Fuzzy apllication (1): fuzzy cluster using fuzzy equivalence relation

as known to all, in fuzzy theory, after define fuzzy set, the next is introduce fuzzy relation. in general form is like this:

R:XxY->[0,1]

I igored the detail of that, and the follow will introduc is: reflexive, symmetric, and transitive property. and what will involve about cluster?

in this approach that is based on equivalence relation, that means all of three propert which have mentioned will be satisfy. so:

equivalence relation: a relation satisfies reflexive, symmetric, and transitive property

actully, this way is also using a-cut as cluster.

 

take instance:

1. Assume a example of relation, which i create it in Matlab. it is an equivalence relation.

 

 R = [1 0 0 0 0; 0.4 1 0 0 0; 0.8 0.4 1 0 0; 0.5 0.4 0.5 1 0; 0.5 0.4 0.5 0.6 1]

 

P.s I let zero to igore half of matrix, actully the matrix is symmetric.

2. so, a-cut, for example a=6 than it will generate a matrix like this:

 

R = [1 0 0 0 0; 0 1 0 0 0; 1 0 1 0 0; 0 0 0 1 0; 0 0 0 1 1]

 

category will cluster to:{A,C},{B},{D,E}.

a=1: {A},{B},{C},{D},{E},

a=8: {A,C},{B},{D},{E}

a=7: ….

….(it will get all kinds of cluster)

 

conclusion

it is an easy way, when exert in real problem, it should be have some polish. by the way, this cluster is not resemble other method in data ming, such as k-means etc. but it is an application of fuzzy relation.

 

P.s if you want to see the matrix, execute the command in matlab. the reason is that youkuaiyun.com pause upload picture in spring festival period.