Receive a 20% Discount on All Purchases Directly Through IGI Global's Online Bookstore

Hung T. Nguyen (New Mexico State University, USA), Vladik Kreinovich (University of Texas at El Paso, USA) and Gang Xiang (University of Texas at El Paso, USA)

Source Title: Intelligent Data Analysis: Developing New Methodologies Through Pattern Discovery and Recovery

Copyright: © 2009
|Pages: 27
DOI: 10.4018/978-1-59904-982-3.ch002

Chapter Preview

Top**What is a random set? An intuitive meaning.** What is a random set? Crudely speaking, a random number means that we have different numbers with different probabilities; a random vector means that we have different vectors with different probabilities; and similarly, a random set means that we have different sets with different probabilities.

How can we describe this intuitive idea in precise terms? To provide such a formalization, let us recall how probabilities and random vectors are usually defined.

**How probabilities are usually defined.** To describe probabilities, in general, we must have a set of possible situations , and we must be able to describe the probability *P* of different properties of such situations. In mathematical terms, a *property* can be characterized by the *set* of all the situations which satisfy this property. Thus, we must assign to sets , the probability value .

According to the intuitive meaning of probability (e.g., as frequency), if we have two disjoint sets and , then we must have . Similarly, if we have countably many mutually disjoint sets , we must have .

A mapping which satisfies this property is called *-additive.*

It is known that even in the simplest situations, for example, when we randomly select a number from the interval , it is not possible to have a -additive function which would be defined on *all* subsets of . Thus, we must restrict ourselves to a class of subsets of . Since subsets represent properties, a restriction on subsets means restriction on properties. If we allow two properties and , then we should also be able to consider their logical combinations , , and – which in set terms correspond to union, intersection, and complement. Similarly, if we have a sequence of properties , then we should also allow properties and which correspond to countable union and intersection. Thus, the desired family should be closed under (countable) union, (countable) intersection, and complement. Such a family is called a *-algebra.*

Search this Book:

Reset

Copyright © 1988-2018, IGI Global - All Rights Reserved