Sparse Time Frequency Representations and Dynamical Systems
In this paper, we establish a connection between the recently developed data-driven time-frequency analysis [T.Y. Hou and Z. Shi, Advances in Adaptive Data Analysis, 3, 1–28, 2011], [T.Y. Hou and Z. Shi, Applied and Comput. Harmonic Analysis, 35, 284–308, 2013] and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into the sparsest collection of Intrinsic Mode Functions (IMFs) over the largest possible dictionary via nonlinear optimization. These IMFs are of the form a(t)cos(θ(t)), where the amplitude a(t) is positive and slowly varying. The non-decreasing phase function θ(t) is determined by the data and in general depends on the signal in a nonlinear fashion. One of the main results of this paper is that we show that each IMF can be associated with a solution of a second order ordinary differential equation of the form x+p(x,t)x+q(x,t)=0. Further, we propose a localized variational formulation for this problem and develop an effective l1-based optimization method to recover p(x,t) and q(x,t) by looking for a sparse representation of p and q in terms of the polynomial basis. Depending on the form of nonlinearity in p(x,t) and q(x,t), we can define the order of nonlinearity for the associated IMF. This generalizes a concept recently introduced by Prof. N. E. Huang et al. [N.E. Huang, M.-T. Lo, Z. Wu, and Xianyao Chen, US Patent filling number 12/241.565, Sept. 2011]. Numerical examples will be provided to illustrate the robustness and stability of the proposed method for data with or without noise. This manuscript should be considered as a proof of concept.
© 2015 by International Press of Boston, Inc. Received: August 12, 2013; accepted (in revised form): February 27, 2014. We would like to thank Professor Norden E. Huang for a number of stimulating discussions on the topic of the order of nonlinearity. This work was supported by NSF FRG Grant DMS-1159138, an AFOSR MURI Grant FA9550-09-1-0613 and a DOE grant DE-FG02-06ER25727. The research of Dr. T.Y. Hou was partially supported by NSF Grant DMS-1318377. The research of Dr. Z. Shi was also in part supported by a NSFC Grant 11201257. Special Issue in Honor of George Papanicolaou's 70th Birthday.
Submitted - 1312.0202v1.pdf
Published - Hou_2015p673.pdf