An optimization based empirical mode decomposition scheme

Abstract

The empirical mode decomposition (EMD) has been developed by N.E. Huang et al. in 1998 as an iterative method to decompose a nonlinear and nonstationary univariate function additively into multiscale components. These components, called intrinsic mode functions (IMFs), are constructed such that they are approximately orthogonal to each other with respect to the $L_2$ inner product. Moreover, the components allow for a definition of instantaneous frequencies through complexifying each component by means of the application of the Hilbert transform. This approach via analytic signals, however, does not guarantee that the resulting frequencies of the components are always non-negative and, thus, ‘physically meaningful’, and that the amplitudes can be interpreted as envelopes.

In this paper, we formulate an optimization problem which takes into account important features desired of the resulting EMD. Specifically, we propose a data-adapted iterative method which minimizes in each iteration step a smoothness functional subject to inequality constraints involving the extrema. In this way, our method constructs a sparse data-adapted basis for the input function as well as a mathematically stringent envelope for the function. Moreover, we present an optimization based normalization to extract instantaneous frequencies from the analytic function approach. We present corresponding algorithms together with several examples.