Principal Components of a Fuzzy Class

The principal components of any fuzzy class C are the most important directions along which this cluster is spread out. Let L be the line through V with the direction u  

The proximity between the fuzzy class C and the line L may be given by the number W L,C), defined as 

We are interested in finding the closest line to the cloud C and so need to minimize W L,C). The best line minimizing IV(L,C) passes through — the mean vector of the class C shown by Dumitrescu to be given by 

The principal directions of the fuzzy class C are eigenvectors of S, the scatter matrix of C. 

It gives a global geometric characterization of the class. The information given by principal components may be used for classification purposes and for dimensionality reduction of a data set having varied characteristics. 

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