Higher Order Dynamic Mode Decomposition

• HODMD is a data-driven modal analysis method that uses time-delay embedding to capture multi-step correlations and extract continuous spectral information.
• It constructs augmented snapshot ensembles and applies singular value decomposition for low-rank approximations, enabling accurate extraction of eigenvalues, growth/decay rates, and modal amplitudes.
• The approach is effective in handling noisy, high-dimensional systems such as mechanical vibrations, circuit transients, and biomedical signals, offering enhanced resolution and robustness.

Higher Order Dynamic Mode Decomposition (HODMD) is a data-driven modal analysis and reduced-order modeling methodology that generalizes classical Dynamic Mode Decomposition (DMD) by introducing time-delay embedding to capture multi-step correlations and reveal additional dynamical modes. HODMD has proven effective for systems where the spectral complexity exceeds the instantaneous spatial resolution, such as high-dimensional mechanical vibrations, circuit transients, noisy environmental measurements, reacting flows, power system oscillations, and biomedical signals. The method systematically constructs augmented snapshot ensembles and exploits singular value decompositions (SVD) to compute low-rank approximations of the underlying high-order Koopman operator, allowing for the extraction of modes, continuous-valued frequencies, growth/decay rates, and modal amplitudes. Recent developments include noise-robust extensions, kernel-based spectrification, hierarchical clustering variants, and rigorous theoretical analysis for higher-order dynamical systems.

1. Mathematical Foundations and Augmented Snapshot Construction

HODMD generalizes the classical one-step Koopman ansatz $x_{k+1} \approx A x_k$ to a $d$-step linear recurrence: $$x_{k+d} = A_1 x_k + A_2 x_{k+1} + \cdots + A_d x_{k+d-1}, \qquad k=1,\dots,K-d$$ This recurrence can be rewritten in terms of delay-embedded (Hankel-augmented) snapshots: $$Z_k = [x_k ;\ x_{k+1} ;\ \ldots ;\ x_{k+d-1}] \in \mathbb{R}^{Md}$$ forming matrices: $$\mathcal{H} = [Z_1,\ Z_2,\ \ldots,\ Z_{K-d}], \qquad \mathcal{H}' = [Z_2,\ Z_3,\ \ldots,\ Z_{K-d+1}]$$ The augmented Koopman operator $A_*$ is a block-companion matrix in $\mathbb{R}^{Md \times Md}$ and encodes the $d$-step autoregressive dynamics (Tuor et al., 2023, Li et al., 10 Feb 2025, Liu, 5 Mar 2025).

The practical dimensionality of $d$ is governed by the number of desired modes $r$ and the number of measured channels $M$, requiring $Md \geq r$. Time-delay embedding artificially increases the effective row dimension, crucial for applications with limited observables (Liu et al., 5 Aug 2025).

2. Algorithmic Pipeline and Key Equations

The essential HODMD pipeline involves:

1. Delay Embedding: Construction of augmented snapshots.
2. Hankel Matrix Formation: Assemble $\mathcal{H}, \mathcal{H}'$ or domain-specific block Hankel analogs.
3. SVD Truncation: Economy-size decomposition of $\mathcal{H}$ (or its reduced variant after prior spatial truncation), retaining $r$ largest singular values. Truncation thresholds (e.g. energy or hard-thresholding) regularize the spectrum and mitigate noise.

   $\mathcal{H} \approx U_r \Sigma_r V_r^*$

4. Low-Rank Koopman Projection: Define projected operator

   $A_r = U_r^* \mathcal{H}' V_r \Sigma_r^{-1}$

5. Eigenvalue Problem: Solve $A_r W = W \Lambda$ to obtain eigenvalues $\lambda_j$ and eigenvectors $W$.
6. Dynamic Modes and Amplitudes: Modes in original space:

   $\Phi = \mathcal{H}' V_r \Sigma_r^{-1} W$

   Modal amplitudes via least-squares projection:

   $b = \Phi^{\dagger} Z_1$

   Extract continuous-time growth rates and frequencies:

   $\lambda_j = \exp[(\sigma_j + i \omega_j) \Delta t]$

   $$\sigma_j = \frac{\text{Re}(\ln \lambda_j)}{\Delta t}, \quad \omega_j = \frac{\text{Im}(\ln \lambda_j)}{\Delta t}$$

Reconstruction formula for arbitrary $t$: $$x(t) \approx \sum_{j=1}^r b_j \phi_j\, e^{(\sigma_j + i \omega_j)\, t}$$ Selection of significant modes uses amplitude or energy thresholds (Conti et al., 2023, Corrochano et al., 2022).

3. Spectral Kernelization: Kernel Density Spectrum

Raw HODMD yields a sparse spectrum of Dirac impulses at modal frequencies. To enable high-resolution spectral visualization and comparison with PSD approaches, kernel density smoothing is applied: $KDS(F) = \sum_{k=1}^r K(F - F_k; A_k, T_k, h)$ Common kernels:

• Gaussian:

  $$K_G(F) = \sum A_k^p\, \exp\left[-\frac{(F - F_k)^2}{2 h^2}\right]$$

  with $h$ controlling bandwidth and $p$ for amplitude/power-weighting.

• Lorentzian (for damped oscillators):

  $$K_L(F) = \sum \frac{A_k T_k}{1 + 4 \pi^2 T_k^2 (F - F_k)^2}$$

Choice of $h$ tunes spectral smoothing and resolution (Tuor et al., 2023).

