202-207.
Neighborhoods Systems: Measure,
Probability and Belief Functions
T.Y.Lin1 2 *
tylin@cs.sjsu.edu Y, Y, Yao
yyao@flash.lakeheadu.ca
1 Berkeley Initiative in Soft Computing, Department of
Electrical Engineering and Computer Science, University of
California, Berkeley, California 94720
2 Department of Computer Science, Lakehead
Univeristy Thunder Bay, Ontario, Canada P7b
5E1
Abstract: The notion of neighborhood system is a
mathematical formalism for “negligible quantity.” It formulate the mathematical concept of neighborhoods in the
context of advanced computing. Neighborhood systems, by definition, include topology (topological neighborhood
systems), rough sets (S5 -neighborhood systems) and binary relations (basic neighborhood systems). In this
paper, real valued functions based on neighborhood systems are studied. The study covers many important
quantities in uncertainty, such as belief functions, measure, and probability; fuzzy sets are not included here,
because it was reported elsewhere. It seems that neighborhood systems are an effective underlying data
structure for managing uncertainty.
Keywords: binary relation, measure, neighborhood, probability, qualitative fuzzy set, rough set, topology.
---------------- * This research is partially supported by Electric Power Research Institute, Palo Alto,
California
* On leave from San Jose State University (tylin@cs.sjsu.edu)
uncertainty from a geometric point of view,
namely, neighborhood systems. Roughly, a
1.Introduction
neighborhood system assigns each object a
Representing and measuring uncertainty are critical
(possibly empty, finite, or infinite) family of
non-empty subsets. Such subsets, called
in advanced computing. Various theories have
neighborhoods, represent the semantics of
been proposed. The most notable theories are
“negligible,” which is the essence of
Zadeh’s fuzzy theories [24] and Pawlak’s
uncertainty handling: neglect the negligible.
rough set theory [1 6]. In this paper, we discuss
Using neighborhoods, one can define open
one of the most encompassing notions of
sets, hence the interior and closure of any
subset (as in topological spaces) [6], [19],
[20]. By taking equivalence classes as
neighborhoods, the lo wer and upper
approximations are precisely the interior and
closure, respectively. Rough set approximation
is a special for m of neighborhood theor y; it
was called cate gory [2]. Generalized rough sets
based on various modal logic [21] are all
special forms of neighborhood systems, called
basic neighborhood systems. The notion of
neighborhood systems is one of the ‘correct’
mathematical formalisms for expressing the semantics of
approximation and uncertainty in the context of advanced
computing. Our interests stemmed from database retrieval and
mining [15], [3], [7], [8] [9], [2], [10], [21]. One may view it
as a first step toward the granulate mathematics described by
Lotfi Zadeh [23]. Some basic notions and its application to
qualitative fuzzy theory was reported in [13], [14]. In this
paper, we turn out attention to the quantitative aspect of
uncertainty. One could develop a full fledge measure and
probability theory as in classical mathematics [5]. At this
early stage, we focus on its applications; belief functions are
formulated using the notion of neighborhood systems [18].
2. Neighborhood Systems
Since the systematic study of neighborhood system
is relatively recent, we recall some of our motivation
from [14]. Neighborhood systems are abstracted from
numerical analysis. In any standard procedure of
finding approximate solutions, the very first step is to
choose a
“small” number e, or equivalently, an e-neighborhood for each point on the
real line. During the process of finding
approximate solutions, this particular family of
neighborhoods never changes. In other words, the only
relevant notion of the real line topology [6] is this particular
family of chosen neighborhoods. We can view the first step
as the step of setting up a proper context for discussions, that
is, one step up the “standard” for what it means by “near” for
this special circumstance. Such a family of chosen
neighborhoods, not the full topology, is the essential
formalism for approximation. The neighborhood is a
fundamental notion in mathematical analysis. It is also a
common notion in many other areas. It appears in logic [1],
in a text of genetic algorithm [4], rough sets [16], generalized
rough sets [21], and databases. A systematic study in the
context of advanced computing was started by the first author
and his students. The study was motivated from database
retrieval and mining [15], [3], [7], [8] [9], [2], [10].
