Complete Ensemble EMD with Adaptive Noise (CEEMDAN)

• CEEMDAN is a noise-assisted time-series decomposition technique that adaptively extracts intrinsic mode functions ensuring exact signal reconstruction.
• It improves upon EMD and EEMD by mitigating mode mixing through adaptive noise scaling and stage-wise ensemble averaging.
• CEEMDAN is widely applied in fields like oceanography, biomedical engineering, and machinery diagnostics for analyzing complex, nonstationary signals.

Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) is a data-driven, noise-assisted signal decomposition method designed to separate nonlinear and nonstationary time series into a set of well-separated Intrinsic Mode Functions (IMFs) and a monotonic residual. CEEMDAN overcomes key limitations inherent in the Empirical Mode Decomposition (EMD) and its earlier noise-assisted improvement, Ensemble EMD (EEMD), by ensuring exact signal reconstruction while adaptively suppressing mode mixing and noise leakage. CEEMDAN's staged, ensemble-based architecture and adaptive noise scaling offer unique advantages for applications involving complex, multiscale, real-world signals, such as those found in engineered and natural systems (Geng et al., 17 Apr 2024, Luukko et al., 2017).

1. Theoretical Foundations and Improvements over EMD/EEMD

EMD decomposes a univariate signal $x(t)$ into additive IMFs, which represent simple oscillatory modes with frequency and amplitude varying in time. EMD's core limitations are mode mixing—where a single IMF contains signals of widely disparate frequency—or mode splitting, stemming from signal intermittency or closely spaced frequencies. EEMD was introduced to reduce these artifacts by averaging the results of multiple EMD runs, each with added white noise. However, EEMD's averaging process of all IMFs across noise realizations generally results in an incomplete decomposition: the sum of extracted IMFs does not recover the original signal exactly due to residual noise.

CEEMDAN addresses these issues by adaptively injecting noise at each IMF extraction stage and ensuring that the ensemble-averaged IMFs, together with the final residual, reconstruct the signal exactly (to numerical precision), thus upholding the completeness property (Luukko et al., 2017).

2. Formal Specification and Algorithmic Workflow

The CEEMDAN workflow can be summarized by the following structured procedure (Geng et al., 17 Apr 2024, Luukko et al., 2017):

• Inputs: $\ x(t)\$ (signal, length $T$), ensemble size $N$ (number of noise realizations), relative noise amplitude $\alpha$, standard deviation threshold $SD_{thr}$ for sifting stoppage, maximum sifting iterations per IMF $L_{max}$.
• Initialization: Set the initial residual $r_0(t) \leftarrow x(t)$.
• Iterative IMF Extraction (for $k=1,2,\dots,K$):
  • Draw $w_i(t)\sim\mathcal{N}(0,1)$ (white Gaussian noise).
  • Generate perturbed signal: $X_i^k(t) = r_{k-1}(t) + \epsilon_{k-1} w_i(t)$.

  • Apply EMD sifting to $X_i^k(t)$ to extract its first IMF $c_i^k(t)$, using:

    $$SD^{\ell} = \frac{\sum_t |h^{\ell-1}(t) - h^{\ell}(t)|^2}{\sum_t [h^{\ell-1}(t)]^2}$$

    Stopping when $SD^\ell < SD_{thr}$ or $\ell > L_{max}$.

  • Compute the $k$-th IMF as the average across noise realizations:

    $C_k(t) = \frac{1}{N} \sum_{i=1}^N c_i^k(t)$

1. Update the residual: $r_k(t) = r_{k-1}(t) - C_k(t)$.
2. Terminate if $r_k(t)$ is monotonic (fewer than two extrema); else, continue.

• Final Representation:

  $x(t) = \sum_{k=1}^K C_k(t) + r_K(t)$

In all recent studies, including (Geng et al., 17 Apr 2024) and (Luukko et al., 2017), typical parameter settings are ensemble size $N=100$, noise amplitude $\alpha=0.2$, sifting stoppage $SD_{thr}=0.2$, and $L_{max}=50$.

