Construction of the membership surface of imprecise vector
© Das and Baruah; licensee Springer. 2014
Received: 1 August 2014
Accepted: 27 November 2014
Published: 10 December 2014
In this article, a method has been developed to construct the membership surface of imprecise vector based on Randomness-Impreciseness Consistency Principle. The Randomness-Impreciseness Consistency Principle leads to define a normal law of impreciseness using two different laws of randomness. The Dubois-Prade left and right reference functions of an imprecise number are distribution function and complementary distribution function respectively. In this article, based on the Randomness-Impreciseness Consistency Principle we have successfully obtained the membership surface of imprecise vector and demonstrated with the help of numerical examples.
KeywordsMembership function Imprecise vector Dubois-Prade left and right reference functions Distribution function Density function
L(x) being a continuous non-decreasing function in the interval [a, b], and R(x) being a continuous non-increasing function in the interval [b, c], with L(a) = R(c) = 0 and L(b) = R(b) = 1. Dubois and Prade named L(x) as left reference function and R(x) as right reference function of the concerned fuzzy number. A continuous non-decreasing function of this type is also called a distribution function with reference to a Lebesgue-Stieltjes measure (De Barra 1987).
In this article, on the simple assumption that the Dubois-Prade left reference function is a distribution function, and similarly the Dubois-Prade right reference function is a complementary distribution function, we are going to demonstrate the method of obtaining the membership surface of an imprecise vector. Here the term imprecise is used instead of fuzzy because, in the Zadehian theory of fuzzy sets there are two flaws (Baruah 2011a, Baruah 2012). First, it had been accepted that the fuzzy sets do not in any way conform to the classical measure theoretic formalisms. Secondly, it had been agreed upon that given a fuzzy set neither its intersection with its complement is the null set, nor its union with the complement is the universal set. The Zadehian definition of complement of a fuzzy set is defective (Baruah 1999). In the Zadehian definition of complementation, fuzzy membership function and fuzzy membership value have been taken to be the same, and that is where the defect lies. Indeed fuzzy membership function and fuzzy membership value are two different things for the complement of a normal fuzzy set (Baruah 2011a). Instead of saying (Baruah 2011b; Baruah 2011c) that the mathematics of fuzziness has been incorrectly explained, Baruah has started the whole process anew, introducing the theory of imprecise sets, which might initially look similar to the theory of fuzzy sets.
2. The mathematical explanation of imprecise vector
We can easily visualize in Figure 1, that the probability of the shaded area is 1 and the probability for the unshaded area of the plates will be ½. But if the number of superimposition is large then it will be very difficult to obtain the probabilities by simply observing the imposition of the plates. So, in that situation a different technique can be used to obtain the probabilities when the number of operation of superimposition is very large. At first, it is discussed, about the operation of superimposition in the two dimensional case when the variable X is imprecise but Y is not imprecise.
The operation of superimposition
where (a(1), 0) = min[(a1, 0), (a2, 0)] , (a(2), 0) = max[(a1, 0), (a2, 0)] , (b(1), 0) = min[(b1, 0), (b2, 0)] and (b(2), 0) = max[(b1, 0), (b2, 0)]. Here we have assumed without any loss of generality that [(a1, 0), (b1, 0)] ∩ [(a2, 0), (b2, 0)] is not void or in other words that max[(a i , 0)] ≤ min[(b i , 0)], i = 1, 2.
Here, in the case of two dimensions we have considered that the value of y is zero. But instead of zero if we consider any precise value of y, then in the above membership surface only the value of y will be changed.
Here, we have also considered the value of x is zero. But instead of zero if we consider any precise value of x, then in the above membership surface only the value of x will be changed.
Now, we are going to discuss about the method how to obtain the membership surface of the vector (X, Y), where the variables x and y both are imprecise.
3. Numerical Examples
According to Randomness- Impreciseness Consistency Principle the left reference functions L(x) = x - 1; 1 ≤ x ≤ 2, 3 ≤ y ≤ 6 and , are distribution functions and the right reference functions and R(y) = 6 - y; 5 ≤ y ≤ 6, 1 ≤ x ≤ 4 are complementary distribution functions.
Now, to get the surface section of the membership surface if we cut the membership surface of the imprecise vector, which is in the two dimensions is nothing but the membership function of a subnormal imprecise number. If we cut the membership surface through the point on which the presence level is one, which is in the two dimensions is nothing but the membership function of a normal imprecise number.
In this article, the method has been shown successfully how to obtain the membership surface of the imprecise vector based on the Randomness- Impreciseness Consistency Principle. Here nothing has been done heuristically. The theory has been successfully developed and demonstrated with the help of numerical examples. Here the method of construction of the membership surface has been studied only for two and three dimensional vectors, but with the help of this method one can easily obtain the membership surface of n-dimensional vector too.
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