2.1 Definitions and
Properties
In this section, we recall some notions of
neighborhood system from [14]. Let U be the universe of discourse
and p be an object or point in U. 1. A neighborhood, denoted by
N(p), or simply N, of p
is a no n-empty subset o f U, which may or may
not contain the object p. Any subset that
contains a ( non -empty) neighborhood is a
neighborhood.
2. A neighborhood system of an object p, denoted by NS(p), is a
maximal family of neighborhoods of p. If p has no neighborhood,
then NS(p) is an empty family; in this case, we simply say that p
has no neighborhood.
3 .
A n e i g h b o r h o o d s y s t e m o f
U , d e n o t e d b y N S ( U ) i s t h e
c o l l e c t i o n o f N S ( p ) f o r a l l p
i n U . S u c h a n e i g h b o r h o o d
s y s t e m m a y a l s o b e c a l l e d
F - t o p o l o g y ( r e a d a s f i n i t e
t y p e t o p o l o g y ) . F o r
s i m p l i c i t y a s e t U t o g e t h e r
w i t h N S ( U ) i s c a l l e d a
n e i g h b o r h o o d s y s t e m s p a c e o r
a n e i g h b o r h o o d s y s t e m . A
n e i g h b o r h o o d s y s t e m i s c a l l e d
a F r e c h e t ( V ) s p a c e , i f e v e r y
N S ( p ) i s n o n - e m p t y [ 1 9 ] .
4. N is open, if N is a neighborhood of every object in N.
5. More generally, a subset X of U is open if for every object in X,
there is a neighborhood N(p) ˝ X. A
subset X is closed if its complement is open. 6.
NS(p) and NS(U) are open if every neighborhood is
open. NS(U) is topological, if U is the usual
topological space [6]. Both NS(U) and the
collection of open sets is called topology if U is a
topological space.
7. Let X be a subset of U. The lower approximation of X
can be defined by
I[X] = { p: there is a N(p) ˝ X} = interior of X,
that is, I[X] is the largest open set contained in X. 8.
Similarly, the upper approximation of X can be
defined by ( 0 is the empty set)
C[X] = {p: " N(p), X ˙ N(p) „ 0}= closure of X. that is, C[X]
is the smallest closed set contains X.
I[X] and C[X] are precisely the lower and upper approximation
in rough set theory.
9. A topological space is a neighborhood system
space, but not the converse.
10 . Intersectio ns and finite unio ns o f clo sed sets
are clo sed .
11. In topological spaces, unions and finite
intersections of open sets are open. In neighborhood
systems, unions is open, but intersections may not be
open.
2.2 Basic Neighborhoods and Binary Relations
13. A minimal neighborhood of p, denoted by MN(p), is
a m i n i m a l m e m b e r o f N S ( p ) i n
t h e s e n s e t h a t M N ( p ) c o n t a i n s
n o m e m b e r o f N ( p ) a s p r o p e r
s u b s e t s . I n g e n e r a l s u c h
M N ( p ) m a y o r m a y n o t e x i s t .
T h e m a x i m a l f a m i l y o f a l l
M N ( p ) a t p wi l l b e d e n o t e d b y
M N S ( p ) . T h e f a m i l y o f
M N S ( p ) f o r a l l p wi l l b e
d e n o t e d b y M N S ( U ) . L e t n ( p )
b e t h e n u m b e r o f ( d i s t i n c t )
M N S ( p ) ’ s a t p . I f , f o r a l l p ,
n ( p ) = n i s a c o n s t a n t i n t e g e r ,
M N S ( U ) i s a n n - m i n i m a l
n e i g h b o r h o o d s y s t e m , a n d
d e n o t e d b y n - M N S ( U ) ; we wi l l
b e i n t e r e s t e d i n 1 - m i n i m a l
n e i g h b o r h o o d s y s t e m s , c a l l e d
b a s i c ( b i n a r y ) n e i g h b o r h o o d
s y s t e m B S ( U ) . B S ( U ) c a n b e
d e f i n e d b y a b i n a r y r e l a t i o n
a n d v i c e v e r s a - s e e b e l o w. S o
“ B ” i n B S ( U ) m a y b e r e f e r r e d
t o a s
a b a s i c n e i g h b o r h o o d
o r a b i n a r y n e i g h b o r h o o d .
