Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 707-736, January 2020.
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 multiresolution intrinsic mode function (MIMF) of the form $ f_k(t)=\sum\nolimits_{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\nolimits_{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)) $ 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 MIMFs (including their multiresolution expansion coefficients and shape functions series) from their superposition. However, due to the lack of efficient algorithms to solve the MMD problem, the application of MMD for large-scale data science is prohibitive, especially for real-time data analysis. 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; 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.

中文翻译：

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一种多分辨率模式分解的快速算法

多尺度建模与仿真，第18卷，第2期，第707-736页，2020年1月。
多分辨率模式分解（MMD）是一种自适应工具，可以分析时间序列$ f（t）= \ sum_ {k = 1} ^ K f_k（t）$，其中$ f_k（t）$是多分辨率固有模式函数（ MIMF）的形式为$ f_k（t）= \ sum \ nolimits_ {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 \ nolimits_ {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））$，其幅度取决于时间，频率和波形。多分辨率扩展系数$ \ {a_ {n，k} \} $，$ \ {b_ {n_k} \} $和形状函数系列$ \ {s_ {cn，k}（t）\} $和$ \ {s_ {sn，k}（t）\} $提供了用于自适应时间序列分析的创新功能。MMD旨在通过叠加来识别这些MIMF（包括其多分辨率扩展系数和形状函数系列）。然而，由于缺乏有效的算法来解决MMD问题，MMD在大规模数据科学中的应用受到了限制，尤其是对于实时数据分析。本文提出了一种基于递归微分态的频谱分析（RDSA）的MMD问题快速求解算法。RDSA允许通过非均匀快速傅立叶变换进行高效的数值实现；从理论上可以保证其收敛性和准确性。给出了来自合成数据和自然现象的数值例子，以证明该方法的有效性。RDSA允许通过非均匀快速傅立叶变换进行高效的数值实现；从理论上可以保证其收敛性和准确性。给出了来自合成数据和自然现象的数值例子，以证明该方法的有效性。RDSA允许通过非均匀快速傅立叶变换进行高效的数值实现；从理论上可以保证其收敛性和准确性。给出了来自合成数据和自然现象的数值例子，以证明该方法的有效性。