A Fast Algorithm for Multiresolution Mode Decomposition
This work addresses the computational bottleneck of MMD, enabling efficient adaptive time series analysis for applications in signal processing and data analysis.
The paper proposes a fast algorithm for multiresolution mode decomposition (MMD) to decompose a time series into multiresolution intrinsic mode functions (MIMFs) with time-dependent amplitudes, frequencies, and waveforms. The algorithm, based on recursive diffeomorphism-based spectral analysis (RDSA) and nonuniform fast Fourier transform (NUFFT), achieves high efficiency and theoretical guarantees on convergence and accuracy.
\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(2πnϕ_k(t))s_{cn,k}(2πN_kϕ_k(t))\\&&+\sum_{n=-N/2}^{N/2-1}b_{n,k} \sin(2πnϕ_k(t))s_{sn,k}(2πN_kϕ_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.