Dynamic Multi-resolution Spectral Decomposition

• Dynamic Multi-resolution Spectral Decomposition (DMSD) is a framework that decomposes complex signals into interpretable spectral components across multiple temporal scales.
• It employs recursive strategies—such as RDSA and mrDMD—to separate slow background phenomena from fast transients, ensuring efficient spectral recovery.
• DMSD’s adaptive, data-driven approach enhances forecasting and signal analysis in fields like bio-signal processing, climate modeling, and beyond.

Dynamic Multi-resolution Spectral Decomposition (DMSD) is an adaptive, hierarchical framework for decomposing complex signals into interpretable spectral components on multiple temporal scales. DMSD encompasses several prominent algorithmic families—including Multiresolution Mode Decomposition (MMD) with Recursive Diffeomorphism-based Spectral Analysis (RDSA), Multi-resolution Dynamic Mode Decomposition (mrDMD), and adaptive multi-scale sparse spectral estimators—united by recursive decomposition strategies, local spectral identification, and the separation of background (slow) phenomena from foreground (fast or intermittent) dynamics. DMSD provides closed-form, data-driven representations that are highly structured, interpretable, and well-suited for analysis and forecasting of nonlinear, nonstationary, or multiscale systems.

1. Mathematical Foundations and Definitions

DMSD frameworks are built upon the principle of recursively decomposing realizations of dynamical systems or time-series data by explicitly modeling their local spectra at multiple resolutions.

1.1 Multiresolution Intrinsic Mode Function (MIMF)

A central object in DMSD as introduced by Tang and Yang is the MIMF, defined on normalized time $t\in[0,1]$ and parametrized by a phase function $\phi_k(t)$ and a principal frequency $N_k$:

$$f_k(t) = \sum_{n=-N/2}^{N/2-1} a_{n,k}\cos\big(2\pi n \phi_k(t)\big) s_{c n,k}\big(2\pi N_k \phi_k(t)\big) + \sum_{n=-N/2}^{N/2-1} b_{n,k}\sin\big(2\pi n \phi_k(t)\big) s_{s n,k}\big(2\pi N_k \phi_k(t)\big)$$

where $a_{n,k}, b_{n,k}$ are multiresolution expansion coefficients and $s_{c n,k}, s_{s n,k}$ are time-varying shape functions in the Wiener algebra class $\mathcal{M}_M$. The overall signal is modeled as a superposition of $K$ such components with known or estimated phases:

$f(t) = \sum_{k=1}^K f_k(t)$

1.2 Multi-resolution DMD Expansion

In the DMD-based lineage, DMSD recursively applies DMD on shorter time bins. At level $\ell$, within each bin $\phi_k(t)$0,

$\phi_k(t)$1

Here $\phi_k(t)$2 are indicator functions for bin boundaries, $\phi_k(t)$3 are DMD modes, $\phi_k(t)$4 their continuous-time frequencies, and $\phi_k(t)$5 modal amplitudes selected for slow or transient content.

2. Algorithmic Architectures

DMSD algorithms instantiate these principles through specific recursive or hierarchical schemes.

2.1 Recursive Diffeomorphism-based Spectral Analysis (RDSA)

RDSA, developed for MIMF models, operates as follows (Tang et al., 2017):

• For a single MIMF with known phase, modulate and warp the signal, interpolate onto a uniform grid, apply the nonuniform FFT (NUFFT), and isolate spectral bands through aliasing/downsampling.
• For multiple components, iteratively estimate each $\phi_k(t)$6 by applying the above procedure to the residual, updating and normalizing shape functions and coefficients.
• Convergence is linear under well-differentiated phases and sufficient separation.

2.2 Multi-resolution Dynamic Mode Decomposition (mrDMD)

mrDMD (Kutz et al., 2015, Manohar et al., 2017, Kong et al., 2024, Climaco et al., 2021, Dylewsky et al., 2019) recursively partitions the time domain, performing DMD (or delay-embedded DMD for univariate series) in each bin. At each level, slow modes (low-frequency content) are retained; the fast residual is further decomposed in finer bins. The decomposition is:

$\phi_k(t)$7

Algorithmic steps for both RDSA and mrDMD are formalized in the cited works as explicit pseudocode.

2.3 Adaptive Sparse and Frequency-Domain Extensions

Recent variants incorporate adaptive frequency grids, smooth sparsity-promotion (e.g., $\phi_k(t)$8 relaxation), and multi-stage refinement (Han et al., 2024). These allow for:

1. Frequency grid initialization and candidate selection.
2. Local grid refinement and majorization-minimization for sparse recovery.
3. Off-grid nonlinear optimization and adaptive degrees of freedom control.
4. Merge and cull insignificant frequency atoms based on model-driven and statistical thresholds.

These methods rigorously guarantee exact recovery under minimum-separation and noiseless conditions.

3. Hierarchical and Multiscale Structure

All DMSD approaches implement a hierarchical process, separating content by temporal scale. At each successive level:

• The time axis is split into $\phi_k(t)$9 bins of decreasing width, $N_k$0.
• Spectral resolution increases with level, allowing for localization of fast transients and harmonics at fine scales.
• Bins are analyzed independently, and local modes are aggregated for global reconstruction.
• The structure supports decomposition of nonstationary and intermittent signals, with explicit control over frequency content by resolution level.

