Random Sets pp 209-242 | Cite as

# Extension of Relational and Conditional Event Algebra to Random Sets with Applications to Data Fusion

- 5 Citations
- 367 Downloads

## Abstract

Conditional event algebra (CEA) was developed in order to represent conditional probabilities with differing antecedents by the probability evaluation of well-defined individual “conditional” events in a single larger space extending the original unconditional one. These conditional events can then be combined logically before being evaluated. A major application of CEA is to data fusion problems, especially the testing of hypotheses concerning the similarity or redundancy among inference rules through use of probabilistic distance functions which critically require probabilistic conjunctions of conditional events. Relational event algebra (REA) is a further extension of CEA, whereby functions of probabilities formally representing single event probabilities — not just divisions as in the case of CEA — are shown to represent actual “relational” events relative to appropriately determined larger probability spaces. Analogously, utilizing the logical combinations of such relational events allows for testing of hypotheses of similarity between data fusion models represented by functions of probabilities. Independent of, and prior to this work, it was proven that a major portion of fuzzy logic — a basic tool for treating natural language descriptions — can be directly related to probability theory via the use of one point random set coverage functions. In this paper, it is demonstrated that a natural extension of the one point coverage link between fuzzy logic and random set theory can be used in conjunction with CEA and REA to test for similarity of natural language descriptions.

## Key words

Conditional Event Algebra Conditional Probability Data Fusion Functions of Probabilities Fuzzy Logic One Point Coverage Functions Probabilistic Distances Random Sets Relational Event Algebra## Preview

Unable to display preview. Download preview PDF.

