An interpolation method taking into account inequality constraints: II. Practical approach

Kostov, C.; Dubrule, Olivier

doi:10.1007/bf00897655

Cited by 33 publications

(10 citation statements)

References 7 publications

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“…where α opt ∈ R N +1 is the solution of the problem (P N ), whereĨ andC are defined in (15) and (16) respectively. Figure 3 shows the convergence of the proposed algorithm.…”

Section: Monotonicity In One Dimensionmentioning

confidence: 99%

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A new method for interpolating in a convex subset of a Hilbert space

Bay¹,

Grammont²,

Maatouk³

2017

Comput Optim Appl

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“…where α opt ∈ R N +1 is the solution of the problem (P N ), whereĨ andC are defined in (15) and (16) respectively. Figure 3 shows the convergence of the proposed algorithm.…”

Section: Monotonicity In One Dimensionmentioning

confidence: 99%

“…, n) and the p inequality constraints given in (17) and (18). This form is generalized to any kernel or semi-kernel, stationary covariance function or generalized covariance function, see [8] and [15].…”

Section: Case Of a Finite Number Of Constraintsmentioning

confidence: 99%

A new method for interpolating in a convex subset of a Hilbert space

Bay¹,

Grammont²,

Maatouk³

2017

Comput Optim Appl

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“…For example, the "thin-plate splines", one of the most commonly used interpolators, are derived by coefficients which minimize the bending energy of a thin plate, while honoring all the data values. The extended version of this smoothing spline, what is called "constrained spline problems", can be also solved in the framework of spline theory by minimizing various objective functions Galli et al, 1984;Kostov and Dubrule, 1986;Wong, 1980).…”

Section: Area-to-point Kriging Interpolatormentioning

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%

“…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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1999

Geostatistics

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An interpolation method taking into account inequality constraints: II. Practical approach

Cited by 33 publications

References 7 publications

A new method for interpolating in a convex subset of a Hilbert space

A new method for interpolating in a convex subset of a Hilbert space

Area-to-point Kriging with inequality-type data

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