Skip to main content
SpringerLink
Log in

Iterative posterior inference for Bayesian Kriging

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

We propose a method for estimating the posterior distribution of a standard geostatistical model. After choosing the model formulation and specifying a prior, we use normal mixture densities to approximate the posterior distribution. The approximation is improved iteratively. Some difficulties in estimating the normal mixture densities, including determining tuning parameters concerning bandwidth and localization, are addressed. The method is applicable to other model formulations as long as all the parameters, or transforms thereof, are defined on the whole real line, \((-\infty, \infty).\) Ad hoc treatments in the posterior inference such as imposing bounds on an unbounded parameter or discretizing a continuous parameter are avoided. The method is illustrated by two examples, one using digital elevation data and the other using historical soil moisture data. The examples in particular examine convergence of the approximate posterior distributions in the iterations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Abramowitz M, Stegun IA (eds) (1965) Handbook of mathematical functions. Dover, New York

  • Banerjee S, Carlin BP, Gelfand AE (2004) Hierarchical modeling and analysis of spatial data. Chapman & Hall/CRC, Boca Raton

  • Berger JO, De Oliveira V, Sansó B (2001) Objective Bayesian analysis of spatially correlated data. J Am Stat Assoc 96(456):1361–1374

    Article  Google Scholar 

  • Bosch DD, Sheridan JM, Lowrance RR, Hubbard RK, Strickland TC, Feyereisen GW, Sullivan DG (2007) Little River Experimental Watershed database. Water Resour Res 43:W09470. doi:10.1029/2006WR005844

    Article  Google Scholar 

  • Bosch DD, Sheridan JM, Marshall LK (2007) Precipitation, soil moisture, and climate database, Little River Experimental Watershed, Georgia, United States. Water Resour Res 43:W09472. doi:10.1029/2006WR005834

    Article  Google Scholar 

  • Christensen OF, Diggle PJ, Ribeiro Jr PJ (2001) Analysing positive-valued spatial data: the transformed Gaussian model. In: Monestiez P, Allard D, Froidevaux R (eds) geoENV III—Geostatistics for Environmental Applications, Kluwer, pp 287–298

  • Cowles MK, Yan J, Smith B (2009) Reparameterized and marginalized posterior and predictive sampling for complex Bayesian geostatistical models. J Comp Graph Stat 18(2):262–282. doi:10.1198/jcgs.2009.08012

    Article  Google Scholar 

  • de Oliveira V, Kedem B, Short DA (1997) Bayesian prediction of transformed Gaussian random fields. J Am Stat Assoc 92(440):1422–1433

    Article  Google Scholar 

  • Diggle PJ, Ribeiro PJ Jr (2002) Bayesian inference in Gaussian model-based geostatistics. Geograph Environ Model 6(2):129–146

    Article  Google Scholar 

  • Diggle PJ, Ribeiro Jr PJ (2007) Model-based geostatistics. Springer, New York

  • Ecker MD, Gelfand AE (1999) Bayesian modeling and inference for geometrically anisotropic spatial data. Math Geol 31(1):67–83

    Google Scholar 

  • Emery X (2007) Reducing fluctuations in the sample variogram. Stoch Environ Res Risk Assess 21:391–403

    Article  Google Scholar 

  • Gelman A (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Anal 1(3):515–533

    Google Scholar 

  • Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian Data Analysis, 2nd edn. Chapman & Hall/CRC, Boca Raton

  • Givens GH, Raftery AE (1996) Local adaptive importance sampling for multivariate densities with strong nonlinear relationships. J Am Stat Assoc 91(433):132–141

    Article  Google Scholar 

  • Hall P, Sheather SJ, Jones MC, Marron JS (1991) On optimal data-based bandwidth selection in kernel density estimation. Biometrika 78(2):263–269

    Article  Google Scholar 

  • Handcock MS, Stein ML (1993) A Bayesian analysis of Kriging. Technometrics 35(4):403–410

    Article  Google Scholar 

  • Hristopulos DT (2002) New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity. Stoch Environ Res Risk Assess 16:43–62

    Article  Google Scholar 

  • Jones MC (1990) Variable kernel density estimates and variable kernel density estimates. Austral J Statist 32(3):361–371

