Skip to main content
SpringerLink
Log in

Point Pattern Analysis of Spatial Deformation and Blurring Effects on Exceedances

  • Published:
Journal of Agricultural, Biological, and Environmental Statistics Aims and scope Submit manuscript

Abstract

Structural characteristics of random field threshold exceedance sets (e.g., size, connectivity, and boundary regularity) are used in practice for definition of different indicators in spatial and spatio-temporal risk analysis. In this work, point process techniques are applied to study the structural changes derived from random field deformations and blurring transformations, meaningful from both physical and methodological points of view in a variety of contexts. Specifically, based on simulations from a flexible random field model class, features such as aggregation/inhibition of patterns defined by centroids of connected components, as well as by boundary A-exit points, are investigated in relation to the local contraction/dilation effects of deformation and the smoothing properties of blurring. Supplementary materials accompanying this paper appear online.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Adler RJ (1981) The Geometry of Random Fields. Wiley, Chichester

    MATH  Google Scholar 

  • —— (2008) Some new random field tools for spatial analysis. Stochastic Environmental Research and Risk Assessment 22: 802-822

  • Adler RJ, Taylor JE (2007) Random Fields and Geometry. Springer, New York

    MATH  Google Scholar 

  • —— (2011) Topological Complexity of Smooth Random Functions. Lecture Notes in Mathematics, vol. 2019. Springer, Berlin Heidelberg

  • Amit Y, Grenander U, Piccioni M (1991) Structural image restoration through deformable templates. Journal of the American Statistical Association 86: 376-387

  • Anderes EB, Stein ML (2008) Estimating deformations of isotropic Gaussian random fields on the plane. The Annals of Statistics 36: 719-741

  • Angulo JM, Madrid AE (2010) Structural analysis of spatio-temporal threshold exceedances. Environmetrics 21: 415-438

  • ——– (2014) A Deformation/Blurring-Based Spatio-Temporal Model Stochastic Environmental Research and Risk Assessment 28: 1061-1073

  • Baddeley AJ, Moller J, Waagepetersen R (2000). Non- and semi-parametric estimation of interaction in homogeneous point patterns. Statistica Neerlandica 54: 329-350

  • Banerjee S, Carlin BP, Gelfand AE, (2014, 2ed) Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC, Boca Raton

  • Brown PE, Karesen KF, Roberts GO, Tonellato S (2000) Blur-generated non-separable space-time models. Journal of the Royal Statistical Society, Series B 62: 847-860

  • Christensen GE, Rabbit RD, Miller MI (1996) Deformable templates using large deformation kinematics. IEEE Transations on Image Processing 5: 1435-1447

  • Clerc M, Mallat S (2003) Estimating deformations of stationary processes. Annals of Statistics 31: 1772-1821

  • Cox DR, Isham V (1980) Point Processes. Chapman & Hall, London.

    MATH  Google Scholar 

  • Cressie N (1993) Statistics for Spatial Data. Wiley, New York

    MATH  Google Scholar 

  • Daley DJ, Vere-Jones D (1988) An Introduction to the Theory of Point Processes. Springer, New York

    MATH  Google Scholar 

  • —— (2003) An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods. Springer, New York

  • —— (2008) An Introduction to the Theory of Point Processes. Volume II: General Theory and Structure. Springer, New York

  • Diggle PJ (2014) Statistical Analysis of Spatial and Spatio-Temporal Point Patterns. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Fouedjio F, Desassis N, Romary T (2015) Estimation of space deformation model for non-stationary random functions. Spatial Statistics 13: 45-61

  • Glasbey CA, Mardia KV (1998) A review of image warping methods. Journal of Applied Statistics 25: 155-171

  • ——– (2001) A penalized likelihood approach to image warping. Journal of the Royal Statistical Society. Series B 63: 465-514

  • Gneiting T, Schlather M (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review 46: 269-282

  • Goitía A, Ruiz-Medina MD, Angulo JM (2005) Joint estimation of spatial deformation and blurring in environmental data. Stochastic Environmental Research and Risk Assessment 19: 1-17

  • Madrid AE, Angulo JM, Mateu J (2012) Spatial threshold exceedance analysis through marked point processes. Environmetrics 23: 108-118

  • Mardia KV, Angulo JM, Goitía A (2006a) Synthesis of image deformation strategies. Image and Vision Computing 24: 1-12

  • Mardia KV, Bookstein FL, Kent JT, Meyer CR (2006b) Intrinsic random fields and image deformations. Journal of Mathematical Imaging and Vision 26: 59-71

  • Perrin O, Monestiez, P (1999) Modelling of non-stationary spatial structure using parametric radial basis deformation. In: geoENV II - Geostatistics for Environmental Applications, Gómez-Hernández, J (ed), 175-186. Springer, Dordrecht

  • Perrin O, Senoussi R (1999) Reducing non-stationary stochastic processes to stationary by a time deformation. Statistics & Probability Letters 43: 393-397

  • Reiss RD (1993) A Course on Point Processes. Springer, New York

    Book  MATH  Google Scholar 

  • Piterbarg VI (1996) Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Ripley BD (1977) Modelling spatial patterns. Journal of the Royal Statistical Society, Series B 39: 172-212

  • Sampson PD, Guttorp P (1992) Nonparametric estimation of nonstationary spatial covariance structure. Journal of the American Statistical Association 87: 108-119

  • Storvik G, Frigessi A, Hirst D (2002) Stationary space-time Gaussian fields and their time autoregressive representation. Statistical Modelling 2: 139-161

  • Stoyan D, Kendall WS, Mecke J (1995) Stochastic Geometry and its Applications. Wiley, New York

    MATH  Google Scholar 

  • van Lieshout MNM (2000) Marked Point Processes and Their Applications. Imperial College Press, London

    Book  MATH  Google Scholar 

  • VanMarcke E (2010) Random Fields: Analysis and Synthesis. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Vera JF, Angulo JM, Roldán JA (2016) Stability analysis in nonstationary spatial covariance estimation. Stochastic Environmental Research and Risk Assessment. doi:10.1007/s00477-016-1228-4

Download references

Acknowledgments

J.M. Angulo and A.E. Madrid have been partially supported by grants MTM2012-32666 and MTM2015-70840-P (MINECO/FEDER). J. Mateu has been partially supported by grants MTM2013-43917-P (MINECO/FEDER) and P1-1B2015-40.

Author information

Authors and Affiliations

  1. Department of Sciences and Informatics, University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, C/ Coronel López Peña s/n, Santiago de la Ribera, 30720, Murcia, Spain

    A. E. Madrid

  2. Department of Statistics and Operations Research, Faculty of Sciences, University of Granada, Campus Fuente Nueva s/n, 18071, Granada, Spain

    J. M. Angulo

  3. Department of Mathematics, Universitat Jaume I, 12071, Castellón, Spain

    J. Mateu

Authors
  1. A. E. Madrid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Angulo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Mateu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. M. Angulo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Madrid, A.E., Angulo, J.M. & Mateu, J. Point Pattern Analysis of Spatial Deformation and Blurring Effects on Exceedances. JABES 21, 512–530 (2016). https://doi.org/10.1007/s13253-016-0262-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13253-016-0262-5

Keywords

Search

Navigation