Kriging with external drift in a Birnbaum–Saunders geostatistical model

  • Fabiana Garcia-Papani
  • Víctor Leiva
  • Fabrizio Ruggeri
  • Miguel A. Uribe-Opazo
Original Paper


Spatial models to describe dependent georeferenced data are applied in different fields and, particularly, are used to analyze earth and environmental data. Most of these applications are carried out under Gaussian spatial models. The Birnbaum–Saunders distribution is a unimodal and positively skewed model which has received considerable attention in several areas, including earth and environmental sciences. In addition, theoretical arguments have been provided to justify its usage in the data modeling from these sciences, at least in the same settings where the lognormal distribution can be employed. This paper presents kriging with external drift based on a Birnbaum–Saunders spatial model. The maximum likelihood method is considered to estimate its parameters. The results obtained in the paper are illustrated by an experimental data set related to agricultural management. Specifically, in this illustration, the spatial variability of magnesium content in the soil as a function of calcium content is analyzed.


Agricultural data analysis Cross-validation Maximum likelihood estimation Non-normal distributions R software Variogram models 



The authors thank the Editors and referees for their constructive comments on an earlier version of this manuscript which resulted in this improved version. This research work was partially supported by CNPq and CAPES grants from the Brazilian government, and by FONDECYT 1160868 grant from the Chilean government.


