Statistical Papers

, Volume 60, Issue 1, pp 199–221 | Cite as

An area-specific stick breaking process for spatial data

  • Mahdi Hosseinpouri
  • Majid Jafari KhalediEmail author
Regular Article


Most of the existing Bayesian nonparametric models for spatial areal data assume that the neighborhood structures are known, however in practice this assumption may not hold. In this paper, we develop an area-specific stick breaking process for distributions of random effects with the spatially-dependent weights arising from the block averaging of underlying continuous surfaces. We show that this prior, which does not depend on specifying neighboring schemes, is noticeably flexible in effectively capturing heterogeneity in spatial dependency across areas. We illustrate the methodology with a dataset involving expenditure credit of 31 provinces of Iran.


Spatial modeling Bayesian nonparametrics Stick-breaking process Geostatistical approach Fixed rank model Basis function 



The editor-in-cheif and referees are gratefully acknowledged. Their valuable comments have improved the manuscript.


  1. Banerjee A, Dunson DB, Tokdar ST (2013) Efficient Gaussian process regression for large datasets. Biometrika 100:75–89MathSciNetCrossRefzbMATHGoogle Scholar
  2. Banerjee S, Gelfand AE, Finley AO, Sang H (2008) Gaussian predictive process models for large spatial data sets. J R Stat Soc B 70:825–848MathSciNetCrossRefzbMATHGoogle Scholar
  3. Barnard J, McCulloch R, Meng X (2000) Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Statistica Sinica 10:1281–1312MathSciNetzbMATHGoogle Scholar
  4. Besag J (1974) Spatial interactions and the statistical analysis of latice systems. J R Stat Soc B 36:192–225MathSciNetzbMATHGoogle Scholar
  5. Besag J, York J, Mollie A (1991) Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cai B, Lawson A, Hossain M, Choi J, Kirby R, Liu J (2013) Bayesian semiparametric model with spatially–temporally varying coefficients selection. Stat Med 32:3670–3685MathSciNetCrossRefGoogle Scholar
  7. Cressie NC (1993) Statistics for spatial data, revised edn. Wiley, New YorkzbMATHGoogle Scholar
  8. Cressie NC, Johannesson G (2008) Fixed rank kriging for very large spatial data sets. J R Stat Soc B 70:209–226MathSciNetCrossRefzbMATHGoogle Scholar
  9. Duan J, Guindani M, Gelfand AE (2007) Generalized spatial Dirichlet process models. Biometrika 94:809–825MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dunson DB, Park JH (2008) Kernel stick-breaking processes. Biometrika 95:307–323MathSciNetCrossRefzbMATHGoogle Scholar
  11. Finley AO, Sang H, Banerjee S, Gelfand AE (2009) Improving the performance of predictive process modelling for large datasets. Comput Stat Data Anal 53:2873–2884CrossRefzbMATHGoogle Scholar
  12. Fuentes M, Reich B (2013) Multivariate spatial nonparametric modelling via kernel processes mixing. Stat Sinica 23:75–97MathSciNetzbMATHGoogle Scholar
  13. Geisser S (1993) Predictive inference: an introduction. Chapman and Hall/CRC Press, LondonCrossRefzbMATHGoogle Scholar
  14. Gelfand AE, Dey DK (1994) Bayesian model choice: asymptotics and exact calculations. J R Stat Soc B 56:501–514MathSciNetzbMATHGoogle Scholar
  15. Gelfand AE, Dey DK, Chang H (1992) Model determination using predictive distributions with implementation via sampling based methods (with discussion). In: Bernardo J et al (eds) Bayesian statistics. Oxford University Press, Oxford, pp 147–167Google Scholar
  16. Gelfand AE, Kottas A, MacEachern SN (2005) Bayesian nonparametric spatial modeling with Dirichlet process mixing. J Am Stat Assoc 100:1021–1035MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gelfand AE, Guindani M, Petrone S (2007) Bayesian nonparametric modeling for spatial data analysis using Dirichlet processes. In: Bernardo J et al (eds) Bayesian statistics. Oxford University Press, OxfordGoogle Scholar
  18. Gelfand AE, Diggle P, Guttorp P, Fuentes M (eds) (2010) Handbook of spatial statistics. Handbooks of modern statistical methods, Chapman and Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  19. Gelfand AE, Banerjee S, Finley A (2013) Spatial design for knot selection in knotbased dimension reduction models. In: Mateu J, Muller W (eds) Spatio-temporal design: advances in efficient data acquisition. Wiley, Chichester, pp 142–169 (Chap. 7)Google Scholar
