CEEMDAN: Adaptive Noise-Assisted Decomposition

• CEEMDAN is an adaptive, noise-assisted algorithm designed to decompose nonstationary signals into intrinsic mode functions and a residual component.
• The method uses ensemble averaging and adaptive noise injection at each stage to reduce mode mixing and ensure exact signal reconstruction.
• Applications include geophysical, biomedical, and financial time series analysis, enhancing forecasting, denoising, and feature extraction tasks.

Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) is an adaptive, noise-assisted algorithm designed to decompose nonlinear and nonstationary time series into a finite set of intrinsic mode functions (IMFs) and a residue, mitigating longstanding issues in empirical mode decomposition (EMD) such as mode mixing, non-orthogonality, and incomplete reconstruction. CEEMDAN iteratively adds adaptive noise at each residual stage and averages EMD results over an ensemble, producing robust, physically interpretable components that facilitate downstream tasks such as time-frequency analysis, forecasting, denoising, and feature extraction.

1. Algorithmic Foundations and Motivation

The CEEMDAN method was developed in response to two major failure modes of EMD and its ensemble variant EEMD: persistent mode mixing and loss of reconstruction integrity in noisy, nonstationary signals (Luukko et al., 2017, Santander et al., 2019). In classical EMD, successive sifting isolates IMFs by enforcing conditions on extrema and zero crossings, but practical applications often yield mixtures of disparate frequency content (mode mixing) and physically ambiguous components, particularly under noise or abrupt transitions.

EEMD addresses some mixing by adding white noise to multiple signal realizations, performing EMD, and averaging the corresponding IMFs. However, EEMD does not guarantee exact decomposition—i.e., the sum of IMFs and the trend does not reconstruct the original signal precisely—and residual mode mixing often persists, especially when noise statistics or ensemble sizes are poorly chosen (Santander et al., 2019).

CEEMDAN achieves completeness and greater robustness by injecting adaptive noise at each stage of decomposition, computing the ensemble mean IMF for each residual, and subtracting it before proceeding to the next mode. This stagewise approach regularizes mode splitting, localizes noise influence, and, through averaging, attenuates the stochasticity introduced by noise (Luukko et al., 2017).

2. CEEMDAN Methodology

The CEEMDAN decomposition of a signal $x(t)$ proceeds as follows, with an explicit emphasis on adaptive noise and ensemble averaging at every extraction stage (Santander et al., 2019, Jiang et al., 2020, Geng et al., 17 Apr 2024):

1. Initialization: Set initial residual $r_0(t) = x(t)$. For each realization $i \in 1, \ldots, N$ add Gaussian noise $w_i(t)$.
2. First IMF Extraction: For each realization,

$x_i^{(0)}(t) = r_0(t) + s_0 w_i(t),$

compute its first EMD IMF, $c_{i,1}(t)$. Average across the ensemble:

$c_1(t) = \frac{1}{N}\sum_{i=1}^N c_{i,1}(t).$

1. Residue Update: Update residue:

$r_1(t) = r_0(t) - c_1(t).$

1. Iterative Steps: At each stage $k \geq 1$,
   • For each realization,

   $r_{k,i}(t) = r_k(t) + s_k w_i(t),$

   extract first IMF $c_{i,k+1}(t)$. - Ensemble average:

   $c_{k+1}(t) = \frac{1}{N}\sum_{i=1}^N c_{i,k+1}(t).$

• Update residue:

  $r_{k+1}(t) = r_k(t) - c_{k+1}(t).$

1. Termination: Stop when $r_k(t)$ becomes monotonic or contains no further meaningful oscillatory component.

This process ensures that, to within numerical round-off, the summation over all IMFs and the residue reconstructs $x(t)$ exactly: $x(t) \approx \sum_{k} c_k(t) + r_K(t)$ where $K$ is determined adaptively (Luukko et al., 2017, Santander et al., 2019).

Adaptive selection of noise amplitude $s_k$ at each stage can be tuned with respect to local signal characteristics or left at a fixed empirical value for generality (Santander et al., 2019, Luukko et al., 2017).

Key improvements over EMD and EEMD include:

• Mitigation of Mode Mixing: Ensemble averaging and adaptive noise reduce the overlap between physically distinct oscillatory processes.
• Exactness (Completeness): By sequentially subtracting ensemble mean IMFs, the decomposition is complete.
• Robustness to Noise: Ensemble statistics average out random fluctuations, and adaptive noise enables separation under significant external noise (Jiang et al., 2020, Geng et al., 17 Apr 2024).

3. Mathematical and Algorithmic Properties

CEEMDAN’s theoretical core lies in its noise-regularized sifting procedure. Mathematically, at each IMF extraction, ensemble averaging of EMD under noise realisations acts as a stochastic filter, producing modes with reduced mixing.

Local maximal-minimal envelope averaging remains the basis for the EMD sifting in each ensemble trial: $m(t) = \frac{e_{\max}(t) + e_{\min}(t)}{2}$ with the iterative sifting condition that the component must exhibit nearly equal numbers of extrema and zero crossings, and have a near-zero local mean (Santander et al., 2019, Luukko et al., 2017).

The stopping criteria at each sifting step are often based on $S$-number checks—requiring matching counts of extrema and zero crossings for several consecutive stages (Luukko et al., 2017).

