Abstract
We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behaviour is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the Hölder class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Abramowicz K, Seleznjev O (2011a) Multivariate piecewise linear interpolation of a random field. arXiv:11021871
Abramowicz K, Seleznjev O (2011b) Spline approximation of a random process with singularity. J Statist Plann Inference 141:1333–1342
Benhenni K, Cambanis S (1992) Sampling designs for estimating integrals of stochastic processes. Ann Statist 20(1):161–194
Berman SM (1974) Sojourns and extremes of Gaussian process. Ann Probab 2:999–1026
Cambanis S, Masry E (1992) Trapezoidal stratified Monte Carlo integration. SIAM J Numer Anal 29(1):284–301
Haber S (1966) A modified Monte-Carlo quadrature. Math Comp 20(95):361–368
Hüsler J, Piterbarg V, Seleznjev O (2003) On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann Appl Probab 13(4):1615–1653
Masry E, Vadrevu A (2009) Random sampling estimates of Fourier transforms: antithetical stratified Monte Carlo. IEEE Trans Signal Process 57:194–204. doi:10.1109/TSP.2008.2007340
Peixin Y (2005) Computational complexity of the integration problem for anisotropic classes. Adv Comput Math 23:375–392
Ripley B (2004) Spatial statistics. Wiley-Blackwell, New Jersey
Ritter K (2000) Average-case analysis of numerical problems. Springer-Verlag, Berlin
Ritter K, Wasilkowski GW, Woźniakowski H (1995) Multivariate integration and approximation for random fields satisfying Sacks-Ylvisaker conditions. Ann Appl Probab 5(2):518–540
Sacks J, Ylvisaker D (1966) Designs for regression problems with correlated errors. Ann Math Statist 37:66–89
Schoenfelder C, Cambanis S (1982) Random designs for estimating integrals of stochastic processes. Ann Statist 10(2):526–538
Seleznjev O (2000) Spline approximation of stochastic processes and design problems. J Statist Plann Inference 84:249–262
Wasilkowski GW (1994) Integration and approximation of multivariate functions: average case complexity with isotropic Wiener measure. J Approx Theory 77(2):212–227
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abramowicz, K., Seleznjev, O. Stratified Monte Carlo Quadrature for Continuous Random Fields. Methodol Comput Appl Probab 17, 59–72 (2015). https://doi.org/10.1007/s11009-013-9347-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-013-9347-6