Date of Award


Degree Type


Degree Name

Master of Science (MS)


Mathematics and Statistics

First Advisor

James C. Byden

Second Advisor

David S. Watkins


Since the introduction of Shannon's Entropy Function in a non-probabilistic setting [l], there has been an expectancy that it will provide a useful measure for the fuzzy cluster validity problem. We hope to demonstrate here (empirically) by Monte Carlo simulation that the entropy Hc(U) of fuzzy c-partitions (U) defined below has expectaction

E(Hc(U)) = ∑ck=2 1/k

and variance

Var(Hc(U)) = 1/n(∑ck=2 1/k2 - (c-1)/(c+1)(π2/6) - 1).

Furthermore, for sufficiently large n, that H is approximately normal.

In this report we propose a method which generates the required fuzzy matrix needed for the simulation. After we finish the simulation, we shall study the empirical distribution of Hc(U) at c=2 and 3 using statistical hypothesis testing.

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