## 2 Theory
### Empirical Mode Decomposition  
The Empirical Mode Decomposition (EMD) is the fundamental part of a technique known as Hilbert-Huang Transform that is an empirical data analysis technique, developed in late 1990s by [PERSON] et al. [17; 18; 19]. Unlike other techniques such as Fourier or Wavelet Transforms, that use prior knowledge or fixed basis to decompose the signal, EMD derives its basis adaptively from the data itself and does not rely on any prior knowledge [20]. This technique is based on an assumption that any data is consisting of a set of simple intrinsic modes of oscillations [21]. The EMD decomposition process, decomposes the original signal into a set of signals known as Intrinsic Mode Functions (IMFs). This is accomplished by a novel process known as sifting, which is repeatedly applied to the signal until it converges on criteria that define an IMF [20]. The EMD sifting process extracts the fastest varying component first and continues to the next IMF. The process produces a finite number of IMFs before converging itself. The extracted IMFs represent simple oscillatory functions with different time-scale. Each IMF by definition will have the same number of extrema and zero-crossings or differ at most by one. Furthermore, the mean value of the envelopes of local maxima and local minima of IMFs are zero at any given point [21]. The technique to calculating the IMFs are briefly explained here but for more details and derivations on the technique one can refer to [17; 21]. The first step to calculate the IMF is to find all the local maxima points of the signal and connect them using cubic spline to obtain the upper envelope of the signal. The same procedure is applied to the local minima points to obtain the lower envelope. Next, the mean value of the two envelopes is calculated. The first component of the signal is calculated by subtracting the obtained mean from the original signal  
\[h_{1}(t)=x(t)-m_{1}(t) \tag{7}\]  
where \(x(t)\) is the original signal and \(m_{1}\) is the calculated mean. If the calculated component \(h_{1}\) meets the IMF criteria, then it will be the first IMF. However if the calculated component does not meet the requirement of an IMF, the procedure is repeated again by replacing the original signal \(x(t)\) with \(h_{1}\)This cycle is repeated until the calculated component meets the IMF criteria. It is then marked as the first IMF (\(c_{1}\)) and subtracted from the original signal. \[x_{new}(t)=x(t)-c_{1}(t) \tag{8}\]  
After this step, the original signal \(x(t)\) is replaced with the new signal \(x_{new}(t)\) and the whole procedure is repeated to calculate the next IMFs. The remaining signal after calculating the last IMF is called the residual signal \(r(t)\) and can be used to reconstruct the original signal using the calculated IMFs using  
\[x(t)=\sum_{i=1}^{n}c_{i}(t)+r(t) \tag{9}\]  
where \(n\) represent the number of calculated IMFs. This process is summarised in Figure 1.  
Figure 1: The procedure to obtain Intrinsic Mode Functions (IMFs) using Empirical Mode Decomposition (EMD) technique, adapted from [22]. Refer to the text for more details. Although this technique is very effective in decomposing non-linear and non-stationary signals, it suffers from certain problems such as end effect or mode mixing (aliasing) which was noted by [PERSON] himself [19]. One solution to avoid the mode mixing problem is to use a newly proposed technique known as Ensemble EMD (EEMD) [23]. This technique uses noise assisted analysis to solve the problem of mode mixing in the EMD algorithm. This can be done by adding White Gaussian noise to the signal. Consider adding noise to the proposed signal for \(M\) times and each time decomposing the signal using EMD technique. Each new signal can be presented as  
\[x_{i}(t)=x(t)+w_{i}(t) \tag{10}\]  
where \(w_{i}(t)\) represents the i-th added white noise to the original signal \(x(t)\) and \(x_{i}(t)\) is the generated noisy signal.