Abstract :
[en] We present a novel algorithm for Ensemble Empirical Mode Decomposition (EEMD) that splices the different noise-added realisations of a signal using even-odd extension, making it computationally more efficient than simply iterating the Empirical Mode Decomposition (EMD). Furthermore, the noise added is computed to cancel out perfectly, reducing the size of the ensemble to be performed, and making the resulting decomposition more representative of the initial signal. This algorithm is available in the R package DecomposeR (https://CRAN.R-project.org/package=DecomposeR), under the name ‘extricate’.
We propose a new methodology to further document the quality of any decomposition based on different concepts that we introduce:
- Integrity quantifies to what extent the sum of the components is equal to the signal. It is defined as the averaged difference between (1) the signal, and (2) the summed components of the decomposition. EMD fulfils integrity by design, except for errors caused by floating-point arithmetic. Ensemble Empirical Mode Decomposition (EEMD) may fail to satisfy integrity unless noisy realisations are carefully chosen to cancel each other when averaging the realisations, which is performed in our ‘extricate’ algorithm.
- Parsimony checks that the decomposition does not generate components that heavily cancel each other out. We propose to quantify it as the ratio between (1) the cumulated absolute values of each component (except the trend), and (2) the cumulated absolute values of the signal (minus the trend). The trend should be ignored in the calculation, because an added trend decreases the parsimony estimation of a similar decomposition.
- IMF departure (IMFD) quantifies the departure of each component to the definition of intrinsic mode functions (IMF), from which instantaneous frequency can reliably be computed. We define it as the exponential of the mean of the absolute differences of the logarithms of frequencies obtained using (1) a robust generalized zero-crossing method and (2) a more local method such as the Hilbert Transform.
- Reversibility is the concept that all initial data points are preserved, even after linear interpolation of irregularly sampled data points. This allows to revert back to the original signal and discuss the significance of each data point. To facilitate reversibility we introduced the concept of quanta (smallest significative resampling interval) and an algorithm computing the highest common rational divisor of given values in R: ‘divisor’.
These concepts can be used to check any decomposition independently of how it was performed (i.e. a posteriori). Once the above-mentioned concepts are taken into account, the instantaneous frequencies, ratios of frequencies and amplitudes of the components can be computed and used to understand signals. These concepts are particularly useful to process highly complex signals such as deep-time paleoclimatic ones, which can be affected by high levels of red noise, frequency modulation, and overprinting of parasitic signals, and are usually sampled irregularly.
We further present a new theoretical concept: exchanging intensity between different components of a first prototype of decomposition, to further refine it. This would enable changing the characteristics of components, without compromising the integrity of a decomposition. We present this as an alternative to sifting, reducing the presence of riding wave without necessarily smoothing the amplitude modulation of components. This is useful in signals when amplitude modulation is of critical importance.