## References

- [1]E. Adams
*The Logic of Conditionals*D. Reidel, Dordrecht, Holland, 1975.zbMATHGoogle Scholar - [2]D.E. Bamber
*Personal communications*Naval Command Control Ocean Systems Center, San Diego, CA, 1992.Google Scholar - [3]G. Dall’aglio, S. Kotz, and G. Salinetti (eds.)
*Advances in Probability Distributions with Given Marginals*Kluwer Academic Publishers, Dordrecht, Holland, 1991.zbMATHCrossRefGoogle Scholar - [4]D. Dubois and H. Prade
*Fuzzy Sets and Systems*Academic Press, New York, 1980.zbMATHGoogle Scholar - [5]E. Eells and B. Skyrms (eds.)
*Probability and Conditionals*Cambridge University Press, Cambridge, U.K., 1994.zbMATHGoogle Scholar - [6]M.J. Frank
*On the simultaneous associativity of F(x**y) and x + y - F(x*y), Aequationes Math., 19 (1979), pp. 194–226.MathSciNetzbMATHCrossRefGoogle Scholar - [7]I.R. Goodman
*Evaluation of combinations of conditioned information: A history*Information Sciences, 57–58 (1991), pp. 79–110.CrossRefGoogle Scholar - [8]I.R. Goodman
*Algebraic and probabilistic bases for fuzzy sets and the development of fuzzy conditioning*Conditional Logic in Expert Systems (I.R. Goodman, M.M. Gupta, H.T. Nguyen, and G.S. Rogers, eds.), North-Holland, Amsterdam (1991), pp. 1–69.Google Scholar - [9]I.R. Goodman
*Development of a new approach to conditional event algebra with application to operator requirements in a CS setting*Proceedings of the 1993 Symposium on Command and Control Research, National Defense University, Washington, DC, June 28–29, 1993, pp. 144–153.Google Scholar - [10]I.R. Goodman
*A new characterization of fuzzy logic operators producing homomorphic-like relations with one-point coverages of random sets*Advances in Fuzzy Theory and Technology, Vol. II (P. P. Wang, ed.), Duke University, Durham, NC, 1994, pp. 133–159.Google Scholar - [11]I.R. Goodman
*A new approach to conditional fuzzy sets*Proceedings of the Second Annual Joint Conference on Information Sciences, Wrightsville Beach, NC, September 28 - October 1, 1995, pp. 229–232.Google Scholar - [12]I.R. Goodman
*Similarity measures of events**relational event algebra**and extensions to fuzzy logic*Proceedings of the 1996 Biennial Conference of North American Fuzzy Information Processing Society-NAFIPS, University of California at Berkeley, Berkeley, CA, June 19–22, 1996, pp. 187–191.CrossRefGoogle Scholar - [13]I.R. Goodman and G.F. Kramer
*Applications of relational event algebra to the development of a decision aid in command and control*Proceedings of the 1996 Command and Control Research and Technology Symposium, Naval Postgraduate School, Monterey, CA, June 25–28, 1996, pp. 415–435.Google Scholar - [14]I.R. Goodman and G.F. Kramer
*Extension of relational event algebra to a general decision making setting*Proceedings of the Conference on Intelligent Systems: A Semiotic Perspective, Vol. I, National Institute of Standards and Technology, Gaithersberg, MD, October 20–23, 1996, pp. 103–108.Google Scholar - [15]I.R. Goodman and G.F. Kramer
*Comparison of incompletely specified models in C41 and data fusion using relational and conditional event algebra*Proceedings of the^{3rd}International Command and Control Research and Technology Symposium, National Defense University, Washington, D.C., June 17–20, 1997.Google Scholar - [16]I.R. Goodman and H.T. Nguyen
*Uncertainty Models for Knowledge-Based Systems*North-Holland, Amsterdam, 1985.zbMATHGoogle Scholar - [17]I.R. Goodman and H.T. Nguyen, A
*theory of conditional information for probabilistic inference in intelligent systems: II product space approach; III mathematical appendix*Information Sciences, 76 (1994), pp. 13–42; 75 (1993), pp. 253–277.MathSciNetzbMATHCrossRefGoogle Scholar - [18]I.R. Goodman and H.T. Nguyen
*Mathematical foundations of conditionals and their probabilistic assignments*International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 3 (1995), pp. 247–339.MathSciNetCrossRefGoogle Scholar - [19]I.R. Goodman, H.T. Nguyen, and E.A. Walker
*Conditional Inference and Logic for Intelligent Systems*North-Holland, Amsterdam, 1991.Google Scholar - [20]T. Hailperin
*Probability logic*Notre Dame Journal of Formal Logic, 25 (1984), pp. 198–212.MathSciNetzbMATHCrossRefGoogle Scholar - [21]D.A. Kappos
*Probability Algebra and Stochastic Processes*Academic Press, New York, 1969, pp. 16–17 et passim.Google Scholar - [22]D. Lewis
*Probabilities of conditionals and conditional probabilities*Philosophical Review, 85 (1976), pp. 297–315.CrossRefGoogle Scholar - [23]V. Mcgee
*Conditional probabilities and compounds of conditionals*Philosophical Review, 98 (1989), pp. 485–541.CrossRefGoogle Scholar - [24]C.V. Negoita and D.A. Ralescu
*Representation theorems for fuzzy concepts*Kybernetes, 4 (1975), pp. 169–174.zbMATHCrossRefGoogle Scholar - [25]H.E. Robbins
*On the measure of a random set*Annals of Mathematical Statistics, 15 (1944), pp. 70–74.MathSciNetzbMATHCrossRefGoogle Scholar - [26]N. Rowe
*Artificial Intelligence through PROLOG*Prentice-Hall, Englewood Cliffs, NJ, 1988 (especially, Chapter 8).Google Scholar - [27]G. Shafer
*A Mathematical Theory of Evidence*Princeton University Press, Princeton, NJ, 1976, p. 48 et passim.zbMATHGoogle Scholar - [28]A. Sklar
*Random variables**joint distribution functions**and copulas*Kybernetika, 9 (1973), pp. 449–460.MathSciNetzbMATHGoogle Scholar - [29]R. Stalnaker
*Probability and conditionals*Philosophy of Science, 37 (1970), pp. 64–80.MathSciNetCrossRefGoogle Scholar - [30]B. Van Fraasen
*Probabilities of conditionals*Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, (W.L. Harper and C.A. Hooker, eds.), D. Reidel, Dordrecht, Holland (1976), pp. 261–300.CrossRefGoogle Scholar