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Data-driven time–frequency analysis

Hou, Thomas Y. and Shi, Zuoqiang (2013) Data-driven time–frequency analysis. Applied and Computational Harmonic Analysis, 35 (2). pp. 284-308. ISSN 1063-5203. http://resolver.caltech.edu/CaltechAUTHORS:20130725-100732399

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Abstract

In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t)cos(θ(t))}, where a∈V(θ), V(θ) consists of the functions smoother than cos(θ(t)) and θ′⩾0. This problem can be formulated as a nonlinear l^0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l^0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.acha.2012.10.001DOIArticle
http://www.sciencedirect.com/science/article/pii/S1063520312001546PublisherArticle
http://arxiv.org/abs/1202.5621arXivDiscussion Paper
Additional Information:© 2012 Elsevier Inc. Received 25 February 2012; Revised 18 August 2012; Accepted 18 October 2012; Available online 24 October 2012; Communicated by Stephane G. Mallat. This work was in part supported by the AFOSR MURI grant FA9550-09-1-0613. We would like to thank Professors Norden E. Huang and Zhaohua Wu for many stimulating discussions on EMD/EEMD and topics related to the research presented here. We would also like to thank Professors Ingrid Daubechies, Stanley Osher, and Zuowei Shen for their interest in this work and for a number of valuable discussions. Prof. Hou would like to express his gratitude to the National Central University (NCU) for their support and hospitality during his visits to NCU in the past two years. We also like to thank the two anonymous reviewers for their constructive comments and suggestions which help to improve the quality of this paper.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-09-1-0613
Subject Keywords:Time–frequency analysis; Instantaneous frequency; Sparse decomposition; Matching pursuit
Record Number:CaltechAUTHORS:20130725-100732399
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20130725-100732399
Official Citation:Thomas Y. Hou, Zuoqiang Shi, Data-driven time–frequency analysis, Applied and Computational Harmonic Analysis, Volume 35, Issue 2, September 2013, Pages 284-308, ISSN 1063-5203, http://dx.doi.org/10.1016/j.acha.2012.10.001.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39587
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:25 Jul 2013 21:25
Last Modified:15 Mar 2016 19:40

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