14. Let R be a binary relation defined on U, then
B(p) = {x : pRx}
is a neighborhood of p. So a binary relation R gives rise
to a basic (binary) neighborhood system. Conversely, one
can use the basic neighborhoods to define the binary
relation. From the implementation point of view, we can
rephrase basic neighborhood systems as follows:
15. A basic neighborhood system BS(U) is a data
structure that assigns to each datum a list of data.
Since E-F = (E’¨ F)’), it follows that every algebra
is a ring.
16. Given a neighborhood system NS(p) at p. A
3. A s-Ring(or s-Boolean Ring) of sets is a non-empty class
minimal member of NS(p) may or may not exit. For S of sets such that
example, a neighborhood system of a real number
(a) if E ˘ S and F ˘ S, then E - F ˘ S (b) if Ei ˘ S then ¨i{Ei | I =1, 2,…} ˘ S
has no minimal neighborhood.
17. A binary relation on U defines one and only
basic (binary) neighborhood system; they are
summarized in the table below [3].
A s-algebra is s-ring containing U. We are interested in
finite universes, s-Ring (sBinary
Relatio
Basic (Binary)
Algebra) is the same as Ring and Algebra. 4. A
Relations
nships
Neighborhoods
non-empty class H of sets is hereditary if,
reflexive,transitive
« S4 , (topological)
H(E),
m *(E ) ” inf {S n
m( En ) | E ˝ ¨n
En andEn ˘ R},
then m * is an outer measure on H(R); if m is
equivalence « clopen topology, S 5 ,
Similar ly the n -grad ed b inar y relatio ns [ 21]
f i n i t e o r s - fi n i t e s o i s m *, wh e r e i n f i s t h e
correspo nd to n -minimal neighbo r hood systems. l e a s t u p p e r
bound. We are interested in finite universes only,
3. Measure and Probability
so the infinite sum is in fact a finite sum.
Let U be the universe, we will be interested in the
6.
Let m be a measure on a ring R. For every E in
following notions [5]. 1. A ring (or Boolean ring)
H(E),
of sets is a non-empty
we define
class R of sets such that if
m * (E ) ” sup {S n
m( En ) | E ° ¨n
En andEn ˘ R},
E˘R and F˘R, then E¨F ˘Rand E-F˘R
Then m * is called an inner measure induced by m,
In other words, a ring is a non -empty class of sets
where sup is greatest lower bound. Infinite sum is finite;
which is closed under the formation of unions and
see item 4.
differences.
Let E be a class of sets. It is not difficult to show that
7.
From [5], pp. 50, item 5 can be improved to
there exists a unique ring R(E), the smallest ring
m *(E ) ” inf { m ( F)|E ˝FandF˘R},
containing E; it will called the ring generated by E.
S i n c e i t i s fi n i t e S( R ) = R a n d e x t e n d e d
me a s u r e i s m i t s e l f.
2. An algebra (or Boolean algebra) of sets is
a nonempty class R of sets such that
4. Borel Sets for Neighborhood Systems
(a) if E ˘ R and F ˘ R , then E¨F ˘ R (b) if E ˘ R, then
8. Traditionally a Borel set is defined on a topological
E’
space, we will extend it to a neighborhood system space.
Let C be the class of all compact and closed
sets. As usual the Borel set is the s-ring generated
wheneverE˘SandF˝ E,thenF˘S.ThepowersetofXisahereditaryclass.For serial « serial reflexive « refle
everyringR,H(R)isthesmallesthereditaryring
symmetric « symmetric symm
generated by R. 5. A measure is an extended real Euclidean fi open
valued , non-negative,
transitive « transitive Euclidean «
and countably additive set function m, defined
reflexive, symmetric « Brower
on a ring, and such that m(0)=0. If m is a
measure on a ring R and if, for every E in
by the class of all compact and closed sets; it will be
denoted by BOrel(U).
9. Since we consider finite neighborhood systems
only, all closed sets are compact; and s-ring is a ring; a
s-algebra is an algebra.
1 0 .
B O r e l ( U )
i s
a n
a l g e b r a
g e n e r a t e d
b y
c l o s e d
s e t s ;
B O r e l ( U ) i s a n a l g e b r a
g e n e r a t e d
b y
o p e n
s e t s .
1 1 .
