Phase Variants of the Common Spatial Patterns Method

Kenneth P. Camilleri, Owen Falzon, Tracey Camilleri, and Simon G. Fabri

Abstract

Numerous studies on EEG data indicate that various brain processes are characterized by phaserelationships between different regions of the brain. The development of analytic techniques that canprovide a better detection of these phase relationships can lead to a better understanding of such brainprocesses. In this chapter two variants of the common spatial patterns (CSP) method which are designed tocapture such phase relationships, namely, the “phase synchronization”-based CSP (P-CSP) algorithm andthe analytic CSP (ACSP) algorithm, are presented. The P-CSP and ACSP methods are analyzed and testedon real EEG data. The nature of the results of the two methods is then discussed, highlighting thedifferences between them.

Key words Electroencephalogram (EEG), Brain–computer interfacing (BCI), Phase synchronization,Common spatial patterns (CSP), Phase-locking value (PLV), Spatial filtering, Steady-state visualevoked potential (SSVEP), Analytic common spatial patterns (ACSP), “Phase synchronization”-based common spatial patterns (P-CSP)

1 Introduction

Significant phase relationships within and between different regionsof the brain have been observed in electroencephalographic (EEG)data recorded during various mental states [1–8].

In [1], Rodriguez et al. detected consistent patterns of phasesynchronization during a visual perception task. Specifically, sub-jects were presented with ambiguous visual stimuli that could beperceived as either faces or meaningless shapes. During instances offace recognition, consistent phase synchronization links weredetected between the occipital, parietal, and frontotemporal areas.In another visual processing study carried out by Busch et al. [2, 3],subjects were presented with brief flashes of light, the luminance ofwhich was thresholded such that subjects could detect approxi-mately half of the visual stimuli. In this case, a correlation betweenthe detection of the visual stimuli and the phase of EEG signals inthe theta and alpha bands was found. Most of the detectioninstances were found to occur at a particular phase value for eachsubject, and minimal detection was found to take place at the

NeuromethodsDOI 10.1007/7657_2013_66# Springer Science+Business Media New York 2014

opposite phase, indicating that the hits and misses were correlatedwith opposite phase values.

A number of studies indicate that the disruption of normalphase synchronization patterns is linked to abnormal mental states.In [4], Bhattacharya analyzed the EEG signals of individualssuffering from seizures and mania and observed a significant reduc-tion in the phase synchronization levels of pathological subjectswhen compared to a control group. Mormann [5] also detected adecrease in phase synchronization activity across EEG recordingsites in epileptic patients prior to seizure onset. In another study byGysels et al. [6], large variations in the phase synchronization levelwere observed in the pathological brain hemisphere of epilepticpatients when the subjects transitioned from the awake state tothe sleep state.

Ito et al. [7] also reported strong phase synchronization activitybetween EEG signals recorded from the frontal and occipital areasof the brain during an eyes closed resting activity. During theseintervals, two spatial phase patterns emerged repeatedly across thesubjects considered in the study. One of these patterns consisted ofgradual phase variations, characteristic of travelling waves, movingfrom the posterior to the anterior region of the scalp or vice versa.The second typical spatial phase pattern that was observed exhib-ited sudden phase variations of approximately 180∘, characteristicof a standing wave, between electrode sites C3 and FC3. Similarstanding and travelling wave phenomena, distinguished by suddenand gradual spatial phase changes, respectively, have also beenobserved in EEG recordings obtained from subjects who wereexposed to repetitive flashing visual stimuli [8].

These studies indicate that several brain processes are charac-terized by consistent spatial phase relationships in EEG signals.Therefore, analytic techniques that can provide a better detectionof phase relationships in EEG activity may lead to a better under-standing of various brain processes. In this chapter, two variants ofthe widely used common spatial patterns (CSP) method are dis-cussed. The proposed variants consist of the “phase synchroniza-tion”-based CSP (P-CSP) algorithm [9, 10] and the analytic CSP(ACSP) algorithm [11–13], each of which takes into account adifferent aspect of the phase relationships in the EEG data. In thischapter, these two algorithms are discussed, with a focus on thedifferent information that the two methods can provide in relationto EEG data analysis.

1.1 Phase Estimation

from EEG Signals

Phase components can be extracted from EEG signals using thediscrete Hilbert transform [14], such that

xhðkÞ ¼ 1

π

X1l¼�1

xðlÞ½1� ejπðk�lÞ�k � l

for l 6¼ k (1)

Kenneth P. Camilleri et al.

