Random-fuzzy variable: Difference between revisions

Revision as of 08:12, 5 July 2019

Revision as of 08:29, 5 July 2019

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<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 ''<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_min'' 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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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 ''<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" />

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