# Adaptive Feature Extraction of Motor Imagery EEG with Optimal Wavelet Packets and SE-Isomap
## 1 Introduction
These manifold learning methods have achieved surprising results in the research of MI-EEG. However, they cannot obtain the explicit mapping relationship from the high-dimensional space to the low-dimensional space while acquiring low-dimensional embedding representations. Therefore, it is impossible to directly extract the features from a new sample when carrying out a pattern recognition task. To solve the problem and enhance the resolvability of multi-class datasets, [PERSON] et al. [14; 17] extended the original isomap and proposed the supervised explicit isomap (SE-isomap) algorithm, in which the geodesic distance matrix is calculated with respect to the class label information, and MDS with explicit transformation is adopted instead of the classical MDS used in the isomap. It is regretful that the SE-isomap has not been applied to the feature extraction of the MI-EEG signal. As we know, MI-EEG is non-stationary and time varying, and it has a clear time-frequency distribution. Both wavelet transform (WT) and wavelet packet decomposition (WPD) are common and effective time-frequency analysis methods [18; 19; 20], and WPD, which is an extension of WT, can simultaneously decompose the signal into high-frequency components and low-frequency components; thus, the range of time-frequency analysis is broadened, and more time-frequency domain information embedded in a signal is then obtained. Additionally, WPD has been proven to be able to obtain the dynamic characteristics of a signal effectively [21]. Consequently, WPD has attracted increasing attention from the BCI field in recent years. [PERSON] et al. [19] applied WPD to extracting the time-frequency features of MI-EEG, and the wavelet packet coefficients were input to the CSP algorithm to obtain a set of six-dimensional eigenvectors. The probabilistic neural network (PNN) was further used to classify the features, and the average recognition accuracy is 88.66%; [PERSON] et al. [22] extracted the MI-EEG time-frequency information by using WPD and then applied the dynamic frequency feature selection (DFFS) algorithm to select the sub-WPD coefficients associated with the imaging task. The random forests algorithm was used to evaluate the features, and the average classification accuracy from nine different subjects reached 84.06%. It is obvious that WPD has been playing an important role in the feature extraction of MI-EEG. However, the frequency range of each wavelet packet subspace (WPS) is relatively fixed; the same WPSs are selected for different subjects by using the above WPD-based methods, namely as the time-frequency features; the WPD coefficients are covered by the same frequency ranges. This is not good for the extraction of subject-specific features and results in the poor adaptability of WPD-based feature extraction methods. In this paper, an adaptive feature extraction method is proposed for MI-EEG based on WPD and SE-isomap. The optimal wavelet packets (OWPs), reflecting the subject-based features, are autonomously selected, and the coefficients of OWPs are utilized to statistically calculate time-frequency features. In the meantime, SE-isomap is applied to obtain the nonlinear manifold structure features, as well as the explicit nonlinear mapping. The combined features show the perfect performance in classification accuracy and computation efficiency. The rest of the paper is organized as follows: In Section 2, the basic principles of WPD, the SE-isomap algorithm and explicit-MDS are briefly introduced. Section 3 describes the feature extraction method with WPD and SE-isomap algorithm in detail. In the following section, extensive experiments are conducted on a publicly available dataset. Section 5 concludes the paper.