> Fuzziness and probability are thus > inextricably linked in my opinion, and neither can be an alternative > to the other, any more than that likelihood could be an alternative > to probability. They are dualistic flip sides of the same uncertainty > coin.
I am sorry for troubling you. If we have a set of point measurements we can build a PDF function. If we have only interval estimation of the measurements we can get only upper and lower probability. If the family of intervals is nested [1] then we can build a fuzzy membership function. How in this situation we can convert PDF to fuzzy membership function and vice versa? Reference [1] Dubois D., Prade H., Random sets and fuzzy interval analysis. Fuzzy Sets and System, 38, 1991, 309-312 Second problem. Let's consider function z=f(x,y). If x is equal to "not exactly x" and y is equal to "maybe y", what we can say about z? Way we have to use fuzzy methodology? As far as I know this problem is very general. When we apply same mathematical theory (e.g. subjective probability, PDF or random sets) we can build experimental procedures to estimate x and y. Additionally we can calculate z. What we can do, if we know only "not exactly x" and "maybe y"? Reference [2] Kreinovich V., 1997, Random Sets Unify, Explain, and Aid Known Uncertainty Methods in Expert Systems, in John Goutsias, Ronald P.S. Mahler, and Hung T. Nguyen (eds.), Random Sets: Theory and Applications, Springer-Verlag, 1997, s. 321-345 [3] Ferrari P., Savoia M., 1998, Fuzzy number theory to obtain conservative results with respect to probability. Computer methods in applied mechanics and engineering, Vol.160, s.205-222 [4] Bilgiç T., Turksen I.B., 1999, Measurement of Membership Functions: Theoretical and Empirical Work, Chapter 3 in D. Dubois, H. Prade (eds) Handbook of Fuzzy Sets and Systems Vol. 1, Fundamentals of Fuzzy Sets, Kluwer, s.195-232 Andrzej Pownuk ------------------------------------------ MSc. Andrzej Pownuk Chair of Theoretical Mechanics Silesian University of Technology E-mail: pownuk@zeus.polsl.gliwice.pl URL: http://zeus.polsl.gliwice.pl/~pownuk ------------------------------------------