A Fast Algorithm for Multiresolution Mode Decomposition

TL;DR

This paper introduces a fast, recursive diffeomorphism-based spectral analysis algorithm for multiresolution mode decomposition, enabling efficient and accurate analysis of complex time series with multiscale intrinsic modes.

Contribution

The paper presents a novel recursive spectral analysis method that significantly improves the efficiency and accuracy of multiresolution mode decomposition in time series analysis.

Findings

The proposed RDSA algorithm is highly efficient due to NUFFT implementation.

The method guarantees convergence and theoretical accuracy.

Numerical examples demonstrate superior performance on synthetic and real data.

Abstract

\emph{Multiresolution mode decomposition} (MMD) is an adaptive tool to analyze a time series $f(t)=\sum_{k=1}^K f_k(t)$, where $f_k(t)$ is a \emph{multiresolution intrinsic mode function} (MIMF) of the form \begin{eqnarray*} f_k(t)&=&\sum_{n=-N/2}^{N/2-1} a_{n,k}\cos(2\pi n\phi_k(t))s_{cn,k}(2\pi N_k\phi_k(t))\\&&+\sum_{n=-N/2}^{N/2-1}b_{n,k} \sin(2\pi n\phi_k(t))s_{sn,k}(2\pi N_k\phi_k(t)) \end{eqnarray*} with time-dependent amplitudes, frequencies, and waveforms. The multiresolution expansion coefficients $\{a_{n,k}\}$, $\{b_{n,k}\}$, and the shape function series $\{s_{cn,k}(t)\}$ and $\{s_{sn,k}(t)\}$ provide innovative features for adaptive time series analysis. The MMD aims at identifying these MIMF's (including their multiresolution expansion coefficients and shape functions series) from their superposition. This paper proposes a fast algorithm for solving the MMD problem based on recursive diffeomorphism-based spectral analysis (RDSA). RDSA admits highly efficient numerical implementation via the nonuniform fast Fourier transform (NUFFT); its convergence and accuracy can be guaranteed theoretically. Numerical examples from synthetic data and natural phenomena are given to demonstrate the efficiency of the proposed method.