Random-fuzzy variable: Difference between revisions

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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> 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 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 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|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> 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|url=http://ieeexplore.ieee.org/document/836316/|journal=IEEE Transactions on Instrumentation and Measurement|volume=49|issue=1|pages=89–93|doi=10.1109/19.836316|via=}}</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|url=http://www.sciencedirect.com/science/article/pii/S0263224103000216|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|url=http://ieeexplore.ieee.org/document/1232364/|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|via=}}</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> 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<center>]]

A Random-fuzzy Variable (RFV) is defined as a type 2 fuzzy variable which satisfies the following conditions:<ref name = "saliconebook">{{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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RFV can also be built from the internal and random distributions by considering the ''α''-cuts of the two possibility distributions(PDs).

An ''α''-cut of a fuzzy variable F can be defined as <ref name="zadeh1" /><ref name = "kaufman" />

<center><math>F_{\alpha } = \{a\,\vert\,\mu _{\rm F} (a) \geq \alpha\}\qquad\textit{where}\qquad0\geq\alpha\geq1</math></center>

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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 ''x*'' 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" />

<center><math>X_{a}^{\alpha} = X_{LI}^{\alpha}-(x^{*}-X_{LR}^{\alpha})</math></center>

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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 name="zadeh1">{{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 name = "kaufman">{{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_external'') and the RFV from a random PD and an internal PD can be seen in the following figure.