, of entities
  f,g,h,I,j,k,l,m
• and is partitioned into
• A, B, C equivalence classes

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Representation Theory(information table)

• All classes A, B, C are named Yellow, Red, Blue.
  First tuple means (see next page)
• f is a member of A, so its
• COLOR is Yellow
• Here COLOR is the name of the given Partition
  (equivalence relation)

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Summarized in Table Format
f A Yellow
g A Yellow
h A Yellow
i B Red
j B Red
k B Red
l C Blue
m C Blue
n C Blue
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One columns Information Table
f Yellow The left hand side is a
g Yellow table in which the
h Yellow Universe V of entities
i Red is represented by one
j Red column, called COLOR.
k Red Each row represent
l Blue the color of one entity
m Blue
n Blue
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Representation Theory for granulation

• Lat few pages explain how RS has handled
  Knowledge representations
• We will
• Extend the idea to binary relations

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Representation Theory

• Given a granulation(has overlapping)

Neighborhood B
i, j, k, l
m, n

f, g, h
Neighborhood C
Neighborhood A
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Neighborhood of x, N(x)
x has N(x) Is in N(x) Name of N(x)
f A A U
g A A U
h A A U
i B A, B V
j B B V
k B B V
l C B,C W
m C C W
n C C W
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The Center set of N(x)
x has N(x)
f A f, g, h have the neighborhood A
g A Each of f, g, h is the center of A
h A The set of all centers is called
i B the center set of A and is
j B denoted by C(A)
k B So i,j,k form the center set C(B)
l C l, m, n form the center set C(C)
m C
n C
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Topological Information Table I
x Name of N(x) The left hand side
f U is a table in which
g U the Universe V of entities
h U is represented by one
i V column, called ?????.
j V Each row represent one
k V entity Syntactically is the
l W same as information table
m W However U, V, W are
n W Related
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Binary Relation B1 (approach one)
Name Name The left hand side is a table that
U U Represents the binary relation of the
U V interactions among attribute values
V V (See Lin 1998)
V U (U, V) is in B1, if U??V ? ?
V W
W W
W V (W, V) is in B1, if W??V ? ?
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Topological Information Table II
x Center set Name of C(-) The left hand side
f C(A) U is a table in which the first
g C(A) U row means an entity f is in
h C(A) U a unique center set C(A) and
i C(B) V C(A) has unique name U
j C(B) V
k C(B) V
l C(C) W
m C(C) W
n C(C) W
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Topological Information Table II
x Name of C(-) The left hand side is a table in which
f U the Universe V of entities is represented
g U by one column
h U Each row represents an entity and
i V the name of its center set
j V
k V
l W
m W
n W
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Binary Relation B2 (approach Two)
Name Name The left hand side is a table that
U U represents the binary relation of the
U V interactions among attribute values
V V Lin 2004 (IEEE-CIS news letter)
V U (U, V) is in B2, if A??C(B) ? ?
V W (A is neighborhood of member in U)
W W V is name of C(B)
W V (V, U) is in B2, if B??C(A) ? ?
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B1 and B2

• 1.B1 is symmetric, and B1 does not describe the
  interactions among U, V, W completely.
• 2.B2 may not be symmetric,moreover

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B1 and B2

• If the given granulation under consideration is
  symmetric
• then B2 does describe the interactions among U,
  V, W completely.
• (In fact B2 is the quotient of the granulation)