New insights and best practices for the successful use of Empirical Mode Decomposition, Iterative Filtering and derived algorithms

Algorithms based on Empirical Mode Decomposition (EMD) and Iterative Filtering (IF) are largely implemented for representing a signal as superposition of simpler well-behaved components called Intrinsic Mode Functions (IMFs). Although they are more suitable than traditional methods for the analysis of nonlinear and nonstationary signals, they could be easily misused if their known limitations, together with the assumptions they rely on, are not carefully considered. In this work, we examine the main pitfalls and provide caveats for the proper use of the EMD- and IF-based algorithms. Specifically, we address the problems related to boundary errors, to the presence of spikes or jumps in the signal and to the decomposition of highly-stochastic signals. The consequences of an improper usage of these techniques are discussed and clarified also by analysing real data and performing numerical simulations. Finally, we provide the reader with the best practices to maximize the quality and meaningfulness of the decomposition produced by these techniques. In particular, a technique for the extension of signal to reduce the boundary effects is proposed; a careful handling of spikes and jumps in the signal is suggested; the concept of multi-scale statistical analysis is presented to treat highly stochastic signals.


Boundary Conditions Synthetic Example
We decompose the signal given in Section 2.1 using the EEMD code written by Zhaohua Wu in 2009 2 , which can be downloaded from the official website of the Taiwanese Research Center for Adaptive Data Analysis https://in.ncu. edu.tw/~ncu34951/research1.htm (the EEMD code we tested is contained in the repository https://in.ncu. edu.tw/~ncu34951/Matlab_runcode.zip). In the first test, we set the number of elements in the ensemble to 800 and the standard deviation to 0.2, as suggested in 2 . The returned decomposition is shown in the left panel of Figure S1. Issues at the boundaries are clearly evident. In the second test, we pre-extend the signal symmetrically up to five times the length of the original signal and we make it periodical, as described in Section 2. The returned decomposition is shown in the right panel of Figure S1. The end effects are now clearly reduced.  Figure S1, we observe the typical issues related to the use of EEMD algorithm: mode splitting and noise-related IMFs.
In Table 1, we report the total computational time for the two EEMD decompositions and the 2-norm of the relative differences between the ground truth and each IMF as well as the trend. This 2-norm quantifies the misfit between ground truth and IMF components produced via EEMD. We now decompose the signal by means of the FIF algorithm (available at www.cicone.com). Results are shown in the left panel of Figure S2. As explained in Section 1, FIF algorithm automatically enforces a periodical extension at the edges. The derived IMFs are affected by errors at the boundaries, to an extent which is greater for the lower frequency components. This is clearly due to the fact that the original signal is not periodical at the boundaries. If, however, we symmetrically pre-extend the signal as in the previous example, the errors at the boundaries are drastically reduced, as shown in the right panel of Figure S2.
In Table 2, we report the total computational time for the two FIF decompositions and the 2-norm of the relative differences between the ground truth and each IMF and the trend. We stress that FIF algorithm does not have issues related to noise-related IMFs, for it does not require to perturb the original signal, like the EEMD algorithm. This characteristic makes the FIF method extremely faster than both EEMD and EMD methods 3 .
We remark that, regardless the method implemented to decompose the signal, it is advisable to pre-extend it symmetrically and to make it periodical, as explained in Section 2.

Boundary Conditions Real Life Example
Here we implement the EMD and FIF algorithms to decompose the signal analyzed in 1 Figure S3, left panel.
The decomposition returned by the FIF algorithm is show in Figure S3, right panel. In order to reduce the boundary effects, we first extend the signal at the boundaries and then we make it periodical. In this example, given the apparent absence of low frequency oscillations, we decide to extend it up to three times the length of the original signal. Furthermore, the symmetric-type extension proves to be the best option in this case 6 .