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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|lastlast1=Pietrosanto|firstfirst1=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|lastlast1=Betta|firstfirst1=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|issn=0263-2241}}</ref><ref>{{Cite journal|lastlast1=Ferrero|firstfirst1=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|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|lastlast1=Mauris|firstfirst1=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|lastlast1=Urbanski|firstfirst1=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|lastlast1=Ferrero|firstfirst1=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|lastlast1=Castillo|firstfirst1=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.
==Definition==
[[File:Random-Fuzzy Variable.png|thumb|right|upright=2.0|{{center|Random-Fuzzy Variable}}]]
A Random-fuzzy Variable (RFV) is defined as a type 2 fuzzy variable which satisfies the following conditions:<ref name = "saliconebook">{{Cite book|title=Measuring uncertainty within the theory of evidence|last=Salicone, Simona|others=Prioli, Marco|date=23 April 2018 |isbn=9783319741390|___location=Cham, Switzerland|oclc=1032810109}}</ref>
* Both the internal and the external functions of the RFV can be identified.
* Both the internal and the external functions are modeled as possibility distributions(pd).
But, there can also be random contributions from other probability distributions such as a [[Uniform distribution (continuous)|uniform distribution]], [[gamma distribution]] and so on.
The probability distribution can be modeled from the measurement data. Then, the probability distribution can be used to model an equivalent possibility distribution using the maximally specific probability-possibility transformation.<ref>{{Cite journal|lastlast1=KLIR†|firstfirst1=GEORGE J.|last2=PARVIZ|first2=BEHZAD|date=1992-08-01|title=Probability-Possibility Transformations: A Comparison|journal=International Journal of General Systems|volume=21|issue=3|pages=291–310|doi=10.1080/03081079208945083|issn=0308-1079}}</ref>
Some common probability distributions and the corresponding possibility distributions can be seen in the figures.
===The construction of the external distribution(''r<sub>external</sub>'') and the RFV===
After modeling the random and internal possibility distribution, the external membership function, '''''r<sub>external</sub>''''', of the RFV can be constructed by using the following equation:<ref>{{Cite journal|lastlast1=Ferrero|firstfirst1=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|pages=1723–1728|doi=10.1109/I2MTC.2015.7151540|isbn=9781479961146|s2cid=22811201 }}</ref>
<div class="center"> <math>r_{\textit{external}}(x)=\sup_{x^\prime}T_{min}[r_{\textit{random}}(x-x^\prime+x^{*}), r_{\textit{internal}}(x^\prime)] </math></div>
where <math>x^{*}</math> is the mode of <math>r_{\textit{random}}</math>, which is the peak in the membership function of <math>r_{random}</math> and ''T<sub>min</sub>'' is the minimum [[t-norm|triangular norm]].<ref>{{Cite journal|lastlast1=Klement|firstfirst1=Erich Peter|last2=Mesiar|first2=Radko|last3=Pap|first3=Endre|date=2004-04-01|title=Triangular norms. Position paper I: basic analytical and algebraic properties|journal=Fuzzy Sets and Systems|series=Advances in Fuzzy Logic|volume=143|issue=1|pages=5–26|doi=10.1016/j.fss.2003.06.007|issn=0165-0114}}</ref>
RFV can also be built from the internal and random distributions by considering the ''α''-cuts of the two possibility distributions(PDs).
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