¹Universidad de Valparaíso, Departamento de Estadística, Valparaíso, Chile. Professor. Email: victor.leiva@uv.cl
²Universidad de Talca, Instituto de Matemáticas y Física, Talca, Chile. Assistant profesor. Email: gponce@utalca.cl
³Universidad de Valparaíso, Departamento de Estadística, Valparaíso, Chile. Assistant profesor. Email: carolina.marchant@uv.cl
⁴Universidad de Talca, Departamento de Producción Forestal, Talca, Chile. Assistant professor. Email: obustos@utalca.cl

Mortality processes and the distribution of the diameter at breast height (DBH) of trees are two important problems in forestry. Trees die due to several factors caused by stress according to a phenomenon similar to material fatigue. Specifically, the force (rate) of mortality of trees quickly increases at a first stage and then reaches a maximum. In that moment, this rate slowly decreases until stabilizing at a constant value in the long term establishing a second stage of such a rate. Birnbaum-Saunders (BS) distributions are models that have received considerable attention currently due to their interesting properties. BS models have their genesis from a problem of material fatigue and present a failure or hazard rate (equivalent to the force of mortality) that has the same behavior as that of the DBH of trees. Then, BS distributions have arguments that transform them into models that can be useful in forestry. In this paper, we present a methodology based on BS distributions associated with this forest thematic. To complete this study, we perform an application of five real DBH data sets (some of them unpublished) that provides statistical evidence in favor of the BS methodology in relation to the forestry standard methodology. This application provides valuable financial information that can be used for making decisions in forestry.

Key words: data analysis, force of mortality, forestry, hazard rate.

Los procesos de mortalidad y la distribución del diámetro a la altura del pecho (DAP) de árboles son dos problemas importantes en el área forestal. Los árboles mueren debido a diversos factores causados por estrés mediante un fenómeno similar a la fatiga de materiales. Específicamente, la fuerza (tasa) de mortalidad de árboles crece rápidamente en una primera fase y luego alcanza un máximo, momento en el que comienza una segunda fase en donde esta tasa decrece lentamente estabilizándose en una constante en el largo plazo. Distribuciones Birnbaum-Saunders (BS) son modelos que han recibido una atención considerable en la actualidad debido a sus interesantes propiedades. Modelos BS nacen de un problema de fatiga de materiales y poseen una tasa de fallas (equivalente a la fuerza de mortalidad) que se comporta de la misma forma que ésa del DAP de árboles. Entonces, distribuciones BS poseen argumentos que las transforman en modelos que puede ser útiles en las ciencias forestales. En este trabajo, presentamos una metodología basada en la distribución BS asociada con esta temática forestal. Para finalizar, realizamos una aplicación con cinco conjuntos de datos reales (algunos de ellos no publicados) de DAP que proporciona una evidencia estadística en favor de la metodología BS en relación a la metodología estándar usada en ciencias forestales. Esta aplicación entrega información que puede ser valiosa para tomar decisiones forestales.

Palabras clave: análisis de datos, fuerza de mortalidad, silvicultura, tasa de riesgo.

Texto completo disponible en PDF

References

1. Aarset, M. V. (1987), 'How to identify a bathtub hazard rate', IEEE Transaction on Reliability 36, 106-108.         [ Links ]

2. Azevedo, C., Leiva, V., Athayde, E. & Balakrishnan, N. (2012), 'Shape and change point analyses of the Birnbaum-Saunders-t hazard rate and associated estimation', Computational Statistics and Data Analysis 56, 3887-3897.         [ Links ]

3. Bailey, R. & Dell, T. (1973), 'Quantifying diameter distributions with the Weibull function', Forest Science 19, 97-104.         [ Links ]

4. Birnbaum, Z. & Saunders, S. (1968), 'A probabilistic interpretation of miner's rule', SIAM Journal of Applied Mathematics 16, 637-652.         [ Links ]

5. Birnbaum, Z. & Saunders, S. (1969), 'A new family of life distributions', Journal of Applied Probability 6, 319-327.         [ Links ]

6. Bliss, C. & Reinker, K. (1964), 'A lognormal approach to diameter distributions in even-aged stands', Forest Science 10, 350-360.         [ Links ]

7. Borders, B., Souter, R., Bailey, R. & Ware, K. (1987), 'Percentile-based distributions characterize forest stand tables', Forest Science 33, 570-576.         [ Links ]

8. Clutter, J. & Bennett, F. (1965), Diameter distributions in old-field slash pine plantation, Report 13, US Forest Service.         [ Links ]

9. Díaz-García, J. & Leiva, V. (2005), 'A new family of life distributions based on elliptically contoured distributions', Journal of Statistical Planning and Inference 128, 445-457. (Erratum: Journal of Statistical Planning and Inference, 137, 1512-1513).         [ Links ]

10. Ferreira, M., Gomes, M. & Leiva, V. (2012), 'On an extreme value version of the Birnbaum-Saunders distribution', Revstat-Statistical Journal 10, 181-210.         [ Links ]

11. García-Güemes, C., Cañadas, N. & Montero, G. (2002), 'Modelización de la distribución diamétrica de las masas de Pinus pinea de Valladolid (España) mediante la función Weibull', Investigación Agraria-Sistemas y Recursos Forestales 11, 263-282.         [ Links ]

12. Gavrilov, L. & Gavrilova, N. (2001), 'The reliability theory of aging and longevity', Journal of Theoretical Biology 213, 527-545.         [ Links ]

13. Guiraud, P., Leiva, V. & Fierro, R. (2009), 'A non-central version of the Birnbaum-Saunders distribution for reliability analysis', IEEE Transaction on Reliability 58, 152-160.         [ Links ]

