# Belief functions induced by multimodal probability density functions, an application to the search and rescue problem

P.-E. Doré; A. Martin; I. Abi-Zeid; A.-L. Jousselme; P. Maupin

RAIRO - Operations Research - Recherche Opérationnelle (2010)

- Volume: 44, Issue: 4, page 323-343
- ISSN: 0399-0559

## Access Full Article

top## Abstract

top## How to cite

topDoré, P.-E., et al. "Belief functions induced by multimodal probability density functions, an application to the search and rescue problem." RAIRO - Operations Research - Recherche Opérationnelle 44.4 (2010): 323-343. <http://eudml.org/doc/245966>.

@article{Doré2010,

abstract = {In this paper, we propose a new method to generate a continuous belief functions from a multimodal probability distribution function defined over a continuous domain. We generalize Smets' approach in the sense that focal elements of the resulting continuous belief function can be disjoint sets of the extended real space of dimension n. We then derive the continuous belief function from multimodal probability density functions using the least commitment principle. We illustrate the approach on two examples of probability density functions (unimodal and multimodal). On a case study of Search And Rescue (SAR), we extend the traditional probabilistic framework of search theory to continuous belief functions theory. We propose a new optimization criterion to allocate the search effort as well as a new rule to update the information about the lost object location in this latter framework. We finally compare the allocation of the search effort using this alternative uncertainty representation to the traditional probabilistic representation.},

author = {Doré, P.-E., Martin, A., Abi-Zeid, I., Jousselme, A.-L., Maupin, P.},

journal = {RAIRO - Operations Research - Recherche Opérationnelle},

keywords = {continuous belief function; multimodal probability density function; consonant belief function; optimal search; search and rescue (SAR)},

language = {eng},

number = {4},

pages = {323-343},

publisher = {EDP-Sciences},

title = {Belief functions induced by multimodal probability density functions, an application to the search and rescue problem},

url = {http://eudml.org/doc/245966},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Doré, P.-E.

AU - Martin, A.

AU - Abi-Zeid, I.

AU - Jousselme, A.-L.

AU - Maupin, P.

TI - Belief functions induced by multimodal probability density functions, an application to the search and rescue problem

JO - RAIRO - Operations Research - Recherche Opérationnelle

PY - 2010

PB - EDP-Sciences

VL - 44

IS - 4

SP - 323

EP - 343

AB - In this paper, we propose a new method to generate a continuous belief functions from a multimodal probability distribution function defined over a continuous domain. We generalize Smets' approach in the sense that focal elements of the resulting continuous belief function can be disjoint sets of the extended real space of dimension n. We then derive the continuous belief function from multimodal probability density functions using the least commitment principle. We illustrate the approach on two examples of probability density functions (unimodal and multimodal). On a case study of Search And Rescue (SAR), we extend the traditional probabilistic framework of search theory to continuous belief functions theory. We propose a new optimization criterion to allocate the search effort as well as a new rule to update the information about the lost object location in this latter framework. We finally compare the allocation of the search effort using this alternative uncertainty representation to the traditional probabilistic representation.

LA - eng

KW - continuous belief function; multimodal probability density function; consonant belief function; optimal search; search and rescue (SAR)

