Abstract
Procedures are developed in order to obtain optimal estimates of linear functionals for a wide class of nonstationary spatial functions. These procedures rely on well-established constrained minimum-norm criteria, and are applicable to multidimensional phenomena which are characterized by the so-called hypothesis of inherentity. The latter requires elimination of the polynomial, trend-related components of the spatial function leading to stationary quantities, and also it generates some interesting mathematics within the context of modelling and optimization in several dimensions. The arguments are illustrated using various examples, and a case study computed in detail.
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References
Wiener, N.,Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, Wiley, New York, New York, 1949.
Yaglom, A. M.,Some Classes of Random Fields in N-Dimensional Space, Related to Stationary Random Processes, Theory of Probability and Its Applications, Vol. 3, pp. 273–320, 1957.
Matheron, G.,La Théorie des Fonctions Aléatoires Intrinsèques Généralisées, Ecole des Mines de Paris, Centre de Géostatistique, Report No. 252, 1971.
Schagen, I. P.,Software for Interpolation and Modeling Unknown Functions, Applied Mathematical Modeling, Vol. 8, pp. 193–196, 1984.
Rice, J. R.,The Approximation of Functions, Addison-Wesley, Reading, Massachusetts, 1964.
Meinardus, G.,Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, New York, New York, 1967.
Davis, P. J.,Interpolation and Approximation, Dover, New York, New York, 1975.
Box, G. E. P., andJenkins, G. M.,Time Series Analysis Forecasting and Control, Holden-Day, San Francisco, California, 1970.
Christakos, G.,Modeling and Estimation of Geological Signals, National Technical University of Athens, Athens, Greece, 1986.
Gelfand, I. M., andVilenkin, N.,Generalized Functions, Vol. 4, Academic Press, New York, New York, 1964.
Gradshteyn, T. S., andRyzhik, I. M.,Tables of Integrals, Series, and Products, Academic Press, New York, New York, 1965.
Christakos, G.,On the Problem of Permissible Covariance and Variogram Models, Water Resources Research, Vol. 20, pp. 251–265, 1984.
Christakos, G.,The Space Transformations and Their Applications in Systems Modeling and Simulation, Proceedings of the International AMSE Conference on Modeling and Simulation, Athens, Greece, Vol. 1.3, pp. 49–68, 1984.
Watson, G. S.,Trend Surface Analysis, Journal of the International Association for Mathematical Geology, Vol. 3, pp. 215–226, 1971.
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Communicated by L. C. W. Dixon
The authors acknowledge with appreciation the comments and criticisms made by Dr. L. C. W. Dixon of the Numerical Optimization Centre, Hatfield Polytechnic, Hatfield, Hertfordshire, England.
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Christakos, G., Paraskevopoulos, P.N. On the functional optimization of a certain class of nonstationary spatial functions. J Optim Theory Appl 52, 191–208 (1987). https://doi.org/10.1007/BF00941280
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DOI: https://doi.org/10.1007/BF00941280