Random-fuzzy variable: Difference between revisions

Revision as of 18:03, 28 February 2020

Revision as of 01:02, 16 December 2020

Line 7:

[[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|volume=|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==

Line 51:

===The construction of the external distribution(''r_external'') and the RFV===

After modeling the random and internal possibility distribution, the external membership function, '''''r_external''''', of the RFV can be constructed by using the following equation:<ref>{{Cite journal|last=Ferrero|first=Alessandro|last2=Prioli|first2=Marco|last3=Salicone|first3=Simona|date=2015|title=Uncertainty propagation through non-linear measurement functions by means of joint Random-Fuzzy Variables|journal=2015 IEEE International Instrumentation and Measurement Technology Conference (I2MTC) Proceedings|___location=Pisa, Italy|publisher=IEEE|volume=|pages=1723–1728|doi=10.1109/I2MTC.2015.7151540|isbn=9781479961146}}</ref>

<center> <math>r_{\textit{external}}(x)=\sup_{x^\prime}T_{min}[r_{\textit{random}}(x-x^\prime+x^{*}), r_{\textit{internal}}(x^\prime)] </math></center>