# Hybrid processing of stochastic and subjective uncertainty data

## Abstract

Uncertainty analyses typically recognize separate stochastic and subjective sources of uncertainty, but do not systematically combine the two, although a large amount of data used in analyses is partly stochastic and partly subjective. We have developed methodology for mathematically combining stochastic and subjective data uncertainty, based on new ``hybrid number`` approaches. The methodology can be utilized in conjunction with various traditional techniques, such as PRA (probabilistic risk assessment) and risk analysis decision support. Hybrid numbers have been previously examined as a potential method to represent combinations of stochastic and subjective information, but mathematical processing has been impeded by the requirements inherent in the structure of the numbers, e.g., there was no known way to multiply hybrids. In this paper, we will demonstrate methods for calculating with hybrid numbers that avoid the difficulties. By formulating a hybrid number as a probability distribution that is only fuzzy known, or alternatively as a random distribution of fuzzy numbers, methods are demonstrated for the full suite of arithmetic operations, permitting complex mathematical calculations. It will be shown how information about relative subjectivity (the ratio of subjective to stochastic knowledge about a particular datum) can be incorporated. Techniques are also developed for conveying uncertainty informationmore »

- Authors:

- Sandia National Labs., Albuquerque, NM (United States)
- Applied Biomathematics, Setauket, NY (United States)
- State Univ. of New York, Stony Brook, NY (United States)

- Publication Date:

- Research Org.:
- Sandia National Labs., Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 200708

- Report Number(s):
- SAND-95-2450

ON: DE96003603

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: Nov 1995

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; PROBABILISTIC ESTIMATION; DATA COVARIANCES; STATISTICS; STOCHASTIC PROCESSES; PROBABILITY; FUZZY LOGIC

### Citation Formats

```
Cooper, J.A., Ferson, S., and Ginzburg, L.
```*Hybrid processing of stochastic and subjective uncertainty data*. United States: N. p., 1995.
Web. doi:10.2172/200708.

```
Cooper, J.A., Ferson, S., & Ginzburg, L.
```*Hybrid processing of stochastic and subjective uncertainty data*. United States. doi:10.2172/200708.

```
Cooper, J.A., Ferson, S., and Ginzburg, L. Wed .
"Hybrid processing of stochastic and subjective uncertainty data". United States.
doi:10.2172/200708. https://www.osti.gov/servlets/purl/200708.
```

```
@article{osti_200708,
```

title = {Hybrid processing of stochastic and subjective uncertainty data},

author = {Cooper, J.A. and Ferson, S. and Ginzburg, L.},

abstractNote = {Uncertainty analyses typically recognize separate stochastic and subjective sources of uncertainty, but do not systematically combine the two, although a large amount of data used in analyses is partly stochastic and partly subjective. We have developed methodology for mathematically combining stochastic and subjective data uncertainty, based on new ``hybrid number`` approaches. The methodology can be utilized in conjunction with various traditional techniques, such as PRA (probabilistic risk assessment) and risk analysis decision support. Hybrid numbers have been previously examined as a potential method to represent combinations of stochastic and subjective information, but mathematical processing has been impeded by the requirements inherent in the structure of the numbers, e.g., there was no known way to multiply hybrids. In this paper, we will demonstrate methods for calculating with hybrid numbers that avoid the difficulties. By formulating a hybrid number as a probability distribution that is only fuzzy known, or alternatively as a random distribution of fuzzy numbers, methods are demonstrated for the full suite of arithmetic operations, permitting complex mathematical calculations. It will be shown how information about relative subjectivity (the ratio of subjective to stochastic knowledge about a particular datum) can be incorporated. Techniques are also developed for conveying uncertainty information visually, so that the stochastic and subjective constituents of the uncertainty, as well as the ratio of knowledge about the two, are readily apparent. The techniques demonstrated have the capability to process uncertainty information for independent, uncorrelated data, and for some types of dependent and correlated data. Example applications are suggested, illustrative problems are worked, and graphical results are given.},

doi = {10.2172/200708},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Wed Nov 01 00:00:00 EST 1995},

month = {Wed Nov 01 00:00:00 EST 1995}

}