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

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

zðkÞ ¼ xðkÞ þ jxhðkÞ ¼ AðkÞejϕðkÞ (2)

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

AðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðkÞ2 þ xhðkÞ2

q(3)

and

ϕðkÞ ¼ arctanxhðkÞxðkÞ

� �; (4)

respectively. The analytic signal representation allows the computa-tion of the instantaneous phase values from frequency bands deter-mined by the user according to the spectral filtering carried out onthe original signal x(k). In the case of EEG signals consisting ofmultiple frequency components, the phase values that result fromthe analytic signals are influenced by all the considered frequencycomponents. In such a case, although the instantaneous amplitudeand phase values are mathematically tractable, a physical interpreta-tion 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 theincreasing number of publications on the subject matter [17–19].However, with the exception of a few phase measures, the develop-ment of feature extraction methods that utilize explicit phase infor-mation in the EEG data remains largely unexplored [20, 21]. Thephase-locking value (PLV), which quantifies the level of phase syn-chronization between two EEG channels, is one of the most widelyused phase-based measures in EEG signal analysis [17, 22–25], andalso forms the basis of our proposed P-CSP method [9, 10]. Thedetails 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 ofthe phase synchronization between two signals. In contrast withclassical measures of connectivity, such as the magnitude squaredcoherence (MSC), the PLV is not influenced by variations in theamplitudes of two signals but depends solely on the phase differ-ences between the two considered signals.

Phase Variants of the Common Spatial Patterns Method

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

ϕxyðkÞ ¼ ϕxðkÞ � ϕyðkÞ (5)

where ϕx(k) and ϕy(k) are the phase values for x(k) and y(k), respec-tively, obtained using the Hilbert transform.

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

S-PLVT ðkÞ ¼ 1

T

XkþT2

j¼k�T2

ejϕxy ðjÞ

������������ (6)

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

2 CSP Incorporating Phase Information

The CSP method is a data analysis technique for two-class discrimi-nation yielding spatial patterns of potential interest [28, 29]. TheCSP method works by decomposing multichannel EEG data into aset of uncorrelated components that can be used to optimallydiscriminate between two classes of data in the least-squaressense. The CSP method will be briefly outlined next, and subse-quently our proposed “phase-synchronization”-based CSP(P-CSP) and analytic CSP (ACSP) methods that may be used toanalyze phase information in EEG signals will be presented anddiscussed.

2.1 The CSP Method Let X1, X2 2 ℝN�T represent EEG trials for two classes of data,where N is the number of EEG channels and T is the number ofsamples per channel. The N � N averaged covariance matricesobtained by averaging the covariance matrices across the trials foreach class of data can be denoted byC1 andC2, respectively. TheCSPmethod can then be used to determine a set of spatial filters w thatextremise the following generalized Rayleigh quotient [30, 31]

Kenneth P. Camilleri et al.

wTC1w

wTC2w(7)

By simultaneously diagonalizing the averaged covariance matri-ces C1 and C2 this expression can be extremised yielding a matrix

WT ¼ w1 w2 . . . wN½ �T [29, 32]. Specifically, ifCc :¼ C1 þC2, theeigenvalue decomposition of Cc can then be represented by

Cc ¼ UcΛcUTc (8)

where the columns ofUc are the eigenvectors corresponding to theeigenvalues in the diagonal matrix Λc. Cc is then whitened by

P ¼ Λ�12

c UTc such that

I ¼ PCcPT

¼ PC1PT þ PC2P

T

:¼ S1 þ S2

(9)

An eigenvalue decomposition is subsequently carried out on S1 toobtain U1Λ1U

T1 , where U1 and Λ1 represent the eigenvector and

eigenvalue matrices, respectively. Pre-multiplying and post-multiplying Eq. (9) by U1

T and U1, respectively, results in

I ¼ UT1S1U1 þUT

1S2U1

I ¼ Λ1 þUT1S2U1

I� Λ1 ¼ UT1S2U1

I� Λ1 ¼ Λ2

(10)

The corresponding real-valued diagonal elements in Λ1 and Λ2

must therefore add up to 1 and are ordered such that when thediagonal elements in Λ1 decrease, those in Λ2 increase, or viceversa. The projection matrix WT can then be obtained from

WT ¼ UT1P (11)

where the first and last rows of WT extremise Eq. (7), therebyproviding a projection that maximizes the variance for one class ofdata while minimizing the variance for the other. In addition to thespatial filters, W, a set of spatial patterns, A ¼ W�T, can also beobtained from the CSP method, where the columns ofA ¼ a1 a2 . . . aN½ � represent a set of spatial patterns which consistof a reprojection of the most discriminative components onto thescalp electrode locations. The first and last columns of A can thusprovide a spatial map of the discriminative EEG activity for theconsidered classes of data.

