Uncertainty-based and fuzzy information
As George J. Klir and Mark J. Wierman wrote in their book Uncertainty-based information: elements of generalized information theory (Physica-Verlag, Heidelberg, 1999):
"A host of problems regarding relative measures of nonspecificity for bodies of evidence defined on infinite universal sets remains unresolved at this time".
The work described in this section was partly motivated by this. Just as in the section on Sorting and the analog median filter, the concept of rearrangement or sorting plays a central role.
Measures of information based on fuzzy sets have been defined for finite and for continuous universes. In the continuous case, the measure of information I(f) depends on the concept of sorting or nonincreasing rearrangement of f.
The paper studies the extent to which this measure of information can be approximated as a limit of discrete distributions with N distinct values, as N tends to infinity. It gives conditions under which there is convergence, and the order of approximation in terms of N and the Lipschitz exponent of the possibility function f.
Part of the material in this book chapter was presented at the NATO Advanced Research Workshop Systematic Organisation of Information in Fuzzy Systems, Oct. 2001.