Adaptive Data Analysis via Sparse Time-Frequency Representation

Abstract

We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed 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 ≥ 0 is assumed to be smoother than cos (θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method.

A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms

“…In recent years, we have witnessed a surge of interests in exploring sparse structures prevailing in many physical and engineering problems. These methods include compressed sensing in signal reconstruction [6,10], hierarchical matrix in discretization of integral operators [13], adaptive data analysis in signal processing [17,18], signal processing for speech and music via l 1 minimization [23,34], proper orthogonal decomposition (POD) methods [3,30], reduced basis (RB) methods [4,24,27] in solving parameterized PDEs, and the dynamically Orthogonal (DO) method in solving SPDEs [28,29]. Most of these methods emphasize the use of spatial basis, but ignore stochastic basis.…”

Two-Dimensional Compact Variational Mode Decomposition

“…The problem is inspired by the one-dimensional empirical mode decomposition (EMD) algorithm [43] and its more recent derivates, such as [24,[38][39][40]48,[54][55][56]61,66,71]. We are interested in the two-dimensional (2D) and higher ndimensional analogs and extensions of such decomposition problems.…”

Robust spectrotemporal decomposition by iteratively reweighted least squares

“…To impose stochastic continuity, we assume a joint prior probability density function over w 1 ; w 2 ; ⋯; w N , which in turn imposes a joint probability density function on ðx n Þ N n=1 . Motivated by the empirical mode decomposition (13) and its variants (11,16), we choose prior densities that enforce sparsity in the frequency domain and smoothness in time. In logarithmic form, the priors we propose are log p 1 ðw 1 ; w 2 ; ⋯;…”

“…We assume a Gaussian or a point-process observation model for the time series and introduce prior distributions on the time-frequency plane that yield maximum a posteriori (MAP) spectral estimates that are smooth (continuous) in time yet sparse in frequency. Our choice of prior distributions is motivated by EMD (13), and its variants (11,16), which decompose signals into a small number of oscillatory components. We term our procedure "spectrotemporal pursuit.…”