4. Computational Complexity and Parameter Selection

The complexity and parameter regimes depend on the core algorithm:

Method	Per-Level Cost	Key Parameters
RDSA	$N_k$1	Bands $N_k$2, $N_k$3, $N_k$4, $N_k$5
mrDMD	$N_k$6	Levels $N_k$7, modes $N_k$8, thresholds $N_k$9
DMRA adaptive	$$f_k(t) = \sum_{n=-N/2}^{N/2-1} a_{n,k}\cos\big(2\pi n \phi_k(t)\big) s_{c n,k}\big(2\pi N_k \phi_k(t)\big) + \sum_{n=-N/2}^{N/2-1} b_{n,k}\sin\big(2\pi n \phi_k(t)\big) s_{s n,k}\big(2\pi N_k \phi_k(t)\big)$$0	Grid width, MM convergence, $$f_k(t) = \sum_{n=-N/2}^{N/2-1} a_{n,k}\cos\big(2\pi n \phi_k(t)\big) s_{c n,k}\big(2\pi N_k \phi_k(t)\big) + \sum_{n=-N/2}^{N/2-1} b_{n,k}\sin\big(2\pi n \phi_k(t)\big) s_{s n,k}\big(2\pi N_k \phi_k(t)\big)$$1, thresholding

Parameter selection is theory-guided: bin size must allow capture of frequency bands of interest, the number of levels $$f_k(t) = \sum_{n=-N/2}^{N/2-1} a_{n,k}\cos\big(2\pi n \phi_k(t)\big) s_{c n,k}\big(2\pi N_k \phi_k(t)\big) + \sum_{n=-N/2}^{N/2-1} b_{n,k}\sin\big(2\pi n \phi_k(t)\big) s_{s n,k}\big(2\pi N_k \phi_k(t)\big)$$2 is limited by minimum bin occupancy, and thresholds are set by model dynamics or desired spectral separation (Kong et al., 2024, Manohar et al., 2017).

5. Applications and Empirical Results

DMSD and its variants have demonstrated efficacy across a range of domains:

• Bio-signal analysis: ECG and PPG decompositions reveal time-varying morphological components, with RDSA achieving accurate recovery of physiological mode structure; residuals approach white noise at higher resolution (Tang et al., 2017).
• Climate and oceanography: mrDMD isolates multiyear, annual, and subannual modes, robustly identifying ENSO (El Niño/La Niña) events using only a handful of optimized samples (Manohar et al., 2017).
• Power electronics: MR-DMD detects low-frequency oscillatory and harmonic instabilities in converter data, even amidst missing or transient-corrupted observations (Kong et al., 2024).
• Vibration and health monitoring: Time-delay mrDMD facilitates separation of damage-sensitive vibration features in wind turbine gearbox sensor data, outperforming classical Fourier and EMD-based techniques (Climaco et al., 2021).
• Sparse spectral estimation: DMRA-based DMSD achieves super-resolution of dense, closely-spaced spectral lines, with strict recovery guarantees and computational efficiency compared to SDP or greedy algorithms (Han et al., 2024).
• Forecasting and data assimilation: DMSD models yield closed-form, cluster-wise evolution equations, providing efficient trajectory simulation for ensemble Kalman filters or similar applications (Dylewsky et al., 2019).

6. Theoretical Guarantees and Limitations

Convergence theorems specify that, under suitable separation conditions for component phases or frequencies, DMSD algorithms produce error bounds that decay linearly in iterative schemes, with final error contingent on partition bias and variance, modal interference, and numerical approximations (Tang et al., 2017). For adaptive sparse methods, uniqueness and relaxation equivalence explicitly tie recovery success to model-based minimum separation and statistical thresholds (Han et al., 2024).

However, practical constraints remain:

• Bin size, frequency selection, and resolution trade off against statistical noise and overfitting.
• Excessively deep hierarchies (large $$f_k(t) = \sum_{n=-N/2}^{N/2-1} a_{n,k}\cos\big(2\pi n \phi_k(t)\big) s_{c n,k}\big(2\pi N_k \phi_k(t)\big) + \sum_{n=-N/2}^{N/2-1} b_{n,k}\sin\big(2\pi n \phi_k(t)\big) s_{s n,k}\big(2\pi N_k \phi_k(t)\big)$$3) may leave bins with insufficient samples for accurate local DMD, degrading performance.
• Fully automated tuning of spectral cutoffs, bin thresholds, or degrees of freedom remains an area of continuing research.

7. Extensions and Future Directions

Potential avenues for future development include incorporation of smooth (wavelet-like) window functions in place of hard bin indicators, streaming updates for real-time processing, robustification via bi-directional or regularized DMD, and integration with modern deep learning architectures (e.g., multi-scale wavelet transformers for dynamic graphs (Feng et al., 4 Mar 2026)). There is also interest in scaling DMSD to extremely high-dimensional or ultra-long time series and in automated, data-driven selection of all resolution and sparsity parameters.

DMSD thus serves as a unifying paradigm for interpretable, multi-scale spectral analysis in a wide array of data-driven scientific and engineering applications, combining rigorous system identification with computational practicalities and adaptability to complex data regimes (Tang et al., 2017, Kutz et al., 2015, Manohar et al., 2017, Kong et al., 2024, Han et al., 2024, Climaco et al., 2021).