UR - http://eudml.org/doc/245966

ER -

## References

top- [1] I. Abi-Zeid and J.R. Frost, SARPlan: A decision support system for Canadian Search and Rescue Operations. Eur. J. Oper. Res. 162 (2004) 630–653. Zbl1065.90515
- [2] S.S. Brown, Optimal Search for a moving target in discret time and space. Oper. Res. 28 (1980) 1275–1289. Zbl0447.90046
- [3] F. Caron, B. Ristic, E. Duflos and P. Vanheeghe, Least committed basic belief density induced by a multivariate Gaussian: Formulation with applications. Int. J. Approx. Reason. 48 (2008) 419–436. Zbl1185.68703
- [4] J. de Guenin, Optimum Distribution of Effort: an Extension of the Koopman Basic Theory. Oper. Res. 9 (1961) 1–7.
- [5] A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38 (1967) 325–339. Zbl0168.17501
- [6] T. Denoeux, Extending stochastic ordering to belief functions on the real line. Inform. Sci. 179 (2009) 1362–1376,. Zbl1171.68044
- [7] P.-E. Doré, A. Fiche and A. Martin, Models of belief functions – Impacts for patterns recognition,13th International Conference on Information Fusion. Edinburgh, Scotland (2010).
- [8] P.-E. Doré, A. Martin and A. Khenchaf, Constructing consonant belief function induced by a multimodal probability. Proceedings of symposium COGnitive systems with Interractive Sensors (COGIS 2009), Espace Hamelin at Paris (2009).
- [9] P.-E. Doré, A. Martin, I. Abi-Zeid, A.-L. Jousselme and P. Maupin, Theory of belief functions for information combination and update in search and rescue operations, 12th International Conference on Information Fusion (2009). Zbl1308.62010
- [10] D. Dubois and H. Prade, The principle of minimum specificity as a basis for evidential reasoning, Processing and Management of Uncertainty in Knowledge-Based Systems on Uncertainty in knowledge-based systems. International Conference on Information table of contents. Springer-Verlag London, UK (1987) 75–84. Zbl0643.94008
- [11] B. Koopman, The theory of search. I. Kinematic bases. Oper. Res. 4 (1956) 324–346.
- [12] B. Koopman, The Theory of Search II, Target Detection. Oper. Res. 4 (1956) 503–531.
- [13] B. Koopman, The theory of search III. The optimum distribution of searching effort. Oper. Res. 5 (1957) 613–626. Zbl0995.90543
- [14] L. Liu. A theory of Gaussian belief functions. Int. J. Approx. Reason. 14 (1996) 95–126. Zbl0941.68763
- [15] A. Martin and C. Osswald, Toward a combination rule to deal with partial conflict and specificity in belief functions theory, 10th International Conference on Information Fusion (2007) 1–8.
- [16] S. Pollock, Search detection and subsequent action: some problems on the interfaces. Oper. Res. 19 (1971) 559–586. Zbl0217.26701
- [17] H.R. Richardson and B. Belkin, Optimal Search with Uncertain Sweep Width. Oper. Res. 20 (1972) 764–784. Zbl0335.90031
- [18] B. Ristic and Ph. Smets, Belief function theory on the continuous space with an application to model based classification. Proceedings of Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU (2004) 4–9.
- [19] S. Schvartz, I. Abi-Zeid and N. Tourigny, Knowledge Engineering for Modelling Reasoning in a Diagnosis Task: Application to Search and Rescue. Can. J. Adm. Sci. 24 (2007) 196.
- [20] G. Shafer, A mathematical theory of evidence. Princeton University Press Princeton, NJ (1976). Zbl0359.62002
- [21] Ph. Smets, The combination of evidence in the transferable belief model. IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 447–458.
- [22] Ph. Smets, Constructing the pignistic probability function in a context of uncertainty. Uncertainty in Artificial Intelligence 5 (1990) 29–39. Zbl0721.68065
- [23] Ph. Smets, Belief functions: the disjunctive rule of combination and the generalized, Int. J. Approx. Reason. 9 (1993) 1–35. Zbl0796.68177
- [24] Ph. Smets, Belief functions on real numbers. Int. J. Approx. Reason. 40 (2005) 181–223. Zbl1110.68149
- [25] Ph. Smets and R. Kennes, The transferable belief model, Artif. Intell. 66 (1994) 191–234. Zbl0807.68087
- [26] Ph. Smets and B. Ristic, Kalman filter and joint tracking and classification based on belief functions in the TBM framework. Inform. Fusion 1 (2007) 16–27.
- [27] T.M. Strat, Continuous belief functions for evidential reasoning. Proceedings of the National Conference on Artificial Intelligence, University of Texas at Austin (1984).
- [28] G. Souris and J.-P. Le Cadre, Un panorama des méthodes d'optimisation, Traitement du Signal 16 (1999) 403–424. Zbl1017.90053
- [29] L.D. Stone, Theory of Optimal Search, Mathematics in Science and in Engineering 118. 1st edition, Academic Press (1975). Zbl0343.90030
- [30] E. Vincent and V.Q.C. Defence, R&D Canada-Valcartier, Measures of Effectiveness for Airborne Search and Rescue Imaging Sensors, DRDC Valcartier TM 301 (2005).
- [31] E. Vincent and D. Valcartier, Searching performance at the 2005 National SAREX, DRDC Valcartier TM 110 (2006).
- [32] Y. Yang, A. Minai and M. Polycarpou, Evidential map-building approaches for multi-UAV cooperative search, American Control Conference, Proceedings of the 2005 (2005) 116–121.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.