腦電波之小波轉換係數分佈的量化分析

Abstract

在分析具有隨時間變化之頻率成份(Time-varying frequency components)的訊號時，時間－頻率表示法(Time-Frequency representation)已被廣泛的使用。在本篇論文中，我們將利用一種時間 －頻率表示法，也就是小波轉換(Wavelet Transform)，來分析性質複雜 的腦波訊號(EEG Signals)。除了利用小波轉換來分析腦波訊號，我們要 進一步的對小波轉換所得到的係數來做量化分析。進行量化分析的工具分 別是奇異值分解(Singular Value Decomposition)與標準差等高線(STD Contour)分析。奇異值分解可將一個矩陣分解成三個矩陣，這三個矩陣分 別具有原矩陣的固有向量(eigenvectors)、固有值(eigenvalues)、與加 權係數的分佈。在本研究中，所分析的矩陣中的數值就是小波分析後的係 數。藉由對小波分析之係數做奇異值分解，我們可以從對應於主要固有值 的固有向量來得到小波分析係數中重要的特性與特徵。標準差等高線則可 以顯示出小波分析中時間－頻率平面波動變化的特性。標準差等高線是由 計算出每一個分割區域中小波分析係數的標準差，並畫出這些標準差的等 高線來完成。我們可以從等高線來觀察到腦波中nonstationary之現象並 研究其spatial correlation。另一方面，我們利用以奇異值分解與標準 差等高線為基礎的方法來顯現出多頻道腦波訊號中各個頻道間spatial correlation的特性。此一方法將有助於未來在腦波訊號上的研究。 Time-frequency representation is widely used in the area of analyzing non-stationary signals. In this thesis, we will introduce a time-frequency representation method, wavelet transform(WT), to analyze the EEG signals with complicated spectral properties. To further characterize the WT mapping, two methods are used to quantify results of WT analysis, singular value decomposition (SVD) and STD contour. SVD decomposes a matrix into three matrices including the eigenvalues, eigenvectors and weight distribution of the original matrix. In this research,entries of the original matrix are the WT coefficients. By applying SVD on thecoefficients, we can obtain the significant features in WT coefficients from the eigenvectors corresponding to dominant eigenvalues. The STD contour characterizes the fluctuations and vigorousness on the time-frequency plane of WT. The STD contours are accomplished by computing the STD of WT coefficients for each sectioned block and make a contour of these STD's. By means of contour plot, wemay observe the nonstationary characteristic of EEG signal and investigate the spatial correlations. To analyze multi-channel EEG signals, we develop an approach to characterizethe spatialcorrelation based on the SVD and STD contour analysis. This approachmay contribute to thefuture research on EEG signals.