## 2 Improved Ensemble Empirical Mode Decomposition
### Improved Ensemble Empirical Mode Decomposition  
It should, however, be noted that, in the EEMD method, each \(x^{i}[n]\) is decomposed independently from the other realizations and, for each one residue, \(r_{k}^{i}[n]=r_{k-1}^{i}[n]-IMF_{k}^{i}[n]\) is obtained. In the so-called improved ensemble empirical mode decomposition (EEEMD) method presented, the decomposition modes will be noted as \(\widetilde{IMF}_{k}\) and the first residue \(r_{1}[n]\) and \(\widetilde{IMF}_{1}[n]\) is obtained by the EEMD method. Then the added Gaussian white noise \(w^{i}(t)\) (\(i=1,2,\cdots,N\)) is decomposed by the EMD method and the \(\widetilde{IMF}_{k}^{i}(k=1,2,\cdots,K)\) of the \(w^{i}(t)\) is obtained first. Taking the other residue \(r_{i}[n]\) as the signal to be decomposed, add the corresponding \(\widetilde{IMF}_{k}^{i}\) of \(w^{i}(t)\) and perform EMD. Let us define the operator \(E_{j}(\cdot)\) which, given a signal, produces the j-th mode obtained by EMD. Let \(w^{i}\) be the zero mean unit variance Gaussian white noise. \(\overline{IMF}_{j}\) stands for the averaging j-th mode of the signal \(x[n]\) and \(\epsilon_{i}\) is the signal noise ratio (SNR) coefficients at each stage. The IEEMD algorithm can be described using the information below. 1. Add \(N\) Gaussian white noise \(w^{i}(t)\) (\(i=1,2,\cdots,N\)) in signal \(x(t)\): \[x^{i}(t)=x(t)+w^{i}(t)\ (i=1,2,\cdots,N)\] (2)
2. Obtain the first residue \(r_{1}(t)\) and the first mode \(\overline{IMF}_{1}\) by EEMD: \[\overline{IMF}_{1}=\frac{1}{N}\sum_{i=1}^{N}E_{j}\Big{(}x^{i}(t)\Big{)}\] (3) \[r_{1}(t)=x(t)-\overline{IMF}_{1}\] (4)
3. Perform EMD on Gaussian white noise \(w^{i}(t)\) and generate the IMFs and residue of \(w^{i}(t)\): \[w^{i}(t)=\sum_{j=1}^{m}E_{j}\Big{(}w^{i}(t)\Big{)}+r^{\prime}_{mi}(t)\] (5) where \(E_{j}\big{(}w^{i}(t)\big{)}\) is the j-th mode of \(w^{i}(t)\) decomposed by the EMD method and the total number is \(m\). \(r^{\prime}_{mi}(t)\) is the residue of \(w^{i}(t)\) decomposed by the EMD method. 4. Take \(r_{1}(t)\) as the signal to be decomposed, add \(E_{1}\big{(}w^{i}(t)\big{)}\) (\(i=1,2,\cdots,N\)) and generate \(r^{k}_{1i}(t)\). \[r^{k}_{1i}(t)=r_{1}(t)+\varepsilon_{1}E_{1}\Big{(}w^{i}(t)\Big{)}\] (6)
5. Perform EMD on \(r^{k}_{1i}(t)\) and obtain the \(\overline{IMF}_{2}\) of signal \(x(t)\) by averaging the \(N\) IMFs. \[\overline{IMF}_{2}=\frac{1}{N}\sum_{i=1}^{N}E_{j}\Big{(}r^{k}_{1i}(t)\Big{)}\] (7)
6. For \(k=2,\cdots,K\), calculate the \(k\)-th residue. \[r_{k}(t)=r_{(k-1)}(t)-\overline{IMF}_{k}(t)\] (8)
7. Decompose \(r_{k}(t)+\varepsilon_{k}E_{k}\big{(}w^{i}(t)\big{)}\), \(i=1,\cdots,N\), until their first EMD mode and define the (\(k\)+1)-th mode using the equation below. \[\overline{IMF}_{k+1}=\frac{1}{N}\sum_{i=1}^{N}E_{1}\Big{(}r_{k}(t)+\varepsilon _{k}E_{k}\Big{(}w^{i}(t)\Big{)}\Big{)}\] (9)
8. Go to step 6 for the next \(k\). 9. Step 6 to 8 are performed until the obtained residue is no longer feasible to be decomposed (the residue does not have at least two extrema). The final residue is expressed using the equation below. \[R(t)=x(t)-\sum_{j=1}^{K}\overline{IMF}_{j}\] (10) where \(K\) is the total number of modes. 10. Let us define that each IMF contains two parts (i.e., the useful characteristic signal and noise signal), and set \(\overline{IMF}_{j}=s_{j}(t)+n_{j}(t)\). \(s_{j}(t)\), which stands for the characteristic signal component, and \(n_{j}(t)\) stands for the noise component. The cross-correlation function between different IMFs is shown below. \[R_{ij}(\tau)=R_{s_{i}s_{j}}(\tau)+R_{s_{i}n_{j}}(\tau)+R_{s_{j}n_{i}}(\tau)+ R_{n_{i}n_{j}}(\tau)\qquad(i\
eq j)\] (11)
11. It is known that corresponding IMFs of different series of noise have no correlation with each other. Then, we can set \(R_{s_{i}n_{j}}(\tau)=R_{s_{j}n_{i}}(\tau)=R_{n_{i}n_{j}}(\tau)=0\), and Equation (11) can be expressed by the formula below.