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Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.21 n.2 San José Jul./Dec. 2014
Geodesic distribution in graph theory: Kullback-Leibler-Symmetric
Distribución geodésica en teoría de Grafos: Kullback-Leibler-Simétrica
Distribución geodésica en teoría de Grafos: Kullback-Leibler-Simétrica
Abstract
Kullback-Leibler information allow us to characterize a family of distributions denominated Kullback-Leibler-Symmetric, which are distance functions and, under some estrictions, generate the Jensen’s equality shown by [1], in this paper denominated Jensen-Equal. On the other hand, [5] and [7] showed that graph theory gives conditions to define a new measurable space and, therefore, new distances, in particular, the distance characterized by [2], denominated Geodesic Distance. The interaction of these ideas allow us to define a new distribution, denominated Geodesic Distributionwhich, under graph theory as center and radius of a graph, we can to develop optimization methodologies based in probabilities of attendance. We obtain many applications and the proposal method is very adaptive. To illustrate, we apply this distribution in spatial statistics.
Keywords: Kullback-Leibler information; graph theory; geodesic distance; geodesic distribution.
Resumen
La información de Kullback-Leibler permite caracterizar una familia de distribuciones que denominamos Kullback-Liebler-Simétricas de las cuales tenemos distribuciones que son funciones de una distancia que bajo restricciones genera la igualdad en la relación de Jensen mostrados por [1], las que denominamos Jensen-Igual. Por otra parte, [5] y [7] presentan que la teoría de grafos permite definir un espacio medible y por tanto nuevas distancias, en particular la caracterizada por [2] denominada distancia Geodésica. La interacción de las dos ideas permite inducir una distribución que denominaremos Geodésica, la cual bajo técnicas de la teoría de grafos, como el centro y el radio de un grafo, permite desarrollar metodologías de optimización en función de las probabilidades de atendimiento. Obtenemos muchas áreas de aplicación y muchas adaptaciones, en las cuales, por ejemplo, aplicamos en un problema de estadística espacial.
Palabras clave: informaciónKullback-Leibler; teoría de grafos; distancia geodésica; distribución geodésica.
Mathematics Subject Classification: 62P99.
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References
[1] Beckenbach, E. (1937) “Generalized convex functions”, Bulletin of American Mathematical Society, 43(6): 363–371. [ Links ]
[2] Bouttier, J.; Di Francesco, P; Guitter, E. (2003) “Geodesic distance in plan argraphs”, Nuclear Physics B 663(3): 535–567. [ Links ]
[3] Chartrand, G.; Oellermann, O.R. (1993) Applied and Algorithmic Graph Theory. McGraw-Hill, New York. [ Links ]
[4] Chartrand, G.; Lesniak, L. (1996) Graphs and Digraphs. Chapman & Hall/CRC, New York. [ Links ]
[5] Chung, F.; Horn, P.; Lu, L. (2012) “Diameter of random spanning trees in a given graph”, Journal of Graph Theory 69(3): 223–240. [ Links ]
[6] Lachos, V.H.; Bandyopadhyay, D.; Dey, D.K. (2011) “Linear and non-linear mixed-effects models for censored HIV viral loads using normal/independent distributions”, Biometrics 67(4): 1594–1604. [ Links ]
[7] Montenegro, E.; Cabrera, E. (2001) “Attractors points in the autosubstitution”, Proyecciones (Antofagasta) 20(2): 193–204. [ Links ]
[8] Montenegro, E.; González, J.; Cabrera, E.; Nettle A.; Robres, R. (2010) “Graphs r-polar spherical realization”, Proyecciones (Antofagasta) 29(1): 31–39. [ Links ]
[9] Schervish, M.J. (1995) Theory of Statistics. Springer, New York. [ Links ]
[2] Bouttier, J.; Di Francesco, P; Guitter, E. (2003) “Geodesic distance in plan argraphs”, Nuclear Physics B 663(3): 535–567. [ Links ]
[3] Chartrand, G.; Oellermann, O.R. (1993) Applied and Algorithmic Graph Theory. McGraw-Hill, New York. [ Links ]
[4] Chartrand, G.; Lesniak, L. (1996) Graphs and Digraphs. Chapman & Hall/CRC, New York. [ Links ]
[5] Chung, F.; Horn, P.; Lu, L. (2012) “Diameter of random spanning trees in a given graph”, Journal of Graph Theory 69(3): 223–240. [ Links ]
[6] Lachos, V.H.; Bandyopadhyay, D.; Dey, D.K. (2011) “Linear and non-linear mixed-effects models for censored HIV viral loads using normal/independent distributions”, Biometrics 67(4): 1594–1604. [ Links ]
[7] Montenegro, E.; Cabrera, E. (2001) “Attractors points in the autosubstitution”, Proyecciones (Antofagasta) 20(2): 193–204. [ Links ]
[8] Montenegro, E.; González, J.; Cabrera, E.; Nettle A.; Robres, R. (2010) “Graphs r-polar spherical realization”, Proyecciones (Antofagasta) 29(1): 31–39. [ Links ]
[9] Schervish, M.J. (1995) Theory of Statistics. Springer, New York. [ Links ]
* Departamento de Estadística, Universidad de Playa Ancha. Valparaiso, Chile. E-Mail: jgonzalez@upla.cl
† IMECC, UNICAMP. Campinas, Brasil. E-Mail: marcos.cascone@gmail.com
Received: 4/Sep/2013; Revised: 4/Jun/2014; Accepted: 10/Jun/2014
