Extrapolation of stationary random fields via level sets
Authors:
A. Das, V. Makogin and E. Spodarev
Journal:
Theor. Probability and Math. Statist. 106 (2022), 85-103
MSC (2020):
Primary 54C40, 14E20; Secondary 46E25, 20C20
DOI:
https://doi.org/10.1090/tpms/1166
Published electronically:
May 16, 2022
MathSciNet review:
4438445
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Abstract: In this paper, we use the concept of excursion sets for the extrapolation of stationary random fields. Doing so, we define excursion sets for the field and its linear predictor, and then minimize the expected volume of the symmetric difference of these sets under the condition that the univariate distributions of the predictor and of the field itself coincide. We illustrate the new approach on Gaussian random fields.
References
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- Evgeny Spodarev, Elena Shmileva, and Stefan Roth, Extrapolation of stationary random fields, Stochastic geometry, spatial statistics and random fields, Lecture Notes in Math., vol. 2120, Springer, Cham, 2015, pp. 321–368. MR 3330581, DOI 10.1007/978-3-319-10064-7_{1}1
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References
- R. Adler and J. Taylor, Random fields and geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516
- F. Alizadeh and D. Goldfarb, Second-order cone programming, vol. 95, 2003, ISMP 2000, Part 3 (Atlanta, GA), pp. 3–51. MR 1971381
- A. Z. Averbuch, P. Neittaanmaki, and V. A. Zheludev, Spline and spline wavelet methods with applications to signal and image processing. Vol. I. Periodic splines, Springer, Dordrecht, 2014. MR 3243559
- J.-M. Azaïs and M. Wschebor, Level sets and extrema of random processes and fields, John Wiley & Sons, Inc., Hoboken, NJ, 2009. MR 2478201
- D. Azzimonti, J. Bect, C. Chevalier, and D. Ginsbourger, Quantifying uncertainties on excursion sets under a Gaussian random field prior, SIAM/ASA J. Uncertain. Quantif. 4 (2016), no. 1, 850–874. MR 3531737
- D. Azzimonti and D. Ginsbourger, Estimating orthant probabilities of high-dimensional Gaussian vectors with an application to set estimation, J. Comput. Graph. Statist. 27 (2018), no. 2, 255–267. MR 3816262
- A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Academic Publishers, Boston, MA, 2004, With a preface by Persi Diaconis. MR 2239907
- M. E. Biancolini, Fast radial basis functions for engineering applications, Springer, Cham, 2017. MR 3753637
- D. Bolin and F. Lindgren, Excursion and contour uncertainty regions for latent Gaussian models, J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015), no. 1, 85–106. MR 3299400
- S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, 2004. MR 2061575
- C. Chevalier, D. Ginsbourger, J. Bect, E. Vazquez, V. Picheny, and Y. Richet, Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set, Technometrics 56 (2014), no. 4, 455–465. MR 3290615
- J. P. Chilés and P. D. Delfiner, Geostatistics, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. MR 1679557
- N. A. C. Cressie, Statistics for spatial data, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1991. MR 1127423
- A. Das, V. Makogin, and E. Spodarev, R code for the extrapolation of Gaussian random fields with minimal error in level sets, https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/mitarbeiter/spodarev/publications/Software/extrapolation_code.R, 2021.
- P. J. Diggle and P. J. Ribeiro, Jr., Model-based geostatistics, Springer Series in Statistics, Springer, New York, 2007. MR 2293378
- C. Gaetan and X. Guyon, Spatial statistics and modeling, Springer Series in Statistics, Springer, New York, 2010. MR 2569034
- A. Genz and F. Bretz, Computation of multivariate normal and $t$ probabilities, Lecture Notes in Statistics, vol. 195, Springer, Dordrecht, 2009. MR 2840595
- K. Höllig and J. Hörner, Approximation and modeling with B-splines, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. MR 3155467
- W. Karcher, E. Shmileva, and E. Spodarev, Extrapolation of stable random fields, J. Multivariate Anal. 115 (2013), 516–536. MR 3004573
- R. Klette and A. Rosenfeld, Digital geometry, Morgan Kaufmann Publishers, San Francisco, CA; Elsevier Science B.V., Amsterdam, 2004, Geometric methods for digital picture analysis. MR 2095127
- M.-J. Lai and L. L. Schumaker, Spline functions on triangulations, Encyclopedia of Mathematics and its Applications, vol. 110, Cambridge University Press, Cambridge, 2007. MR 2355272
- C. Lantuéjoul, Geostatistical simulation: Models and algorithms, Springer, Berlin, 2002.
- G. Matheron, Matheron’s theory of regionalized variables, International Association for Mathematical Geosciences. Studies in Mathematical Geosciences, Oxford University Press, Oxford, 2019. MR 3932139
- M. Mohammadi, Prediction of $\alpha$-stable GARCH and ARMA-GARCH-M models, J. Forecast. 36 (2017), no. 7, 859–866. MR 3714411
- M. Mohammadi and A. Mohammadpour, Best linear prediction for $\alpha$-stable random processes, Statist. Probab. Lett. 79 (2009), no. 21, 2266–2272. MR 2591983
- Yu. A. Rozanov, Stationary random processes, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1967. MR 0214134
- G. Samorodnitsky and M. Taqqu, Stable non-Gaussian random processes, Chapman & Hall/CRC, 1994. MR 1280932
- M. Scheuerer, A comparison of models and methods for spatial interpolation in statistics and numerical analysis, Ph.D. thesis, Georg-August Universität, Göttingen, 2009.
- M. Scheuerer, R. Schaback, and M. Schlather, Interpolation of spatial data—a stochastic or a deterministic problem?, European J. Appl. Math. 24 (2013), no. 4, 601–629. MR 3082868
- M. Schlather, A. Malinowski, P. J. Menck, M. Oesting, and K. Strokorb, Analysis, simulation and prediction of multivariate random fields with package randomfields, Journal of Statistical Software 63 (2015), no. 8, 1–25.
- Albert N. Shiryaev, Probability. 1, third ed., Graduate Texts in Mathematics, vol. 95, Springer, New York, 2016. MR 3467826
- E. Spodarev, E. Shmileva, and S. Roth, Extrapolation of stationary random fields, Stochastic geometry, spatial statistics and random fields, Lecture Notes in Math., vol. 2120, Springer, Cham, 2015, pp. 321–368. MR 3330581
- M. L. Stein, Interpolation of spatial data: Some theory for kriging, Springer Series in Statistics, Springer-Verlag, New York, 1999. MR 1697409
- H. Tomita, Statistics and geometry of random interface systems, Formation, dynamics and statistics of patterns, Vol. 1, World Sci. Publ., River Edge, NJ, 1990, pp. 113–157. MR 1158478
- E. Vazquez and M. P. Martinez, Estimation of the volume of an excursion set of a Gaussian process using intrinsic kriging, arxiv:math/0611273, Preprint, 2006.
- H. Wackernagel, Multivariate geostatistics: An introduction with applications, Springer Berlin Heidelberg, 2013. MR 3074883
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Additional Information
A. Das
Affiliation:
Institute of Stochastics, University of Ulm, Germany
Email:
abhinabdas7@gmail.com
V. Makogin
Affiliation:
Institute of Stochastics, University of Ulm, Germany
Email:
vitalii.makogin@uni-ulm.de
E. Spodarev
Affiliation:
Institute of Stochastics, University of Ulm, Germany
Email:
evgeny.spodarev@uni-ulm.de
Keywords:
Stationary random field,
Gaussian random field,
extrapolation,
linear prediction,
excursion,
level set,
second order cone programming,
quadratically constrained quadratic problem
Received by editor(s):
July 31, 2021
Accepted for publication:
October 18, 2021
Published electronically:
May 16, 2022
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv