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Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context

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Abstract

This paper derives conditions under which a stable solution to the least-squares linear estimation problem for multifractional random fields can be obtained. The observation model is defined in terms of a multifractional pseudodifferential equation. The weak-sense and strong-sense formulations of this problem are studied through the theory of fractional Sobolev spaces of variable order, and the spectral theory of multifractional pseudodifferential operators and their parametrix. The Theory of Reproducing Kernel Hilbert Spaces is also applied to define a stable solution to the direct and inverse estimation problems. Numerical projection methods are proposed based on the construction of orthogonal bases of these spaces. Indeed, projection into such bases leads to a regularization, removing the ill-posed nature of the estimation problem. A simulation study is developed to illustrate the estimation results derived. Some open research lines in relation to the extension of the derived results to the multifractal process context are also discussed.

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Acknowledgements

This work has been supported in part by projects MTM2009-13393, MTM2012-32674, MTM2009-13250, and MTM2012-32666 of the DGI, MEC, and P09-FQM-5052, P08-FQM-03834 of the Andalusian CICE, Spain.

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Authors and Affiliations

  1. Faculty of Sciences, University of Granada, Campus Fuente Nueva s/n, 18071, Granada, Spain

    M. D. Ruiz-Medina, R. M. Espejo & J. M. Angulo

  2. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Queensland, 4001, Australia

    V. V. Anh

  3. Sciences and Helth Faculty, University of Jaén, Campus Las Lagunillas s/n, 23071, Jaén, Spain

    M. P. Frías

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  1. M. D. Ruiz-Medina
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  3. R. M. Espejo
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  4. J. M. Angulo
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  5. M. P. Frías
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Correspondence to V. V. Anh.

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Ruiz-Medina, M.D., Anh, V.V., Espejo, R.M. et al. Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context. J Optim Theory Appl 167, 888–911 (2015). https://doi.org/10.1007/s10957-013-0423-4

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  • DOI: https://doi.org/10.1007/s10957-013-0423-4

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