I f
U
i s
S 5 -neighborhood system space, then BOrel(U) is
the collections of all definable sets, namely, finite
unions of equivalence classes; definable sets are the
clopen sets.
P
r
o
p
o
s
i
t
i
o
n
1
.
L
e
t
U
b
e
a
f
i
n
i
t
e
S
5 - n e i g h b o r h o o d
s y s t e m s p a c e a n d
m
i s
a
m e a s u r e
( f o r
e x a m p l e ,
t h e
counting measure that is the cardinal number of a finite
set) on BOrel(U). Then the outer measure and inner
measure
m*(E ) = m (C(E ))
m* ( E ) = m ( I ( E ) )
a r e
t h e
m e a s u r e
o f
l o w e r
a n d
u p p e r
a p p r o x i m a t i o n .
C o r o l l a r y
2 .
U
i s
a
f i n i t e
S 5 -neighborhood system space and m is
a measure. r(E)=m(E)/TOTAL, where TOTAL= m(U). Then P
is a probability measure and its
outer and inner probability measure are the probability of
upper and lower approximation; they are belief and
plausibility functions respectively; see [17]
r is important measure, so we will give a formal definition in next.
12. Let U be a finite se t and m is the counting
measure. For simplicity, the probability measure
r=m(E)/TOTAL will be called counting probability
measure.
1 3 . I f U i s S 4 - n e i g h b o r h o o d s ys t e m s p a c e , t h en
B O r el ( U) i s t h e c l a s s o f a l l f i n i t e , d i sj o i n t
unions of
proper differences o f sets of clo sed sets, and
BOrel(U) is the class o f all finite, disjoint unio ns
of
proper differences of sets of closed
sets.
Proposition 3. U is a finite
S 4 -neighborhood system space and m is a measure
(for example, the counting
measure that is the cardinal number of a finite set) on
BOrel(U). Then the outer measure and inner measure
m * (E)=sup{m(F ) | E ° F and F are closed},
m*(E ) = inf { m * (F ) | E ˝ F and Fare open},
5. Belief Functions
Let us recall some notions from [18]. Let U be a finite
set and POwer(U) be its power set. If we use a full word
as a notation, we cap the first two characters; so that one
can distinguish between notations and words.
14. Belief function: A unit interval valued
function
Bel : POwer(U) fi [0, 1]
is called a belief function if
(a)
Bel (0 )=0 (b)
Bel(U)=1
(c)
For every finite collection
E j , j=1,2,…n of subsets of U,
Bel(¨j n Ej ) ‡ Ss n (-1)t Bel ( Es)
w h e r e s r e p r e s e n t a l l
p o s s i b l e f i n i t e s u b s e t s
o f { 1 , 2 , . . n } a n d t i s t h e
n - | s | + 1 , w h e r e | s | d e n o t e
t h e c a r d i n a l n u m b e r o f s .
Bel can be constructed from basic
probability (see next item):
Bel (A) = S m(B),
where B varies through all subsets of A.
15. Basic probability: A unit interval valued
function
m : POwer(U) fi [0, 1]
is called basic probability if
(a) m(0) =0 (0 is also used to denote empty set) (b)
S n
m(En ) =1, where E n varies through POwer(U).
This definition is somewhat deceiving, what we really
have here is a generalization of probability mass function
from points to subsets. If we assign basic probability to
each basic neighborhood (and zero to all other sets), we
get immediately a belief function on U. More generally if
we assign basic probability to all minimal neighborhoods
(and zero to all other sets), we again get a belief
function on U. Neighborhood systems (of finite space)
are the most natural underlying structure for belief
functions. Conversely, if a space has a belief function,
then there is a very natural neighborhood system
associated to the belief function: Given a
idefined by LEarn(x)
, w h e r e C is the unique basic neighborhood of x, i
belief function, there is a basic probability. The
=C
equivalently the concept learned by x. Some
collection of sets on which the basic probability
comments are in order. For each x there is a unique
are non-zero is a neighborhood system. Namely,
basic neighborhood, so LEarn is a well defined map
we have the following:
(we also consider multi-valued learning [11]).
