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Nonlinear Mode Decomposition: a new noise-robust, adaptive decomposition method
Dmytro Iatsenko, Peter V. E. McClintock, Aneta Stefanovska
TL;DR
Signal decomposition must separate individual oscillations from one another and from noise, while existing methods have recognized limitations. NMD combines time-frequency analysis, surrogate data tests, and harmonic identification, with most settings adapted automatically to the signal. NMD recovers full underlying oscillations of any waveform and returns physically meaningful modes, stopping when the residual is noise.
Problem
Signal decomposition must separate individual oscillations from one another and from noise, while existing methods have recognized limitations.
Source: §1 · pp. 1–2Method
NMD combines time-frequency analysis, surrogate data tests, and harmonic identification, with most settings adapted automatically to the signal.
Source: §1 · p. 3, §3.3 · p. 8Results
NMD recovers full underlying oscillations of any waveform and returns physically meaningful modes, stopping when the residual is noise.
Source: §8 · p. 36Takeaways & Limitations
The authors conclude NMD can be applied to many multicomponent signals, including those in life sciences, geophysics, astrophysics, and econometrics.
Source: §8 · p. 36Failure to pass the surrogate test does not rule out physiologically meaningful activity: oscillations may be small and masked by noise, or rapid amplitude/frequency changes may be poorly captured at the chosen resolution.
Source: §7.1 · p. 32
Abstract
from arXiv · showhide
We introduce a new adaptive decomposition tool, which we refer to as Nonlinear Mode Decomposition (NMD). It decomposes a given signal into a set of physically meaningful oscillations for any waveform, simultaneously removing the noise. NMD is based on the powerful combination of time-frequency analysis techniques - which together with the adaptive choice of their parameters make it extremely noise-robust - and surrogate data tests, used to identify interdependent oscillations and to distinguish deterministic from random activity. We illustrate the application of NMD to both simulated and real signals, and demonstrate its qualitative and quantitative superiority over the other existing approaches, such as (ensemble) empirical mode decomposition, Karhunen-Loeve expansion and independent component analysis. We point out that NMD is likely to be applicable and useful in many different areas of research, such as geophysics, finance, and the life sciences. The necessary MATLAB codes for running NMD are freely available at http://www.physics.lancs.ac.uk/research/nbmphysics/diats/nmd/.
1. Introduction
Reliable signal decomposition must separate meaningful oscillations from one another and from noise, despite their complex waveforms and changing amplitudes and frequencies. The introduction presents NMD as an adaptive approach intended to address shortcomings in existing methods.
1. Introduction
- Recovering individual oscillations from noisy measured signals is necessary to study their properties, which can inform applications such as earthquake prediction and health assessment.The introduction frames decomposition as separating modes from each other and from background noise.Source: §1 · p. 1
- Existing methods may use sensitive, non-adaptive parameters, lack noise robustness, split complex waveforms, or return modes that are not physically meaningful.The authors state that current methods have at least two of these flaws.Source: §1 · p. 2
- EMD has noise sensitivity and can split some complex waveforms, while EEMD adds white Gaussian noise and averages repeated EMD results but remains non-adaptive and is not considered noise-robust.EEMD’s added-noise standard deviation is a non-adaptive parameter, though its choice is reported to have little influence on results.Source: §1 · p. 2
- Single-channel PCA/Karhunen–Loève expansion and ICA depend on how time-shifted blocks form a multivariate signal and share the physically-meaningful-mode shortcoming.The introduction also notes that these methods contain another of the listed flaws, without specifying it in this passage.Source: §1 · p. 2
- Time-frequency decompositions identify curves of amplitude peaks and recover their oscillations, but non-sinusoidal modes can appear as multiple curves.The introduction presents high noise-robustness as an advantage of these approaches.Source: §1 · p. 2
- NMD combines time-frequency analysis, surrogate data tests, and harmonic identification; the authors report adaptive settings, noise robustness, and recovery of physically meaningful oscillations across waveforms.The paper evaluates NMD on simulated and real data.Source: §1 · p. 3
2. Background and terminology
The background defines oscillatory components and nonlinear modes, then introduces time-frequency representations for tracking signal content. It distinguishes the WFT and WT by their frequency and time resolution and notes a tunable resolution parameter.
