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In measurements, the measurement obtained can suffer from two types of uncertainties<ref>{{Cite book|url=https://www.worldcat.org/oclc/34150960|title=An introduction to error analysis : the study of uncertainties in physical measurements|last=Taylor, John R. (John Robert), 1939-|date=1997|publisher=University Science Books|isbn=0935702423|edition=2nd ed|___location=Sausalito, Calif.|oclc=34150960}}</ref>. The first is the random uncertainty which is due to the noise in the process and the measurement. The second contribution is due to the systematic uncertainty which may be present in the measuring instrument. Systematic errors, if detected, can be easily compensated as they are usually constant throughout the measurement process as long as the measuring instrument and the measurement process are not changed. But it can not be accurately known while using the instrument if there is a [[systematic error]] and if there is, how much? Hence, systematic uncertainty could be considered as a contribution of a fuzzy nature.▼
▲In measurements, the measurement obtained can suffer from two types of uncertainties.<ref>{{Cite book|url=https://www.worldcat.org/oclc/34150960|title=An introduction to error analysis : the study of uncertainties in physical measurements|last=Taylor, John R. (John Robert), 1939-|date=1997|publisher=University Science Books|isbn=0935702423|edition=
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|last=Pietrosanto|first=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|last=Betta|first=Giovanni|last2=Liguori|first2=Consolatina|last3=Pietrosanto|first3=Antonio|date=2000-06-01|title=Propagation of uncertainty in a discrete Fourier transform algorithm|url=http://www.sciencedirect.com/science/article/pii/S0263224199000688|journal=Measurement|volume=27|issue=4|pages=231–239|doi=10.1016/S0263-2241(99)00068-8|issn=0263-2241}}</ref><ref>{{Cite journal|last=Ferrero|first=A.|last2=Lazzaroni|first2=M.|last3=Salicone|first3=S.|date=2002|title=A calibration procedure for a digital instrument for electric power quality measurement|url=http://ieeexplore.ieee.org/document/1044714/|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|via=}}</ref>
[[Lotfi A. Zadeh|L.A.Zadeh]] introduced the concepts of fuzzy variables and fuzzy sets.<ref>{{Cite journal|last=Zadeh|first=L. A.|date=1965-06-01|title=Fuzzy sets|url=http://www.sciencedirect.com/science/article/pii/S001999586590241X|journal=Information and Control|volume=8|issue=3|pages=338–353|doi=10.1016/S0019-9958(65)90241-X|issn=0019-9958}}</ref><ref>{{Cite journal|last=Zadeh|first=Lotfi A.|date=1973|title=Outline of a New Approach to the Analysis of Complex Systems and Decision Processes|url=http://ieeexplore.ieee.org/document/5408575/|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>
'''Random-fuzzy variable (RFV)''' is a [[Type-2 fuzzy sets and systems|type 2 fuzzy variable]],<ref>{{Cite journal|last=Castillo|first=Oscar|last2=Melin|first2=Patricia|last3=Kacprzyk|first3=Janusz|last4=Pedrycz|first4=Witold|date=2007|title=Type-2 Fuzzy Logic: Theory and Applications|url=http://ieeexplore.ieee.org/document/4403084/|journal=|volume=|pages=|doi=10.1109/grc.2007.118|via=}}</ref>
==Definition==
[[File:Random-Fuzzy Variable.png|thumb|right|upright=2.0|<center>Random-Fuzzy Variable<center>]]
A Random-fuzzy Variable (RFV) is defined as a type 2 fuzzy variable which satisfies the following conditions:<ref>{{Cite book|url=https://www.worldcat.org/oclc/1032810109|title=Measuring uncertainty within the theory of evidence|last=Salicone, Simona,|others=Prioli, Marco,|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).
