This paper demonstrates how an integrated theory can be built on the foundation of possibility theory. Information and uncertainty were considered in 'fuzzy' literature since 1982. Our departing point is the model proposed by Klir for the discrete case. It was elaborated axiomatically by Ramer, who also introduced the continuous model. Specificity as a numerical function was considered mostly within Dempster-Shafer evidence theory. An explicity definition was given first by Yager, who has also introduced it in the context of possibility theory. Axiomatic approach and the continuous model have been developed very recently by Ramer and Yager. They also establish a close analytical correspondence between specificity and information. In literature to date, specificity and uncertainty are defined only for the discrete finite domains, with a sole exception. Our presentation removes these limitations. We define specificity measures for arbitrary measurable domains

Ramer, Arthur

English

NASA Technical Reports Server

N93-29519
Comparison of Specificity and Information for
Fuzzy Domains
Arthur Ramer
Computer Science, University of New South Wales
PO Box 1, Kensington, NSW 2033, Australia
May 1, 1992
A > - • ' - • . ^j -\ X • -•i/
1 Overview
Given a universe of discourse X—a domain of possible outcomes—an ex-
periment may consist of selecting one of its elements, subject to operation
of chance, or of observing the elements, subject to imprecision.
A priori uncertainty about the actual result of the experiment may be
quantified, representing either the likelihood of the choice of x 6 X or the
degree to which any such x € X would be suitable as a description of the
outcome. The former case corresponds to probability distribution, while the
latter gives a possibility assignment on X.
Study of such assignments and thier properties comes under the purview
of possibility theory [1]. It, like probability theory, assigns values in between
0 and 1 to express likelihoods of outcomes. Here, however, similarity ends.
Possibility theory uses maximum and minimum functions to combine uncer-
tainty, where probability theory uses plus and times operations. This leads
to a very dissimilar theory in its analytical framework, even though they
share several semantic concepts.
One of them consists of expressing quantitatively the uncertainty asso-
ciated with a given distribution [2, 3]. Its value corresponds to the gain
of information that would result from conducting an experiment and ascer-
taining its actual result. This gain becomes simutaneously a decrease in
uncertainty about the outcome of an experiment.
The other concept we consider in depth is one of specificity. Although
it has been introduced previously in a few different forms, a closer analysis
shows that they share main epistemic features. We follow here the presen-
tation of Ramer and Yager [10].
14
https://ntrs.nasa.gov/search.jsp?R=19930020330 2020-03-17T04:46:46+00:00Z
Fuzzy set (X, p) can be considered as a form of a likelihood function,
with the elements of X where p reaches its maximum playing privileged role.
When selecting x : P(ZO) = maxp(i) is important to ask how definite has
been such decision, and whether another element would offer a close choice.
In this interpretation, specificity becomes an attribute of the complete
set of possibilities, the attribute assuming either numeric or linguistic values.
Here we develop a comprehensive model of such specificity, expressed as a
numerical function of a possibility assignment.
2 Introduction
ifThis paper demonstrates how an integrated theory can be built on the foun-
dation of possibility theory. Information and uncertainty were cosidered in
'fuzzy' literature since 1982. Our departing point is the model proposed by
Klir [4, 5] for the discrete case. It was elaborated axiomatically by Ramer
[9], who also introduced the continuous model [7].
Specificity as a numerical function was considered mostly within Dempster-
Shafer evidence theory. An explicit definition was given first by Yager [11],
who has also introduced it in the context of possibility theory [12]. Ax-
iomatic approach and the continuous model have been developed very re-
cently by Ramer and Yager [10]. They also establish a close analytical
correspondence between specificity and information.
In literature to date, specificity and uncertainty are defined only for
the discrete finite domains, with a sole exception of [10]. Our presentation
removes these limitations. We define specificity measures for arbitrary mea-
surable domainseWhen discrete, they can be finite or infinite or, in general
have==7/(":X:i) < oo or n(X) = oo. prespecified pattern. By abuse of the
language we refer to this model as a continuous one.
We adopt the convention of avoiding, whenever possible, subscripts and
indices. We do not specify explicitly basis of logarithms, as its change would
simply amount to a multiplying all expressions by the same constant. Fol-
lowing tradition, binary logarithms—log-i—are assumed for the discrete dis-
tributions, and natural—In— for the continuous cases. We use (p) for the
decreasing rearrangement of the sequence (p,-). For finite sequences, rear-
rangements are permutations of their elements. For infinite sequences and
functions we construct rearrangements using cuts. To define /, given / on
X, we want all their a—cuts to be of the same measure. We put
P(y) = M{* : /(*) > y}),
15
Now for the discrete rearrangements we associate with the sequence (p) =
(pi, . . .,pn, • • •) a step function / : x i-» p\x~\, where [x] denotes the greatest
integer no less than x. Then the descending rearrangement / corresponds
