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MalnadachBot (talk | contribs) Fixed Lint errors - replaced obsolete center tags. (Task 12) |
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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|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|pages=1723–1728|doi=10.1109/I2MTC.2015.7151540|isbn=9781479961146}}</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></
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|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|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>
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An ''α''-cut of a fuzzy variable F can be defined as <ref name="zadeh1" /><ref name = "kaufman" />
<div class="center"><math>F_{\alpha } = \{a\,\vert\,\mu _{\rm F} (a) \geq \alpha\}\qquad\textit{where}\qquad0\leq\alpha\leq1</math></
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.
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To build the RFV, let us consider the ''α''-cuts of the two PDs i.e., ''r<sub>random</sub>'' and ''r<sub>internal</sub>'' 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 <math>x^{*}</math> 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" />
<div class="center"><math>X_{a}^{\alpha} = X_{LI}^{\alpha}-(x^{*}-X_{LR}^{\alpha})</math></
<div class="center"><math>X_{b}^{\alpha} = X_{LI}^{\alpha}</math></
<div class="center"><math>X_{c}^{\alpha} = X_{UI}^{\alpha}</math></
<div class="center"><math>X_{d}^{\alpha} = X_{UI}^{\alpha}-(X_{UR}^{\alpha}-x^{*})</math></
Using the above equations, the ''α''-cuts are calculated for every value of ''α'' which gives us the final plot of the RFV.
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