Systems, Man & Cybernetics - January 2015 - 24

ability theory, it later came to
be known as the "probabilistic
constraint." Later definitions,
based on the application of the
max conorm, led to "possibilistic" clustering methods [13].
This characterization was,
by itself, enough to send me
into what Jim correctly describes as the probabilistic
maelstrom. Although it was
clear to the mathematical economists whom I was helping
that degrees of belongingness
were not related to probabilities both for fundamental as
well as theoretical reasons. At
the fundamental level, it was
clear that these values repre(a)
(b)
sented a resemblance between
certain prototypical situations
Figure 3. the Ph.D. dissertation cover pages of (a) Jim bezdek and (b) Enrique
rather than their probabilities
ruspini. [Image (a) courtesy of L. a. zadeh, and image (b) courtesy of Enrique
of being those ideal situations.
ruspini.]
The basic observation was
that,
in
the
problem in question, a samin
different
subsets."
Whenever
the
600 pages. I did not favor the fuzzy
ple point may be classified as resemmetric structure of the sample set could
approach-I more or less fell into it by
bling an "inflation" to some degree, but
be visualized, as with subsets of the
chance. For me, the controversy was
it did not make sense to characterize it
Euclidean plane, it was clear that such
attractive-new science should be
as the probability of that economical
a goal was impossible to attain in the
controversial: it was, and (somewhat
state. (I must confess that I had helped
vast majority of situations (as shown in
surprisingly, 50 years later), it still is.
to stimulate this reading of degrees
Figure 4), which was chosen to include
ER: My choice of the theory of
of belongingness as probabilities by
the various problems besetting the
fuzzy sets had both conceptual and
my choice of notation. The difference
application of existing computational
practical motivations.
between these interpretations was,
classification techniques. (This data
At the conceptual level, it was clear
nonetheless, clear from the time that I
set, now commonly described as the
that the clustering problem or, at least,
started to think about these problems.)
Ruspini data set, had curious origins.
its formulation in terms of similarities
Further examination of the theoretical
It was one of several chosen from the
or distances could not be solved within
underpinnings of the theories of fuzzy
literature by a young intern who did not
the framework of classical set theory.
sets and probability strongly suggested
record its provenance other than the
The typical formulation of this problem
that the latter was not suited to the
label "Bees." I have not yet been able to
was to "partition a finite set of points
descriptive goals that I was seeking.
find its source.)
into nonintersecting subsets so that
Fuzzy sets, which I had met fortuitously
Reformulating the problem as that
points that are close to each other are in
on a browsing spree at my departmenof finding a fuzzy partition of the samthe same subset of the partition, while
tal library, were clearly the appropriate
ple set overcame this problem as the
points that are far from each other are
bases for a "new approach."
points could be classified as belongFinally, at the practical level, existing to a subset to a degree expressed
ing classification methods were largeby a number between zero and one.
ly heuristic in nature and, in general,
This reformulation required the
not driven by principled approaches
definition of fuzzy partition. In the
to the characterization of the clusteroriginal paper, this definition was
ing problem.
made by extending the original notion
Reflecting the confusion about the
of the fuzzy set complement of Zadeh
nature of the problem that existed at
[5]. Because this definition, essentially
that time, a pioneering book [14] pubbased on the use of addition as the triFigure 4. a data set employed
lished a few years later presented a
angular
conorm
modeling
set
union,
in the original fuzzy clustering
number of methods on the basis of
resembled the additive axiom of probexperiments.
24

IEEE SyStEmS, man, & CybErnEtICS magazInE Janu ar y 2015



Table of Contents for the Digital Edition of Systems, Man & Cybernetics - January 2015

Systems, Man & Cybernetics - January 2015 - Cover1
Systems, Man & Cybernetics - January 2015 - Cover2
Systems, Man & Cybernetics - January 2015 - 1
Systems, Man & Cybernetics - January 2015 - 2
Systems, Man & Cybernetics - January 2015 - 3
Systems, Man & Cybernetics - January 2015 - 4
Systems, Man & Cybernetics - January 2015 - 5
Systems, Man & Cybernetics - January 2015 - 6
Systems, Man & Cybernetics - January 2015 - 7
Systems, Man & Cybernetics - January 2015 - 8
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Systems, Man & Cybernetics - January 2015 - 12
Systems, Man & Cybernetics - January 2015 - 13
Systems, Man & Cybernetics - January 2015 - 14
Systems, Man & Cybernetics - January 2015 - 15
Systems, Man & Cybernetics - January 2015 - 16
Systems, Man & Cybernetics - January 2015 - 17
Systems, Man & Cybernetics - January 2015 - 18
Systems, Man & Cybernetics - January 2015 - 19
Systems, Man & Cybernetics - January 2015 - 20
Systems, Man & Cybernetics - January 2015 - 21
Systems, Man & Cybernetics - January 2015 - 22
Systems, Man & Cybernetics - January 2015 - 23
Systems, Man & Cybernetics - January 2015 - 24
Systems, Man & Cybernetics - January 2015 - 25
Systems, Man & Cybernetics - January 2015 - 26
Systems, Man & Cybernetics - January 2015 - 27
Systems, Man & Cybernetics - January 2015 - 28
Systems, Man & Cybernetics - January 2015 - 29
Systems, Man & Cybernetics - January 2015 - 30
Systems, Man & Cybernetics - January 2015 - 31
Systems, Man & Cybernetics - January 2015 - 32
Systems, Man & Cybernetics - January 2015 - 33
Systems, Man & Cybernetics - January 2015 - 34
Systems, Man & Cybernetics - January 2015 - 35
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Systems, Man & Cybernetics - January 2015 - 37
Systems, Man & Cybernetics - January 2015 - 38
Systems, Man & Cybernetics - January 2015 - 39
Systems, Man & Cybernetics - January 2015 - 40
Systems, Man & Cybernetics - January 2015 - 41
Systems, Man & Cybernetics - January 2015 - 42
Systems, Man & Cybernetics - January 2015 - 43
Systems, Man & Cybernetics - January 2015 - 44
Systems, Man & Cybernetics - January 2015 - 45
Systems, Man & Cybernetics - January 2015 - 46
Systems, Man & Cybernetics - January 2015 - 47
Systems, Man & Cybernetics - January 2015 - 48
Systems, Man & Cybernetics - January 2015 - Cover3
Systems, Man & Cybernetics - January 2015 - Cover4
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