to (p).
3 Information and uncertainty
We use the model of possibility theory introduced by Zadeh [13]. We view
mapping p as assigning a degree of assurance or certainty that an element of
X is the outcome of an experiment. A priori we know only the distribution
p; to determine x 6 X means to remove uncertainty about the result, thus
entailing a gain of information. We would be particularly interested in quan-
tifying that gain of information, which would also express the uncertainty
inherent in the complete distribution p.
Following established principles of information theory [3], we stipulate
that such information function satisfies certain standard properties. For pi
on X and pj on Y we define a noninterracting, joint distribution pi ® p2
on X x Y as
If p was already defined on a product domain X x Y, we construct its
projections (marginal distributions) using maximum operation
p' : x i-»- maxp(x,y), p" : y i-> maxp(x,j/).
There is often a need to consider a given assignment p as defined on on a
larger domain, without, however, making any essential change to the possi-
bility values it represents. We do so by defining pY for y D X , as agreeing
with p on the elements of X, and 0 otherwise. Lastly, the elements of the
domain of discourse could be permuted; if s : X — » X is one-to-one, we
define
s(p) : x I-* p(s(x)).
We now postulate [5]
additivity /(pi ® p2) = /(pi) +
subadditivity /(p) < 7(p') + /(p")
symmetry
expansibility
It turns out that these properties essentially characterize the admissible
information functions [6, 9]. Subject to the normalization of parameters, for
16
the discrete case of X — {x\ , . . . , xn}
u(p) = 5Z(w~p'v
which can be also written using finite differences notation
We observe that the distribution which carries the highest uncertainty
value consists of assigning possibility 1 to all the events in X . It states that,
a priori, every event is fully possible. This distribution, carrying no prior
information, can be considered the most uninformed one.
We shall now extend previous definitions to arbitrary measurable do-
mains [7]. To avoid technical complications, we consider only a typical case
of the unit interval.
As a first step, the discrete formula U(x) = ]T)p,Vlogi suggests forming
Jo f(x)dlnx, = $ ^jp-dx as a candidate expression for the value of informa-
tion. Unfortunately, f(x) is equal to 1 at 0, and the integral above diverges.
A solution can be found through a technique (used also in probability) of in-
formation distance between a given distribution and the most 'uninformed'
one — where U- uncertainty attains its maximum. Our final formula becomes
This integral is well defined and avoids the annoying singularity at 0. It can
be used for a very wide class of functions, including all polynomials.
4 Principles of specificity
The discussion will be conducted in terms of a discrete countable distribution
(p,-), with finite distributions viewed as the initial segments. Our objective
is to capture formally the informal intuition about specificity. The main
premise is the principle of juxtaposition:
5p(p) expresses the preference for a certain maximal po over any
and all the remaining p,.
Now let us consider how, having selected p0 = max(p), its informal speci-
ficity is estimated. We look first for the next largest p, and estimate how its
presence diminishes the specificity. The process is then iterated in the order
of decreasing values of p,, every next value lowering the estimated speci-
ficity. We can picture it as a sequential process, its input the decreasing
17
rearrangement (pi). We may also surmise that, for a given i, the drop in
specificity caused by pi will not depend on the earlier inputs p~i, . . . ,p,_i.
This assumption of independent influence is consistent with the juxtaposition
interpretation of specificity.
Let us consider the effect of a uniform modification of (p). For a scal-
ing ap = (api, . . . ,apn, . . .),0 < a < 1, we may assume that the rela-
tive specificities remain unchanged, while with a shift of values p — j3 =
(Pi — /9, ... ,pn — /?,. . .) no change should occur.
Last item considered will be the effect of offering yet another choice,
identical in value to several choices already provided. The common percep-
tion of specificity is that the change due to such n-th choice will be ever less
as n increases — a diminishing return. For its relative effect, we can postulate
taking away the same proportion of the specificity still available. After all,
we consider yet another identical choice; only we consider it at stage n and
not sooner.
We can extract an analytical representation from the rules elaborated
above. The result is a linear formula
Sp(p) = pi -
t>2
with J3,>2ttJ,- = 1- i-From here we can conclude that lim.^oo w, = 0, and
1 > w-2 > u>3 > • • -, in agreement with the 'diminishing returns'.
We shall consider the linear form of Sp(p) as general specificity function.
It is general enough to fit most applications and, if tu, are supplied, it offers
a comparison scale among the distributions.
Coefficients wt can be established precisely if we assume the rule of con-
stant influence of equal choices. After more calculations
for some u>, 0 < w < 1, producing a definite form of specificity. Choosing
ijj = \ (in spirit of binary logarithms) gives Sp(p) = Pi - ]C s^r- In tne
above formulas the role of pi is manifestly different from that of p,, i > 2. A
more symmetric expression can be obtained defining Wt- = 1 — w2 — . . . — w^,
resulting in a general expression
and the definite one
18


Comparison of specificity and information for fuzzy domains

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server