Vorobyev, Oleg Yu. and Vorobyev, Alexey O. (2003): On the New Notion of the SetExpectation for a Random Set of Events. Published in: Proc. of the II AllRussian FAM'2003 Conference , Vol. 1, (27. April 2003): pp. 2337.

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Abstract
The paper introduces new notion for the setvalued mean set of a random set. The means are defined as families of sets that minimize mean distances to the random set. The distances are determined by metrics in spaces of sets or by suitable generalizations. Some examples illustrate the use of the new definitions.
Item Type:  MPRA Paper 

Original Title:  On the New Notion of the SetExpectation for a Random Set of Events 
Language:  English 
Keywords:  mean random set, metrics in set space, mean distance, Aumann expectation, Frechet expectation, Hausdorff metric, random finite set, mean set, setmedian, setexpectation 
Subjects:  C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics 
Item ID:  17901 
Depositing User:  Oleg Vorobyev 
Date Deposited:  16. Oct 2009 07:09 
Last Modified:  12. Feb 2013 21:49 
References:  [1] I.E. Abdou and W.K. Pratt, Quantitative design and evalualion of enhancement thresholding edge detectors, in: Proc. of the IEEE 67 (1979) 753763. [2] L.S. Andersen, Inference for hidden Markov models, in: A. Possolo ed., Proc. of the Joint IMSAMSSIAM Summer Res. Conf. on Spatial Statistics and lmaging (Brunswick. Maine. 1988) 113. [3] Z. Artstein and R.A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab. 3 (1975) 879882. [4] A.J. Baddeley, An error metric for binary images, in: W. Forstner, H. Ruwiedel, eds., Robust Computer Vision: Quality of Vision Algorithms (Wichmann: Karlsruhe, 1992) 5978. [5] A.J. Baddeley, Errors in binary images and an Lp version of the Hausdorff metric, Nieuw Archief voor Wiskunde 10 (1992) 157183. [6] A. Baddeley and I. Molchanov, Averaging of Random Sets Based on Their Distance Functions, Journal of Mathematical Imaging and Vision 8 (1998) 7992. [7] C. Beer, On convergence of closed sets in a metric space and distance functions, Bull Austral. Math. Soc. 31 (1985) 421432. [8] M. Frechet, Les elements aleatoires de nature quelconque dans un espace distancie, Ann. Inst. H. Poincare 10 (1948) 235310. [9] N. Friel and I. Molchanov, A New Thresholding Technique Based on Random Sets, Pattern Recognition 32 (1999) 15071517. [10] A. Frigessi and H. Rue, Bayesian image classification with Baddeley's Delta loss, J. Comput. Graph. Statist. 6 (1997) 5573. [11] S Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Analysis and Machine Intelligence 6 (1984) 721741. [12] J. Haslett and G. Horgan, Linear discriminant analysis in image restoration and the prediction of error rate, in: A. Possolo Ed., Proc. of the Joint IMSAMSSIAM Summer Res. Conf. on Spatial Statisitics and Imaging (Brunswick, Maine, 1988) 112119. [13] I. Molchanov, Random Sets in View of Image Filtering Application, in: Dougherty E., Astola J., eds., Nonlinear Filter for Image Processing 11 (1999) 419447. [14] H. Rue and A.R. Syversveen, Bayesian object recognition with Baddeley's Delta loss, Preprint Statistics 8 Department of Mathematics, The University of Trondheim, (Trondheim, Norway, 1995). [15] D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields (Wiley, Chichester, 1994). [16] C.C. Taylor, Measures of similarity between two images, in: A. Possolo ed., Proc. of the Joint IMSAMSSIAM Summer Res. Conf. On Spatial Statistics and Imaging (Brunswick, Maine, 1988) 382391. [17] R.A. Vitale, An alternate formulation of mean value for random geometric figures, J. Microscopy 151 (1988) 197204. [18] O.Yu. Vorobyev, About SetCharacteristics of States of Distributed Probabilistic Processes, Izvestiya of SB USSR AS 3 (1977) 37. [19] O.Yu. Vorobyev, Mean Measure Modeling (Moscow, Nauka, 1984). [20] O.Yu. Vorobyev, Random Set Models of Fire Spread, Fire Technology, USA National Fire Protection Association, 32 (2) (1996) 137173. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/17901 