Download Fundamentals of Statistics with Fuzzy Data by Hung T. Nguyen PDF

By Hung T. Nguyen

This examine monograph provides easy foundational elements for a conception of data with fuzzy facts, including a collection of functional purposes. Fuzzy info are modeled as observations from random fuzzy units. Theories of fuzzy common sense and of random closed units are used as simple constituents in construction statistical suggestions and techniques within the context of vague information, together with coarse info research. The monograph additionally goals at motivating statisticians to examine fuzzy facts to magnify the area of applicability of information generally.

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In summary, there is a variety of fuzzy logics, each of which is formed by a choice of basic connectives and implication operators. 3 Fuzzy Relations An ordinary n-array relation is a subset R of the Cartesian product U 1 u U u " u U n of n sets. e. R: V o [0,1] . For example, R= much more important than is a 2 -array fuzzy relation . Let us elaborate a little bit about this fuzzy relation in decision- 28 Chapter 3 Modeling of Fuzzy Data making, say, by ranking alternatives using Saaty’s method of Analytic Hierarchy Process (AHP), see [74].

Thus, we create a random mechanism, just like a game of chance, to select samples. A sample is a subset of the population U. The random process leading to selections of samples is formally a random set. Samples are selected at random. T. Nguyen and B. com 36 Chapter 4 Random Fuzzy Sets size n, then a probability sampling design could be a random set who probability density function on 2U (the power set of U) is f : 2U o >0,1@ f ( A) ­ §N· °1 / ¨¨ ¸¸ if | A | n ® ©n¹ ° 0 otherwise ¯ Here a man-made randomization is introduced to make inductive logic valid.

Now the true input–output y = f(x) is unknown, and we have only some fuzzy information about it via the fuzzy rule base. Thus a good design is a good approximation procedure for f. , x d ) o y* We need to investigate to what extend f* will be a good approximation for f. Basically, this is the problem of approximation of functions. So a design methodology will lead to a good approximation for f if for any H > 0, we can find an f * such that || f  f * ||d H , where || ˜ || denote a distance between f and f * .

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