Fast empirical mode decompositions of multivariate data based on adaptive spline-wavelets and a generalization of the Hilbert-Huang-transform (HHT) to arbitrary space dimensions

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*Title:*	Main Title: Fast empirical mode decompositions of multivariate data based on adaptive spline-wavelets and a generalization of the Hilbert-Huang-transform (HHT) to arbitrary space dimensions
*Description:*	Abstract: 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 infinite 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.
*Identifier:*	10.1142/S1793536910000513 (DOI)

Responsible Party

*Creators:*	Gabriela Jager (Author), Robin Koch (Author), Angela Kunoth (Author), Roland Pabel (Author)
*Publisher:*	World Scientic Publishing Company
*Publication Year:*	2013

Topic

*TR32 Topic:*	Other
*Related Subprojects:*	C1, C7
*Subjects:*	Keywords: Multivariate Data Analysis, HHT, Tensor Product Spline Wavelets, Wavelet, Monogenic Function, Clifford Algebra, Riesz Transform

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*Filename:*	2010_Jager_AADA.pdf
*Data Type:*	Text - Article
*Size:*	22 Pages
*File Size:*	3.1 MB
*Date:*	Issued: 01.07.2010
*Mime Type:*	application/pdf
*Data Format:*	PDF
*Language:*	English
*Status:*	Completed

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*Download Permission:*	Only Project Members
*General Access and Use Conditions:*	For internal use only
*Access Limitations:*	For internal use only
*Licence:*	[TR32DB] Data policy agreement

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*Publication Status:*	Published
*Review Status:*	Peer reviewed
*Publication Type:*	Article
*Article Type:*	Journal
*Source:*	Advances in Adaptive Data Analysis (AADA)
*Issue:*	3
*Volume:*	2
*Number of Pages:*	22 (337 - 358)

Metadata Details

*Metadata Creator:*	Roland Pabel
*Metadata Created:*	02.12.2013
*Metadata Last Updated:*	02.12.2013
*Subproject:*	C1
*Funding Phase:*	1
*Metadata Language:*	English
*Metadata Version:*	V50

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