2.2 The P-CSP

Method

In this section the details of the P-CSP proposed in [9, 10] arepresented. The P-CSP method combines the discriminative featureextraction and dimensionality reduction properties of the

Phase Variants of the Common Spatial Patterns Method

CSP framework with phase synchronization measures from thesingle-trial PLV approach. The P-CSP method utilizes the CSPframework on phase synchronization signals determined by com-puting the S-PLV between EEG signals. The P-CSP algorithm canthus provide a set of features which are related to the most discrim-inative phase synchronization links in the data, without requiringprior knowledge or additional processing steps for channel pairselection.

2.2.1 Method Given two classes of data, X1, X2 2 ℝN�T, where N is the numberof EEG channels and T is the number of samples per channel, thefirst step of the P-CSP method involves the computation of S-PLVsignals for each possible pairwise combination of channels. For thispurpose, the real-valued EEG signals are first converted into ana-lytic signals by using the Hilbert transform, and the instantaneousphase component is then extracted from the computed analyticsignals. Subsequently, the instantaneous phase difference for eachpossible combination of channel pairs is determined such that for

EEG data from N channels, N ðN�1Þ2 phase difference signals are

obtained. These signals can in turn be used to determine S-PLVsignals representing the varying level of phase locking in the con-sidered data as described in Sect. 1.2.

While the conventional CSP method uses the covariance matri-ces of the original EEG recordings to extremise a generalizedRayleigh quotient, the P-CSP method makes use of the computedS-PLV signals to extremise this quotient. However, in contrast withthe CSP method, where the mean value of the EEG signals is not ofrelevance, the S-PLV signals used in the P-CSP method carrysignificant information in their mean value which would be lostduring the computation of the covariance matrices. Therefore, inorder to retain the mean component of the S-PLV signals, the“second moment” matrices C(i, j) are computed from the sig-nals [10] as follows:

Cði; jÞ ¼ 1

T

XTt¼1

siðtÞsj ðtÞ (12)

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

From the computed spatial filters for the P-CSP method, a setof spatial patterns A ¼ W�T can also be derived. These constitutepatterns of phase synchronization which can yield information onthose synchronization links associated with the most discriminativecomponents for the considered classes.

Kenneth P. Camilleri et al.

2.3 The Analytic

Common Spatial

Patterns Method

Another variant of the CSP method that considers phase informa-tion in the EEG data is the analytic common spatial patterns(ACSP) method proposed in [11–13]. The ACSP method consid-ers an analytic representation of the EEG signals in order to obtainan explicit representation of the amplitude and phase componentsof the data, thereby addressing the limitations of the conventionalCSP technique when dealing with phase relationships betweenEEG signals from different spatial locations.

2.3.1 Method The P-CSP method presented earlier was designed to represent thelevel of phase-locking between channels without taking intoaccount signal amplitude and without providing informationabout the actual phase differences between the channels. TheACSP method [11] we have proposed follows the standard CSPmethod [29], with the distinct difference that the input signals aretransformed into their analytic counterparts and the joint diagona-lization process is then computed on the resulting complex-valuedcovariance matrices. The ACSP method thus involves carrying outthe joint diagonalization computations with complex-valued matri-ces rather than the real-valued matrices that are typical for thestandard CSP implementation. It is this complex representationand the resulting complex-valued spatial patterns and spatial filtersthat lead to a number of advantages of the proposed technique overthe standard CSP method. In particular, this complex representa-tion allows for the representation of class-specific phase differencespatial maps apart from the magnitude spatial maps typical for CSP.

