Journal
ADVANCES IN DATA SCIENCE AND ADAPTIVE ANALYSIS
Volume 2, Issue 3, Pages 337-358Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S1793536910000513
Keywords
Multivariate data analysis; arbitrary space dimension; Hilbert-Huang-Transform; (HHT); empirical mode decomposition (EMD); intrinsic mode functions; (IMFs); tensor product spline wavelets; adaptive wavelet-nD-EMD; monogenic function; Riesz transform; Clifford algebra
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Funding
- Deutsche Forschungsgemeinschaft (DFG)
- Pattern in SoilVegetation-Atmosphere Systems: Monitoring, Modelling, and Data Assimilation
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The Hilbert-Huang-Transform (HHT) has proven to be an appropriate multiscale analysis technique specifically for nonlinear and nonstationary time series on non-equidistant grids. It is empirically adapted to the data: first, an additive decomposition of the data (empirical mode decomposition, EMD) into certain multiscale components is computed, denoted as intrinsic mode functions. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time-frequency spectrum and instantaneous (time-dependent) frequencies. For the first step, the empirical decomposition of the data, a different method based on local means has been developed by Chen et al. (2006). In this paper, we extend their method to multivariate data sets in arbitrary space dimensions. We place special emphasis on deriving a method which is numerically fast also in higher dimensions. Our method works in a coarse-to-fine fashion and is based on adaptive (tensor-product) spline-wavelets. We provide some numerical comparisons to a method based on linear finite elements and one based on thin-plate-splines to demonstrate the performance of our method, both with respect to the quality of the approximation as well as the numerical efficiency. Second, for a generalization of the Hilbert transform to the multivariate case, we consider the Riesz transformation and an embedding into Clifford-algebra valued functions, from which instantaneous amplitudes, phases and orientations can be derived. We conclude with some numerical examples.
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