**3. Canonical correlation analysis (CCA)**

This algorithm has been developed by *H*. Hotelling and is considered as a way to measure the linear relationship between two multidimensional variables. It detects two bases with the correlation matrix between the variables of interest that are in diagonal form and the correlations on the diagonal get maximized, in such a way that the dimensionality of these new bases is either less than or equal to the smallest dimension of the two variables [5].

Canonical correlation analysis (CCA) is first proposed by Hotelling. CCA is an algorithm for the determination of the linear association between two set variables. This is done with the help of the variance and covariance matrix of the data [6].

A set of linear combinations named A and B are considered as:

$$A\_P = \begin{bmatrix} a\_{11}, a\_{12}, \dots, a\_{1m} \end{bmatrix}^T \tag{14}$$

$$B\_Q = \begin{bmatrix} b\_{11}, b\_{12}, \dots, b\_{1n} \end{bmatrix}^T \tag{15}$$

Let *Cpp* and *Cqq* be the variance of the *Ap* and *BQ* respectively and *Cpq* is the covariance between AP and BQ. Then the above equation can be rewritten as:

$$P^\* = \frac{A\_p^T \mathbf{C}\_{pp} B\_Q}{\sqrt{A\_p^T \mathbf{C}\_{pp} A\_p \sqrt{B\_q^T \mathbf{C}\_{qq} B\_q}}} \tag{16}$$

This *P*<sup>∗</sup> should be maximum to achieve the best self-correlation. Therefore, this optimization can be solved by

$$\mathbf{C}\_{pp}^{-1}\mathbf{C}\_{pq}\mathbf{C}\_{qq}^{-1}\mathbf{C}\_{qp}A\_P = \rho A\_P \tag{17}$$

$$\mathbf{C}\_{qq}^{-1}\mathbf{C}\_{qp}\mathbf{C}\_{pp}^{-1}\mathbf{C}\_{pq}\mathbf{B}\_Q = \rho\mathbf{B}\_Q\tag{18}$$

This *ρ*represents the Eigenvalue which is equal to the square of *P*<sup>∗</sup> .

$$
\rho = \sqrt{\mathcal{P}^\*} \tag{19}
$$

This canonical pair will be calculated and separated by calculating self-correlation and a mutual decorrelation between input sources.

### **4. Wavelet transform**

Wavelet Transform (WT) has good localization properties in the time and frequency domain [6], and so it is a widely accepted and successful method being used for de-noising [11]. Currently, so many approaches are available at the algorithmic level to de-noise using Wavelet Transform, which is mainly based on shrinkage, where the EEG signals get decomposed in the form of wavelets and then noise removal is performed using shrinkage and thresholding. The quality of Wavelet Transform in transforming a time-domain signal into time and frequency localization assists in comprehending the signal's behavior in a much better way.

The Wavelet Transform could be defined as the following equation, which is the inner product or cross-correlation of {*xn*[*m*]} signal with scaled and time-shifted wavelet Ψ*a,b*[*m*], that is:

$$WT\_{\mathfrak{x}\_n}[a, b] = \left(\mathfrak{x}\_n, \Psi\_{a,b}\right) \tag{20}$$

where,

$$\Psi\_{a,b}[m] = |a|^{-\frac{1}{2}} \Psi^{\frac{m-b}{a}}$$

*a*—Scale parameters. *b*—Translation parameters. ψ*<sup>a</sup>*,*<sup>b</sup>*½ � *m* - Appropriate wavelet function.

### **5. Empirical mode decomposition**

Empirical mode decomposition is a non-linear method to represent a non-stationary signal into the sum of zero-mean sub-components. This method decomposes a signal into several intrinsic mode functions through an iterative method known as sifting. At the first level, the Intrinsic Mode function (IMF1) is the mean of the upper and lower envelop of the original EEG signal x(t). Then the residual signal is obtained by subtracting IMF1 from x(t). This process is iterated till the stopping criterion is fulfilled (Residual signal energy content is close to zero). The remaining residual signal is

$$P\_n(\mathbf{t}) = P\_{n-1}(\mathbf{t}) - I\mathbf{M}F\_n(\mathbf{t})\tag{21}$$

where, *Pn*ðÞ¼ *t x t*ð Þ.

Finally, the signal is reconstructed by adding all IMFs and residual signals as

$$\varkappa(t) = P\_n(t) + \sum\_{i=1}^{N} \text{IMF}\_i(t) \tag{22}$$

The method of detecting IMFs is sensitive to the amalgam of undesired signal components present in surroundings. These noises affect the EMD process. Thus, mode mixing is used to overcome the disparate scale oscillations with amplitude in the near range of the IMFs peaks which can be available randomly in the whole dataset. Consequently, a more powerful and noise-assisted version of the EMD algorithm was presented termed as Ensemble Empirical Mode Decomposition (EEMD), which solves this mode mixing quandary and employs the average value of EMD ensembles that filters out the IMFs for the given signal. Moreover, this method also depends on the added noise amplitude to the input signal and the number of trials [6, 9].