Similar to the CSP method, the scope of the ACSP method isto discriminate between two classes of data by determining a set ofspatial filters that maximize the variance for one class of data whileminimizing the variance for the other. However, the first step in the

The P-CSP Algorithm

EEG dataClass 1

X1

S-PLV signalsClass 1

S1

S-PLVClass 1 data

‘second moment’matrixC1 Simultaneous

diagonalisation of

C1,C2

Spatial filters

W

EEG dataClass 2

X2

S-PLV signalsClass 2

S2

S-PLV Class 2 data

‘second moment’matrixC2

Fig. 1 The block diagram provides a representation of the P-CSP algorithm, where a set of spatial filtersW thatdiscriminate two classes of EEG data X1 and X2 based on the most discriminative phase synchronizationinformation in the data are determined

Phase Variants of the Common Spatial Patterns Method

ACSP procedure involves a conversion of the real-valued EEGsignals into complex-valued signals such that the EEG trials X1

and X2 are transformed into the complex-valued trials Z1 and Z2.This can be achieved by computing the Hilbert transform of theEEG signals filtered within the frequency band of interest. Thecovariance matrix of the complex-valued EEG data X1 is thengiven by

C1 ¼ E Z1 � E½Z1�ð Þ Z1 � E½Z1�ð Þ�½ � (13)

where E[�] represents the expectation operator. The same proce-dure can be applied for Z2 to obtain C2. Thus, in this case C1 andC2 constitute a pair of N � N Hermitian positive semi-definitematrices, with real-valued elements along the main diagonal andgenerally complex-valued off-diagonal elements.

For the ACSP method, the problem consists in finding a vectorw that extremises

w�C1w

w�C2w(14)

where the principal difference from the formulation in (7) for theCSPmethod lies in the complex-valued covariance matrices that areused and the complex-valued spatial filters w∗ that are determinedthrough a complex-valued implementation of the simultaneousdiagonalization approach for the ACSP method [11, 12]. A projec-tion matrix W� ¼ w1 w2 . . . wN½ �� is obtained where the rows ofW∗ correspond to a set of complex-valued spatial filters that maxi-mize the variance for one class of data and minimize the variance forthe other class. The block diagram in Fig. 2 shows a representationof the steps involved in determining spatial filters using the ACSPalgorithm.

The ACSP Algorithm

EEG DataClass 1

X1

Analytic signalsClass 1

Z1

Complex-valuedcovariance matrix

Class1C1

Complex-valuedcovariance matrix

Class2C2

Simultaneousdiagonalisation of

C1,C2

Complex-valuedSpatial filters

WEEG Data

Class 2

X2

Analytic signalsClass 2

Z2

Fig. 2 The block diagram provides a representation of the ACSP algorithm where a set of complex-valuedspatial filtersW that discriminate two classes of EEG data X1 and X2 by considering an analytic representationof the data are determined

Kenneth P. Camilleri et al.

In addition to these spatial filters W∗, a set of complex-valuedspatial patterns, A ¼ ðW�Þ�1 can also be determined from theACSP method. Since for the ACSP method, the matrix A is acomplex-valued array, it may be fashioned into a magnitude and aphase array. The spatial patterns associated with the most discrimi-native components, if cautiously considered, may thus give usefulneurophysiological insight into the discriminative brain activityrelated to a given set of tasks [30] as described in the discussionin Sect. 3.2.

3 Results and Discussion

While the P-CSP and ACSP methods have been used for two-classEEG discrimination tasks, such as in brain–computer interfaces,this contribution focuses on the spatial patterns yielded by thetwo methods.

3.1 P-CSP Scalp

Patterns for Real EEG

Data

The P-CSP method was applied to motor imagery data to obtain atransformation that maximizes the difference between two motorimagery classes based on the PLV in order to obtain spatial patternsthat distinctly characterize each class. Specifically, the P-CSP algo-rithm was tested on EEG data obtained from BCI Competition IVdataset IIa [33]. EEG recordings from 22 scalp locations from ninesubjects, sampled at 250 Hz were considered. For each subject,data from 72 left and 72 right imagined hand movements wereanalyzed. The electrode placement and experimental protocol usedfor data acquisition are shown in Fig. 3. Additional details of therecording protocol can be found in [33].

Fig. 3 The recording setup with 22 scalp electrodes. A Laplacian filter was applied to reduce volumeconduction and reference effects. Boundary electrodes (shaded) were therefore excluded and the tenremaining electrodes were used for further processing. (Diagrams adapted from [33])

Phase Variants of the Common Spatial Patterns Method

A spatial Laplacian filter was applied to the EEG data in orderto reduce reference signal [34, 35] and volume conduction [36]effects, both of which can lead to spurious phase synchronizationlevels and misleading interpretations of results. Due to the Lapla-cian filtering, edge electrodes were subsequently ignored and sig-nals from the ten remaining electrodes were considered for furtherprocessing.

