# A Study on Sensitive Bands of EEG Data under Different Mental Workloads
## 3 Data Analysis Method
### Data Preprocessing Method  
Independent Component Analysis  
(1) Principle  
The noise-free model of ICA used for eliminating artifacts is represented by Figure 4, and the relationship can be expressed by Equation (1):  
\[x=AS \tag{1}\]  
where \(A\in R^{N\times M}\) is the signal transfer matrix, \(x\) is the \(N\)-dimensional observation vector, and \(S\) is the \(M\)-dimensional (\(N\geq M\)) original signal. Independent component analysis is to design a matrix \(W\) to find \(y=Wx\) and solve the independent components \(y\). We remove \(y^{\prime}\) by one or a part of the component of \(y\), which is represented by \(y^{\prime}=py\). Then \(x^{\prime}=Ay^{\prime}\) is restored, where \(x^{\prime}\) is the useful signal left after we eliminate the interference. It can be seen from the model that ICA estimates the source component \(S_{i}\) from the mixed signal \(x=(x_{1},x_{2},...,x_{n})\) and the mixing matrix \(A\) as well. It is based on statistically independent signals from different sources. (2) Description of the ICA issue  
Figure 4: Independent component analysis eliminates the artifact model. Figure 3: Part of the original EEG signal of the subject 5 under low load. Due to the statistically independent conditions of the source signal, it can be separated from the mixed signal, and the separated \(y(t)\) components are also independent of each other. Researchers in this field have proposed different criterias from different application perspectives. Currently, the criterias based on the minimization of mutual information (MMI) and the maximization of entropy are the most widely used. The basic theory is derived from the probability density function of independence. In actual work, the probability density is generally unknown and difficult to estimate. The commonly used method is to expand the probability density as a series and convert it into high-order statistics. The estimation of the high-order statistics, or the introduction of nonlinear links at the output to establish optimization criteria, also implies the pursuit of higher-order statistics. The principle of the criterion is explained by the minimum mutual information criterion. The minimum mutual information criterion is defined as: Let \(y\) be an \(M\)-dimensional random variable, \(p(y)\) be its probability density, \(p(y_{i})\) be the edge density of component \(I\) in \(y\), then the mutual information entropy of \(y\) is defined as:  
\[I(Py)=\int p(y)log\frac{p(y)}{\prod p(y_{i})}dy \tag{2}\]  
Obviously, \(p(y)=\prod\limits_{i=1}^{M}p(y_{i})\), \(I(p_{y})=0\) when it is assumed that the signal source is independent, that is, the components are independent of each other. Therefore, the mutual information is extremely small as a criterion for independent components. In practical applications, the Kullback-Leibler divergence between \(p(y)\) and \(\prod\limits_{i=1}^{M}p(y_{i})\) is generally used as a quantitative measure of the degree of independence:  
\[I(y)=KL[P(y),\prod\limits_{N=i=1}^{N=i=1}p_{i}(y_{i})] \tag{3}\]  
\[I(y)=\int p(y)log[\frac{P(Y)}{\prod\limits_{N=i=1}^{N=i=1}p_{i}(y_{i})}]dy \tag{4}\]  
Obviously, \(I(y)\geq 0\), when each component is independent, \(I(y)=0\). Therefore, the direct form of the minimum mutual information criterion is: looking for \(B\) under \(y=bx\), which is the minimum of \(I(y)\) of (3). In order to be practically usable, the probability density in \(I(y)\) needs to be expanded into a series. Since the entropy of the Gaussian distribution is the largest in the probability density distribution with equal covariance, the Gaussian distribution of covariance is commonly used as a reference standard in the expansion.