## 2 Theoretical Descriptions  
### Regenerated Phase-Shifted Sinusoid-Assisted EMD Theory  
EEMD is an improved EMD algorithm, which is known as empirical mode decomposition with additive noise. It is presented as a means of inhibiting the mode mixing phenomenon (MMP) and to ensure that the decomposed IMFs have a certain physical meaning. The small amplitude of white noise is introduced into the signal, which is analyzed during the whole process according to the conventional EMD. The uniform distribution features of white noise can make the extreme distribute more evenly in all scales, which has a positive function in inhibiting the discontinuity of IMFs. The EEMD algorithm is described as follows:  
1. Gaussian white noise with mean zero and standard deviation is added to the decomposed signal \(x(t)\). Meanwhile, the normalization treatment is carried out;
2. The IMFs of all the scales are obtained by using EMD algorithm to decompose the normalized signal;
3. Repeat the above two steps (a)-(b) for \(n\) times, where the added random white noise for every time is required to obey normal distribution;
4. Make an average of \(n\)-groups totality of IMF components by EMD decomposition and obtain \(x(t)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}c_{ij}(t)+\sum\limits_{j=1}^{n }r_{j}(t)\), where \(c_{ij}(t)\) is the \(i\)-th IMF components obtained from the \(j\)-th decomposition and \(r_{j}(t)\) is the remainder. From the above analysis, it can be seen that EEMD can reduce the mode mixing and energy leakage to a certain degree. However, it is unnecessary to add white noise for unlimited times and reduce the effects of white noise on signal decomposition by calculating the average every time, as this is a very time-consuming process. Moreover, the amplitude of the additive noise is selected by principles from experience, which causes some difficulties in real applications. For instance, if there is a signal with complex composition, the added noise with small amplitude is of no use, while adding noise with a larger amplitude may instead lead to several spurious modes. The novel points about the introduced RPSEMD method is summarized into two aspects. First, the sinusoid-assisted signal is set as the auxiliary signal:  
\[s_{k}(t|a_{k},f_{k},w_{k})=a_{k}\cos(2\pi f_{k}t+w_{k}) \tag{1}\]  
where \(a_{k}\), \(f_{k}\), and \(w_{k}\) denote the amplitude, frequency and phase, respectively. Following this, the sinusoid-assisted signal \(s_{k}\) is obtained by the intrinsic mode (IM) selected using clustering analysis. Secondly, varied \(w_{k}\) is performed to change the positions of the extreme point and to enhance the decomposition effect. The main computational process of RPSEMD is described as follows:1. Initialize \(k=1\);
2. Apply standard EMD algorithm to \(x(t)\) and then determine \(a_{k}\) and \(f_{k}\) with the resulting IMFs. \(w_{ki}\) is acquired by uniformly sampling in \([0,2\pi]\) with the phase shifting number \(I(1\leq i\leq I)\). After this, \(s_{k}(t|a_{k},f_{k},w_{k})\) is obtained. 3. The standard EMD algorithm of \(x(t)+s_{k}(t|a_{k},f_{k},w_{k})\) is performed, which aims to obtain the first IMF. The final IMF \(c_{k}(t)\) is calculated by averaging all these first IMFs. 4. Remove \(c_{k}(t)\) from \(x(t)\): \(x(t)\gets x(t)-c_{k}(t)\). Let \(k=k+1\);
5. Repeat the above steps (b)-(d) multiple times until no more IMF can be produced. Consequently, the final \(x(t)\) is regarded as the residue \(r(t)\). Basically, the determination of the parameter (\(a_{k}\),\(f_{k}\),\(w_{k}\)) is significant for the RPSEMD method. Under these circumstances, the initial IMFs obtained by EMD are represented as \(ci_{k^{\prime}}(t)\) (\(1\leq k^{\prime}\leq K^{\prime}\)). For the extreme point of \(ci_{1}(t)\), it is necessary to obtain its instantaneous amplitudes \(ai_{1}(e)\) and instantaneous frequency \(fi_{1}(e)\), where \(e\) indicates the index of an extreme point. Hierarchical clustering is executed according to the distribution characteristics for the instantaneous frequencies of an extreme point.