Random-fuzzy variable: Difference between revisions

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[[Lotfi A. Zadeh|L.A.Zadeh]] introduced the concepts of fuzzy variables and fuzzy sets.<ref name = "zadeh2">{{Cite journal|last=Zadeh|first=L. A.|date=1965-06-01|title=Fuzzy sets|journal=Information and Control|volume=8|issue=3|pages=338–353|doi=10.1016/S0019-9958(65)90241-X|issn=0019-9958|doi-access=free}}</ref><ref name = "zadeh3">{{Cite journal|last=Zadeh|first=Lotfi A.|date=1973|title=Outline of a New Approach to the Analysis of Complex Systems and Decision Processes|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-3|issue=1|pages=28–44|doi=10.1109/TSMC.1973.5408575|issn=0018-9472}}</ref> Fuzzy variables are based on the theory of possibility and hence are possibility distributions. This makes them suitable to handle any type of uncertainty, i.e., both systematic and random contributions to the total uncertainty.<ref>{{Cite journal|last=Mauris|first=G.|last2=Berrah|first2=L.|last3=Foulloy|first3=L.|last4=Haurat|first4=A.|date=2000|title=Fuzzy handling of measurement errors in instrumentation|journal=IEEE Transactions on Instrumentation and Measurement|volume=49|issue=1|pages=89–93|doi=10.1109/19.836316}}</ref><ref>{{Cite journal|last=Urbanski|first=Michał K.|last2=Wa̧sowski|first2=Janusz|date=2003-07-01|title=Fuzzy approach to the theory of measurement inexactness|journal=Measurement|series=Fundamental of Measurement|volume=34|issue=1|pages=67–74|doi=10.1016/S0263-2241(03)00021-6|issn=0263-2241}}</ref><ref>{{Cite journal|last=Ferrero|first=A.|last2=Salicone|first2=S.|date=2003|title=An innovative approach to the determination of uncertainty in measurements based on fuzzy variables|journal=IEEE Transactions on Instrumentation and Measurement|language=en|volume=52|issue=4|pages=1174–1181|doi=10.1109/TIM.2003.815993|issn=0018-9456}}</ref>

'''Random-fuzzy variable (RFV)''' is a [[Type-2 fuzzy sets and systems|type 2 fuzzy variable]],<ref>{{Cite book|last=Castillo|first=Oscar|last2=Melin|first2=Patricia|last3=Kacprzyk|first3=Janusz|last4=Pedrycz|first4=Witold|date=2007|chapter=Type-2 Fuzzy Logic: Theory and Applications|pages=145|doi=10.1109/grc.2007.118|title=2007 IEEE International Conference on Granular Computing (GRC 2007)|isbn=978-0-7695-3032-1}}</ref> defined using the mathematical possibility theory,<ref name = "zadeh2" /><ref name = "zadeh3" />, used to represent the entire information associated to a measurement result. It has an internal possibility distribution and an external possibility distribution called membership functions. The internal distribution is the uncertainty contributions due to the systematic uncertainty and the bounds of the RFV are because of the random contributions. The external distribution gives the uncertainty bounds from all contributions.

==Definition==

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So, essentially an ''α''-cut is the set of values for which the value of the membership function <math>\mu _{\rm F} (a)</math> of the fuzzy variable is greater than ''α''. So, this gives the upper and lower bounds of the fuzzy variable F for each ''α''-cut.

The ''α''-cut of an RFV, however, has 4 specific bounds and is given by <math>RFV^{\alpha} = [X_{a}^{\alpha}, X_{b}^{\alpha}, X_{c}^{\alpha}, X_{d}^{\alpha}]</math>.<ref name = "saliconebook" />. <math>X_{a}^{\alpha}</math> and <math>X_{d}^{\alpha}</math> are the lower and upper bounds respectively of the external membership function(''r_external'') which is a fuzzy variable on its own. <math>X_{b}^{\alpha}</math> and <math>X_{c}^{\alpha}</math> are the lower and upper bounds respectively of the internal membership function(''r_internal'') which is a fuzzy variable on its own.

To build the RFV, let us consider the ''α''-cuts of the two PDs i.e., ''r_random'' and ''r_internal'' for the same value of ''α''. This gives the lower and upper bounds for the two ''α''-cuts. Let them be <math>[X_{LR}^{\alpha}, X_{UR}^{\alpha}]</math> and <math>[X_{LI}^{\alpha}, X_{UI}^{\alpha}]</math> for the random and internal distributions respectively. <math>[X_{LR}^{\alpha}, X_{UR}^{\alpha}]</math> can be again divided into two sub-intervals <math>[X_{LR}^{\alpha}, x^{*}]</math> and <math>[x^{*}, X_{UR}^{\alpha}]</math> where <math>x^{*}</math> is the mode of the fuzzy variable. Then, the ''α''-cut for the RFV for the same value of ''α'', <math>RFV^{\alpha} = [X_{a}^{\alpha}, X_{b}^{\alpha}, X_{c}^{\alpha}, X_{d}^{\alpha}]</math> can be defined by <ref name = "saliconebook" />