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REVIEW 4 major objections 5 minor 12 references

A novel algorithm for the decomposition of non-stationary multidimensional and multivariate signals

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper introduces MdMvFIF, an algorithm that decomposes non-stationary signals varying over both space and time into separate spatial and temporal Intrinsic Mode Functions, allowing independent frequency filtering in each domain.

desk verdict A plausible spatiotemporal extension of fast iterative filtering with a real novelty—separate space and time filtering—but underspecified stopping criteria make the current version not fully reproducible. read the letter →

arxiv 2412.00553 v1 pith:K5JRKUZQ submitted 2024-11-30 math.NA cs.NAeess.SPstat.ML

Roberto Cavassi , Antonio Cicone , Enza Pellegrino , Haomin Zhou This is my paper · ORCID
classification math.NAcs.NAeess.SPstat.ML MSC 65T5094A12
keywords signaldecompositionintrinsicmodefunctionsiterativefilteringmultivariatesignalsmultidimensionalnon-stationaryfastempirical
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) algorithm, which decomposes signals that vary simultaneously across space and time into Intrinsic Mode Functions (IMFs), simple oscillatory components. Unlike earlier extensions of empirical mode decomposition that treat time as an extra spatial dimension, MdMvFIF keeps time separate, so it can filter different frequency ranges in space and in time independently. The algorithm alternates between extracting a spatial IMF and a temporal IMF, with filter sizes chosen adaptively from the signal's own extrema and rotation behaviour. The paper demonstrates the method on synthetic signals and on a year of global air-temperature measurements.

What carries the argument

The method's engine is the alternating inner-loop iteration $f_{k+1}=f_k - M(f_k)$, where the moving average $M$ is a convolution with a compactly supported, nonnegative, even Fokker-Planck filter. In space, the filter support $\hat{\Omega}$ is set to the minimum over all time steps of the adaptive support sizes used by FIF2; in time, the filter half-length $L$ is twice the average distance between successive extrema of the rotation angle $\tilde{\theta}(t)$ between consecutive multivariate signal vectors. All convolutions are evaluated in the frequency domain using the FFT, making each inner step cheap. The alternate extraction of space then time IMFs is the mechanism that keeps the two domains decoupled.

What would settle it

Run MdMvFIF on a synthetic signal with a known non-separable coupling, such as $f(v,t)=\sin(2\pi (v + ct))$ for a known speed $c$, and check whether the spatial IMF and temporal IMF together reproduce the signal without cross-contamination and without depending on the stopping threshold.

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Extended reading notes

Core claim

The central claim is that a non-stationary signal $f(v,t)$ defined on space $v \in \mathbb{R}^n$ and time $t$ can be decomposed into separate spatial and temporal IMFs by alternating iterations: first filter in space with a multidimensional Fokker-Planck window whose support is the minimum over time of the adaptive FIF2 filter supports, then filter in time with a window whose length is set from the rotation angles of the multivariate vectors $f(\cdot,t)$. The alternating procedure is continued until no oscillations remain in either domain. Because the space and time filters are computed independently, the method can remove high-frequency content in time while keeping low-frequency structure in space, or the reverse, which earlier multidimensional EMD-based algorithms cannot do.

Load-bearing premise

The alternating space-then-time iteration is assumed to converge to a stable decomposition for any input, but no convergence proof, error bound, or explicit stopping threshold is given.

Editorial extensions

If this is right

  • The algorithm can filter different frequency ranges in space and time independently, which earlier multidimensional EMD variants cannot do.
  • Signals with non-stationary behaviour in both domains can be decomposed without a priori assumptions about the number of components or a chosen basis.
  • The method applies to data of any spatial dimension, and its FFT-based convolutions keep each iteration efficient for large tensors.
  • The demonstrated application to Earth's air temperature shows the method producing interpretable spatial and temporal IMFs from a real 3D dataset.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not state it, but if the alternating iteration reliably converges, the same space-time separation could be applied to higher-dimensional data such as volumetric climate output or 4D microscopy, where filtering each axis independently would be valuable.
  • A natural testable extension, not explored in the paper, is to apply MdMvFIF to signals with known separable components and measure the cross-talk between the extracted spatial and temporal IMFs.
  • The rotation-angle rule for setting the temporal filter length could be transferred to other adaptive decomposition methods that currently rely on one-dimensional extrema statistics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces MdMvFIF, an extension of Fast Iterative Filtering for signals that vary simultaneously in space and time. The proposed algorithm alternately applies spatial FIF2-style filtering and temporal MvFIF-style filtering to extract spatial and temporal Intrinsic Mode Functions (IMFs), using FFT-based convolution and Fokker-Planck filters with adaptively computed support sizes. The method is demonstrated on two synthetic examples with known ground-truth components and on global daily air-temperature data. The authors claim that MdMvFIF is the first decomposition algorithm that treats time as a dimension separate from space, allowing independent frequency filtering in each domain.

