The Uncertainty Measure of Hierarchical Quotient Space Structure

Algorithm 1

Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛}.

Input: A fuzzy similarity relation matrix 𝑀(𝑅)=(𝑟̃_𝑖𝑗)_𝑛×𝑛

Output: A hierarchical quotient space structure 𝜋_𝑋(𝑅) (Where 𝑅 is a fuzzy equivalence

relation derived from 𝑅)

Step  1 Using the matrix 𝑀(𝑅)=(𝑟̃_𝑖𝑗)_𝑛×𝑛, a similarity degree sequence 𝜆={𝜆₁,𝜆₂,…,𝜆_𝑡}

can be obtained.

Step  2 For 𝑘=1 to 𝑛

   {If 𝑟̃_𝑖𝑗≥𝜆_𝑘, 𝑥_𝑖 and 𝑥_𝑗 are located in the same subblock, else 𝑥_𝑖 and 𝑥_𝑗 are located

   in the different subblock respectively. A quotient space 𝑋(𝜆_𝑘) is constructed}

   {if 𝑋(𝜆_𝑘)={𝑋}, go to Step 𝟑}

Step  3 End

Step  4 A hierarchical quotient space structure 𝜋_𝑋(𝑅) is obtained.