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===The random distribution(''r<sub>random</sub>'')===
''r<sub>random</sub>'' is the possibility distribution of the random contributions to the uncertainty. Any measurement instrument or process suffers from [[
This is completely random in nature and is a normal probability distribution when several random contributions are combined according to the [[Central limit theorem]].<ref>{{Cite book|url=https://www.worldcat.org/oclc/761646775|title=Introduction to Probability and Statistics for Engineers and Scientists.|last=Ross, Sheldon M.|date=2009|publisher=Elsevier Science|isbn=9780080919379|edition=
But, there can also be random contributions from other probability distributions such as a [[uniform distribution]], [[gamma distribution]] and so on.
The probability distribution can be modeled from the 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|last=KLIR†|first=GEORGE J.|last2=PARVIZ|first2=BEHZAD|date=1992-08-01|title=Probability-Possibility Transformations: A Comparison|url=https://doi.org/10.1080/03081079208945083|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.
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[[File:Uniform distribution in probability and possibility.png|thumb|right|upright=2.0|<center>Uniform distribution in probability and possibility.<center>]]
[[File:Triangular distribution in probability and possibility.png|thumb|center|upright=2.0|<center>Triangular distribution in probability and possibility.<center>]]
===The internal distribution(''r<sub>internal</sub>'')===
''r<sub>internal</sub>'' is the internal distribution in the RFV which is the possibility distribution of the systematic contribution to the total uncertainty. This distribution can be built based on the information that is available about the measuring instrument and the process.
The largest possible distribution is the uniform or rectangular possibility distribution. This means that every value in the specified interval is equally possible. This actually represents the state of total ignorance according to the [[Dempster–Shafer theory|theory of evidence]]<ref>{{Cite book|url=https://www.worldcat.org/oclc/1859710|title=A mathematical theory of evidence|last=Shafer, Glenn, 1946-|date=1976|publisher=Princeton University Press|isbn=0691081751|___location=Princeton, N.J.|oclc=1859710}}</ref> which means it represents a scenario in which there is maximum lack of information.
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===The construction of the external distribution(''r<sub>external</sub>'') and the RFV===
After modelling 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|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|url=http://ieeexplore.ieee.org/document/7151540/|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|via=}}</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>
where ''x*'' is the mode of <math>r_{\textit{random}}</math>, which is the peak in the membership function of ''r_{random}'' and ''T<sub>min</sub>'' is the minimum [[t-norm|triangular norm]].<ref>{{Cite journal|last=Klement|first=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|url=http://www.sciencedirect.com/science/article/pii/S0165011403004950|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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Using the above equations, the ''α''-cuts are calculated for every value of ''α'' which gives us the final plot of the RFV.
A Random-Fuzzy variable is capable of giving a complete picture of the random and systematic contributions to the total uncertainty from the ''α''-cuts for any confidence level as the confidence level is nothing but ''1-α''.<ref>{{Cite journal|last=Zadeh|first=L. A.|date=1975-09-01|title=Fuzzy logic and approximate reasoning|url=https://doi.org/10.1007/BF00485052|journal=Synthese|language=en|volume=30|issue=3|pages=407–428|doi=10.1007/BF00485052|issn=1573-0964}}</ref><ref>{{Cite book|url=https://www.worldcat.org/oclc/24309785|title=Introduction to fuzzy arithmetic : theory and applications|last=Kaufmann, A. (Arnold), 1911-|date=1991|publisher=Van Nostrand Reinhold Co|others=Gupta, Madan M.|isbn=0442008996|edition= [New ed.]|___location=New York, N.Y.|oclc=24309785}}</ref>
An example for the construction of the corresponding external membership function(''r<sub>external</sub>'') and the RFV from a random PD and an internal PD can be seen in the following figure.
[[File:Construction of an RFV.png|thumb|center|upright=3.0|<center>Construction of an external membership function and the RFV from internal and random possibility distributions.</center>]]
==See
* [[Fuzzy set]]
* [[T-norm]]
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* [[Probability theory]]
* [[Probability distribution]]
==References==
<references/>
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