Formulation of the . . . Our Explanation Towards a Natural Interval Main Reference Interpretation of Pythagorean and Complex Home Page Title Page Degrees of Confidence ◭◭ ◮◮ ◭ ◮ Jose Perez, Eric Torres, and Vladik Kreinovich Page 1 of 7 University of Texas at El Paso Go Back El Paso, TX 79968, USA Full Screen jmperez6@miners.utep.edu Close emtorres6@miners.utep.edu vladik@utep.edu Quit
1. Formulation of the Problem Formulation of the . . . • Often: Our Explanation Main Reference – we only know the expert’s degrees of confidence a, b ∈ [0 , 1] in statements A and B , – and we need to estimate the expert’s degree of con- Home Page fidence in A & B . Title Page • The algorithm f & ( a, b ) providing the corresponding es- ◭◭ ◮◮ timate is known as an “and”-operation , or a t-norm. ◭ ◮ • One of the most frequently used “and”-operation is Page 2 of 7 min( a, b ) . Go Back • Similarly, one of the most frequently used “or”- Full Screen operation is Close max( a, b ) . Quit
2. Formulation of the Problem (cont-d) Formulation of the . . . • Often, it is difficult for an expert to describe his/her Our Explanation degree of certainty by a single number a . Main Reference • An expert is more comfortable describing it by range (interval) [ a, a ] of possible values. Home Page • An alternative way of describing this is as an intuition- Title Page istic fuzzy degree , i.e., a pair of values a and 1 − a . ◭◭ ◮◮ • If we know: ◭ ◮ – intervals [ a, a ] and [ b, b ] corresponding to a and b , Page 3 of 7 – then the range of possible degree of confidence in Go Back A & B is formed by values min( a, b ) corresponding to all a ∈ [ a, a ] and b ∈ [ b, b ]. Full Screen • Since min( a, b ) is monotonic, this range has the form Close [min( a, b ) , min( a, b )] . Quit • Similarly, the range for A ∨ B is [max( a, b ) , max( a, b )].
3. Formulation of the Problem (cont-d) Formulation of the . . . • A recent paper describes extensions of the above defi- Our Explanation nitions from a, b ∈ [0 , 1] to a, b ∈ [ − 1 , 1]. Main Reference • These extensions are denoted by Home Page [absmin( a, b ) , absmin( a, b )] and [absmax( a, b ) , absmax( a, b )] . Title Page • Here, absmin( a, b ) = a if | a | < | b | . ◭◭ ◮◮ • absmin( a, b ) = b is | a | > | b | , and ◭ ◮ • absmin(a,b)= −| a | if | a | = | b | and a � = b . Page 4 of 7 • absmax( a, b ) = a if | a | > | b | . Go Back • absmax( a, b ) = b if | a | < | b | , and Full Screen • absmax( a, b ) = | a | if | a | = | b | and a � = b . Close • These operations have nice properties – associativity, Quit distributivity – but what is their meaning?
4. Our Explanation Formulation of the . . . • In addition to closed intervals, let us consider open and Our Explanation semi-open ones. Main Reference • An open end will be then denoted by the negative num- ber: Home Page – for example, (0 . 3 , 0 . 5] is denoted as [ − 0 . 3 , 0 . 5], and Title Page – the interval (0 . 3 , 0 . 5) is denoted as [ − 0 . 3 , − 0 . 5]. ◭◭ ◮◮ • By considering all possible cases, one can show that: ◭ ◮ – for two intervals A = [ a, a ] and B = [ B, B ], Page 5 of 7 – the range of possible values Go Back { min( a, b ) : a ∈ A, b ∈ B } Full Screen Close – is indeed equal to Quit [abs min( a, b ) , abs min( a, b )] .
5. Our Explanation (cont-d) Formulation of the . . . • For example, for A = (0 . 3 , 0 . 7] = [ − 0 . 3 , 0 . 7] and B = Our Explanation [0 . 2 , 0 . 6) = [0 . 2 , − 0 . 6], we have Main Reference { min( a, b ) : a ∈ A, b ∈ B } = [0 , 2 , 0 . 6) = [0 . 2 , − 0 . 6] . Home Page • Here indeed: Title Page – we have absmin( − 0 . 3 , 0 . 2) = 0 . 2 and ◭◭ ◮◮ – we have absmin(0 . 7 , − 0 . 6) = − 0 . 6 . ◭ ◮ • Similarly, the range of possible values of max( a, b ) Page 6 of 7 forms the interval Go Back [abs max( a, b ) , abs max( a, b )] . Full Screen • Thus, we get a natural interval explanation of the ex- Close tended operations. Quit
6. Main Reference Formulation of the . . . • S. Dick, R. Yager, and O. Yazdanbakhsh, “On Our Explanation Pythagorean and complex fuzzy set operations”, IEEE Main Reference Transactions on Fuzzy Systems , 2016, Vol. 24, No. 5, pp. 1009–1021. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 7 Go Back Full Screen Close Quit
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