3. Completeness and Mode Separation

A central property of CEEMDAN is its completeness: the sum of all ensemble-averaged IMFs plus the last residue exactly reconstructs the input signal. This property holds at each stage by induction, and final reconstruction is

$x(t) = \sum_{j=1}^K C_j(t) + r_K(t)$

ensuring no leakage or loss, subject only to numerical precision limits. Unlike EEMD, which introduces accumulation of noise and imperfect signal recovery, CEEMDAN compartmentalizes noise at each stage, utilizes adaptively scaled perturbation, and cancels it in the ensemble average.

CEEMDAN also demonstrates improved mode separation: due to noise-assisted, staged extraction, IMFs are more uniformly and physically interpretable, and transmission of high/low-frequency content into incorrect IMFs (mode mixing) is greatly reduced. In engineered signals such as unmanned surface vessel motion attitudes, this results in IMFs that are well-aligned with distinct physical disturbance components (Geng et al., 17 Apr 2024).

4. Computational Complexity and Optimization

For a length-$T$ signal, CEEMDAN with $K$ extracted modes, $N$ ensemble members, and mean $r$ sifting iterations per IMF, the cumulative complexity is $O(N K r T)$. Each noise realization per mode only requires extraction of the first IMF, providing significant efficiency compared to naive ensemble runs that would extract all IMFs each time.

The "libeemd" library provides an efficient implementation, exploiting OpenMP for parallel processing across ensemble members and employing C99-optimized routines for extrema detection, cubic spline interpolation, and robust handling of flat signal segments to safeguard against envelope artifacts (Luukko et al., 2017). Boundary effects are managed through fallback interpolation schemes, and pathological oversifting is capped at a user-defined maximum number of iterations.

5. Parameterization and Practical Recommendations

The canonical parameter choices, as established by Wu & Huang (2009), Torres et al. (2011), and operationalized in libeemd and experimental settings (Luukko et al., 2017), are:

Parameter	Typical Value/Range	Notes
Ensemble size ($N$)	100–500 (default 100–250)	Larger $N$ lowers variance, increases run-time
Noise amplitude ($\alpha$)	0.2	Relative to current residual std
Sifting threshold ($SD_{thr}$)	0.2	Standard “Huang toolbox” value
Max siftings ($L_{max}$)	50–100	Prevents pathologies in sifting

These settings achieve a balance between decomposition fidelity, computational cost, and mode separation. The implementation in (Geng et al., 17 Apr 2024) utilizes these defaults, inferring parameter values from the original algorithm unless otherwise specified.

6. Applications and Performance in Modern Signal Processing

CEEMDAN serves as a general-purpose, data-adaptive, nonlinear time-frequency analysis tool. Notable research, such as "Prediction of Unmanned Surface Vessel Motion Attitude Based on CEEMDAN-PSO-SVM" (Geng et al., 17 Apr 2024), deploys CEEMDAN as a preprocessing stage in a predictive pipeline. Here, each IMF is separately modeled using a support vector machine (SVM) optimized by particle swarm optimization (PSO), and predictions are recombined for final output. This approach achieved a 17% reduction in mean absolute error compared to an EMD-PSO-SVM approach, and a 43% improvement over CEEMDAN-SVM, demonstrating direct benefits of the enhanced decomposition.

Additionally, the open-source implementation, "libeemd," enables scalable application of CEEMDAN in Python, R, and C environments (Luukko et al., 2017). Applications span oceanography, biomedical engineering, machinery diagnostics, and financial data analysis, all domains characterized by highly multicomponent and nonstationary signals.

7. Limitations and Perspectives

CEEMDAN, while addressing the key limitations of its predecessors, introduces increased computational cost proportional to ensemble size and the number of sifting iterations. However, this increase is mitigated by parallelizability and staged computation. No fundamental methodological enhancement to CEEMDAN itself was introduced in (Geng et al., 17 Apr 2024); innovation centered on integration with downstream learning and optimization models rather than modification of CEEMDAN's structure.

A plausible implication is that future work may investigate adaptive or data-driven determination of optimal noise amplitude, stopping criteria, or ensemble size to further enhance efficiency or decomposition quality.

• Prediction of Unmanned Surface Vessel Motion Attitude Based on CEEMDAN-PSO-SVM (Geng et al., 17 Apr 2024)
• Introducing libeemd: A program package for performing the ensemble empirical mode decomposition (Luukko et al., 2017)