Significance. If the algorithm is well-defined and its output is reliable, the paper addresses a real gap: prior multidimensional EMD-type methods treat time as an additional spatial dimension and therefore cannot independently filter frequency ranges in space and time. The conceptual separation of spatial and temporal IMFs is a useful contribution, and the synthetic examples suggest the method can recover known components. The paper also provides an online Matlab implementation, which is a positive step for reproducibility. However, the absence of a specified stopping criterion, the lack of convergence or error analysis, and the purely qualitative validation currently limit the significance of the claims.

major comments (4)
  1. [Section 2, Algorithm 1] The inner loops stop 'as soon as a prefixed threshold is reached' based on the relative change in f_k, but no threshold value, formula, or rule is given, and the supporting reference is incomplete ('[2, ?]'). As written, the output of Algorithm 1 is not well-defined: different thresholds will generally produce different IMF counts and waveforms. The authors must specify the stopping rule precisely, state the threshold used in all numerical experiments, and discuss sensitivity to that threshold.
  2. [Section 2, Algorithm 1] The outer while loop continues 'until there are no more oscillations left both in space and time', but the paper gives no operational definition of 'oscillations left' or of the detection procedure. Without a precise stopping condition for the outer loop, the total number of IMFs produced is ambiguous. Provide a concrete criterion (for example, a condition on the number of local extrema in each spatial slice and in each temporal series) and prove or at least verify experimentally that the outer loop terminates.
  3. [Section 2 and Section 1.1] No convergence or error analysis is provided for the alternating space-then-time inner loops. In particular, it is not shown that the iteration f_{k+1} = f_k - M(f_k) converges for the chosen filter supports (pΩ computed via Eq. (3) and L computed from θ̃(t) in Eq. (4)), nor that the result is independent of the space-then-time ordering. Because the central claim is that the algorithm extracts intrinsic components, the authors should either prove a convergence result under the stated assumptions on the Fokker-Planck filters or provide a numerical convergence study that demonstrates invariance of the extracted IMFs with respect to the stopping thresholds and the filtering order.
  4. [Section 3, Figures 5 and 10] The validation against ground truth is only visual; the paper reports no quantitative error metrics between the extracted and true spatial or temporal IMFs. This is load-bearing because the abstract and introduction claim that MdMvFIF is 'capable of extracting IMFs'. Add quantitative errors (for example, relative L2 errors, correlation coefficients, or instantaneous-frequency errors for each recovered IMF) and, where possible, compare with FIF2, Serial-EMD, or FA-MVEMD on the same synthetic examples.
minor comments (5)
  1. [Algorithm 1] The pseudocode uses inconsistent variable names: 'IMFss' and 'IMFst' should be unified into a single output structure, and the initialization 'IMFs = tu' appears to be a typographical artifact that should be replaced with the empty-set notation.
  2. [Section 2, Eq. (4)] The sentence describing f(v, t̃) as 'a vector in a R^{n+1} space' is misleading: the angle θ̃(t) is computed between spatial fields, i.e., vectors whose length is the number of spatial grid points, not vectors in R^{n+1}. Reword this passage to avoid confusion.
  3. [Section 2] The incomplete citation '[2, ?]' for the stopping criterion should be corrected to the appropriate reference (likely [2] or [3]).
  4. [Section 3] Figures 5 and 10 display differences between ground truth and recovered components, but no numerical scale or acceptable error level is discussed in the text; a sentence describing the magnitude of the residual errors would be helpful.
  5. [Section 1] The statement that IF is 'the only based on iteration, and therefore not requiring any a priori assumption on the signal' overstates the case, since the stopping criterion, the filter support rule, and the choice of filter all involve assumptions; please soften or qualify this claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algorithm's output is validated against known ground truth and no prediction reduces to a fitted input.

full rationale

The paper proposes MdMvFIF as an algorithmic extension of previously published FIF variants and validates it on synthetic signals with known ground-truth components and on real geophysical data. No parameter is fitted to the target output, and no equation defining the method is equivalent to the claimed result. The self-citations to prior FIF, MvFIF, and FIF2 work are used as building blocks for the algorithm, not as a substitute for validation or as a forbidden uniqueness argument. The stopping criteria are indeed underspecified and lack convergence guarantees, but that is a correctness and well-definedness concern, not circularity: the method can be wrong or threshold-dependent without being self-referential. The numerical comparisons to ground truth are external benchmarks, so the central claim of extracting IMFs is not supported solely by definition or by self-citation. The incomplete reference '[2, ?]' is a citation defect, not a circular step. No circular step can be exhibited under the required standard of quoting a specific reduction or fitted-input-renamed-as-prediction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted constants. The main free choices are algorithmic, namely the stopping threshold and the method for estimating the spatial filter support. The central assumption is that the alternating iteration converges meaningfully, which is not proved.