  1. Allard D, Naveau P (2007) A new spatial skew-normal random field model. Commun Stat Theor Methods 36:1821–1834CrossRefGoogle Scholar
  2. Assumpção RAB, Uribe-Opazo MA, Galea M (2014) Analysis of local influence in geostatistics using Student-t distribution. J Appl Stat 41:2323–2341CrossRefGoogle Scholar
  3. Bishop T, McBratney A (2001) A comparison of predictions methods for creation of the creation of field-extent soil property maps. Geoderma 103:149–160CrossRefGoogle Scholar
  4. Cambardella C, Moorman T, Novak J, Parkin T, Karlen D, Turco R, Konopka A (1994) Field-scale variability of soil properties in central Iowa soils. Soil Sci Soc Am J 58:1501–1511CrossRefGoogle Scholar
  5. Caro-Lopera F, Leiva V, Balakrishnan N (2012) Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum–Saunders distributions. J Multivar Anal 104:126–139CrossRefGoogle Scholar
  6. Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty. Wiley, New YorkCrossRefGoogle Scholar
  7. Cressie N (2015) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  8. De Bastiani F, Cysneiros AHMA, Uribe-Opazo MA, Galea M (2015) Influence diagnostics in elliptical spatial linear models. TEST 24:322–340CrossRefGoogle Scholar
  9. Diggle PJ, Ribeiro PJ (2007) Model-based geoestatistics. Springer, New YorkGoogle Scholar
  10. Dunn P, Smyth G (1996) Randomized quantile residuals. J Comput Graph Stat 5:236–244Google Scholar
  11. Ferreira M, Gomes MI, Leiva V (2012) On an extreme value version of the Birnbaum–Saunders distribution. REVSTAT Stat J 10:181–210Google Scholar
  12. Garcia-Papani F, Uribe-Opazo MA, Leiva V, Aykroyd RG (2018) Birnbaum–Saunders spatial regression models: diagnostics and application to chemical data. Chemom Intell Lab Syst (in press)Google Scholar
  13. Garcia-Papani F, Leiva V, Uribe-Opazo MA, Aykroyd RG (2017) Birnbaum–Saunders spatial modelling and diagnostics applied to agricultural engineering data. Stoch Environ Res Risk Assess 31:105–124CrossRefGoogle Scholar
  14. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, OxfordGoogle Scholar
  15. Hengl T, Heuvelink G, Stein A (2003) Comparison of kriging with external drift and regression-kriging. In: Technical report, International Institute for Geo-information Science and Earth Observation (ITC), Enschede, The NetherlandsGoogle Scholar
  16. Hengl T, Heuvelink G, Stein A (2004) A generic framework for spatial prediction of soil variables based on regression-kriging. Geoderma 120:75–93CrossRefGoogle Scholar
  17. Hu Y, Jia Z, Cheng J, Zhao Z, Chen F (2016) Spatial variability of soil arsenic and its association with soil nitrogen in intensive farming systems. J Soils Sedim 16:169–176CrossRefGoogle Scholar
  18. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New YorkGoogle Scholar
  19. Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New YorkGoogle Scholar
  20. Journel AG (1980) The lognormal approach to predicting local distributions of selective mining unit grades. J Int Assoc Math Geol 12:285–303CrossRefGoogle Scholar
  21. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, New YorkGoogle Scholar
  22. Lange K (2001) Numerical analysis for statisticians. Springer, New YorkGoogle Scholar
  23. Leiva V (2016) The Birnbaum–Saunders distribution. Academic Press, New YorkGoogle Scholar
  24. Leiva V, Athayde E, Azevedo C, Marchant C (2011) Modeling wind energy flux by a Birnbaum–Saunders distribution with unknown shift parameter. J Appl Stat 38:2819–2838CrossRefGoogle Scholar
  25. Leiva V, Ferreira M, Gomes MI, Lillo C (2016) Extreme value Birnbaum–Saunders regression models applied to environmental data. Stoch Environ Res Risk Assess 30:1045–1058CrossRefGoogle Scholar
  26. Leiva V, Marchant C, Ruggeri F, Saulo H (2015) A criterion for environmental assessment using Birnbaum–Saunders attribute control charts. Environmetrics 26:463–476CrossRefGoogle Scholar
  27. Leiva V, Sanhueza A, Angulo JM (2009) A length-biased version of the Birnbaum–Saunders distribution with application in water quality. Stoch Environ Res Risk Assess 23:299–307CrossRefGoogle Scholar
  28. Leiva V, Santos-Neto M, Cysneiros FJA, Barros M (2014) Birnbaum–Saunders statistical modelling: a new approach. Stat Modell 14:21–48CrossRefGoogle Scholar
  29. Leiva V, Saulo H (2017) Environmental applications based on Birnbaum–Saunders models. In: Adhikari A, Adhikari MR, Chaubey YP (eds) Mathematical and statistical applications in life sciences and engineering. Springer, Singapore, pp 283–304CrossRefGoogle Scholar
  30. Lillo C, Leiva V, Nicolis O, Aykroyd RG (2018) L-moments of the Birnbaum–Saunders distribution and its extreme value version: estimation, goodness of fit and application to earthquake data. J Appl Stat 45:187–209CrossRefGoogle Scholar
  31. Lopes AS (1998) International soil fertility manual. Potafos, Piracicaba (in Portuguese)Google Scholar
  32. Marchant C, Leiva V, Cavieres MF, Sanhueza A (2013) Air contaminant statistical distributions with application to PM10 in Santiago, Chile. Rev Environ Contam Toxicol 223:1–31Google Scholar
  33. Marchant C, Leiva V, Cysneiros FJA (2016a) A multivariate log-linear model for Birnbaum–Saunders distributions. IEEE Trans Reliab 65:816–827CrossRefGoogle Scholar
  34. Marchant C, Leiva V, Cysneiros FJA, Liu S (2018) Robust multivariate control charts based on Birnbaum–Saunders distributions. J Stat Comput Simul 88:182–202CrossRefGoogle Scholar
  35. Marchant C, Leiva V, Cysneiros FJA, Vivanco JF (2016b) Diagnostics in multivariate generalized Birnbaum–Saunders regression models. J Appl Stat 43:2829–2849CrossRefGoogle Scholar
  36. Mardia K, Marshall R (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71:135–146CrossRefGoogle Scholar
  37. Nocedal J, Wright S (1999) Numerical optimization. Springer, New YorkCrossRefGoogle Scholar
  38. Pan J, Fei Y, Foster P (2014) Case-deletion diagnostics for linear mixed models. Technometrics 56:269–281CrossRefGoogle Scholar
  39. Pawlowsky-Glahn V, Egozcue JJ (2016) Spatial analysis of compositional data: a historical review. J Geochem Explor 164:28–32CrossRefGoogle Scholar
  40. Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015) Modeling and analysis of compositional data. Wiley, New YorkGoogle Scholar
  41. Podlaski R (2008) Characterization of diameter distribution data in near-natural forests using the Birnbaum–Saunders distribution. Can J For Res 18:518–527CrossRefGoogle Scholar
  42. R Core Team (2016) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaGoogle Scholar
  43. Rendu JM (1979) Normal and lognormal estimation. Math Geol 11:407–422CrossRefGoogle Scholar
  44. Rimstad K, Omre H (2014) Skew-Gaussian random fields. Spat Stat 10:43–62CrossRefGoogle Scholar
  45. Santana L, Vilca F, Leiva V (2011) Influence analysis in skew-Birnbaum–Saunders regression models and applications. J Appl Stat 38:1633–1649CrossRefGoogle Scholar
  46. Saulo H, Leiva V, Ziegelmann FA, Marchant C (2013) A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data. Stoch Environ Res Risk Assess 27:1479–1491CrossRefGoogle Scholar
  47. Severini TA (2000) Likelihood methods in statistics. Oxford University Press, OxfordGoogle Scholar
  48. Soares A (2000) Geostatistics for earth and environmental sciences. IST Press, Lisboa (in Portuguese)Google Scholar
  49. Tolosana-Delgado R, Pawlowsky-Glahn V (2007) Kriging regionalized positive variables revisited: sample space and scale considerations. Math Geol 39:529–558CrossRefGoogle Scholar
  50. Uribe-Opazo MA, Borssoi JA, Galea M (2012) Influence diagnostics in Gaussian spatial linear models. J Appl Stat 39:615–630CrossRefGoogle Scholar
  51. Vilca F, Sanhueza A, Leiva V, Christakos G (2010) An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile. Stoch Environ Res Risk Assess 24:771–782CrossRefGoogle Scholar
  52. Webster R, Oliver M (2009) Geostatistics for environmental scientists. Wiley, ChichesterGoogle Scholar
  53. Wolter KM (2007) Introduction to variance estimation. Springer, New YorkGoogle Scholar
  54. Xia J, Zeephongsekul P, Packer D (2011) Spatial and temporal modelling of tourist movements using semi-Markov processes. Tour Manag 51:844–851CrossRefGoogle Scholar
  55. Zhang H, Zimmerman DL (2005) Towards reconciling two asymptotic frameworks in spatial statistics. Biometrika 92:921–936CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Postgraduate Program in Agricultural Engineering and Center of Exact Sciences and TechnologyUniversidade Estadual do Oeste do ParanáCascavelBrazil
  2. 2.School of Industrial EngineeringPontificia Universidad Católica de ValparaísoValparaísoChile
  3. 3.CNR-IMATIMilanoItaly

Personalised recommendations