An FIR least-squares filter of order 64, with forward andreverse filtering (to avoid phase distortion), was then used to band-pass filter the signals to the frequency range of 8–28 Hz. Themethods tested in this study were applied to the broadband filteredEEG signals and subsequently repeated on data filtered at sub-bands of 4 Hz each. The broadband data gave the best discrimina-tion performance in terms of classification accuracy and thereforeonly results from the broadband EEG data are considered here.From each trial a 2 s interval starting 0.5 s after cue onset (referto Fig. 3) was considered for further analysis using the CSP andP-CSP methods. This interval was selected based on thebest-performing algorithm in BCI Competition IV [33] for thisdataset. For the P-CSPmethod, a 0.5 s moving windowwas used tocompute the S-PLV signals.

3.1.1 CSP Results Figure 4 shows the extreme spatial patterns, a1 and aN, obtainedfrom the CSP method, corresponding to the left- and right-handmotor imagery tasks for the three best-performing subjects (S3, S8,and S9).

A higher weighting is observed over the C3 area for the left-hand motor imagery task for Subject S3. For the other two subjectsa higher weighting is observed over Cz with a lower weightingexhibited on the posterior right-hand side of the scalp. The increas-ing amplitude over the right posterior areas of the scalp maps forthe latter two subjects is mainly due to extrapolation effects.For the right-hand motor imagery task a strong coefficient isnoted on the right-hand side of the scalp over C4 for subjects S3and S9, with the focus shifting slightly posteriorly for subject S8. Inthe case of subjects S8 and S9 a mild central component is alsoobserved for the right-hand motor imagery patterns. Although anyinterpretation of these spatial patterns should be carried out withcaution, the resulting patterns appear to be related to ERD/ERSactivity that are typically observed over the sensorimotor area dur-ing motor imagery tasks [37].

3.1.2 P-CSP Results The four highest coefficients of the phase synchronization patternsobtained from the P-CSP method for the best-performing subjects(also S3, S8, and S9 in this case) are shown in Fig. 5. The computedphase synchronization patterns associated with the most discrimi-native components exhibit synchronization links that are markedlyconsistent across the subjects. For left-hand motor imagery data,

Kenneth P. Camilleri et al.

strong interactions are obtained for channel pairs 4–9, 6–9, and8–10 across most subjects. These prominent links are in fact presentin the highest weighted phase synchronization links for the threebest-performing subjects, shown in Fig. 5. Similarly, for right-handmotor imagery, consistent phase synchronization patterns wereagain determined across most subjects with strong links showingup repeatedly between channels 3–7 and 8–10 in the resultingpatterns. The main differentiating characteristic between the twoclasses thus appears to consist of the contralateral synchronizationlinks in the phase synchronization patterns corresponding to eachmotor imagery task. These links are consistent with a discriminativecontralateral phase synchronization during the considered motorimagery tasks.

The P-CSP method provides an efficient way to analyze thePLV between all channels and obtain a maximally discriminanttransformation allowing for the reduction of the dimensionality ofthe data. It is of particular interest as well that the P-CSP method

Fig. 4 Spatial patterns obtained from the CSP method for the three best-performing subjects, namely, S3, S8, and S9. The absolute values for the spatialpatterns are displayed. The patterns on the left relate to the most significantactivity related to imagined left-hand movement, while the ones on the right arethose associated with imagined right-hand movements

Phase Variants of the Common Spatial Patterns Method

Fig. 5 The phase synchronization patterns obtained from the P-CSP method for the three best-performingsubjects, namely, S3, S8, and S9. For clarity the figures above only show the links related to the channel pairsassociated with the four highest phase synchronization pattern coefficients. A number of links related to eachclass of data show up consistently across subjects indicating the likeliness of common underlying synchroni-zation processes taking place across the subjects

Kenneth P. Camilleri et al.

provides a graph of phase-lock-related measures that are distinct foreach class of data. The method thus complements the informationobtained from the CSP method by providing phase-lock-derivedfeature graphs. However, the P-CSP method does not provide anyinformation regarding the actual phase differences between chan-nels. Conversely, the ACSP method was designed to extend theCSP directly into the complex domain, thus obtaining separatemagnitude and phase spatial maps.

