Random-fuzzy variable: Difference between revisions

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This systematic error can be approximately modeled based on our past data about the measuring instrument and the process.

Statistical methods can be used to calculate the total uncertainty from both systematic and random contributions in a measurement.<ref>{{Cite journal|last1=Pietrosanto|first1=A.|last2=Betta|first2=G.|last3=Liguori|first3=C.|date=1999-01-01|title=Structured approach to estimate the measurement uncertainty in digital signal elaboration algorithms|url=https://digital-library.theiet.org/content/journals/10.1049/ip-smt_19990001|journal=IEE Proceedings - Science, Measurement and Technology|language=en|volume=146|issue=1|pages=21–26|doi=10.1049/ip-smt:19990001|issn=1350-2344}}</ref><ref>{{Cite journal|last1=Betta|first1=Giovanni|last2=Liguori|first2=Consolatina|last3=Pietrosanto|first3=Antonio|date=2000-06-01|title=Propagation of uncertainty in a discrete Fourier transform algorithm|journal=Measurement|volume=27|issue=4|pages=231–239|doi=10.1016/S0263-2241(99)00068-8|bibcode=2000Meas...27..231B |issn=0263-2241}}</ref><ref>{{Cite journal|last1=Ferrero|first1=A.|last2=Lazzaroni|first2=M.|last3=Salicone|first3=S.|date=2002|title=A calibration procedure for a digital instrument for electric power quality measurement|journal=IEEE Transactions on Instrumentation and Measurement|language=en|volume=51|issue=4|pages=716–722|doi=10.1109/TIM.2002.803293|bibcode=2002ITIM...51..716F |issn=0018-9456}}</ref> But, the computational complexity is very high and hence, are not desirable.

[[Lotfi A. Zadeh|L.A.Zadeh]] introduced the concepts of fuzzy variables and fuzzy sets.<ref name = "zadeh2">{{cite q | Q25938993 |last1=Zadeh |first1=L.A. | author-link1 = Lotfi A. Zadeh | journal = [[Information and Computation|Information and Control]] | doi-access = free }}</ref><ref name = "zadeh3">{{cite q | Q56083455 |last1=Zadeh |first1=L.A. | author-link1 = Lotfi A. Zadeh | journal = [[IEEE Systems, Man, and Cybernetics Society#Publications|IEEE Transactions on Systems, Man, and Cybernetics]] }}</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|last1=Mauris|first1=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|bibcode=2000ITIM...49...89M }}</ref><ref>{{Cite journal|last1=Urbanski|first1=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|bibcode=2003Meas...34...67U |issn=0263-2241}}</ref><ref>{{Cite journal|last1=Ferrero|first1=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|bibcode=2003ITIM...52.1174F |issn=0018-9456}}</ref>

'''Random-fuzzy variable (RFV)''' is a [[Type-2 fuzzy sets and systems|type 2 fuzzy variable]],<ref>{{Cite book|last1=Castillo|first1=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|s2cid=1942035 }}</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.