4. Practical Parameter Tuning and Computational Complexity

Critical parameters:

• Order $d$: Should exceed twice the true number of complex-conjugate modes. Empirical guideline: $d \gtrsim 2 r_\text{true} / M$; ensure sufficient data such that $K > 2d$ (Tuor et al., 2023, Li et al., 10 Feb 2025).
• SVD thresholds: For spatial reduction, use energy retention (e.g. $>$99%); for temporal truncation, employ hard-thresholding or relative criteria (e.g. $10^{-6}$).
• Kernel bandwidth $h$: Chosen to match minimal desired modal frequency separation; too small $h$ retrieves raw impulses, too large $h$ merges peaks.
• Computational complexity: Dominated by economy-size SVD $(O[(Md)^2(K-d)])$ and subsequent eigen-decomposition $O(r^3)$. For large-scale simulations, random or parallel SVD is advantageous (Lazpita et al., 29 Jul 2025).

5. Comparative Advantages and Limitations

Method	Frequency Resolution	Damping/Decay Extraction	Noise Robustness
FFT/STFT/PSD	Fixed by window length ($\Delta f = 1/T_\text{window}$); binning	No	Poor (leakage, spread)
Classical DMD	Limited by snapshot rank; discrete sampling	Exponential, single-step	Moderate
HODMD	Continuous spectrum, sub-$\Delta f$; arbitrary time delays	Explicit via $\sigma_j$	Superior (delay embedding, SVD)

HODMD surpasses FFT/STFT/PSD by avoiding window/bandwidth constraints, resolving closely spaced and decaying modes, and reducing spectral leakage. Compared to standard DMD, HODMD can identify more dynamical modes when spatial dimension is limiting and is particularly robust in noisy, high-dimensional, and weakly nonlinear regimes (Tuor et al., 2023, Liu et al., 5 Aug 2025, Li et al., 10 Feb 2025, Conti et al., 2023).

Limitations include sensitivity to over-large $d$ (amplifying noise, higher computational cost), the need to calibrate spectral and amplitude thresholds, and assumptions of approximate linearity over the chosen time window (Liu et al., 5 Aug 2025).

6. Representative Application Domains

• Mechanical Vibrations (Tuor et al., 2023): Modal analysis of complex structures, damped oscillators, railway axle vibration extraction; recovers modal frequencies to sub-Hz accuracy and identifies weak modes invisible to PSD.
• Circuit Model Order Reduction (Liu et al., 5 Aug 2025): HODMD enables equation-free reduction and fast prediction of large-scale circuits, breaking rank limitations due to few observed ports.
• Power Systems (Li et al., 10 Feb 2025): Extracts spatio-temporal modes for frequency oscillations in networks with variable inertia; separates local and global dynamic modes.
• Environmental and Fluid Dynamics (Conti et al., 2023, Lazpita et al., 29 Jul 2025, Corrochano et al., 2022): Dimensionality reduction and physical mode identification for noisy sensor data, cardiac flow, and combustion databases using multi-dimensional and hierarchical extensions.
• Biomedical Imaging (Groun et al., 2022): Modal decomposition and frequency-based pathology classification in echocardiography; robust detection of physiological rhythms and disease biomarker patterns.

7. Extensions, Kernel Theory, and Hierarchical Variants

Theoretical generalizations employ higher-order Liouville operators and signal-valued reproducing kernel Hilbert spaces (RKHS), enabling rigorous analysis for $d$th-order autonomous systems without state augmentation or numerical differentiation (Rosenfeld et al., 2021). Hierarchical HODMD (h-HODMD) iteratively clusters variables sharing dynamic features, improving reconstruction error and yielding interpretable clusters for reduced-order kinetic modeling and control in reacting flows (Corrochano et al., 2023).

• "High Order Dynamic Mode Decomposition for Mechanical Vibrations and Modal Analysis" (Tuor et al., 2023)
• "Model Order Reduction for Large-scale Circuits Using Higher Order Dynamic Mode Decomposition" (Liu et al., 5 Aug 2025)
• "Learning the Frequency Dynamics of the Power System Using Higher-order Dynamic Mode Decomposition" (Li et al., 10 Feb 2025)
• "Energy Modelling and Forecasting for an Underground Agricultural Farm using a Higher Order Dynamic Mode Decomposition Approach" (Conti et al., 2023)
• "Efficient Reduced Order Modeling Based on HODMD to Predict Intraventricular Flow Dynamics" (Lazpita et al., 29 Jul 2025)
• "Higher order dynamic mode decomposition to model reacting flows" (Corrochano et al., 2022)
• "Higher Order Dynamic Mode Decomposition: from Fluid Dynamics to Heart Disease Analysis" (Groun et al., 2022)
• "A parallel-in-time method based on the Parareal algorithm and High-Order Dynamic Mode Decomposition with applications to fluid simulations" (Liu, 5 Mar 2025)
• "Time Series Source Separation using Dynamic Mode Decomposition" (Prasadan et al., 2019)
• "Hierarchical Higher-Order Dynamic Mode Decomposition for Clustering and Feature Selection" (Corrochano et al., 2023)
• "Theoretical Foundations for the Dynamic Mode Decomposition of High Order Dynamical Systems" (Rosenfeld et al., 2021)