# Level sets and minimum volume sets of probability density functions

## Details

Serval ID

serval:BIB_9924C62F1A67

Type

**Article**: article from journal or magazin.

Collection

Publications

Institution

Title

Level sets and minimum volume sets of probability density functions

Journal

International Journal of Approximate Reasoning

ISSN

0888-613X

Publication state

Published

Issued date

2003

Volume

34

Number

1

Pages

25-47

Language

english

Abstract

Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse proposition is verified. The distribution function of the level cuts of a density function is also introduced. It provides a different visualization of the distribution of the probability for the random variable in question. It is also very useful to prove the above proposition. The volume of the minimum volume sets varies according to its probability alpha: smaller volume implies lower probability and vice versa and larger volume implies larger probability and vice versa. In this context, 1 - alpha is the error of an erroneously classification of a new observation inside of the minimum volume set or corresponding level set. To decide the volume and/or the error of the level set that will serve to summarize the support of the random variable, a alpha - lambda plot is defined. We also study the relation of the minimum volume set approach with random set theory when cc is a random variable and extend the most specific probability-possibility transformation proposed in [System Theory, Knowledge Engineering and Problem Solving, in: Fuzzy Logic: State of the Art, vol. 12, Kluwer, 1993, pp. 103-112] to continuous piece-wise strictly monotone probability density functions.

Keywords

minimum volume set, level set, random set, probability-possibility transformation

Web of science

Open Access

Yes

Create date

12/03/2013 13:39

Last modification date

20/08/2019 16:00