# Adaptive Feature Extraction of Motor Imagery EEG with Optimal Wavelet Packets and SE-Isomap
## 1 Introduction  
A brain computer interface (BCI) system is an emerging technology that is regarded as a new way for the human brain to control external devices without the neural pathway [1]. People can control external devices by imagining the relevant action. Currently, BCI-based rehabilitation therapy is focused mainly on the recognition of motor imagery EEG (MI-EEG) [2; 3; 4] that is collected when a subject performs a specific motion imagination. However, the high dimension of the MI-EEG signal, which leads to high computational cost, usually brings an obstacle to the implementation of the on-line recognition algorithm and adversely affects the classification accuracy of MI-EEG. Therefore, how to effectively extract the low-dimensional representation hidden in the high-dimensional dataset has been a hotspot in the BCI and machine learning field in recent years [5; 6; 7]. To solve the computational complexity and data storage problem caused by the high dimension of signals, many dimensionality reduction methods have been used in traditional BCI technology, such as principal component analysis (PCA), independent components analysis (ICA) and multidimensional scaling (MDS). PCA assumes that the samples are not correlated, and its main idea is to calculate a group of new features arranged in the descending order of importance from the original ones that are linear combinations [8]; ICA is similar to PCA, except that the components are designed to be independent of each other; and MDS is an unsupervised data dimensionality reduction method, in which efforts are taken to find the low-dimensional embedding that best preserves the pair-wise distances between the original data points [9]. All of these methods have been widely used in visualization and feature extraction, for they are well understood and easy to implement. Unfortunately, they have a common inherent limitation, i.e., they are all linear methods, while MI-EEG can be regarded as the output of the brain, which contains much nonlinear geometry information [10]. The features extracted by the methods will not be adequate and effective [11], and the classification accuracy will also be affected. The manifold learning theory provides a new way to solve the problem. In 2000, two nonlinear dimensionality reduction methods, isomap and locally linear embedding (LLE) algorithms, were proposed by [PERSON] et al. [12] and [PERSON] et al. [13]. They were published in the famous \"Science\" journal and created a new genre of manifold learning (ML) in machine learning. ML assumes that the high-dimensional spatial data observed in the real world are usually generated by a relatively small number of degrees of freedom [12], and the data can be embedded in a lower dimensional space according to its preserved geometry [14]. Since then, ML has begun to emerge in the BCI field. [PERSON] et al. [15] proposed a new kernel function method for the symmetric positive definite (SPD) matrix of the MI-EEG signal, and the isomap algorithm was applied to the dimension reduction of the SPD matrix. The KNN classifier was used to classify features, and the experiments yielded a promising result when compared with the most popular manifold learning methods, as well as the common spatial pattern (CSP) technique; [PERSON] et al. [16] calculated the power, covariance and a series of the entropy of the EEG signal to obtain its original features, and the LLE algorithm was implemented to reduce the feature dimension. The depth of anesthesia was estimated by quadratic discriminant analysis, and the average classification rate was 88.4%. [PERSON] et al. [10], who came from the Kharkevich Institute, proposed a method to reduce the dimension of the covariance matrix of the MI-EEG signal by using the isomap algorithm, and the concept of Riemannian geometry was introduced. The useful information hidden in MI-EEG was explored from the perspective of spatial structure, and the classification accuracy is as good as the state-of-the-art CSP algorithm.