By Professor Hung T. Nguyen, Professor Berlin Wu (auth.)

ISBN-10: 3540316957

ISBN-13: 9783540316954

ISBN-10: 3540316973

ISBN-13: 9783540316978

This learn monograph offers simple foundational points for a idea of facts with fuzzy facts, including a suite of sensible purposes. Fuzzy info are modeled as observations from random fuzzy units. Theories of fuzzy common sense and of random closed units are used as uncomplicated components in development statistical strategies and tactics within the context of obscure facts, together with coarse facts research. The monograph additionally goals at motivating statisticians to examine fuzzy information to amplify the area of applicability of facts in general.

HUNG T. NGUYEN is a professor of Mathematical Sciences at New Mexico nation college, USA.

BERLIN WU is a professor of Mathematical Sciences at nationwide Chengchi collage, Taipei, Taiwan.

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**Extra info for Fundamentals of Statistics with Fuzzy Data**

**Example text**

1. 3 Fuzzy Set-valued Random Elements 39 (ii) The support {x : A( x) ! 0} of A is bounded. (iii) The D -level set AD of A are closed intervals From the above definition, it is not hard to show that? (a) Real numbers are fuzzy numbers. e. AD is closed, D >0,1@ . e. intervals). As stated above, the use of intervals and their arithmetic is appropriate in some situations involving imprecision. When the intervals themselves are not sharply defined, we arrive at the concept of fuzzy intervals. e. x R such that A( x) 1 ).

3 Fuzzy Relations 27 For negation operators, we can also consider the class of operators K : [0,1] o [0,1] , such that: (a) K (0) 1, K (1) 0 (b) K is non-increasing (c) K ( K (x)) = x similarly, fuzzy implication operators are generalizations of material implication operator of two-valued logic. Specifically, a fuzzy implication is a map : [0,1] u [0,1] o [0,1] , where restriction to {0,1} u {0,1} coincides with the truth table of the material implication. Some examples are 1 if x d y (i) ( x y ) = ® ¯0 if x !

Such a partition is referred to as a coarsening of X. Providing a model for observing X. If we consider the set-valued map S(w) = the element of the partition containing X(w), then S is formally a random set with probability density P( S Ai ) = P ( X Ai ) , i=1,…,k Thus, coarse data are actually observations of random sets which are special cases of random fuzzy sets. The following is an example of statistical estimation in the context of coarse data. For details, see [38]. Suppose the random variable X admits a probability density function f ( x | T ) , where T 4 R .

### Fundamentals of Statistics with Fuzzy Data by Professor Hung T. Nguyen, Professor Berlin Wu (auth.)

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