Spline interpolation for envelope construction is typically applied with "not-a-knot" end conditions, and flat extrema are handled by considering the center of flat regions for stability in the face of plateaus (Luukko et al., 2017).

Parallel implementations can leverage OpenMP or similar frameworks to accelerate ensemble computations, and cross-platform libraries (such as libeemd) offer Python and R interfaces for practical adoption (Luukko et al., 2017).

4. Applications and Empirical Findings

CEEMDAN is used across a range of scientific and engineering disciplines:

• Earth and Environmental Sciences: Separation of oscillatory seasonal components and trends in geophysical or environmental time series (e.g., gas consumption cycles, PM2.5 pollution, monsoon rainfall) (Luukko et al., 2017, Jiang et al., 2020, Niyogi, 2023).
• Biomedical Signal Analysis: Extraction of physiological rhythms from ECG, EMG, or surface electromyography, where nonstationary and nonlinear behaviors dominate (Santander et al., 2019).
• Engineering Signal Denoising: Removal of broad-band mechanical and electromagnetic noise in force measurements for water entry and vessel attitude prediction via hybrid CEEMDAN-SVM-PSO models (Spinosa et al., 2021, Geng et al., 17 Apr 2024).
• Finance and Market Microstructure: Analysis of volatility and time-frequency features in asset prices, including applications extending to adaptive or complementary ensemble techniques (Leung et al., 2021).
• Speech Enhancement: CEEMDAN can remove specific noise types from speech but is limited in separating multiple overlapping speech sources unless their frequency and amplitude ratios satisfy explicit bounds (Melhem et al., 18 Nov 2024). The method generally enhances speech quality (e.g., increasing PESQ and SDR) in high SNR noise cases but is ineffective for typical "cocktail party" separation.

Hybrid prediction frameworks such as CEEMDAN-DeepTCN for air pollution forecasting (Jiang et al., 2020) or CEEMDAN-PSO-SVM for marine vessel dynamics (Geng et al., 17 Apr 2024), consistently demonstrate that CEEMDAN’s ability to split the original signal into scale-specific IMFs improves downstream learning and regression accuracy relative to direct modeling of the raw series.

5. Performance, Limitations, and Interpretability

The experimental literature demonstrates several salient properties:

• Physical Meaningfulness: CEEMDAN decreases mode interference relative to EMD/EEMD, supporting more confident Hilbert spectral analyses, trend extraction, and component-wise modeling (Santander et al., 2019).
• Parameter Sensitivity: Though robust, CEEMDAN performance depends on noise amplitude and ensemble size (e.g., Nstd = 0.2, NR = 500 recommended in (Santander et al., 2019)), with trade-offs between reconstruction error, mode mixing mitigation, and computational load.
• Interpretability: The method ensures each IMF represents unique process dynamics, reducing bias and nonphysical artifacts.
• Computational Overhead: Enhanced robustness comes at greater computational cost, especially when large ensembles are required (Santander et al., 2019).
• Separation Conditions: In audio source separation, CEEMDAN cannot separate signals with frequency ratios in [0.6, 1.6] or amplitude ratios outside [0.3, 3]; repeated subtraction may introduce amplitude deformation (Melhem et al., 18 Nov 2024). The number of IMFs is non-constant and can vary depending on local signal structure, limiting direct streaming or real-time deployment.
• Comparison with Recent Decompositions: For certain applications, variants such as Nonlinear Mode Decomposition (NMD) (Iatsenko et al., 2012), Empirical Mode Modeling (EMM) (Park et al., 2021), Adaptive Complementary EMD (ACE-EMD) (Leung et al., 2021), and NPCEEMD (Kumar et al., 2023) may outperform CEEMDAN in mode orthogonality, noise rejection, or adaptability, typically by either further automating parameter selection, introducing additional regularization, or using more flexible noise models.

6. Implementation Considerations and Open-Source Resources

Efficient CEEMDAN implementations are available, notably libeemd (C99, with Python and R bindings) facilitating integration into analytical pipelines (Luukko et al., 2017). Best practices include:

• Validating extracted modes' physical interpretability rather than relying solely on reconstruction error (Santander et al., 2019).
• Experimenting with noise amplitude and ensemble parameters to balance decomposition granularity with computational feasibility.
• Assessing the completeness and uniqueness of IMFs, especially in denoising and forecasting scenarios where subcomponent independence and non-mixing are critical (Santander et al., 2019).

7. Outlook and Comparisons

Ongoing research in signal decomposition has yielded nonparametric and hybrid methods addressing some of CEEMDAN’s remaining limitations. For example, NPCEEMD (Kumar et al., 2023) uses fractional Gaussian noise rather than white noise and discards parameter tuning, while ACE-EMD (Leung et al., 2021) tailors the noise amplitude over time for heteroskedastic series. In certain forecasting and deep learning contexts, empirical wavelet transforms (EWT) (Niyogi, 2023) or nonlinear mode decompositions (Iatsenko et al., 2012) have been shown to better distribute complexity among subcomponents, providing more uniformly learnable inputs for models such as LSTM or TCNs.

Despite these advances, CEEMDAN remains a preferred, general-purpose method when robust, physically significant adaptive decomposition of complex, nonstationary signals is required and when exact reconstruction and reduction of mode mixing are mandatory. Its efficacy depends on careful parameter selection and is maximized when subsequent processing (prediction, denoising, or feature extraction) leverages its explicit separation of oscillatory modalities.