Least-squares estimation of multifractional random fields in a Hilbert-valued context. (English) Zbl 1342.60079
Summary: 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 derived estimation results. Some open research lines in relation to the extension of the derived results to the multifractal process context are also discussed.
MSC:
| 60G60 | Random fields |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 62M09 | Non-Markovian processes: estimation |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 47G30 | Pseudodifferential operators |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
Keywords:
multifractional random fields; least-squares estimation; multifractional pseudodifferential operators; inverse problem; reproducing kernel Hilbert spacesReferences:
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