*1.6.2 Artifact correction*

1. Principal Component Analysis (PCA).

These methods are based on EEG and artifacts decomposition into spatial components, which is inclusive of recognizing artifactual components and reassembling the EEG without those artifactual components that have been recognized, but it is problematic in the case of PCA. The PCA algorithm first decomposes the EEG signals into uncorrelated, but it is not required that these must be independent of each other which are spatially orthogonal and that's why it cannot deal with higher-order statistical dependencies. Furthermore, it is not practically possible to completely separate artifacts from interested brain signals specifically when both of these signals have comparable amplitudes.

The following expression describes principal component decomposition:

$$
\beta\_1 = \phi\_1 X\_0' \tag{6}
$$

where,

*β*<sup>1</sup> is set of first principal component scores whose mean equals to zero; ϕ<sup>1</sup> represents the first principal component;

ϕ1*X*<sup>0</sup> <sup>0</sup> could be considered as a vector or matrix transposition.

By maximizing the variance ofϕ1*X*<sup>0</sup> 0, ϕ1can be simply calculated as:

$$\boldsymbol{\Phi}\_{1} = \arg\max \mathbf{Var} \left( \phi\_{1} \mathbf{X}\_{0}' \right); \text{where } ||\phi\_{1}|| = \mathbf{1} \tag{7}$$

$$\boldsymbol{\Phi}\_{1} = \arg\max \boldsymbol{\Phi}\_{1} \boldsymbol{X}\_{0}^{\prime} \boldsymbol{X}\_{1} \\ \boldsymbol{\Phi}\_{1}^{\prime}; \text{where } ||\boldsymbol{\Phi}\_{1}|| = \mathbf{1} \tag{8}$$

Successive principal components can easily be obtained iteratively by demising the first k principal components from *X*0, presented as below:

$$X\_k = X\_{k-1} - X\_{k-1} \phi'\_k \phi\_k \tag{9}$$

Now to find ϕk+1, *Xk* has been treated as a data matrix that can be done by maximizing the variance of ϕ*<sup>k</sup>*þ<sup>1</sup>*X*<sup>0</sup> *<sup>k</sup>* using following equation:

$$\boldsymbol{\Phi}\_{k+1} = \arg\max \mathbf{V} \mathbf{a} \mathbf{r} \left( \boldsymbol{\Phi}\_{k+1} \mathbf{X}\_k' \right) \tag{10}$$

Subject to ϕ*<sup>k</sup>*þ<sup>1</sup> � � � � � � � � <sup>¼</sup> sqrt <sup>P</sup>*<sup>p</sup> <sup>j</sup>*¼<sup>1</sup>ϕ<sup>2</sup> *k*þ1 *j* � � <sup>¼</sup> 1 and <sup>ϕ</sup>*<sup>k</sup>*þ<sup>1</sup>⊥ϕ*k*for *<sup>j</sup>* = 1, 2 … *<sup>k</sup>*.

Alternatively, Singular Value Decomposition (SVD) is the simplest and efficient way that can be applied to find a centered data-matrix *X*0, that can be expressed as:

$$X\_0 = \mathbf{U} \mathbf{D} \mathbf{V}'\tag{11}$$

Where *K* ≤ min (*n*, *p*); *U*<sup>0</sup> *U* = *V*<sup>0</sup> *V* = *Ik*;

*D* is a diagonal matrix with *d*<sup>1</sup> > … *d*<sup>2</sup> > *d*<sup>k</sup> on the diagonal.

UD matrix constitutes principal component scores, which are variable coordinates in the case of principal components [3].

### **2. Independent component analysis (ICA)**

This method was developed to handle issues that occurred due to Blind Source Separation, abbreviated as BSS to form the components which must be as independent as possible [8] and can be represented mathematically:

$$X = A\,\,\mathfrak{s} + n\,\,\tag{12}$$

Where *X* is the observed signal, *n* is the noise, *A* is the mixing matrix, and s is the independent components (ICs) or sources. To find linear transformation *W* of *X*, for determining the independent outputs as:

$$
\mu = WX = WAs\tag{13}
$$

Where *u* is the estimated ICs and it is highly required that components must be statistically independent instead of a mixture.

After a thorough investigation and deep analysis and research work conclusion has been drawn that ICA provides much better results for de-noising [6]. A whole chapter has been devoted to describing ICA, which belongs to existing work in Single-Stage Artifact Removal Algorithm.