2.1. Main notation
- An AM/FM component is written as x(t) = A(t) cos φ(t), with positive instantaneous amplitude and frequency ν(t) = φ′(t); the paper restricts attention to components satisfying an analytic approximation.Without this restriction, amplitude and phase are not uniquely defined and cannot in principle be reliably recovered by time-frequency analysis.Source: §2.1 · p. 3
- A nonlinear mode groups components from the same activity, including harmonics that can make one complex oscillation appear as several components.The mode’s phase and frequency are those of its fundamental, and its waveform is represented by a periodic wave-shape function.Source: §2.1 · pp. 3–4
- The signal is treated as a sum of nonlinear modes plus noise; the goal is to recover the modes and their amplitude, phase, frequency, and harmonic parameters.Source: §2.1 · p. 4
2.2. Time-frequency representations
- The WFT and WT provide time-frequency representations from which components can be extracted and reconstructed; their resolution parameter f0 sets a time–frequency tradeoff.The paper uses a Gaussian window for the WFT and a lognormal wavelet for the WT.Source: §2.2 · pp. 4–5
- The WFT has linear frequency resolution and constant time resolution, whereas the WT has logarithmic frequency resolution and increasing time resolution at higher frequencies.The representations are evaluated on discrete frequency values, and predictive padding is used to reduce boundary distortions.Source: §2.2 · p. 5
3. Nonlinear Mode Decomposition: basic idea and ingredients
This section outlines how NMD extracts oscillatory modes by identifying fundamentals and their true harmonics, reconstructing modes, and iterating on the residual. It describes surrogate-based tests that assess harmonic interdependence and whether components or residuals remain distinguishable from noise, with interpretation depending on time-frequency representation reliability and modulation complexity.
3. Nonlinear Mode Decomposition: basic idea and ingredients
- NMD extracts each mode’s fundamental, finds candidate harmonics, tests which are true, sums them into a mode, subtracts it, and repeats on the residual.The procedure can use either the WFT or WT throughout, with a later section combining them for greater adaptivity.Source: §3 · p. 5
3.1. Component extraction and reconstruction
- With adequate time-frequency resolution, components form ridge curves that track their modulation, while nonsinusoidal oscillations appear as multiple components representing a fundamental and harmonics.Components can be reconstructed using ridge estimates, which are more robust to noise and interference, or direct estimates, which are exact when the time-frequency support contains all component power without side influences.Source: §3.1 · pp. 5–8
3.2. Harmonics: extracting the candidates
- Assuming the extracted component is fundamental, candidate harmonic ridges are found by following nearby TFR amplitude peaks from integer multiples of its instantaneous frequency.The candidate harmonic parameters are then reconstructed using the component reconstruction methods.Source: §3.2 · p. 8
3.3. Harmonics: identifying the true ones
- True harmonics are retained only when surrogate testing finds dependence on the fundamental, reducing the risk of accepting nearby independent components.The default dependence measure weights amplitude and phase consistency equally, excludes frequency consistency, and requires at least 95% significance; candidates must also pass a consistency threshold.Source: §3.3 · pp. 8–10
3.4. Harmonics: practical issues
- To improve harmonic extraction, NMD subtracts each confirmed harmonic before testing the next and checks for lower subharmonics if the initial component may not be fundamental.The search for higher harmonics defaults to stopping after three consecutive false candidates; the reverse search likewise stops after three consecutive false subharmonics.Source: §3.4 · p. 11
3.5. Stopping criterion
- NMD stops when surrogate testing indicates that the residual is linear noise rather than meaningful oscillatory activity.FT surrogates randomize Fourier phases while preserving the power spectrum; NMD compares the order in extracted amplitude or frequency modulation, quantified with spectral entropy, between the signal and surrogates.Source: §3.5 · pp. 11–12
- D(αA, αν) is preferred to alternatives such as correlation dimension [38, 39, 40] because it reflects component representation quality in the TFR and the deterministic proportion of extracted amplitude and frequency dynamics.A residual that fails the test may be noise or a component that the TFR cannot reliably resolve from noise.Source: §3.5 · pp. 12–13
- Test power increases when amplitude and frequency modulation have lower spectral entropy and when their spectra are more concentrated, reducing the noise power within the component’s frequency band.Even a pure tone can yield a powerful test because of numerical issues, except when it has an integer number of cycles over the signal length.Source: §3.5 · p. 13
- Because D(1, 0) and D(0, 1) can be more powerful for different modulation strengths, NMD tests both alongside D(1, 1) and uses the maximum significance.D(1, 0) is preferable for relatively small amplitude variability and considerable frequency variability; D(0, 1) is better otherwise.Source: §3.5 · p. 13
- With 40 Fourier-transform surrogates and a 95% significance level, NMD rejects the noise null when at least 38 surrogate statistics exceed the original; rejection for any of three statistics means decomposition continues.The three tests use D(1, 1), D(0, 1), and D(1, 0).Source: §3.5 · p. 13
- At each iteration, detrending with a third-order polynomial helps prevent trends from distorting the surrogate test; zero padding is recommended for matching boundary distortions, while other cases use predictive padding.For the surrogate test, the original statistic is recalculated using component parameters from the zero-padded signal’s TFR.Source: §3.5 · p. 13
- Fourier-transform surrogates are simpler and more natural than IAAFT surrogates for WFT and WT, which use the Fourier transform but not the value distribution; the test also works for rescaled, non-Gaussian, and non-stationary linear noise.Test outcome appears to depend primarily on component representation quality in the TFR, affected by noise power and distribution near the component’s instantaneous frequency.Source: §3.5 · pp. 13–14
4. Improvements
This section describes improvements to harmonic analysis and reconstruction, including restricting time-frequency calculations to relevant component-frequency ranges and using those ranges in surrogate tests for stopping. Some changes increase computational cost, while the stated complexity remains O(N log N).
