# zbMATH — the first resource for mathematics

Indicator kriging without order relation violations. (English) Zbl 1158.86005
Summary: Indicator kriging (IK) is a spatial interpolation technique aimed at estimating the conditional cumulative distribution function (ccdf) of a variable at an unsampled location. Obtained results form a discrete approximation to this ccdf, and its corresponding discrete probability density function (cpdf) should be a vector, where each component gives the probability of an occurrence of a class. Therefore, this vector must have positive components summing up to one, like in a composition in the simplex. This suggests a simplicial approach to IK, based on the algebraic-geometric structure of this sample space: simplicial IK actually works with log-odds. Interpolated log-odds can afterwards be easily re-expressed as the desired cpdf or ccdf. An alternative but equivalent approach may also be based on log-likelihoods. Both versions of the method avoid by construction all conventional IK standard drawbacks: estimates are always within the $$(0,1)$$ interval and present no order-relation problems (either with kriging or co-kriging). Even the modeling of indicator structural functions is clarified.

##### MSC:
 86A32 Geostatistics
GSLIB
Full Text:
##### References:
 [1] Aitchison J (1984) Reducing the dimensionality of compositional data sets. Math Geol 16(6):617–636 [2] Aitchison J (1986) The statistical analysis of compositional data, Monographs on statistics and applied probability. Chapman & Hall, London (Reprinted in 2003 with additional material by The Blackburn Press) · Zbl 0688.62004 [3] Billheimer D, Guttorp P, Fagan W (2001) Statistical interpretation of species composition. J Am Stat Assoc 96(456):1205–1214 · Zbl 1073.62573 [4] Bogaert P (1999) On the optimal estimation of the cumulative distribution function in presence of spatial dependence. Math Geol 3(2):213–239 [5] Bogaert P (2002) Spatial prediction of categorical variables: The Bayesian maximum entropy approach. Stoch Environ Res Risk Assess 16:425–448 · Zbl 1020.86001 [6] Carle S, Fogg G (1996) Transition probability-based indicator geostatistics. Math Geol 28(4):453–476 · Zbl 0970.86552 [7] Carr J (1994) Order relation correction experiments for probability kriging. Math Geol 26(5):605–621 [8] Carr J, Mao N (1993) A general-form of probability kriging for estimation of the indicator and uniform transforms. Math Geol 25(4):425–438 [9] Christakos G (1990) A Bayesian/maximum entropy view to the spatial estimation problem. Math Geol 22(7):763–777 · Zbl 0970.86521 [10] Deutsch C, Journel A (1998) GSLIB–geostatistical software library and user’s guide, 2nd edn. Oxford University Press, New York [11] Egozcue JJ, Pawlowsky-Glahn V (2005) Groups of parts and their balances in compositional data analysis. Math Geol 37(7):795–828 · Zbl 1177.86018 [12] Egozcue JJ, Pawlowsky-Glahn V (2006) Simplicial geometry for compositional data. In: Buccianti A, Mateu-Figueras G, Pawlowsky-Glahn V (eds) Compositional data analysis in the geosciences. Geological Society of London, London, pp 145–169 · Zbl 1156.86307 [13] Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal C (2003) Isometric logratio transformations for compositional data analysis. Math Geol 35(3):279–300 · Zbl 1302.86024 [14] Journel AG (1983) Nonparametric estimation of spatial distributions. Math Geol 15(3):445–468 · Zbl 0598.70013 [15] Juang K, Lee D, Hsiao C (1998) Kriging with cumulative distribution function of order statistics for delineation of heavy-metal contaminated soils. Soil Sci 163(10):797–804 [16] Lindley DV (1964) The Bayesian analysis of contingency tables. Ann Math Stat 35(4):1622–1643 · Zbl 0134.37101 [17] Mateu-Figueras G, Pawlowsky-Glahn V, Barceló-Vidal C (2003) Distributions on the simplex. In: Thió-Henestrosa S, Martín-Fernández JA (eds) Compositional data analysis workshop–CoDaWork’03, proceedings. Universitat de Girona, Girona · Zbl 1302.86024 [18] Matheron G (1976) A simple substitute for the conditional expectation: The disjunctive kriging. In: Guarascio M, David M, Huijbregts C (eds) Advanced geostatistics in the mining industry. NATO advances study institute series; series C: mathematical and physical sciences, vol 24. Reidel, Dordrecht, pp 221–236 [19] Matheron G (1993) Une conjecture sur la covariance d’un ensemble aleatoire. In: Cahiers de geostatistique, vol 3. Ecole de Mines de Paris, Fontainebleau, pp 107–113 [20] Myers DE (1983) Estimation of linear combinations and co-kriging. Math Geol 15(5):633–637 [21] Pardo-Igúzquiza E, Dowd P (2005) Multiple indicator cokriging with application to optimal sampling for environmental monitoring. Comput Geosci 31(1):1–13 [22] Pawlowsky V (1989) Cokriging of regionalized compositions. Math Geol 21(5):513–521 [23] Pawlowsky-Glahn V (2003) Statistical modelling on coordinates. In: Thió-Henestrosa S, Martín-Fernández JA (eds) Compositional data analysis workshop–CoDaWork’03, proceedings. Universitat de Girona, Girona [24] Pawlowsky-Glahn V, Egozcue JJ (2001) Geometric approach to statistical analysis on the simplex. Stoch Environ Res Risk Assess 15(5):384–398 · Zbl 0987.62001 [25] Pawlowsky-Glahn V, Egozcue JJ (2002) BLU estimators and compositional data. Math Geol 34:259–274 · Zbl 1031.86007 [26] Pawlowsky-Glahn V, Olea RA (2004) Geostatistical analysis of compositional data. Studies in mathematical geology, vol 7. Oxford University Press, Oxford · Zbl 1105.86004 [27] Sullivan J (1984) Conditional recovery estimation through probability kriging–theory and practice. In: Geostatistics for natural resources characterization, 2nd edn. NATO-ASI, Stanford [28] Suro-Perez V, Journel A (1991) Indicator principal component kriging. Math Geol 23(5):759–788 [29] Tolosana-Delgado R (2006) Geostatistics for constrained variables: Positive data, compositions and probabilities. Applications to environmental hazard monitoring. PhD Thesis, Universitat de Girona, Girona, Spain, p 198 [30] Tolosana-Delgado R, Pawlowsky-Glahn V (2007) Kriging regionalized positive variables revisited sample space and scale considerations. Math Geol 39(6):529–558 · Zbl 1141.86306 [31] Tolosana-Delgado R, Pawlowsky-Glahn V, J Egozcue J, van den Boogaart KG (2005) A compositional approach to indicator kriging. In: Cheng Q, Bonham-Carter G (eds) 2005 annual conference of the international association for mathematical geology (IAMG), Toronto, Canada, pp 651–656 [32] Vargas-Guzman J, Dimitrakopoulos R (2003) Successive nonparametric estimation of conditional distributions. Math Geol 35(1):39–52 · Zbl 1302.86029
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.