Content deleted Content added
m minor copy edits |
m minor copy edits |
||
Line 17:
* Both the internal and external functions have a unitary value for possibility to the same interval of values.
An RFV can be seen in the figure. The external membership function is the distribution in blue and the internal membership function is the distribution in red. Both the membership functions are possibility distributions. Both the internal and external membership functions have a unitary value of possibility only in the rectangular part of the RFV.
If there are only systematic errors in the measurement, then the RFV simply becomes a [[Fuzzy set|fuzzy variable]] which consists of just the internal membership function. Similarly, if there is no systematic error, then the RFV becomes a fuzzy variable with just the random contributions and therefore, is just the possibility distribution of the random contributions.
Line 23:
==Construction==
A random-fuzzy variable can be constructed using an internal possibility distribution (''r<sub>internal</sub>'') and a random possibility distribution (''r<sub>random</sub>'').
===The random distribution (''r<sub>random</sub>'')===
Line 31:
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|title=Introduction to Probability and Statistics for Engineers and Scientists.|last=Ross, Sheldon M.|date=2009|publisher=Elsevier Science|isbn=9780080919379|edition= 4th|___location=Burlington|oclc=761646775}}</ref>
However, 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|last1=KLIR†|first1=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>
Line 60:
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
<div class="center"><math>F_{\alpha } = \{a\,\vert\,\mu _{\rm F} (a) \geq \alpha\}\qquad\textit{where}\qquad0\leq\alpha\leq1</math></div>
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<sub>external</sub>'') 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<sub>internal</sub>'') 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<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
<div class="center"><math>X_{a}^{\alpha} = X_{LI}^{\alpha}-(x^{*}-X_{LR}^{\alpha})</math></div>
Line 82:
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 q | Q57275767 |last1=Zadeh |first1=L.A. | author-link1 = Lotfi A. Zadeh | publisher = [[Springer Science+Business Media|Springer]] }}</ref><ref name = "kaufman">{{Cite book|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}}]]
| |||