    Article  Google Scholar 

  • Jones MC, Marron JS, Sheather SJ (1996) A brief survey of bandwidth selection for density estimation. J Am Stat Assoc 91(433):401–407

    Google Scholar 

  • Kitanidis PK (1986) Parameter uncertainty in estimation of spatial functions: Bayesian analysis. Water Resour Res 22(4):499–507

    Article  Google Scholar 

  • Kovitz JL, Christakos G (2004) Spatial statistics of clustered data. Stoch Environ Res Risk Assess 18:147–166

    Article  Google Scholar 

  • Liu JS, Chen R, Wong WH (1998) Rejection control and sequential importance sampling. J Am Stat Assoc 93(443):1022–1031

    Article  Google Scholar 

  • Marron JS, Wand MP (1992) Exact mean integrated squared error. Ann Statist 20(2):712–736

    Article  Google Scholar 

  • Michalak AM, Kitanidis PK (2005) A method for the interpolation of nonnegative functions with an application to contaminant load estimation. Stoch Environ Res Risk Assess 19:8–23

    Article  Google Scholar 

  • Palacios MB, Steel MFJ (2006) Non-Gaussian Bayesian geostatistical modeling. J Am Stat Assoc 101:604–618

    Article  CAS  Google Scholar 

  • Paleologos EK, Sarris TS (2011) Stochastic analysis of flux and head moments in a heterogeneous aquifer system. Stoch Environ Res Risk Assess 25:747–759

    Article  Google Scholar 

  • Sain SR (2002) Multivariate locally adaptive density estimation. Comput Stat Data Anal 39(2):165–186

    Article  Google Scholar 

  • Scott DW (1992) Multivariate density estimation. John Wiley & Sons, Inc.

  • Sheather SJ, Jones MC (1991) A reliable data-based bandwidth selection method for kernel density estimation. J R Stat Soc, B 53(3):683–690

    Google Scholar 

  • Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, Boca Raton

  • Stein ML (1999) Interpolation of spatial data: some theory for Kriging. Springer, New York

  • Terrell GR, Scott DW (1992) Variable kernel density estimation. Ann Statist 20(3):1236–1265

    Article  Google Scholar 

  • Wackernagel H (2003) Multivariate geostatistics, 3rd edn. Springer-Verlag, Berlin

    Google Scholar 

  • Wand MP, Jones MC (1995) Kernel Smoothing. Chapman & Hall/CRC

  • Warnes JJ, Ripley BD (1987) Problems with lieklihood estimation of covariance functions of spatial Gaussian processes. Biometrika 74(3):640–642

    Article  Google Scholar 

  • West M (1993) Approximating posterior distributions by mixture. J R Stat Soc B 55(2):409–422

    Google Scholar 

  • Young LJ, Gotway CA (2007) Linking spatial data from different sources: the effects of change of support. Stoch Environ Res Risk Assess 21:589–600. doi:10.1007/s00477-007-0136-z

    Article  Google Scholar 

  • Zhang H (2004) Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J Am Stat Assoc 99(465). doi:10.1198/016214504000000241

  • Zhang Z (2011) Adaptive anchored inversion for Gaussian random fields using nonlinear data. Inverse Problems 27(12):125011. doi:10.1088/0266-5611/27/12/125011

    Article  Google Scholar 

  • Zimmerman DL (1993) Another look at anisotropy in geostatistics. Math Geol 25(4):453–470

    Article  Google Scholar 

  • Zimmerman DL (2010) Likelihood-based methods. In: Gelfand AE, Diggle PJ, Fuentes M, Guttorp P (eds) Handbook of spatial statistics, Chap 4, CRC Press, Boca Raton

Download references

Acknowledgements

The author’s Senior Visiting Scholarship at Tsinghua University was funded by the Excellent State Key Lab Fund no. 50823005, National Natural Science Foundation of China, and the R&D Special Fund for Public Welfare Industry no. 201001080, Chinese Ministry of Water Resources.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, USA

    Zepu Zhang

  2. State Key Lab of Hydroscience and Engineering, Tsinghua University, Beijing, China

    Zepu Zhang

Authors
  1. Zepu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zepu Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Z. Iterative posterior inference for Bayesian Kriging. Stoch Environ Res Risk Assess 26, 913–923 (2012). https://doi.org/10.1007/s00477-011-0544-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-011-0544-y

Keywords

Search

Navigation