Variants of the Common Spatial Patterns Method



(1)

where x h (k) is the discrete Hilbert transform of a signal x(k) [15]. The above convolution effectively introduces a 90∘ phase shift at each frequency component of the signal x(t).

For a discrete signal x(k), the complex-valued representation of the signal can be obtained by


$$\displaystyle{ z(k) = x(k) + jx_{h}(k) = A(k){\rm e}^{j\phi (k)} }$$

(2)
where the term on the right-hand side is the polar form representation of the analytic signal z(k). The instantaneous amplitude A(k) and instantaneous phase ϕ(k) components are computed as


$$\displaystyle{ A(k) = \sqrt{x{(k)}^{2 } + x_{h } {(k)}^{2}} }$$

(3)
and


$$\displaystyle{ \phi (k) =\arctan \left (\frac{x_{h}(k)} {x(k)} \right ), }$$

(4)
respectively. The analytic signal representation allows the computation of the instantaneous phase values from frequency bands determined by the user according to the spectral filtering carried out on the original signal x(k). In the case of EEG signals consisting of multiple frequency components, the phase values that result from the analytic signals are influenced by all the considered frequency components. In such a case, although the instantaneous amplitude and phase values are mathematically tractable, a physical interpretation of the underlying activity may be ambiguous [16].



1.2 Phase-Based Features for EEG Class Data Analysis


In recent years there has been a growing interest in the use of phase-based features for the analysis of EEG data as evidenced by the increasing number of publications on the subject matter [17–19]. However, with the exception of a few phase measures, the development of feature extraction methods that utilize explicit phase information in the EEG data remains largely unexplored [20, 21]. The phase-locking value (PLV), which quantifies the level of phase synchronization between two EEG channels, is one of the most widely used phase-based measures in EEG signal analysis [17, 22–25], and also forms the basis of our proposed P-CSP method [9, 10]. The details of the PLV measure are next described.


1.2.1 Single-Trial Phase-Locking Value


Developed by Lachaux et al. [26, 27], the PLV gives a measure of the phase synchronization between two signals. In contrast with classical measures of connectivity, such as the magnitude squared coherence (MSC), the PLV is not influenced by variations in the amplitudes of two signals but depends solely on the phase differences between the two considered signals.

The first step for determining the PLV between two signals, x(k) and y(k), involves the computation of their instantaneous phase values. This is given by


$$\displaystyle{ \phi _{xy}(k) =\phi _{x}(k) -\phi _{y}(k) }$$

(5)
where ϕ x (k) and ϕ y (k) are the phase values for x(k) and y(k), respectively, obtained using the Hilbert transform.

Assuming a single trial, the phase-locking value is computed along a moving window to obtain a dynamic value capturing the phase-locking variation within that trial. In such a case, the PLV can be applied at each time sample, k, by using a moving window of size T to obtain the single-trial PLV (S-PLV) [27]


$$\displaystyle{ \text{S-PLV}_{T}(k) = \left \vert \frac{1} {T}\sum _{j=k-{\frac{T} {2}} }^{k+\frac{T} {2} }{\rm e}^{j\phi _{xy}(j)}\right \vert }$$

(6)

The size of the considered time window is typically based on the frequency of the signals under consideration. Specifically, the window is chosen to incorporate a selected number of oscillatory cycles. When a large window size (and a large number of cycles) is considered, a smoother S-PLV is obtained at the cost of greater computational requirements. On the other hand, shorter time windows incorporate fewer oscillations and are more likely to lead to spurious phase locking arising by chance alone [27].



2 CSP Incorporating Phase Information


The CSP method is a data analysis technique for two-class discrimination yielding spatial patterns of potential interest [28, 29]. The CSP method works by decomposing multichannel EEG data into a set of uncorrelated components that can be used to optimally discriminate between two classes of data in the least-squares sense. The CSP method will be briefly outlined next, and subsequently our proposed “phase-synchronization”-based CSP (P-CSP) and analytic CSP (ACSP) methods that may be used to analyze phase information in EEG signals will be presented and discussed.