Propositon 4. U is a finite space. U has a belief
The map LEarn gives rise to a partition on U (its
function iff there is a neighborhood system on U.
quotient set is isomorphic to COncept). The probabilit
In the next few paragraphs, we describe some
measure P_m on POwer(COncept)
specific examples on previous theorem. 16. In
induces a probability measure r_m on U as follows:
[11], Lin and Hadjimichael studied non- 1 r_m(LEarn(X))=P_m(X), X ˘ POwer(COncept).
classificatory learning. It is a multilevel learning.
Note that the collection of all inverse image,
Mathematically, it is a sequence of mappings. At first
- 1 LEarn(X)), X ˘ POwer(COncept)
level, it maps each point (a base concept) to its unique
is the s-ring generated by the equivalence classes o
basic neighborhood, called a concept of level 1. The
U. So we have results similar to the Corollary 2.
family of such basic neighborhoods is denoted by
T h e o r e m 5 . U i s a fi n i t e b a s i c n e i g h b o r h o o d
COncept(1) or simply Concept. In general step, it maps
s ys t e m. C O n c e p t i s t h e d i s t i n c t l i s t o f b a s i c
each point in COncept(n) to an element in
neighborhoods.
COncept(n+1), where COncept(n+1) is called concept
of level (n+1) and is the family of basic neighborhoods
of COncept(n). Implicitly the level one learning is also
in [12]; the soft rules in level 0 are hard rules in level
1.
} be the distinct list of basic neighborhoods
* = e_c_Bel (item 17)
19. This is a
i, … , C ,.,Cn
(note that two distinct points may have the
generalization of Pawlak's results. The
same neighborhood.) For example, in rough outer and inner probability measures of r_m are the
set theory, COncept is the set of
probability of lower and upper approximation of the
17. Let COncept
equivalence relation LEarn. This theorem is related to, but
={C 1
different from [22].
equivalence classes. Let m be the “external sum
measure.” Recall that | • | is the cardinal number.
i1 |+|C2 | + …+ |C
n | + … + |C| m (C i)= | Ci |
20. We will call this s-alg
Next, we consider the following basic probability
of
m (COncept)= | C
m : POwer(2 C O n c e p t ) fi [ 0 , 1 ]
the basic neighborhood system
defined by the equations,
probability; the belief function
6. Conclusions
i(a)
)/m(COncept)
(b) m(A) = 0 if Aˇ Concept.
m induces a belief function on U; we call it the external
counting belief function, denoted by e_c_Bel of the basic
neighborhood system. The "same" m, as a probability mass
function, induces a probability measure P_m on
Power(COncept)
18. Now we will consider the learning
map
LEarn : U fi COncept
m(C
i) = m
(C
Neighborhood systems were introduce
arena of advanced computing for mod
retrievals. It turns out to be a very effe
uncertainty. In this paper, we examine
functions," measure, probability, and b
of neighborhood systems; fuzzy sets w
study conclude that neighborhoods ma
mathematical formalism for uncertaint
effective formalism to granulate inform
Acknowledgment
This author would like to express his
Professor Zadeh for his kind guidance
to join the Berkeley Initiative in Soft C
(BISC). Our deepest thanks also go to
Dr. Martin Wildberger at EPRI, Electric Power Research
Institute, for his generous sponsorship.
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TsauYoung(T.Y.)LinreceivedhisPh.DfromYaleUniversity,andnowisaProfessoratSan
JoseStateUniversityandVisitingScholarinBISC,UniversityofCalifornia-Berkeley.Hehasbeen
chairsand membersofprogramcommitteesinvariousconferencesandworkshops,associate
editorsandmembersofeditorialboardsofseveralinternationaljournals.Heisthepresidentof
InternationalRoughSetSociety.Hisinterestsincludeapproximationindatabaseand
knowledge-basesystems,datamining,datasecurity,fuzzysets,intelligentcontrol,Petrinets,and
roughsets(alphabeticalorder).
Yiyu(Y.Y.)YaoreceivedhisPh.DfromUniveristyofReginaandnowisanassociateprofessorat
LkeheadUniversity.Heservedinthe programcommitteesofseveralconferencesand
workshops,alsoisanassociateeditorofaninternationaljournal.Heisthe secretaryofroughcontrol
group.Hisinterestsincludefuzzysets,informationretrieval,androughsets(alphabeticalorder).