4.1. Adaptive representation of the harmonics
- Adaptive per-harmonic resolution addresses how WFT and WT tradeoffs can misrepresent harmonics differently across amplitude- and frequency-modulated modes.For amplitude modulation, WFT represents all harmonics well, while WT can merge higher harmonics; for frequency modulation, WT better captures higher-harmonic variation but may leave some harmonics inseparable. The method searches for each harmonic’s resolution by maximizing its consistency with the first harmonic while requiring it to pass the surrogate test.Source: §4.1 · pp. 15–17
4.2. Improved reconstruction of nonlinear modes
- Refining harmonic parameters through weighted averaging uses their theoretical relationships to reduce cumulative noise and improve nonlinear-mode reconstruction.The refinement assumes a constant waveform and weights harmonics by amplitude; the authors note that reconstruction errors are not generally proportional to amplitude, but are too difficult to estimate for use in the method.Source: §4.2 · pp. 19–20
4.3. Speed improvement
- Computing the time-frequency representation only over the expected harmonic range avoids the cost of evaluating it across the full frequency range.The range is estimated from the first harmonic’s support; using a percentile-based bound helps avoid overly wide ranges caused by noise.Source: §4.3 · p. 20
- The frequency range may remain unsqueezed when noise prevents reliable squeezing, or be stretched when the initial range is too narrow.The estimates in (4.11) and (4.13) define the range for calculating the WFT or WT.Source: §4.3 · p. 21
- Surrogate-test time-frequency representations can be limited to the original component’s estimated frequency range, reducing computational cost.The range is estimated from (4.11) with h = 1.Source: §4.3 · p. 21
5. Parameters and their choice
NMD settings combine adaptive time-frequency and reconstruction choices with established defaults, but the resolution parameter f0 remains signal-dependent and cannot yet be selected universally.
5.1. Resolution parameter f0
- f0 trades temporal for frequency resolution, and its optimal signal-specific value remains the key setting NMD cannot adaptively choose.Increasing f0 improves frequency resolution but reduces time resolution; for the WT, the authors usually use f0 = 1.Source: §5.1 · pp. 21–22
5.2. TFR type: WFT or WT?
- After preliminary WT extraction, NMD selects WT when parameter variability rises with frequency and otherwise re-extracts the component with WFT.The preliminary WT is used because it is usually faster and has a more universal resolution-parameter choice.Source: §5.2 · pp. 22–23
5.3. Reconstruction method: direct or ridge?
- For harmonics, NMD selects the reconstruction method with the highest consistency while adapting the resolution parameter.Source: §5.3 · p. 24
- Adaptive reconstruction uses ridge estimates when noise is high and modulation is low, and direct reconstruction for more accurate parameter variations in the opposite case.Ridge reconstruction is more noise-robust, while direct reconstruction follows time variation more accurately at low noise.Source: §5.3 · pp. 23–24
5.4. Other parameters
- Other settings use established or empirical defaults; tighter numerical-accuracy settings trade greater computation for only slight accuracy gains in most cases.The surrogate-test significance levels use the standard 95% convention.Source: §5.4 · pp. 24–25
6. Simulation examples
In simulated signals with known components, NMD adapts its time-frequency representation and harmonic tests to recover modes and noise. Across two examples, it reconstructs components more faithfully than EMD/EEMD, including under severe noise, while the second example exposes a limitation in choosing the initial resolution parameter.