3.2 ACSP Scalp

Patterns for Real EEG

Data

The ACSP method was applied to real EEG data in order todiscriminate between left-hand and right-hand motor imagerydata. The spatial patterns that result from the ACSP and CSPmethod are analyzed and compared.

3.2.1 Methodology The ACSP algorithm was tested on motor imagery EEG dataobtained from BCI Competition III, Dataset IIIa [38]. Datafrom three subjects, L1, K3, and K6, performing imagined left-hand and imagined right-hand movements were considered. TheEEG signals were sampled at 250 Hz and obtained from 60 elec-trodes according to the recording protocol shown in Fig. 6 [38]. Inour work the data were bandpass filtered from 8 to 30 Hz foranalysis purposes. Further details on the recording protocol andpreprocessing steps used in this study are provided in [11].

3.2.2 Results and

Discussion

Considering the magnitude spatial patterns from the CSP method,and the separate magnitude and phase spatial patterns obtainedusing the ACSPmethod for Subject K3 shown in Fig. 7, it is evidentthat the spatial patterns from the CSP method and the amplitudecomponent for the complex-valued ACSP spatial patterns exhibitsimilar activity. For both methods, the patterns associated with left-hand motor imagery indicate an increased activity around C3, whilethe patterns associated with right-hand motor imagery indicate

Fig. 6 The positions of the scalp EEG electrodes and the recording protocol for the motor imagery experiment.The diagrams are adapted from the description notes of Dataset IIIa from BCI Competition 2005 [38]

Phase Variants of the Common Spatial Patterns Method

increased activity around C4. These spatial patterns agree with theresults obtained in similar motor imagery experiments reported inthe literature involving the application of the CSP method onmotor imagery data [28, 30]. The magnitude patterns obtainedin fact indicate a relative increase in EEG variance over the ipsilat-eral hemisphere due to event-related desynchronization (ERD) onthe contralateral hemisphere.

In addition to the magnitude patterns, phase patterns have alsobeen extracted from the complex-valued spatial patterns obtainedfrom the ACSPmethod. The phase components that result indicateabrupt phase variations of approximately 180∘ spanning from thecontralateral sensorimotor region to the frontal region possiblyrelated to the event-related desynchronization taking place in thecontralateral area. In contrast, the ipsilateral sensorimotor areasexhibited much smoother phase variations possibly reflecting thelack of desynchronization in this region. Any interpretation of thesephase maps should, however, be treated with caution consideringthat they were obtained from broadband data and may therefore beinfluenced by a number of different frequency components. To ourknowledge no studies have taken into account such phase maps inthe context of motor imagery; most studies considering phase in

Fig. 7 The spatial patterns associated with the most discriminative components obtained from the CSP andACSP methods for Subject K3. The top patterns associated with the left-hand imagery task, while the bottompatterns are associated with the right-hand motor imagery task. The angles for the phase components are inradians

Kenneth P. Camilleri et al.

motor imagery were concerned with phase synchronizationbetween channels [22, 24] rather than the phases of the signalsthemselves. Studies on signal phase have, however, been reportedregarding steady state visual evoked potentials (SSVEPs) in [8, 36]where travelling and standing wave patterns were observed. Whenthe ACSP method was applied to SSVEP data, phase patternsexhibiting posterior to anterior phase changes consistent with thephase changes reported in [8, 36] were obtained. Further details onthis application of ACSP may be found in [12, 13].

The ACSP method therefore provides phase difference spatialmaps which are complementary to the P-CSP in terms of its classphase discriminatory information. In cases where class phase dis-criminatory information exists, the CSP attempts to capture thisimplicitly in the magnitude spatial patterns, generally confoundingthe patterns. The complex-valued nature of the ACSP methodkeeps the magnitude and phase characteristics distinct, thus avoid-ing this problem and providing two separate, but complementaryspatial maps to represent the distinct magnitude and phase class-discriminatory patterns.