2.1 The CSP Method


Let X 1 , X 2  ∈ ℝ N×T represent EEG trials for two classes of data, where N is the number of EEG channels and T is the number of samples per channel. The N × N averaged covariance matrices obtained by averaging the covariance matrices across the trials for each class of data can be denoted by C 1 and C 2 , respectively. The CSP method can then be used to determine a set of spatial filters w that extremise the following generalized Rayleigh quotient [30, 31]


$$\displaystyle{ \frac{{\mathbf{ w}}^{\mathbf{ T}}\mathbf{ C}_{\mathbf{ 1}}\mathbf{ w}} {{\mathbf{ w}}^{\mathbf{ T}}\mathbf{ C}_{\mathbf{ 2}}\mathbf{ w}} }$$

(7)

By simultaneously diagonalizing the averaged covariance matrices C 1 and C 2 this expression can be extremised yielding a matrix $${\mathbf{ W}}^{\mathbf{ T}} ={ \left [\mathbf{ w}_{\mathbf{ 1}}\mathbf{ w}_{\mathbf{ 2}}\ldots \mathbf{ w}_{\mathbf{ N}}\right ]}^{\mathbf{ T}}$$  [29, 32]. Specifically, if $$\mathbf{ C}_{\mathbf{ c}} := \mathbf{ C}_{\mathbf{ 1}} + \mathbf{ C}_{\mathbf{ 2}}$$ , the eigenvalue decomposition of C c can then be represented by


$$\displaystyle{ \mathbf{ C}_{\mathbf{ c}} = \mathbf{ U}_{\mathbf{ c}}\boldsymbol{ \Lambda }_{\mathbf{ c}}\mathbf{ U}_{\mathbf{ c}}^{\mathbf{ T}} }$$

(8)
where the columns of U c are the eigenvectors corresponding to the eigenvalues in the diagonal matrix $$\boldsymbol{ \Lambda }_{\mathbf{ c}}$$ . C c is then whitened by $$\mathbf{ P} =\boldsymbol{ \Lambda }_{\mathbf{ c}}^{-\frac{1} {2} }\mathbf{ U}_{\mathbf{ c}}^{\mathbf{ T}}$$ such that


$$\begin{array}{lll} \mathbf{ I} & = \mathbf{ PC}_{\mathbf{ c}}{\mathbf{ P}}^{\mathbf{ T}} \\& = \mathbf{ P}\mathbf{ C}_{\mathbf{ 1}}{\mathbf{ P}}^{\mathbf{ T}} + \mathbf{ P}\mathbf{ C}_{\mathbf{ 2}}{\mathbf{ P}}^{\mathbf{ T}} \\& := \mathbf{ S}_{\mathbf{ 1}} + \mathbf{ S}_{\mathbf{ 2}} \end{array}$$

(9)
An eigenvalue decomposition is subsequently carried out on S 1 to obtain $$\mathbf{ U}_{\mathbf{ 1}}\boldsymbol{ \Lambda }_{\mathbf{ 1}}\mathbf{ U}_{\mathbf{ 1}}^{\mathbf{ T}}$$ , where U 1 and Λ 1 represent the eigenvector and eigenvalue matrices, respectively. Pre-multiplying and post-multiplying Eq. (9) by U 1 T and U 1 , respectively, results in


$$\begin{array}{lll} \mathbf{ I} & = \mathbf{ U}_{\mathbf{ 1}}^{\mathbf{ T}}\mathbf{ S}_{\mathbf{ 1}}\mathbf{ U}_{\mathbf{ 1}} + \mathbf{ U}_{\mathbf{ 1}}^{\mathbf{ T}}\mathbf{ S}_{\mathbf{ 2}}\mathbf{ U}_{\mathbf{ 1}} & \\\mathbf{ I} & =\boldsymbol{ \Lambda }_{\mathbf{ 1}} + \mathbf{ U}_{\mathbf{ 1}}^{\mathbf{ T}}\mathbf{ S}_{\mathbf{ 2}}\mathbf{ U}_{\mathbf{ 1}} & \\\mathbf{ I} -\boldsymbol{ \Lambda }_{\mathbf{ 1}} & = \mathbf{ U}_{\mathbf{ 1}}^{\mathbf{ T}}\mathbf{ S}_{\mathbf{ 2}}\mathbf{ U}_{\mathbf{ 1}} & \\\mathbf{ I} -\boldsymbol{ \Lambda }_{\mathbf{ 1}} & =\boldsymbol{ \Lambda }_{\mathbf{ 2}} &\end{array}$$