6.1. Example 1
- With constant resolution, the WFT separates harmonics near 3–4 Hz but lacks time resolution at 7 Hz, while the WT resolves 7 Hz but merges the lower harmonics.Adaptive harmonic representations address this mismatch.Source: §6.1 · p. 25
- NMD identified the genuine harmonics h = 3, 4, 7 and rejected the others, adapting the resolution parameter for each harmonic.It stopped after three consecutive false higher harmonics, h = 8, 9, 10.Source: §6.1 · p. 26
- In Example 1, NMD extracted one physically meaningful mode and correctly identified the residual as Brownian noise, with the reconstruction and residual closely matching their true counterparts.The next component failed the surrogate test against noise, so extraction stopped.Source: §6.1 · p. 26, Figure 7 · p. 27
- Unlike NMD, EMD/EEMD produced 13 components: only the first harmonic was recovered more-or-less reliably, while other components mixed or corrupted harmonics.The 3rd and 4th harmonics were mixed, and the 7th was badly corrupted.Source: §6.1 · p. 27, Figure 8 · p. 27
6.2. Example 2
- In the severe-noise example, NMD joined all true harmonics into two modes, with extracted amplitude ratios and phase shifts close to their actual values, even for a buried harmonic.Table 1 reports the harmonic tests and extracted-versus-true amplitude ratios and phase shifts; ridge reconstruction was selected for all harmonics.Source: §6.2 · pp. 28–29, Table 1 · p. 29
- In this strong-noise case, NMD reconstructed both NMs with great accuracy and recovered residual noise almost exactly; by comparison, (E)EMD produced 13 components, none reliably representing even a single harmonic.Figure 10 compares NMD’s reconstructed NMs and residual with their true counterparts.Source: §6.2 · p. 29, Figure 10 · p. 30
7. Real life examples
The real-data examples use NMD to separate physiological blood-flow modes and remove cardiac artifacts from EEG. They also illustrate extraction through surrogate testing, adaptive time-frequency settings, and reference-signal phase dynamics.
7.1. Decomposing human blood flow signals
- Across two blood-flow examples, NMD extracted cardiac oscillations in both, sympathetic/myogenic activity in one, and respiratory activity in the other; ranges IV–VI failed noise tests.The extracted modes had complex waveforms, and NMD also returned their amplitudes, phases, and frequencies.Source: Figure 11 · p. 31, §7.1 · p. 31, Figure 12 · p. 32
- For blood flow, NMD processes each physiological frequency range by testing its dominant component against noise, extracting harmonics, reconstructing the mode, and subtracting it.The procedure repeats within a range until the next component fails the noise test.Source: §7.1 · pp. 30–31
- Blood-flow oscillations vary among subjects, and NMD may support their classification and study of relationships between cardiac waveforms and health conditions.The authors found no clear chronotaxicity evidence in the tested modes, though the test method's assumptions leave that possibility open.Source: §7.1 · pp. 31–33
- A failed surrogate test does not establish that physiological activity is absent: weak components or unsuitable resolution can hide it, while rapidly changing modes may need a difficult-to-select time-varying f0(t).The authors note that small respiratory oscillations can be masked by noise.Source: §7.1 · p. 32
7.2. Removing cardiac artifacts from a human EEG signal
- Because EEG cardiac artifacts can differ in waveform from ECG activity, NMD uses ECG phase and frequency as a reference and requires shared phase dynamics rather than matching waveform shapes.The authors contrast this with ICA's linear-mixing assumption, which fails in their example.Source: §7.2 · pp. 33, 35
- With ECG-derived timing, NMD extracted EEG cardiac artifacts whose waveform resembled blood-flow pulses rather than ECG, supporting a ballistocardiogram origin.The cardiac ridge became more visible after adapting the EEG WFT resolution parameter.Source: Figure 13 · p. 34, §7.2 · pp. 34–35
- Reference-based NMD filtering could extract cardiac and respiratory components from blood flow; the authors found no respiratory oscillation in the EEG and inferred little breathing effect on that measurement.They propose this reference-signal approach as potentially more accurate than straightforward blood-flow decomposition, especially for respiratory activity.Source: §7.2 · p. 35
8. Conclusions
NMD is presented as a noise-robust, adaptive method that recovers full oscillations and stops when only noise remains. The authors report successful applications to simulated and real data and argue that the method can apply across many multicomponent-signal fields.
8. Conclusions
- NMD combines time-frequency analysis, surrogate data tests, and harmonic identification, with its noise-test procedure reported as more robust than conventional tests based on correlation dimension or other nonlinearity measures.The component techniques are also presented as useful individually.Source: §8 · p. 35
- NMD recovers full underlying oscillations of any waveform, is extremely noise-robust and adaptive, and returns physically meaningful modes until the residual is noise.The authors report successful applications to simulated and real data and advantages over existing approaches.Source: §8 · p. 36
- The authors propose NMD for multicomponent signals across life sciences, geophysics, astrophysics, and econometrics, including cases where other methods fail.They suggest revisiting cases previously analyzed with (E)EMD.Source: §8 · p. 36
- MATLAB codes, instructions, and examples for NMD are freely available.The examples and instructions are provided in text and video formats.Source: §8 · p. 36