4 Concluding Remarks

In this chapter we have reviewed two recent adaptations of the CSPmethod that, similar to the CSP method, yield spatial maps thatmay provide class discriminatory characteristics. The P-CSPmethod consists of a heuristic approach based on PLV featuresthat provides class-specific phase-lock-derived spatial graphs,whereas the ACSP method consists of a natural complex extensionof the CSP method that provides a class-discriminatory phase mapin addition to the class-discriminatory magnitude map. In a sense,these two methods complement each other by providing spatialpatterns which may give class discriminatory information that isrelated to the channel phase differences in the case of the ACSPmethod and to the degree of phase-lock between channels in thecase of the P-CSP method.

References

1. Rodriguez E, George N, Lachaux JP, Martin-erie J, Renault B, Varela FJ (1999) Perception’sshadow: long-distance synchronization ofhuman brain activity. Nature 397(6718):430–433

2. Busch NA, Dubois J, VanRullen R (2009) Thephase of ongoing EEG oscillations predictsvisual perception. J Neuroscience 29(24):7869–7876

3. Busch NA, VanRullen R (2010) SpontaneousEEG oscillations reveal periodic sampling of

visual attention. Proc Natl Acad Sci 107(37):16048–16053

4. Bhattacharya J (2001) Reduced degree oflong-range phase synchrony in pathologicalhuman brain. Acta Neurobiol Exp 61(4):309–318

5. Mormann F (2003) Synchronization phenom-ena in the human epileptic brain. Ph.D. disser-tation, University of Bonn, March 2003

6. Gysels E, Le Van Quyen M, Martinerie J, BoonP, Vonck K, Lemahieu I, Van De Walle R

Phase Variants of the Common Spatial Patterns Method

(2002) Long-term evaluation of synchroniza-tion between scalp EEG signals in partial epi-lepsy. In: Proceedings of the 9th internationalconference on neural information processing,2002. ICONIP ’02, vol 3, pp 1495–1498

7. Ito J, Nikolaev AR, Van Leeuwen C (2005)Spatial and temporal structure of phase syn-chronization of spontaneous alpha EEG activ-ity. Biol Cybern 92(1):54–60

8. Burkitt G, Silberstein R, Cadusch P, Wood A(2000) Steady-state visual evoked potentialsand travelling waves. Clin Neurophysiol 111(2):246–258

9. Falzon O, Camilleri KP (2009) An algorithmfor brain computer interfacing based on phasesynchronization spatial patterns. In: Thirdinternational conference on advanced engi-neering computing and applications insciences, ADVCOMP ’09, October 2009,pp 142–147

10. Falzon O, Camilleri KP (2012) Phase synchro-nization features and common spatial patternsfor the classification of motor imagery data. In:The Ninth IASTED international conferenceon biomedical engineering, BIOMED 2012,2012

11. Falzon O, Camilleri KP, Muscat J (2010)Complex-valued spatial filters for task discrimi-nation. In: Proceedings of the 2010 annualinternational conference of the IEEE engineer-ing in medicine and biology society (EMBS),September 2010, pp 4707–4710

12. Falzon O, Camilleri KP, Muscat J (2012b) Theanalytic common spatial patterns method forEEG-based BCI data. J Neural Eng 9(4):045009

13. Falzon O, Camilleri KP, Muscat J (2012c)Complex-valued spatial filters for SSVEP-based BCIs with phase coding. IEEE TransBiomed Eng 59(9):2486–2495

14. Gold B, Oppenheim AV, Rader CM (1969)Theory and implementation of the discreteHilbert transform. In: Symposium on com-puter processing in communications, Polytech-nic Institute of Brooklyn, 1969, pp 235–250

15. Cohen L (1995) Time-frequency analysis,Chap 2. Prentice Hall, New Jersey, p 30–36

16. Pockett S, Bold GE, Freeman WJ (2009) EEGsynchrony during a perceptual-cognitive task:Widespread phase synchrony at all frequencies.Clin Neurophysiol 120(4):695–708

17. Krusienski DJ, McFarland DJ, Wolpaw JR(2012) Value of amplitude, phase, and coher-ence features for a sensorimotor rhythm-basedbrain-computer interface. Brain Res Bull87:130–134

18. Wei Q, Wang Y, Gao X, Gao S (2007) Ampli-tude and phase coupling measures for featureextraction in an EEG-based brain-computerinterface. J Neural Eng 4(2):120–129

19. Townsend G, Graimann B, Pfurtscheller G(2006) A comparison of common spatial pat-terns with complex band power features in afour-class BCI experiment. IEEE TransBiomed Eng 53(4):642–651

