A general form of probability kriging for estimation of the indicator and uniform transforms

Carr, James R.; Mao, Nai-hsien

doi:10.1007/bf00894777

Cited by 18 publications

(10 citation statements)

References 9 publications

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“…Thus, another nonparametric approach, called probability kriging which is also based on cokriging estimator has also been proposed. This method represents the information from all the available attribute values by using the order relation of observed values (Carr and Mao 1993;Carr 1994). The uniform value, also called the standardized rank, which denotes the order relation of observed values, is assigned as the only auxiliary variable in probability kriging to improve estimation of the probability of the attribute value being greater/lower than the desired threshold.…”

Section: Introductionmentioning

confidence: 99%

Indicator and probability kriging methods for delineating Cu, Fe, and Mn contamination in groundwater of Najafgarh Block, Delhi, India

Adhikary

Dash

Bej

et al. 2010

Environ Monit Assess

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Two non-parametric kriging methods such as indicator kriging and probability kriging were compared and used to estimate the probability of concentrations of Cu, Fe, and Mn higher than a threshold value in groundwater. In indicator kriging, experimental semivariogram values were fitted well in spherical model for Fe and Mn. Exponential model was found to be best for all the metals in probability kriging and for Cu in indicator kriging. The probability maps of all the metals exhibited an increasing risk of pollution over the entire study area. Probability kriging estimator incorporates the information about order relations which the indicator kriging does not, has improved the accuracy of estimating the probability of metal concentrations in groundwater being higher than a threshold value. Evaluation of these two spatial interpolation methods through mean error (ME), mean square error (MSE), kriged reduced mean error (KRME), and kriged reduced mean square error (KRMSE) showed 3.52% better performance of probability kriging over indicator kriging. The combined result of these two kriging method indicated that on an average 26.34%, 65.36%, and 99.55% area for Cu, Fe, and Mn, respectively, are coming under the risk zone with probability of exceedance from a cutoff value is 0.6 or more. The groundwater quality map pictorially represents groundwater zones as "desirable" or "undesirable" for drinking. Thus the geostatistical approach is very much helpful for the planners and decision makers to devise policy guidelines for efficient management of the groundwater resources so as to enhance groundwater recharge and minimize the pollution level.

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Section: Introductionmentioning

confidence: 99%

Indicator and probability kriging methods for delineating Cu, Fe, and Mn contamination in groundwater of Najafgarh Block, Delhi, India

Adhikary

Dash

Bej

et al. 2010

Environ Monit Assess

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“…None of these methods completely eliminate the order relation problems (Carr and Mao 1993;Carle and Fogg 1996). Even DK, with its high theoretical complexity, usually yields negative estimates in the tails of the cpdf.…”

Section: Introductionmentioning

confidence: 99%

Indicator Kriging without Order Relation Violations

Tolosana‐Delgado

Pawlowsky‐Glahn

Egozcue³

2008

Math Geosci

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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.

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“…Probability kriging is a special form of the cokriging procedure wherein only one variable, the indicator transformation, is estimated using two spatial variables, the indicator and uniform transforms (Carr and Mao, 1993). For probability kriging, the indicator code I ( x ) is assigned as the main variable and the other variable, and the uniform value u ( x ) is assigned as the auxiliary variable in the cokriging estimator.…”

Section: Methodsmentioning

confidence: 99%

Geostatistical Analysis and Risk Assessment on Soil Total Nitrogen and Total Soil Phosphorus in the Dongting Lake Plain Area, China

Lü

Niu

et al. 2007

J of Env Quality

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Nonpoint-source pollution and water body eutrophication have become increasing concerns for scientists and policymakers. Nitrogen and phosphorus affect environmental pollution, especially lake eutrophication. To assess the environmental risk of soil total nitrogen (TN) and total phosphorus (TP) pollution, a typical ecological unit of Dongting Lake plain was selected as the experimental site. To verify the stationary of the data, a moving windows technique was adopted. Our results showed that Box-Cox transformation achieved normality in the data set and dampened the effect of outliers. The best theoretical model for semivariogram of TN and TP was a spherical model. The ordinary kriging estimates of TN and TP concentrations were mapped. The integrative comparisons of semivariogram parameters with different trends to the kriging prediction errors of TN and TP indicated that the two-order trend is preferable. Kriging SDs provided valuable information that will increase the accuracy of TN and TP mapping. The probability kriging method is useful to assess the risk of N and P pollution by providing the conditional probability of N and P concentrations exceeding the threshold concentrations of 3.2 and 1.05 g/kg, respectively. The probability distribution of TN and TP at different levels will be helpful to conduct risk assessment, optimize fertilization, and control the pollution of N and P.

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A general form of probability kriging for estimation of the indicator and uniform transforms

Cited by 18 publications

References 9 publications

Indicator and probability kriging methods for delineating Cu, Fe, and Mn contamination in groundwater of Najafgarh Block, Delhi, India

Indicator and probability kriging methods for delineating Cu, Fe, and Mn contamination in groundwater of Najafgarh Block, Delhi, India

Indicator Kriging without Order Relation Violations

Geostatistical Analysis and Risk Assessment on Soil Total Nitrogen and Total Soil Phosphorus in the Dongting Lake Plain Area, China

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