Systems, Man & Cybernetics - January 2015 - 22

policy and, as output, variables describing the expected behavior of the
simulated system.
Their problem, however, was that
the output of each run was a 1,500dimensional vector. The analysis of
these runs was a daunting problem
(it is even now, despite advances in
big data). Since they were running
simulations much faster than they
could analyze their results, they were
unable to quickly summarize and describe their findings. They had come
across, however, early literature on
numerical taxonomy (then focused on
"biological systematics," which was
mainly concerned with classification
of biological species). The seminal paper of the field, by Sokal and Sneath,
had been published in 1963 [3]. [Robert Reuven Sokal (1926-2012) was
an Austrian-American biostatistician
and entomologist, and Peter Henry
Andrews Sneath (1923-2011) was a
British microbiologist. Sneath began
to work on numerical methods for
classifying bacteria in the late 1950s.
Principles of Numerical Taxonomy
[3] was a widely noticed textbook on
an approach to classification, basing
it on measures of overall similarity
rather than any inference of phylogeny. The book was revised in 1973
and published as Numerical Taxonomy: The Principles and Practice
of Numerical Classification [4].] It
is interesting to note that the field
was so young that, at the point, there
were not even accepted translations
to Spanish of words such as pattern
or clustering.
After trying to understand and
formulate the nature of the problem
(I am a mathematician after all!), it
was clear to me that the stated goal
of clustering procedures (classify
similar objects into the same class
and different objects into different
classes) could not be attained within
the framework of classical set theory.
By sheer accident, one day I walked
into the small library of the Department of Mathematics at the School
of Science. Perusing through the new
arrivals rack, I found the 1965 issue
of Information and Control with
22

Lotfi Zadeh's seminal paper [5]. [Lotfi
A. Zadeh (born in 1921) is an American electrical engineer and professor
emeritus of computer science at the
University of California, Berkeley. In
1964, he founded the theory of fuzzy
sets and systems. For the history of
this theory, see [7].] It was clear to
me and my colleagues that this was
a much better framework to consider
and rigorously pose fuzzy clustering
problems. Drawing also from results
in the field of operations research,
I was soon able to pose the clustering problem in terms of finding the
optimal solution of a continuous variable system with well-defined performance criteria and constraints.
It is interesting to note that the
issue of whether or not we should
think of degrees of memberships as
some form of probabilities came early in our discussions and was quickly
dealt with (it is incredible, after all
these years, to read the silly, repetitive, arguments in exchanges in the
Berkeley Initiative in Soft Computing mailing group [bisc-group@lists.
eecs.berkeley.edu] on the subject). In
the context of the macroeconomics
problem, a simulation output could
be indicative of some well-defined
behavior, e.g., inflation, but, in many
instances, that object could not be associated with any major economical
prototype being similar, to different
extents, to various paradigmatic situations. In such cases, it was possible
to describe the object in terms of a
resemblance to prototypes, but it did
not make sense to describe a clearly
characterized object in terms of its
probability of being an "inflation" or
a "recession."
Jim Bezdek (J B): I became
aware of fuzzy sets and fuzzy clustering simultaneously in 1971 when
I was a graduate student in applied
mathematics at Cornell University.
Joe Dunn, my thesis advisor, along
with H.D. Block and Larry Payne,
handed me copies of two papers, Lotfi's 1965 paper on fuzzy sets [5] and
Enrique's 1969 paper on fuzzy clustering [8], with the admonition, "You
might find these interesting."

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

[Joseph Charles Dunn received
his Ph.D. degree from Adelphi University in 1967. He became a professor of mathematics at North Carolina
State University in 1976. He has been
a member of the editorial board of
Computational Optimization and
Applications since 1992 and served
as editor for Fuzzy Sets and Systems.
His research was funded by the U.S.
National Science Foundation (NSF)
for 23 years. Dunn retired in 2002.]
[Henry David Block (1920-1978)
beca me a professor of applied
mathematics at Cornell University
in 1955, where, after two years in
the Department of Mathematics, he
joined the Department of Theoretical and Applied Mechanics, in which
he served until his death. In his last
years, his work was focused on problems of learning, pattern recognition, AI, self-reproducing machines,
and the design of robots capable of
acquiring natural language. He collaborated with Frank Rosenblatt, the
builder of the perceptron. He derived
mathematical statements analyzing
the machine's behavior and proved
theorems about the convergence of
learning algorithms. Laurence E.
Payne (1923-2011) was a professor of
applied mathematics at Cornell University, where he worked in the area
of partial differential equations as
applied to civil and mechanical engineering until his retirement in 1994.]
My thesis topic was "something in
pattern recognition," yet to be determined, so Joe thought these papers
might point me toward a specific
objective. That's my story.
RS: Jim, you started "something
with pattern recognition" just after
the euphoria that started with Rosenblatt's perceptron and came to an
abrupt halt in 1969, when the mathematicians and AI pioneers Minsky
[born in 1927] and Papert [born in
1928] published their study of perceptron networks in a book [9].
[Frank Rosenblatt (1928-1971) was
an American neurobiologist. Together
with Charles Wightman, he developed
this first artificial neuronal network


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