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Piecewise-Multilinear Interpolation of a Random Field

Part of: Stochastic processes Approximations and expansions

Published online by Cambridge University Press:  04 January 2016

Konrad Abramowicz  and
Oleg Seleznjev
Konrad Abramowicz*
Affiliation:
Umeå University
Oleg Seleznjev*
Affiliation:
Umeå University
*
∗ Postal address: Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden.
∗ Postal address: Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden.
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Abstract

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We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Keywords

Approximationrandom fieldpiecewise-multilinear interpolatorsampling design

MSC classification

Primary: 60G60: Random fields
Secondary: 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 41A05: Interpolation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 945 - 959
Copyright
© Applied Probability Trust 

References

Abramowicz, K. and Seleznjev, O. (2011). Spline approximation of a random process with singularity. J. Statist. Planning Inf. 141, 1333–1342.Google Scholar
Adler, R. and Taylor, J. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Benhenni, K. (2001). Reconstruction of a stationary spatial process from a systematic sampling. In Selected Proceedings of the Symposium on Inference for Stochastic Processes (IMS Lecture Notes Monogr. Ser. 37), Institute of Mathematical Statistics, Beachwood, OH, pp. 271–279.Google Scholar
Berman, S. M. (1974). Sojourns and extremes of Gaussian processes. Ann. Prob. 2, 999–1026.CrossRefGoogle Scholar
Brouste, A., Istas, J. and Lambert-Lacroix, S. (2007). On fractional Gaussian random fields simulations. J. Statist. Software 23, 1–23.Google Scholar
Christakos, G. (1992). Random Field Models in Earth Sciences. Academic Press, London.Google Scholar
De Boor, C., Gout, C., Kunoth, A. and Rabut, C. (2008). Multivariate approximation: theory and applications. An overview. Numer. Algorithms 48, 1–9.Google Scholar
Hüsler, J., Piterbarg, V. and Seleznjev, O. (2003). On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Prob. 13, 1615–1653.Google Scholar
Kuo, F. Y., Wasilkowski, G. W. and Woźniakowski, H. (2009). On the power of standard information for multivariate approximation in the worst case setting. J. Approximation Theory 158, 97–125.CrossRefGoogle Scholar
Lancaster, P. and Šalkauskas, K. (1986). Curve and Surface Fitting. An Introduction. Academic Press, London.Google Scholar
Lifshits, M. A. and Zani, M. (2008). Approximation complexity of additive random fields. J. Complexity 24, 362–379.Google Scholar
Müller-Gronbach, T. (1998). Hyperbolic cross designs for approximation of random fields. J. Statist. Planning Inf. 66, 321–344.CrossRefGoogle Scholar
Müller-Gronbach, T. and Schwabe, R. (1996). On optimal allocations for estimating the surface of a random field. Metrika 44, 239–258.Google Scholar
Nikolskii, S. M. (1975). Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin.CrossRefGoogle Scholar
Pratt, W. K. (2007). Digital Image Processing: PIKS Scientific Inside. John Wiley, New York.CrossRefGoogle Scholar
Ritter, K. (2000). Average-Case Analysis of Numerical Problems. Springer, Berlin.Google Scholar
Ritter, K., Wasilkowski, G. W. and Woźniakowski, H. (1995). Multivariate integration and approximation for random fields satisfying Sacks–Ylvisaker conditions. Ann. Appl. Prob. 5, 518–540.Google Scholar
Sacks, J. and Ylvisaker, D. (1966). Designs for regression problems with correlated errors. Ann. Math. Statist. 37, 66–89.CrossRefGoogle Scholar
Seleznjev, O. (1996). Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments. Adv. Appl. Prob. 28, 481–499.Google Scholar
Seleznjev, O. (2000). Spline approximation of random processes and design problems. J. Statist. Planning Inf. 84, 249–262.Google Scholar
Stein, M. (1999). Interpolation of Spatial Data. Springer, New York.CrossRefGoogle Scholar
Su, Y. (1997). Estimation of random fields by piecewise constant estimators. Stoch. Process. Appl. 71, 145–163.Google Scholar
Su, Y. and Cambanis, S. (1993). Sampling designs for estimation of a random process. Stoch. Process. Appl. 46, 47–89.Google Scholar
Yanjie, J. and Yongping, L. (2000). Average widths and optimal recovery of multivariate Besov classes in L p (R d ). J. Approximation Theory 102, 155–170.Google Scholar
Zhang, X. and Wicker, S. B. (2005). On the optimal distribution of sensors in a random field. ACM Trans. Sensor Networks 1, 301–306.CrossRefGoogle Scholar