free parameters (2)
  • Stopping criterion threshold
    The inner loops in Algorithm 1 iterate until a prefixed threshold is reached, but the threshold value and definition are never specified; the decomposition output depends on it.
  • Spatial filter support computation rule
    Section 2 says Omega_t is estimated as in FIF2 based on extrema count or Fourier spectrum, but no exact formula is given; this determines the global support pOmega = min_t Omega_t.
assumptions (3)
  • domain assumption The signal is periodic at its boundaries or pre-extended to be periodic.
    Section 1.1 assumes periodicity to use FFT-based convolution; the paper states this limitation can be addressed by pre-extending, but the examples do not document this step.
  • ad hoc to paper The alternating space then time filtering converges to meaningful IMFs for arbitrary input.
    Algorithm 1's outer loop repeats until no oscillations remain, but no convergence or uniqueness result is proved; the decomposition depends on the stopping rule and the order of filtering.
  • domain assumption The rotation angle theta(t) in Equation (4) is well-defined for all t, i.e., all spatial frames have nonzero norm.
    If f(v, t) = 0 for some t, the normalized vector is undefined; no handling of zero frames is given.

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Cite this review

Pith. "Pith review of A novel algorithm for the decomposition of non-stationary multidimensional and multivariate signals." pith.science (2026). https://pith.science/paper/K5JRKUZQ

@misc{pith2026241200553,
  author       = {Pith},
  title        = {Pith review of: A novel algorithm for the decomposition of non-stationary multidimensional and multivariate signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5JRKUZQ}},
  note         = {Machine review of arXiv:2412.00553}
}
read the original abstract

The decomposition of a signal is a fundamental tool in many fields of research, including signal processing, geophysics, astrophysics, engineering, medicine, and many more. By breaking down complex signals into simpler oscillatory components we can enhance the understanding and processing of the data, unveiling hidden information contained in them. Traditional methods, such as Fourier analysis and wavelet transforms, which are effective in handling mono-dimensional stationary signals struggle with non-stationary data sets and they require, this is the case of the wavelet, the selection of predefined basis functions. In contrast, the Empirical Mode Decomposition (EMD) method and its variants, such as Iterative Filtering (IF), have emerged as effective nonlinear approaches, adapting to signals without any need for a priori assumptions. To accelerate these methods, the Fast Iterative Filtering (FIF) algorithm was developed, and further extensions, such as Multivariate FIF (MvFIF) and Multidimensional FIF (FIF2), have been proposed to handle higher-dimensional data. In this work, we introduce the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) technique, an innovative method that extends FIF to handle data that vary simultaneously in space and time. This new algorithm is capable of extracting Intrinsic Mode Functions (IMFs) from complex signals that vary in both space and time, overcoming limitations found in prior methods. The potentiality of the proposed method is demonstrated through applications to artificial and real-life signals, highlighting its versatility and effectiveness in decomposing multidimensional and multivariate nonstationary signals. The MdMvFIF method offers a powerful tool for advanced signal analysis across many scientific and engineering disciplines.

Figures

Figures reproduced from arXiv: 2412.00553 by the authors.

Figure 1
Figure 1. Example 1. Left panel: Signal at time t “ 0. Right panel: Time evolution for position v “ p150, 150q. The data are shown in Figures 1 and 2. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Example 1. Left panel: 3D signal evolution in the second spa [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Example 1. Left panel: time decomposition for data at posit [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Example 1. From left to right, first, second IMF and trend [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Example 1. Differences between the ground truth and the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Example 2. Left panel: Signal at time t “ 0. Right panel: Time evolution for position v “ p150, 150q. [9] S. Sfarra, A. Cicone, B. Yousefi, S. Perilli, L. Robol, X.P.V. Maldague, Maximizing the detection of thermal imprints in civil engineering composites via numerical…
Figure 7
Figure 7. Figure 7: Example 2. Left panel: 3D signal evolution in the second spa [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Example 2. Left panel: time decomposition for data at posit [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Example 2. From left to right, first, second IMF and trend [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Example 2. Differences between the ground truth and th [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Example 3. Left panel: air temperature measured on Jan [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Example 3. Decomposition over space of the data into two [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: Example 3. Left panel: time decomposition for data at pos [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

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Reference graph

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