14. Hafley, W. & Schreuder, H. (1977), 'Statistical distributions for fitting diameter and height data in even-aged stands', Canadian Journal of Forest Research 7, 481-487.         [ Links ]

15. Johnson, N., Kotz, S. & Balakrishnan, N. (1995), Continuous Univariate Distributions, Vol. 2, Wiley, New York.         [ Links ]

16. Leiva, V., Athayde, E., Azevedo, C. & Marchant, C. (2011), 'Modeling wind energy flux by a birnbaum-saunders distribution with unknown shift parameter', Journal of Applied Statistics 38, 2819-2838.         [ Links ]

17. Leiva, V., Barros, M., Paula, G. & Sanhueza, D. (2008), 'Generalized Birnbaum-Saunders distributions applied to air pollutant concentration', Environmetrics 19, 235-249.         [ Links ]

18. Leiva, V., Sanhueza, A. & Angulo, J. (2009), 'A length-biased version of the Birnbaum-Saunders distribution with application in water quality', Stochastic Environmental Research and Risk Assessment 23, 299-307.         [ Links ]

19. Leiva, V., Vilca, F., Balakrishnan, N. & Sanhueza, A. (2010), 'A skewed sinh-normal distribution and its properties and application to air pollution', Communications in Statistics - Theory and Methods 39, 426-443.         [ Links ]

20. Lenhart, J. & Clutter, J. (1971), Cubic foot yield tables for old-field loblolly pine plantations in the Georgia Piedmont, Report 22, US Forest Service.         [ Links ]

21. Li, F., Zhang, L. & Davis, C. (2002), 'Modeling the joint distribution of tree diameters and heights by bivariate generalized Beta distribution', Forest Science 48, 47-58.         [ Links ]

22. Little, S. (1983), 'Weibull diameter distributions for mixed stands of western conifers', Canadian Journal of Forest Research 13, 85-88.         [ Links ]

23. Maltamo, M., Puumalinen, J. & Päivinen, R. (1995), 'Comparison of beta and Weibull functions for modelling basal area diameter in stands of Pinus sylvestris and Picea abies', Scandinavian Journal of Forest Research 10, 284-295.         [ Links ]

24. Marchant, C., Leiva, V., Cavieres, M. & Sanhueza, A. (2013), 'Air contaminant statistical distributions with application to PM10 in Santiago, Chile', Reviews of Environmental Contamination and Toxicology 223, 1-31.         [ Links ]

25. McEwen, R. & Parresol, B. (1991), 'Moment expressions and summary statistics for the complete and truncated Weibull dstribution', Communications in Statistics - Theory and Methods 20, 1361-1372.         [ Links ]

26. McGee, C. & Della-Bianca, L. (1967), Diameter distributions in natural yellow-poplar stands, Report 25, US Forest Service.         [ Links ]

27. Meyer, H. (1952), 'Structure, growth, and drain in balanced uneven-aged forests', Journal of Forestry 50, 85-92.         [ Links ]

28. Miner, M. A. (1945), 'Cumulative damage in fatigue', Journal of Applied Mechanics 12, 159-164.         [ Links ]

29. Nelson, T. (1964), 'Diameter distribution and growth of loblolly pine', Journal of Applied Mechanics 10, 105-115.         [ Links ]

30. Palahí, M., Pukkala, T. & Trasobares, A. (2006), 'Modelling the diameter distribution of Pinus sylvestris, Pinus nigra and Pinus halepensis forest stands in Catalonia using the truncated Weibull function', Forestry 79, 553-562.         [ Links ]

31. Pece, M., de Benítez, C. & de Galíndez, M. (2000), 'Uso de la función Weibull para modelar distribuciones diamétricas en una plantación de Melia azedarach', Revista Forestal Venezolana 44, 49-52.         [ Links ]

32. Podlaski, R. (2006), 'Suitability of the selected statistical distribution for fitting diameter data in distinguished development stages and phases of near-natural mixed forests in the Swietokrzyski National Park (Poland)', Forest Ecology and Management 236, 393-402.         [ Links ]

33. Podlaski, R. (2008), 'Characterization of diameter distribution data in near-natural forests using the Birnbaum-Saunders distribution', Canadian Journal of Forest Research 18, 518-527.         [ Links ]

34. Rennolls, K., Geary, D. & Rollinson, T. (1985), 'Characterizing diameter distributions by the use of the Weibull distribution', Forestry 58, 57-66.         [ Links ]

35. Santelices, R. & Riquelme, M. (2007), 'Forest mensuration of Nothofagus alessandri of Coipu\'e provenance', Bosque 28, 281-287.         [ Links ]

36. Schmelz, D. & Lindsey, A. (1965), 'Size-class structure of old-growth forests in Indiana', Forest Science 11, 258-264.         [ Links ]

37. Schreuder, H. & Hafley, W. (1977), 'A useful bivariate distribution for describing stand structure of tree heights and diameters', Biometrics 33, 471-478.         [ Links ]

38. Vilca, F. & Leiva, V. (2006), 'A new fatigue life model based on the family of skew-elliptical distributions', Communications in Statistics - Theory and Methods 35, 229-244.         [ Links ]

39. Vilca, F., Santana, L., Leiva, V. & Balakrishnan, N. (2011), 'Estimation of extreme percentiles in Birnbaum-Saunders distributions', Computational Statistics and Data Analysis 55, 1665-1678.         [ Links ]

40. Wang, M. & Rennolls, K. (2005), 'Tree diameter distribution modelling: introducing the logit-logistic distribution', Canadian Journal of Forest Research 35, 1305-1313.         [ Links ]

41. Zutter, B., Oderwald, R., Murphy, P. & Farrar, R. (1986), 'Characterizing diameter distributions with modified data types and forms of the Weibull distribution', Forest Science 32, 37-48.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX: