×

Interval-valued kriging for geostatistical mapping with imprecise inputs. (English) Zbl 1483.86011

Kriging is one of the most applied tool in geostatistics. Due to various limitations the used data are often imprecise and should be modelled as interval-valued data or fuzzy-valued data. This paper proposes a modification to Diamond’s interval-valued kriging [P. Diamond, Math. Geol. 20, No. 3, 145–165 (1988; Zbl 0970.86500)]. In contrast to Diamond it plays a key role that the authors use not a interval-valued variance and covariance but a real-valued one which goes back to R. Körner [Fuzzy Sets Syst. 92, No. 1, 83–93 (1997; Zbl 0936.60017)]. The resulting interval-valued kriging overcomes many of the mathematical and computational difficulties of previous attempts. The numerical implementations are presented in R packages. The proposed method is applied to kriging of design ground snow loads in Ohio.

MSC:

86A32 Geostatistics
65G40 General methods in interval analysis
62J86 Fuzziness, and linear inference and regression
62M30 Inference from spatial processes
62P12 Applications of statistics to environmental and related topics

Software:

Gstat; intkrige; gstat; fields; R-Geo; R
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Artstein, Z.; Vitale, R., A strong law of large numbers for random compact sets, Ann. Probab., 5, 879-882 (1975) · Zbl 0313.60012
[2] ASCE, Minimum Design Loads and Associated Criteria for Buildings and Other Structures (2017), American Society of Civil Engineers
[3] Aumann, R., Integrals of set-valued functions, J. Math. Anal. Appl., 12, 1-12 (1965) · Zbl 0163.06301
[4] Bandemer, H.; Gebhardt, A., Bayesian fuzzy kriging, Fuzzy Sets Syst., 112, 405-418 (2000) · Zbl 1040.62088
[5] Bardossy, A.; Bogardi, I.; Kelly, W., Kriging with imprecise (fuzzy) variograms. I: theory, Math. Geol., 22, 63-79 (1990) · Zbl 0964.62512
[6] Bardossy, A.; Bogardi, I.; Kelly, W., Kriging with imprecise (fuzzy) variograms. II: application, Math. Geol., 22, 81-94 (1990) · Zbl 0964.62513
[7] Bean, B., intkrige: a numerical implementation of interval-valued kriging (2019), R package version 1.0.0
[8] Bean, B.; Maguire, M.; Sun, Y., Comparing design ground snow load prediction in Utah and Idaho, J. Cold Reg. Eng., 33, Article 04019010 pp. (2019)
[9] Bean, B.; Maguire, M.; Sun, Y.; Wagstaff, J.; Al-Rubaye, S. A.; Wheeler, J.; Jarman, S.; Rogers, M., The 2020 National Snow Load Study (2021), Utah State University Department of Mathematics and Statistics: Utah State University Department of Mathematics and Statistics Logan, UT, Technical Report 276
[10] Billard, L., Dependencies and variation components of symbolic interval-valued data, (Selected Contributions in Data Analysis and Classification (2007), Springer: Springer Berlin), 3-12, chapter · Zbl 05486137
[11] Billard, L.; Diday, E., From the statistics of data to the statistics of knowledge, J. Am. Stat. Assoc., 98, 470-487 (2003)
[12] Bivand, R. S.; Pebesma, E. J.; Gomez-Rubio, V.; Pebesma, E. J., Applied Spatial Data Analysis with R (2013), Springer, 747248717 · Zbl 1269.62045
[13] Blanco-Fernández, A.; Corral, N.; González-Rodríguez, G., Estimation of a flexible simple linear model for interval data based on set arithmetic, Comput. Stat. Data Anal., 55, 2568-2578 (2011) · Zbl 1464.62030
[14] Chang, M.; Pakzad, S. N., Optimal sensor placement for modal identification of bridge systems considering number of sensing nodes, J. Bridge Eng., 19, Article 04014019 pp. (2014)
[15] Chilès, J. P.; Desassis, N., Fifty years of kriging, (Handbook of Mathematical Geosciences (2018), Springer), 589-612
[16] Couso, I.; Dubois, D., A general framework for maximizing likelihood under incomplete data, Int. J. Approx. Reason., 93, 238-260 (2018) · Zbl 1452.68205
[17] Daly, C., The climate data guide: PRISM high-resolution spatial climate data for the United States: max/min temp, dewpoint, precipitation (2020)
[18] Debreu, G., Integration of correspondences, (Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability II (1967)), 351-372 · Zbl 0211.52803
[19] Denoeux, T., Maximum likelihood estimation from uncertain data in the belief function framework, IEEE Trans. Knowl. Data Eng., 25, 119-130 (2013)
[20] D’Esposito, M. R.; Palumbo, F.; Ragozini, G., Interval archetypes: a new tool for interval data analysis, Stat. Anal. Data Min. ASA Data Sci. J., 5, 322-335 (2012) · Zbl 07260333
[21] Diamond, P., Interval-valued random functions and the kriging of intervals, Math. Geol., 20, 145-165 (1988) · Zbl 0970.86500
[22] Diamond, P., Fuzzy kriging, Fuzzy Sets Syst., 33, 315-332 (1989) · Zbl 0679.62088
[23] Diamond, P., Least squares fitting of compact set-valued data, J. Math. Anal. Appl., 147, 531-544 (1990)
[24] Fiacco, A.; McCormick, G., Nonlinear Programming: Sequential Unconstrained Minimization Techniques (1969), John Wiley & Sons: John Wiley & Sons New York · Zbl 0193.18805
[25] Gil, M.; González-Rodríguez, G.; Colubi, A.; Montenegro, M., Testing linear independence in linear models with interval-valued data, Comput. Stat. Data Anal., 51, 3002-3015 (2007) · Zbl 1161.62358
[26] Gil, M.; Lopez, M.; Lubiano, M.; Montenegro, M., Regression and correlation analyses of a linear relation between random intervals, Test, 10, 183-201 (2001) · Zbl 0981.62062
[27] Gil, M.; Lubiano, M.; Montenegro, M.; Lopez, M., Least squares fitting of an affine function and strength of association for interval-valued data, Metrika, 56, 97-111 (2002) · Zbl 1433.60004
[28] Gioia, F.; Lauro, C. N., Principal component analysis on interval data, Comput. Stat., 21, 343-363 (2006) · Zbl 1113.62072
[29] González-Rodríguez, G.; Blanco, A.; Corral, N.; Colubi, A., Least squares estimation of linear regression models for convex compact random sets, Adv. Data Anal. Classif., 1, 67-81 (2007) · Zbl 1131.62058
[30] Goovaerts, P., Geostatistics for Natural Resources Evaluation (1997), Oxford University Press
[31] Gräler, B.; Pebesma, E.; Heuvelink, G., Spatio-temporal interpolation using gstat, The R Journal, 8, 204-218 (2016)
[32] Guillaume, R.; Dubois, D., A min-max regret approach to maximum likelihood inference under incomplete data, Int. J. Approx. Reason., 121, 135-149 (2020) · Zbl 1445.68217
[33] Han, A.; Hong, Y.; Lai, K. K.; Wang, S., Interval time series analysis with an application to the sterling-dollar exchange rate, J. Syst. Sci. Complex., 21, 558-573 (2008) · Zbl 1177.91113
[34] Han, A.; Hong, Y.; Wang, S., Autoregressive conditional models for interval-valued time series data (2012), preprint, available at
[35] Hohn, M., Geostatistics and Petroleum Geology (1998), Springer Science & Business Media
[36] Hörmander, H., Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Ark. Mat., 3, 181-186 (1954) · Zbl 0064.10504
[37] Hüllermeier, E., Learning from imprecise and fuzzy observations: data disambiguation through generalized loss minimization, Int. J. Approx. Reason., 55, 1519-1534 (2014) · Zbl 1407.68396
[38] Jensen, P. A.; Bard, J. F., Operations Research Models and Methods [Supplementary Appendix], vol. 1 (2003), John Wiley & Sons Incorporated
[39] Körner, R., A variance of compact convex random sets (1995), Institut für Stochastik: Institut für Stochastik Bernhard-von-Cotta-Str. 2 09599 Freiberg, Technical Report
[40] Körner, R., On the variance of fuzzy random variables, Fuzzy Sets Syst., 92, 83-93 (1997) · Zbl 0936.60017
[41] Körner, R.; Näther, W., Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates, Inf. Sci., 109, 95-118 (1998) · Zbl 0930.62072
[42] Körner, R.; Näther, W., On the variance of random fuzzy variables, (Bertoluzza, C.; Gil, M.; Ralescu, D., Statistical Model, Analysis and Management of Fuzzy Data (2001), Physica: Physica Heidelberg), 22-39
[43] Liel, A. B.; DeBock, D. J.; Harris, J. R.; Ellingwood, B. R.; Torrents, J. M., Reliability-based design snow loads. II: reliability assessment and mapping procedures, J. Struct. Eng., 143, Article 04017047 pp. (2017)
[44] Lima Neto, E.; dos Anjos, U. U., Regression model for interval-valued variables based on copulas, J. Appl. Stat., 42, 2010-2029 (2015) · Zbl 1514.62515
[45] Lima Neto, E.; Carvalho, F., Centre and range method for fitting a linear regression model to symbolic interval data, Comput. Stat. Data Anal., 52, 1500-1515 (2008) · Zbl 1452.62493
[46] Lima Neto, E.; Carvalho, F., Constrained linear regression models for symbolic interval-valued variables, Comput. Stat. Data Anal., 54, 333-347 (2010) · Zbl 1464.62055
[47] Loquin, K.; Dubois, D., Kriging and epistemic uncertainty: a critical discussion, (Methods for Handling Imperfect Spatial Information (2010), Springer), 269-305
[48] Loquin, K.; Dubois, D., A fuzzy interval analysis approach to kriging with ill-known variogram and data, Soft Comput., 16, 769-784 (2012)
[49] Lyashenko, N., Limit theorem for sums of independent compact random subsets of Euclidean space, J. Sov. Math., 20, 2187-2196 (1982) · Zbl 0489.60041
[50] Maia, A. L.S.; de A. T. de Carvalho, F.; Ludermir, T. B., Forecasting models for interval-valued time series, Neurocomputing, 71, 3344-3352 (2008), Advances in Neural Information Processing (ICONIP 2006) / Brazilian Symposium on Neural Networks (SBRN 2006)
[51] Matheron, G., Principles of geostatistics, Econ. Geol., 58, 1246-1266 (1963)
[52] Matheron, G., The Theory of Regionalized Variables and Its Applications (1971), École Nationale Supérieure des Mines
[53] Matheron, G., The intrinsic random functions and their applications, Adv. Appl. Probab., 5, 439-468 (1973) · Zbl 0324.60036
[54] McKee, M.; Nassar, A.; Torres-Rua, A.; Aboutalebi, M.; Kustas, W., Implications of sensor inconsistencies and remote sensing error in the use of small unmanned aerial systems for generation of information products for agricultural management, (Autonomous Air and Ground Sensing Systems for Agricultural Optimization and Phenotyping III (2018), International Society for Optics and Photonics), 1066402
[55] Molchanov, I., Theory of Random Sets (2005), Springer-Verlag: Springer-Verlag London · Zbl 1109.60001
[56] Montenegro, M.; Casals, M. R.; Colubi, A.; Gil, M.Á., Testing ‘two-sided’ hypothesis about the mean of an interval-valued random set, (Dubois, D.; Lubiano, M. A.; Prade, H.; Gil, M.Á.; Grzegorzewski, P.; Hryniewicz, O., Soft Methods for Handling Variability and Imprecision (2008), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 133-139
[57] Näther, W., Linear statistical inference for random fuzzy data, Statistics, 29, 221-240 (1997) · Zbl 1030.62530
[58] Nowak, A. S.; Collins, K. R., Reliability of Structures (2012), CRC Press
[59] Nychka, D.; Furrer, R.; Paige, J.; Sain, S., Fields: tools for spatial data (2017), R package version 11.6
[60] Pebesma, E. J., Multivariable geostatistics in S: the gstat package, Comput. Geosci., 30, 683-691 (2004)
[61] Pebesma, E. J.; Bivand, R. S., Classes and methods for spatial data in R, R News, 5, 9-13 (2005)
[62] R: A Language and Environment for Statistical Computing (2020), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria
[63] Rådström, H., An embedding theorem for spaces of convex sets, Proc. Am. Math. Soc., 3, 165-169 (1952) · Zbl 0046.33304
[64] Rao, S. T.; Ku, J. Y.; Rao, K. S., Analysis of toxic air contaminant data containing concentrations below the limit of detection, J. Air Waste Manage. Assoc., 41, 442-448 (1991)
[65] Sinova, B.; Colubi, A.; Gil, M.; González-Rodŕiguez, G., Interval arithmetic-based simple linear regression between interval data: discussion and sensitivity analysis on the choice of the metric, Inf. Sci., 199, 109-124 (2012) · Zbl 06094584
[66] Sun, Y., Asymptotic tests for interval-valued means, Stat. Probab. Lett., 121, 70-77 (2017) · Zbl 1437.62095
[67] Sun, Y.; Lian, G.; Lu, Z.; Loveland, J.; Blackhurst, I., Modeling the variance of return intervals toward volatility prediction, J. Time Ser. Anal., 41, 492-519 (2020) · Zbl 1450.62116
[68] Wei, Y.; Wang, S.; Wang, H., Interval-valued data regression using partial linear model, J. Stat. Comput. Simul., 87, 3175-3194 (2017) · Zbl 07192115
[69] Ypma, J.; Borchers, H. W.; Eddelbuettel, D.; Ypma, M. J., Package ‘nloptr’ (2020)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.