Construction of Normal Fuzzy Numbers using the Mathematics of Partial Presence

Abstract

Every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. Indeed, two probability measures are necessary and sufficient to define a normal law of fuzziness. Hence, the measure theoretic matters with reference to fuzziness have to be studied accordingly. In this chapter, we are going to discuss how to construct normal fuzzy numbers using this concept which is based on our mathematics of partial presence. Three case studies have been presented with reference to expressing stock prices in terms of fuzzy numbers. We have shown how exactly a normal fuzzy number appears.

Construction of normal fuzzy numbers: A case study with earthquake waveform data

“…These values are, say (a 1 , a 2 , a 3,….…, a 39 ) and (b 1 , b 2 , b 3,……, b 39 ) respectively. Now, using the operation of set superimposition defined by Baruah (2010Baruah ( , 2011aBaruah ( , 2011bBaruah ( , 2011cBaruah ( , 2012 we may proceed to construct normal fuzzy numbers as discussed in (Das et al, 2013), which would define the uncertainty associated with waveform variations.…”

“…But how exactly to construct the membership function of a fuzzy number mathematically remained a problem. Baruah (2010Baruah ( , 2011aBaruah ( , 2011bBaruah ( , 2011cBaruah ( , 2012 has shown that two laws of randomness are necessary as well as sufficient to define a normal law of fuzziness. In other words, trying to frame one single law of probability from a given law of fuzziness, as had been tried upon while formulating the existing probability-possibility consistency principles, was not mathematically meaningful an exercise, because we need two laws of randomness, probabilistic or otherwise, and not one single law of probability, to define a law of fuzziness.…”

“…Partial presence of an element in a fuzzy set is defined by the name membership function. Based on the Randomness-Fuzziness Consistency Principle (Baruah, 2010(Baruah, , 2011a(Baruah, , 2011b(Baruah, , 2011c(Baruah, , 2012, in this article we shall show how to construct normal fuzzy numbers using the data of minimum and maximum amplitudes of every individual oscillation of the waveform of an earthquake that had occurred in the city of Guwahati on May 25, 1998.…”

Construction of the membership function of normal fuzzy numbers

“…Construction of normal fuzzy number has been discussed in [Baruah, 2011b[Baruah, , 2012] based on the Randomness -Fuzziness Consistency Principle deduced by Baruah [Baruah, 2010[Baruah, , 2011a[Baruah, , 2011b[Baruah, , 2011c[Baruah, , 2012. In this article we shall show how to construct normal fuzzy numbers (Das et al, 2013a(Das et al, , 2013b using the data of minimum and maximum temperature in Guwahati city for the month of December 2012 and up to 30 th January 2013.…”

“…But how exactly to construct the membership function of a fuzzy number mathematically remained a problem. Baruah (2010Baruah ( , 2011aBaruah ( , 2011bBaruah ( , 2011cBaruah ( , 2012 has shown that two laws of randomness are necessary as well as sufficient to define a normal law of fuzziness. This has led to a proper measure theoretic explanation of partial presence, and construction of fuzzy numbers can therefore be based on that.…”