20. Krusienski DJ, Grosse-Wentrup M, Galan F,Coyle D, Miller KJ, Forney E, Anderson CW(2011) Critical issues in state-of-the-art brain-computer interface signal processing. J NeuralEng 8(2):025002

21. Bashashati A, Fatourechi M, Ward RK, BirchGE (2007) A survey of signal processing algo-rithms in brain-computer interfaces based onelectrical brain signals. J Neural Eng 4(2):R32–R57

22. Wang Y, Hong B, Gao X, Gao S (2006) Phasesynchrony measurement in motor cortex forclassifying single-trial EEG during motor imag-ery. In: Proceedings of the 28th IEEE EMBSannual international conference, 2006,pp 75–78

23. Besserve M, Jerbi K, Laurent F, Baillet S, Mar-tinerie J, Garnero L (2007) Classificationmethods for ongoing EEG and MEG signals.Biological Res 40:415–437

24. Gysels E, Celka P (2004) Phase synchroniza-tion for the recognition of mental tasks in abrain-computer interface. IEEE Trans NeuralSyst Rehabil Eng 12(4):406–415

25. Brunner C, Scherer R, Graimann B, Supp G,Pfurtscheller G (2006) Online control of abrain-computer interface using phase synchro-nization. IEEE Trans Biomed Eng 53(12):2501–2506

26. Lachaux JP, Rodriguez E, Martinerie J, VarelaFJ (1999) Measuring phase synchrony in brainsignals. Hum Brain Mapp 8(4):194–208

27. Lachaux J-P, Rodriguez E, Quyen MLV, LutzA, Martinerie J, Varela FJ (1999) Studyingsingle-trials of phase synchronous activity inthe brain. Int J Bifurcat Chaos 10(10):2429–2439

28. Ramoser H, M€uller-Gerking J, Pfurtscheller G(2000) Optimal spatial filtering of single trialEEG during imagined hand movement. IEEETrans Biomed Eng 8(4):441–446

29. M€uller-Gerking J, Pfurtscheller G, Flyvbjerg H(1999) Designing optimal spatial filters forsingle-trial EEG classification in a movementtask. Clin Neurophysiol 110(5):787–798

30. Blankertz B, Tomioka R, Lemm S, KawanabeM, M€uller K-R (2008) Optimizing spatial

Kenneth P. Camilleri et al.

filters for robust EEG single-trial analysis.IEEE Signal Process Mag 25(1):41–56

31. Lotte F, Guan C (2011) Regularizing commonspatial patterns to improve BCI designs: Uni-fied theory and new algorithms. IEEE TransBiomed Eng 58(2):355–362

32. Koles ZJ, Lazar MS, Zhou SZ (1990) Spatialpatterns underlying population differences inthe background EEG. Brain Topogr2:275–284

33. BCI Competition IV Data set IIa (2012) Dataset provided by the institute for knowledge dis-covery (laboratory of brain-computerinterfaces), Graz University of Technology,(Clemens Brunner, Robert Leeb, GernotM€uller-Putz, Alois Schlogl, Gert Pfurtscheller).http://www.bbci.de/competition/iv/desc_2a.pdf Last accessed: June 4, 2012

34. Guevara R, Velazquez JL, Nenadovic V, Wenn-berg R, Senjanovic G, Dominguez LG (2005)Phase synchronization measurements using

electroencephalographic recordings: what canwe really say about neuronal synchrony? Neu-roinformatics 3(4):301–314

35. Schiff SJ (2005) Dangerous phase. Neuroin-formatics 3(4):301–314

36. Nunez PL, Srinivasan R (2006) Electric fieldsof the brain. The neurophysics of EEG, 2nd ed.Oxford University Press, Oxford

37. Pfurtscheller G, Lopes Da Silva FH (2011)EEG event-related desynchronization (ERD)and event-related synchronization (ERS). In:Niedermeyer’s electroencephalography: Basicprinciples, clinical applications, and relatedfields, 6th edn, Chap 45. Lippincott Williams& Wilkins, Philadelphia, USA

38. Pfurtscheller G, Schlogl A BCI Competition III,Data set IIIa (2012) Data set provided by thelaboratory of brain-computer interfaces, GrazUniversity of Technology.http://www.bbci.de/competition/iii/desc_IIIa.pdf Last accessed:June 4, 2012

Phase Variants of the Common Spatial Patterns Method