Conclusiones

The followings are the EEG features for BCI investigated in this project. They include band- power or event related desynchronisation/synchronisation, time domain parameters, adaptive autoregressive parameters and common spatial patterns.

Bandpower and event related desynchronisation/synchronisation

The power of an EEG signal, bandpower, is one of the most commonly used EEG features in motor imagery-based BCI [117, 118, 119]. The bandpower is usually computed within a narrow frequency band. This allows the selection of specific narrow frequency bands that are

3.1. Brain computer interface technologies 40 most relevant for a particular BCI subject since the reactive frequency band may vary for each subject in a motor imagery-based BCI [120]. The computation of bandpower is done by first filtering the EEG data in the required frequency band, squaring and smoothing/averaging the resulting signal. This is shown in Equ. 3.1 where X(t) is the narrow bandpass filtered signal and the h.i is used to represent smoothing/averaging operator. The smoothing is usually done over a time of one second.

Bandpower =X(t)2 . (3.1) Event related desynchronisation/synchronisation: The traditional method of visualising power changes of EEG signal is the event related desynchronisation (ERD) and event related synchronisation (ERS) [32, 121]. The ERD and ERS refer to decrease and increase respec- tively of EEG power relative to a baseline period within a narrow frequency band. Movement related cortical processes like those during motor imagery and physical execution can be quantified with ERD across the sensorimotor cortex. Brain activities like the processing of a visual or auditory stimuli can be quantified with ERS. The methods are sometimes together referred to as event related desynchronisation/synchronisation ( ERD/ERS).

Ref = 1 tref.L tref 2 X t=tref 1 L X i=1 Xi(t)2 ERD/ERS(t) = L P i=1 Xi(t)2 Ref − 1 (3.2) The ERD/ERS method was introduced by Pfurtscheller and colleagues [32, 121]. It is com- puted by subtracting the baseline/reference EEG power from the EEG power in a period containing a response to an event. It is normally averaged over several trials to reduce noise. A formula for computing ERD/ERS is shown in Equ. 3.2 where L is the number of trials, t is the sampling interval, tref = tref 2− tref 1 is the reference period. This equation gives

ERD as negative values and ERS as positive values.

Time domain parameter

Time domain parameter (TDP) is similar to bandpower features except that TDP uses broad band filtered instead of narrow band filtered EEG signal [113]. It is less popular in the field of BCI than the bandpower method even though it has been shown to outperform the bandpower [113]. It is typically calculated within the entire active band of EEG during motor imagination which is about 7-30 Hz [113]. The TDP method used in this thesis is given by

3.1. Brain computer interface technologies 41 Equation 3.3 [113], T DP (t)j =  var dX(t) j dtj  , j = 0, ...p (3.3) where X(t) is a wide band filtered EEG, t is the current sample, j is the derivative or- der (0 ≤ p ≤ 9) [113], var is the variance operator and h.i is used to represent smooth- ing/averaging operator. Note that the variance operator in this equation acts like the band- power operator since the variance of a centered (mean=0) signal is equal to the bandpower [113]. The computation of TDP and its usage in an adaptive BCI system is shown schemat- ically in Fig. 3.1. The squaring and smoothing is part of bandpower calculation while the logarithmic transformation enforces normal distribution of the feature as required by the lin- ear discriminant classifier. The BCI system in Fig. 3.1 is merely used to showcase the TDP method; the details of the entire BCI system will be presented later in this chapter (see 3.1.3).

M I Score: 10 dx(t) dt d2_x(t) dt2 d9_x(t) dt9 x(t) u2 _smooth log LDA Adapt [Bias,weight] Feature 1 1 0 0 1 0 .. .. Buffer Feedback [0|1] TDP EEG PC

Figure 3.1: BCI setup showing the computation of TDP

Autoregressive parameters

The autoregressive parameters [122] in the adaptive form [123, 124] were one of the method considered in this thesis as EEG features for BCI. This section describes the autoregressive model.

Let X(N, t) = X(t) represents an N -channel EEG signal at time t. The autoregressive model describes the signal X(t) as a sum of its p (order) weighted past values with added white noise e(t) [122]. An equation of the autoregressive model is shown in Equ. 3.4 where A(j) (j = {1, 2...p}) is a matrix of size N × N containing the coefficients for weighting the

3.1. Brain computer interface technologies 42 past values. The noise is described by a Normal distribution, thus: G{0, σ2

e}. X(t) = p X j=1 A(j)X(t − j) + e(t) e(t) = p X j=0 A(j)X(t − j) . (3.4) In the second line of Equ. 3.4 the sign of A is assumed to be inverted and A(0) is set to the identity matrix. Applying the z-transformation on the second line of Equ. 3.4 gives Equ. 3.5 where v here is the frequency.

e(f ) = A(v)X(v)

X(v) = A−1(v)e(v) = H(v)e(v) . (3.5) The autoregressive method works well for a stationary signal where the parameters A(j) characterising the model are constants (i.e. they are time invariant). But the EEG signal in BCI is non-stationary. The adaptive autoregressive model is used to describe such a non- stationary signal. Equ. 3.6 describes an adaptive autoregressive model where the noise e(t) is thus: G{0, σ2_e(t)}. This formula is similar to that of Equ. 3.4 except that the A(j, t) and the variance of e(t) are time-varying.

X(t) = p X j=1 A(j, t)X(t − j) + e(t) e(t) = p X j=0 A(j, t)X(t − j) . (3.6) The classical method of dealing with the computation of the time-varying autoregressive parameters is to segment X(t) into short time windows and then compute the parameters individually in each window. Other methods have been used recently. Of these are, the least mean squares approach [124] and the Kalman filtering approach [124, 125]. These methods are capable of estimating the time-varying autoregressive parameters even in real-time. The Biosig and the rtsBCI toolboxes [126] include methods for computing the parameters. Since the parameters are time-varying due to changes in EEG signal, differences between for ex- ample the left and right hand motor activities can be coded in the adaptive autoregressive parameters. The estimated parameters serve as features for a classification algorithm like that described in Section 3.1.2. The adaptive autoregressive parameters were investigated in this thesis but other methods were preferred as BCI features.

3.1. Brain computer interface technologies 43

Common spatial pattern

The method of common spatial pattern (CSP) can be applied to two classes of EEG time series to create a new time series which can have a reduced dimension. The variance of the new time series serves as a feature to a classification algorithm because one of the classes in the new time series has a minimised variance while the other has a maximised variance. The following describes the CSP method.

Let Xi

c(N, t) denotes trial number i of a bandpass filtered EEG of condition/class c = {1, 2}

with N channels and t samples.

The normalised covariance matrix Ri_cfor trial i in class c is given by Ri_c = X i cXciT trace(Xi cXciT) (3.7) where the superscript T denotes the matrix transpose operator. The covariance matrix is then averaged across trials for each class thus:

Rc =

X

i=1

Ri_c. (3.8) Then the composite covariance matrix is given by [127]

R = R1+ R2. (3.9)

The composite matrix can be decomposed to get

R = BλBT (3.10) where B is a matrix eigenvector such that BBT = 1N ×N and λ is a diagonal matrix of eigen-

values sorted in the descending order of magnitude. Next, applying the whitening transform [128] given by

W = √1 λB

T _(3.11)

makes the variance in the B space a unity [128, 129]. Now filtering R, R1 and R2 with W

gives Equ. 3.12 [129].

3.1. Brain computer interface technologies 44 Therefore the individual eigenfunction decomposition of S1and S2should give Equ. 3.13

S1 = U λ1UT

S2 = U λ2UT .