Complete Ensemble EMD with Adaptive Noise (CEEMDAN) in Python

CEEMDAN is available in Python through PyEMD.

There are as many empirical mode decomposition (EMD) variations as many teams are working on it. Everyone notices that in general EMD is very helpful method, yet, there's room for improvement. Few years back I have stopped doing modifications myself in exchange for working on mathematically sound model of coupled oscillator. Nevertheless, since I spent quite a lot of time on EMDs and have enjoy playing with it, from time to time something will catch my eye. This is what happened with Complete Ensemble EMD with Adaptive Noise (CEEMDAN).

As name suggests this is an expansion on the ensemble EMD, which was already covered. What's the difference? In case of CEEMDAN we're also decomposing our perturbation to the system, i.e. added noise. Method creates an ensemble of many perturbations, decomposes them using EMD and resulting IMFs are included to evaluate components of the input. I will refer to these components as cIMF. The actual algorithm was first proposed by Torres et. al [1], but shortly after an improvement in efficiency was proposed[2]. These updates refer mainly to noticing that for their purpose one doesn't need to compute all IMFs and weight parameter can be progressively scaled as well. What exactly is this algorithm?

Let's define operator IMF_i(S) which returns i^th IMF (pure EMD) of its input S and M(S) to provide local mean, i.e. M(S) = S - IMF₁(S), then the algorithm is as follows:

1. Create Gaussian noise ensemble W={wⁱ}, where $i\in[1..N]$, and decompose them using EMD.
2. For input signal S calculate grand average of local means from signal perturbed by scaled noise first IMF $R_{1} = \frac{1}{N}\sum_{1}^{N} M(S+ \beta_0 IMF_{1}(w^{i}))$.
3. Assign first cIMF to be: C₁ = S - R₁.
4. Compute $R_{k}= \frac{1}{N} \sum_{i=1}^{N} M(R_{k-1} + \beta_{k-1} IMF_{k}(w^{i}))$.
5. Calculate k^th cIMF as C_k = R_k-1 - R_k.
6. Iterate 4. and 5. until set of {S, C_k} fulfils EMD stopping criteria.

As it can be seen a family of parameters β has been included in the algorithm. These scalars refer to the amount of decomposed noise used to compute cIMFs. This is what the authors refer to as noise adaptive. These parameters are arbitrary, but it's suggested in improved version [2] to set them as $\beta_{k}= \epsilon_{0} \sigma(R_k)$, where σ is standard divination of argument and ε is another arbitrary parameter. Looking at point 4. one can see that for i^th residue we are using i^th IMF computed from noise. This is a problem, because EMD decomposes signal into a finite set of components and it can happen that there isn't i^thIMF. In this case authors are suggesting to assume component to be equal 0.

Advantage of this variation comes from the fact that created decomposition {C_i} fully reconstructs input. This is in contrast to EEMD which doesn't guarantee such completeness. However, with CEEMDAN questions rise regarding the meaning of added scaled IMFs of noise. Augmenting signal with ensemble of pure noise creates perturbations of input without any distinguished direction. As it has been observed by Flandrin et al. [3] when decomposing white noise EMD acts as a dyadic filter bank. This means that extracted IMFs will have preferred structure and adding them to input will be similar to adding vector with random length but particular direction.

Regardless of all, CEEMDAN is definitely an interesting method. Just purely by the number of citations it seems that I'm not the only one thinking that. I've included it to my Python PyEMD package, so feel free to play with it and leave some feedback.

References

[1] Torres ME, Colominas MA, Schlotthauer G, Flandrin P. A complete ensemble empirical mode decomposition with adaptive noise. InAcoustics, speech and signal processing (ICASSP), 2011 IEEE international conference on 2011 May 22 (pp. 4144-4147). IEEE.

[2] Colominas MA, Schlotthauer G, Torres ME. Improved complete ensemble EMD: A suitable tool for biomedical signal processing. Biomedical Signal Processing and Control.

[3] Flandrin P, Rilling G, Goncalves P. Empirical mode decomposition as a filter bank. IEEE signal processing letters.