  20. Guhaniyogi R, Finley AO, Banerjee S, Gelfand AE (2011) Adaptive Gaussian predictive process models for large spatial datasets. Environmetrics 22:997–1007MathSciNetCrossRefGoogle Scholar
  21. Geisser S, Eddy WF (1979) A predictive approach to model selection. J Am Stat Assoc 74:153–160MathSciNetCrossRefzbMATHGoogle Scholar
  22. Heaton MJ, Kleiber W, Sain SR, Wiltberger M (2015) Emulating and calibrating the multiple-fidelity Lyon–Fedder–Mobarry magnetosphere–ionosphere coupled computer model. J R Stat Soc C 64:93–113MathSciNetCrossRefGoogle Scholar
  23. Hossain M, Lawson A, Cai B, Choi J, Liu J, Kirby R (2013) Space-time stick-breaking processes for small area disease cluster estimation. Environ Ecol Stat 20:91–107MathSciNetCrossRefGoogle Scholar
  24. Hund L, Chen JT, Krieger N, Coull BA (2012) A geostatistical approach to large-scale disease mapping with temporal misalignment. Biometrics 68:849–858MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ishwaran H, James LF (2001) Gibbs sampling methods for stick-breaking priors. J Am Stat Assoc 96:161–173MathSciNetCrossRefzbMATHGoogle Scholar
  26. Katzfuss M (2013) Bayesian nonstationary spatial modeling for very large datasets. Environmetrics 24:189–200MathSciNetCrossRefGoogle Scholar
  27. Katzfuss M, Cressie NC (2012) Bayesian hierarchical spatio-temporal smoothing for very large datasets. Environmetrics 23:94–107MathSciNetCrossRefGoogle Scholar
  28. Katzfuss M, Hammerling D (2014) Parallel inference for massive distributed spatial data using low-rank models. arXiv:1402.1472
  29. Kelsall JE, Wakefield JC (2002) Modeling spatial variation in disease risk: a geostatistical approach. J Am Stat Assoc 97:692–701MathSciNetCrossRefzbMATHGoogle Scholar
  30. Kottas A, Duan J, Gelfand AE (2008) Modelling disease incidence data with spatial and spatio-temporal dirichlet process mixtures. Biom J 50:29–42MathSciNetCrossRefGoogle Scholar
  31. Li P, Banerjee S, Hanson T, McBean A (2015) Bayesian hierarchical models for detecting boundaries in areally referenced spatial datasets. Stat Sinica 25:385–402zbMATHGoogle Scholar
  32. MacEachern SN (2000) Dependent Dirichlet processes. Technical Report, Department of Statistics, The Ohio State UniversityGoogle Scholar
  33. Papaspiliopoulos O, Roberts GO (2008) Retrospective Markov Chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika 95:169–186MathSciNetCrossRefzbMATHGoogle Scholar
  34. Petrone S, Guindani M, Gelfand AE (2008) Hybrid Dirichlet mixture models for functional data. Technical Report, Department of Statistics, Duke UniversityGoogle Scholar
  35. Plummer M (2008) Penalized loss functions for Bayesian model comparison. Biostatistics 9:523–539CrossRefzbMATHGoogle Scholar
  36. Reich B, Fuentes M (2007) A multivariate semiparametric Bayesian spatial modeling framework for hurricane surface wind fields. Ann Appl Stat 1:249–264MathSciNetCrossRefzbMATHGoogle Scholar
  37. Ren L, Du L, Carin L, Dunson D (2011) Logistic stick-breaking process. J Mach Learn Res 12:203–239MathSciNetzbMATHGoogle Scholar
  38. Rodríguez A, Dunson DB (2011) Nonparametric Bayesian models through probit stick-breaking processes. Bayesian Anal 6:145–178MathSciNetCrossRefzbMATHGoogle Scholar
  39. Ruppert D, Wand MP, Caroll RJ (2003) Semiparametric regression. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  40. Sengupta A, Cressie NC (2013) Hierarchical statistical modeling of big spatial datasets using the exponential family of distributions. Spat Stat 4:14–44CrossRefGoogle Scholar
  41. Sethuraman J (1994) A constructive definition of Dirichlet priors. Stat Sinica 4:639–650MathSciNetzbMATHGoogle Scholar
  42. Waller LA, Gotway CA (2004) Applied spatial statistics for public health data. Wiley, HobokenCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of StatisticsTarbiat Modares UniversityTehranIran

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