An interpolation method taking into account inequality constraints: I. Methodology

Dubrule, Olivier; Kostov, C.

doi:10.1007/bf00897654

Cited by 61 publications

(31 citation statements)

References 5 publications

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“…where f m (u) are low-order polynomials corresponding to the drift functions in Kriging, and ϒ(:) is a set of parametric functionals, which is equivalent to the variogram when it is replaced with a generalized covariance of an intrinsic RF (Chilès and Delfiner, 1999;Cressie, 1993;Dubrule and Kostov, 1986). For example, in one-dimensional space R 1 , ϒ(u, u k ) = ϒ(|h|) = |h| 3 , in two-dimensional space R 2 , ϒ(u, u k ) = ϒ(|h|) = |h| 2 log(|h|), where |h| denotes the distance between u and u k .…”

Section: Constrained Thin-plate Splinesmentioning

confidence: 99%

“…The coefficients of constrained thin-plate splines are determined by finding a unique minimizer of the quadratic form of Equation (22), and then constraining the minimization problem via equality and inequality constraints. In two papers Dubrule and Kostov (1986), Kostov and Dubrule (1986), they developed a method for constrained point-to-point Kriging prediction on the analogy of dual Kriging formalism with splines. In essence, the algorithm replaces the strongest inequality data, which violate some inequality constraints when the initial prediction using only areal data is evaluated against bound values, so that all the other constraints are automatically satisfied .…”

Section: Constrained Area-to-point Predictionmentioning

confidence: 99%

“…Local constraints mixed with upper, lower, and two-sided interval constraints are assigned to a line transect data set (see Dubrule and Kostov (1986)), whereas global constraints, i.e., non-negativity, are enforced to entire simulated 2D surface. In both cases, we suppose that areal data are averages of the unknown point support values in any support.…”

Section: Case Studymentioning

confidence: 99%

“…The "soft-Kriging approach", whose formalism allows building a predictor which honors prior constraint intervals, including a constraint interval at all points in the study area, addresses the confidence intervals for the resulting predictions, but requires internal consistency of the covariance models used for the indicator variable, which is hard to obtain in practice. Another approach is to constrain the Kriging predictions via non-linear optimization techniques (Barnes and You, 1992;Dubrule and Kostov, 1986;Kostov and Dubrule, 1986); this approach will be extended in this paper to address constrained area-to-point prediction. Outside geostatitics, pioneering work has been done by Tobler (1979), who proposed a pychnophylactic interpolation method to account for constraints, including mass-preservation and and non-negativity.…”

Section: Introductionmentioning

confidence: 99%

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Area-to-point Kriging with inequality-type data

Yoo

Kyriakidis

2006

J Geograph Syst

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In practical applications of area-to-point spatial interpolation, inequality constraints, such as non-negativity, or more general constraints on the maximum and/or minimum allowable value of the resulting predictions, should be taken into account. The geostatistical framework proposed in this paper deals with area-to-point interpolation problems under such constraints, while: (i) explicitly accounting for support differences between sample data and unknown values, (ii) guaranteeing coherent predictions, and (iii) providing a measure of reliability for the resulting predictions. The analogy between the dual form of area-to-point interpolation and a spline allows to solve constrained area-to-point interpolation problems via a constrained quadratic minimization algorithm, after accounting for the following three issues: (i) equality and inequality constraints could be applied to different supports, and such support differences should be considered explicitly in the problem formulation, (ii) if inequality constraints are enforced on the entire set of points discretizing the areal data, it is impossible to obtain a solution of the quadratic programming problem, and (iii) the uniqueness and existence of the solution has to be diagnosed. In this work, stable and efficient computation of point predictions is achieved through the following two steps: (i) initial prediction at all locations via unconstrained area-to-point interpolation, and (ii) constrained area-to-point interpolation with inequality information only at those points whose initial predicted values violate the inequality constraints. Last, the application of the proposed method to area-to-point spatial interpolation with inequality constraints in one and two dimensions is demonstrated using realistically simulated data.

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Section: Constrained Thin-plate Splinesmentioning

confidence: 99%

Section: Constrained Area-to-point Predictionmentioning

confidence: 99%

Section: Case Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Area-to-point Kriging with inequality-type data

Yoo

Kyriakidis

2006

J Geograph Syst

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“…This is because, if this approach is used in generating conditional realizations of the modeled process, the ensemble of realizations yields a finite probability for elements of s being equal to the constraint value, which is contrary to the definition of continuous random variables. A related approach, based on spline formalism, was presented in Dubrule and Kostov [1986].…”

Section: Constrained Interpolation and Inverse Modelingmentioning

confidence: 99%

A Gibbs sampler for inequality‐constrained geostatistical interpolation and inverse modeling

Michalak

2008

Water Resources Research

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[1] Interpolation and inverse modeling problems are ubiquitous in environmental sciences. In many applications, the parameters being estimated or mapped have physical constraints, such as nonnegativity (e.g. concentration, hydraulic conductivity), solubility limits, censored data (e.g. due to dry wells or detection limits), and other physical boundaries or missing data. Geostatistical interpolation and inverse modeling techniques have often been applied for estimating such parameters, but these methods typically cannot enforce physical constraints. This paper describes a statistically rigorous and computationally efficient Gibbs sampler, a Markov chain Monte Carlo technique, based on an a priori truncated Gaussian distribution model, which allows for multiple and variable physical constraints to be enforced within a geostatistical framework. Sample interpolation and inverse modeling applications confirm that estimates, uncertainty bounds and conditional simulations reflect the specified constraints, leading to conclusions that are more consistent with the underlying conceptual model, and provide a more accurate measure of the posterior uncertainty of the parameters being estimated. In addition, especially in inverse modeling applications, a posteriori confidence bounds are narrower even in areas where constraints are not imposed. The method is applicable in multiple dimensions, for data with or without measurement error, and with any variogram model.

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References

1999

Geostatistics

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An interpolation method taking into account inequality constraints: I. Methodology

Cited by 61 publications

References 5 publications

Area-to-point Kriging with inequality-type data

Area-to-point Kriging with inequality-type data

A Gibbs sampler for inequality‐constrained geostatistical interpolation and inverse modeling

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