Abstract
Least-squares linear functional reconstruction and extrapolation of a random field defining the random input of a linear system represented by an integral equation is considered. This problem is solved for a class of random fields with reproducing kernel Hilbert space norm equivalent to the norm of a Sobolev space of an appropriate fractional order. More specifically, functional reconstruction and extrapolation formulae are derived from generalized wavelet-based orthogonal expansions of the input and output random fields in the class considered (see Angulo and Ruiz-Medina, 1999, for the ordinary case). In the Gaussian and ordinary case, the results derived also provide sample-path functional reconstruction and extrapolation formulae. Simulation studies are carried out for systems defined in terms of fractional integration of fractional Brownian motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, L., Hall, N., Jawerth, B., andPeters, G. (1993). Wavelets on closed subsets of the real line. In L. L. Schumaker and G. Webb, eds.,Recent Advances in Wavelets Analysis, pp. 345–369, Academic Press, New York.
Angulo, J. M., andRuiz-Medina, M. D. (1999). Multiresolution approximation to the stochastic inverse problem.Advances in Applied Probability, 31:1039–1057.
Angulo, J. M., Ruiz-Medina, M. D., andAnh, V. V. (2000). Estimation and filtering of fractional generalized random fields.Journal of Australian Mathematical Society. Series A, 69:1–26.
Anh, V. V., Angulo, J. M., andRuiz-Medina, M. D. (1999). Possible long-range dependence in fractional random fields.Journal of Statistical Planning and Inference, 80:95–110.
Anh, V. V., Angulo, J. M., Ruiz-Medina, M. D., andTieng, Q. (1998). Long-range dependence and second-order intermittency of two-dimensional turbulence. Environmental Modelling and Software, 13:233–238.
Benassi, A., Jaffard, S., andRoux, D. (1997). Elliptic Gaussian random processes.Revista Matemática Iberoamericana, 13:19–90.
Berliner, L. M., Wikle, C. K., andMilliff, R. F. (1999) Multiresolution wavelet analyses in hierarchical Bayesian turbulence models. In P. Muller and B. Vidakovic, eds.,Lecture Notes in Statistics 141, pp. 341–359. Springer-Verlag, New York.
Campi, S. (1980). An inverse problem related to the travel time of seismic waves.Bolletino dell'Unione Matematica Italiana B, 17:661–674.
Christakos, G. (2000).Modern Spatiotemporal Geostatistics. Oxford University Press, Oxford.
Christakos, G., andHristopulos, D. T. (1998)Spatio-Temporal Environmental Health Modelling: A Tractatus Stochasticus. Kluwer Academic Publishers. Boston.
Chui, C. K. (1992).An Introduction to Wavelets. Academic Press, London.
Cohen, A., Dahmen, W., andDevore, R. (2000). Multiscale decomposition on bounded domains.Transactions of the American Mathematical Society, 352:3651–3685.
Cohen, A., Daubechies, I., andVial, P. (1993). Wavelets on the interval and fast wavelet transforms.Applied Computational an Harmonic Analysis, 1:54–81.
Daubechies, I. (1992).Ten Lectures on Wavelets. SIAM, Philadelphia.
Dietrich, C. D. andNewsam, G. N. (1989). A stability analysis of geostatistical approach to aquifer transmissivity identification.Stochastic Hydrology and Hydraulics, 3:293–316.
Donoho, D. L. (1995) Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition.Applied and Computational Harmonic Analysis, 2:101–126.
Ekstrom, M. (1982). Realizable Wiener filtering in two dimensions.IEEE Transactions on Acoustics, Speech, and Signal Processing, 30:31–40.
Gel'fand, I. M. andVilenkin, N. Y. (1964).Generalized Functions 4. Academic Press, New York.
Huang, H. C. andCressie, N. (1996). Spatio-temporal prediction of snow water equivalent using the Kalman filter.Computational Statistics and Data Analysis, 22:159–175.
Jaffard, S. andMeyer, Y. (1989). Bases d'ondelettes dans des ouverts de ℝn.Journal de Mathématiques Pures et Appliquées, 68:95–108.
Kitanidis, P. K. andVomvoris, E. G. (1983). A geostatistical approach to the inverse problem for generalized random variables.Inverse Problems, 5:599–612.
Louis, A. K. (1992). Medical imaging: state of the art and future development.Inverse Problems, 8:709–738.
Lubbig, H. (1995).The Inverse Problem. Academie Verlag, Berlin.
Meyer, Y. (1992).Wavelets and Operators, Cambridge University Press, Cambridge.
Miller, E. L. andWillsky, A. S. (1995). A multiscale approach to sensor fusion and the solution of linear inverse problems.Applied and Computational Harmonic Analysis, 2:127–147.
Ramm, A. G. (1990)Random Fields Estimation Theory. Logman Scientific & Technical, New York.
Resnikoff, H. L. andWells, R. O. (1998).Wavelet Analysis. The Scalable Structure of Information. Springer Verlag, New York.
Ruiz-Medina, M. D. andAngulo, J. M. (2002). Spatio-temporal filtering using wavelets.Stochastic Environmental Research and Risk Assessment, 16:241–266.
Ruiz-Medina, M. D., Angulo, J. M., andAnh, V. V. (2002). Stochastic fractional-order differential models on fractals.Theory of Probability and Mathematical Statistics, 67:130–146.
Ruiz-Medina, M. D., Angulo, J. M., andAnh, V. V. (2003a). Fractional generalized random fields on bounded domains.Stochastic Analysis and Applications, 21:465–492.
Ruiz-Medina, M. D., Angulo, J. M., andAnh, V. V. (2003b). Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields.Journal of Multivariate Analysis, 85:192–216.
Ruiz-Medina, M. D., Anh, V. V., andAngulo, J. M. (2001). Stochastic fractional-order differential models with fractal boundary conditions.Statistics and Probability Letters, 54:47–60.
Saied, E. A. (2000). Anomalous diffusion on fractal objects: additional analytic solutions.Chaos, Solitons and Fractals, 11:1369–1376.
Samorodnitsky, G. andTaqqu, M. S. (1994).Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman & Hall, New York.
Santosa, F. andSymes, W. W. (1998). Computation of the Hessian for least-squares solutions of inverse problems of reflection seismology.Inverse Problems, 4:211–233.
Sun, N. Z. (1994).Inverse Problems in Groundwater Modeling. Theory and Applications of Transport in Porous Media. Kluwer Academic Publishers, Boston.
Trampert, J., Leveque, J. J., andCara, M. (1992). Inverse problems in seismology. In M. Bertero and E. Pike, eds.,Inverse Problems in Scattering and Imaging. Adam Hilger, Bristol.
Triebel, H. (1978).Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Co., Amsterdam.
Wikle, C. K. andCressie, N. (1999). A dimension-reduction approach to space-time Kalman filtering.Biometrika, 86:815–829.
Wornell, G. W. (1990). A Karhumen-Loève-like expansion for 1/f processes via wavelets.IEEE Transactions on Information Theory, 36:859–861.
Yanovsky, V. V., Chechkin, A. V., Schertzer, D., andTur, A. V. (2000). Lévy anomalous diffusion and fractional Fokker-Planck equation.Physica A. Statistical Mechanics and its Applications, 282:13–34.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported in part by projects BFM2000-1465 and BFM-2002-01836 of the DGI, Spain.
Rights and permissions
About this article
Cite this article
Fernández-Pascual, R., Ruiz-Medina, M.D. & Angulo, J.M. Wavelet-based functional reconstruction and extrapolation of fractional random fields. Test 13, 417–444 (2004). https://doi.org/10.1007/BF02595780
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02595780