(10)

The corresponding real-valued diagonal elements in $$\boldsymbol{ \Lambda }_{\mathbf{ 1}}$$ and $$\boldsymbol{ \Lambda }_{\mathbf{ 2}}$$ must therefore add up to 1 and are ordered such that when the diagonal elements in $$\boldsymbol{ \Lambda }_{\mathbf{ 1}}$$ decrease, those in $$\boldsymbol{ \Lambda }_{\mathbf{ 2}}$$ increase, or vice versa. The projection matrix W T can then be obtained from


$$\displaystyle{{ \mathbf{ W}}^{\mathbf{ T}} = \mathbf{ U}_{\mathbf{ 1}}^{\mathbf{ T}}\mathbf{ P} }$$

(11)
where the first and last rows of W T extremise Eq. (7), thereby providing a projection that maximizes the variance for one class of data while minimizing the variance for the other. In addition to the spatial filters, W, a set of spatial patterns, $$\mathbf{ A} ={ \mathbf{ W}}^{-\mathbf{ T}}$$ , can also be obtained from the CSP method, where the columns of $$\mathbf{ A} = \left [\mathbf{ a}_{\mathbf{ 1}}\mathbf{ a}_{\mathbf{ 2}}\ldots \mathbf{ a}_{\mathbf{ N}}\right ]$$ represent a set of spatial patterns which consist of a reprojection of the most discriminative components onto the scalp electrode locations. The first and last columns of A can thus provide a spatial map of the discriminative EEG activity for the considered classes of data.


2.2 The P-CSP Method


In this section the details of the P-CSP proposed in [9, 10] are presented. The P-CSP method combines the discriminative feature extraction and dimensionality reduction properties of the CSP framework with phase synchronization measures from the single-trial PLV approach. The P-CSP method utilizes the CSP framework on phase synchronization signals determined by computing the S-PLV between EEG signals. The P-CSP algorithm can thus provide a set of features which are related to the most discriminative phase synchronization links in the data, without requiring prior knowledge or additional processing steps for channel pair selection.


2.2.1 Method


Given two classes of data, X 1 , X 2  ∈ ℝ N×T , where N is the number of EEG channels and T is the number of samples per channel, the first step of the P-CSP method involves the computation of S-PLV signals for each possible pairwise combination of channels. For this purpose, the real-valued EEG signals are first converted into analytic signals by using the Hilbert transform, and the instantaneous phase component is then extracted from the computed analytic signals. Subsequently, the instantaneous phase difference for each possible combination of channel pairs is determined such that for EEG data from N channels, $$\frac{N(N-1)} {2}$$ phase difference signals are obtained. These signals can in turn be used to determine S-PLV signals representing the varying level of phase locking in the considered data as described in Sect. 1.2.

While the conventional CSP method uses the covariance matrices of the original EEG recordings to extremise a generalized Rayleigh quotient, the P-CSP method makes use of the computed S-PLV signals to extremise this quotient. However, in contrast with the CSP method, where the mean value of the EEG signals is not of relevance, the S-PLV signals used in the P-CSP method carry significant information in their mean value which would be lost during the computation of the covariance matrices. Therefore, in order to retain the mean component of the S-PLV signals, the “second moment” matrices C(i, j) are computed from the signals [10] as follows:


$$\displaystyle{ \mathbf{ C}(i,j) = \frac{1} {T}\sum _{t=1}^{T}s_{ i}(t)s_{j}(t) }$$

(12)

Similar to the CSP method, the computed matrices are used as inputs for a simultaneous diagonalization procedure from which a transformation matrix W is obtained. The block diagram in Fig. 1 shows a representation of the steps involved in determining spatial filters using the P-CSP algorithm.

A303611_1_En_66_Fig1_HTML.gif


Fig. 1
The block diagram provides a representation of the P-CSP algorithm, where a set of spatial filters W that discriminate two classes of EEG data X 1 and X 2 based on the most discriminative phase synchronization information in the data are determined

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Jun 25, 2017 | Posted by in PATHOLOGY & LABORATORY MEDICINE | Comments Off on Variants of the Common Spatial Patterns Method

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