An introduction to random sets by Hung T. Nguyen

By Hung T. Nguyen

The learn of random units is a big and swiftly turning out to be sector with connections to many components of arithmetic and functions in greatly various disciplines, from economics and determination thought to biostatistics and picture research. the downside to such variety is that the study stories are scattered in the course of the literature, with the end result that during technological know-how and engineering, or even within the statistics group, the subject isn't renowned and masses of the big strength of random units continues to be untapped. An advent to Random units presents a pleasant yet good initiation into the idea of random units. It builds the root for learning random set facts, which, considered as obscure or incomplete observations, are ubiquitous in brand new technological society. the writer, well known for his best-selling a primary path in Fuzzy good judgment textual content in addition to his pioneering paintings in random units, explores motivations, reminiscent of coarse info research and uncertainty research in clever platforms, for learning random units as stochastic types. different themes comprise random closed units, similar uncertainty measures, the Choquet necessary, the convergence of capability functionals, and the statistical framework for set-valued observations. An abundance of examples and workouts strengthen the recommendations mentioned. Designed as a textbook for a path on the complex undergraduate or starting graduate point, this booklet will serve both good for self-study and as a reference for researchers in fields akin to records, arithmetic, engineering, and desktop technology.

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Let (Ω, A, P ) be a probability space and U = {x1 , x2 , x3 , x4 }. Let X : Ω → U be a random variable with probability measure PX and S : Ω → 2U \ {∅} be a random set with density f (A) = P (S = A) given by f ({x1 }) = f ({x2 }) = f ({x1 , x2 , x3 }) = © 2006 by Taylor & Francis Group, LLC 1 , 6 1 , 6 f ({x3 }) = f (U ) = 1 , 6 1 , 4 f ({x4 }) = f (A) = 0, 1 , 12 Finite Random Sets 51 where A is any other subset of U . Suppose that S is a CAR model for X. Then 9 27 1 PX (x1 ) = PX (x2 ) = , PX (x3 ) = PX (x4 ) = .

Sn ) = j=1 Sj Thus, the maximum likelihood estimator of θ can be computed using only the observed S1 , S2 , . . , Sn . However, the investigation of large sample properties of the estimator requires also distributional aspects of the random set model S. 6 An Introduction to Random Sets Perception-Based Information Coarse data analysis can be viewed as a special procedure in perceptionbased information gathering process by humans. This type of data is used in the field of artificial intelligence to imitate remarkable human intelligent behavior, say, in decision making and control.

For example, a perception-based information of the form “Tony is young” has the following structure. The underlying variable of interest is X = age(of Tony) with range U = [0, 100]. , as a function A : U → [0, 1]. Students interested in the theory of fuzzy sets and logics can read a book like [89]. Since the value of X cannot be observed with accuracy, A = “young” is taken as an observation value instead. This fuzzy value A is in fact one of the possible fuzzy values, not of X, but of some coarsening S of X.

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