Date of Award

1979

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

James C. Byden

Second Advisor

David S. Watkins

Abstract

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.

Included in

Mathematics Commons

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