Steven Lemm, Christin Schafer and Gabriel Curio- Aggregating Classification Accuracy across Time: Application to Single Trial EEG

 

Aggregating Classification Accuracy across Time: Tim e: Applica Application tion to Single Single Trial EEG

Steven Lemm and Christin Sch¨ afer afer and Gabriel Curio   ∗†

Abstract We present a method for binary on-line classification of triggered but temporally blurred events that are embedded in noisy time series in the context of on-line discrimination between left and right imaginary hand-movement. In particular the goal of the binary classification problem is to obtain the decision, as fast and as reliably as possible from the recorded EEG single trials. tria ls. To provide provide a probabili probabilistic stic decisi decision on at every every time-poin time-pointt   t   the presented method gathers information from two distinct sequences of features across acro ss time. In order to inco incorporate rporate deci decisions sions from prior time time-poin -points ts we suggest an appropriate weighting scheme, that emphasizes time instances, providing a higher discriminatory power between the instantaneous class distributions of each feature, where the discriminatory power is quantified in terms of the Bayes error of misclassification. The effectiveness of this procedure is verified by its successful application in the 3rd BCI competition. competition. Disclosure Disclosure of the data after after the competition competition revealed this approach to be superior with single trial error rates as low as 10.7,, 11.5 and 16.7% 10.7 16.7% for the three different different subjects under study. study.

1

Intr Introdu oduct ctio ion n

The ultimate goal of brain-computer interfacing interfacing (BCI) is to translate human intentions intentions into a control signal for a device, such as a computer application, a wheelchair or a neuroprosthesis (e.g. [20]). Most pursued approaches utilize the accompanying EEG-rhythm perturbation in order to distinguish between single trials (STs) of left and right hand imaginary movements e.g. [8,11, 14, 21]. Up to now there are just a few publ published ished appr approac oaches hes utilizing utilizing additional additional features, such as slow cortical potential, e.g. [3,4,9] This paper describes the algorithm that has been successfully applied in the 2005 international data analysis competition on BCI-tasks [2] (data set IIIb) for the on-line discrimination between imagined left and right hand movement. The objective of the competition was to detect the respective motor intention as early and as reliably as possible. Consequently, the competing algorithms have to solve the on-line discrimination task as based on information mati on on the event event onset. Thus it is not with within in the scope of the competition competition to solve solve the problem of detecting the event onset itself. We approach this problem by applying an algorithm that combines the different characteristics teris tics of two two feat features: ures: the modulati modulations ons of the ongoing rhythmic rhythmic activity activity and the slow cortical Movement Related Potential (MRP). Both features are differently pronounced over time and exhibit a large trial to trial variability and can therefore be considered as temporally porall y blurred. blurred. Conseq Consequen uently tly,, the proposed method combin combines es at one hand the MRP with ∗

S. Lemm and C. Sch¨ afer afer are with the Intelligent Data Analysis group at the Fraunhofer Institute FIRST, Berlin, Germany - e-mail:   {steven.lemm,christin.schaefer}@first.fhg.de † G. Cur Curio io is wit with h the Neu Neurop rophy hysic sicss Gro Group, up, Dep Depart artmen mentt of Neu Neurol rology ogy,, Cam Campus pus Benjam Benjamin in Franklin, Charit´ Cha rit´ e,University e,Univers ity Medi Medicine cine Be Berlin, rlin, Ge Germany rmany

 

the oscillatory oscillatory feature and on the other hand gather informa information tion across time as int introduced roduced in [8,16]. More precisely, at each time point we estimate probabilistic models on the labeled training data - one for each class and feature - yielding a sequence of weak instantaneous classifiers class ifiers,, i.e. posterior posterior class distri distributi butions. ons. The classificatio classification n of an unlabeled unlabeled ST is then derived by weighted combination of these weak probabilistic classifiers using linear combination according to their instantaneous discriminatory power. The paper is organized organized as follows follows:: secti section on II describes describes the feature feature and its extractio extraction, n, In section III introduces the probabilistic model as well as the combinatorial framework to gather information from the different features across time. In section III the results on the competition data are given, followed by a brief conclusion.

2 2.1

Featu eature re Neuroph Neurophysio ysiology logy

The human perirolandic sensorimotor cortices show rhythmic macroscopic EEG oscillations (µ-rhythm) [6], with spectral peak energies around 10 Hz (localized predominantly predominantly over the postcentral postcen tral somatosenso somatosensory ry cortex) and 20 Hz (over the precent precentral ral motor cortex) cortex).. Modulations of the   µ-rhythm have been reported for different physiological manipulations, e.g., by motor activity activity, both actual and imag imagined ined [7, 13, 18], as well well as by somatosens somatosensory ory stimulation ulati on [12]. Standard Standard trial ave averages rages of   µ-rhythm power show a sequence of attenuation, termed event-related desynchronization (ERD) [13], followed by a rebound (event-related synchronization: ERS) which often overshoots the pre-event baseline level [15]. In case of sensorimotor cortical processes accompanying finger movements Babiloni et al. [1] demonstrated that movement related potentials (MRPs) and ERD indeed show up with different spatio-temporal activation patterns across primary (sensori-)motor cortex (M1), supplementary motor area (SMA) and the posterior parietal cortex (PP). Most importantly, the ERD response magnitude did not correlate with the amplitude of the negative MRPs slope. In the subsequen subsequentt we will combi combine ne both feat features. ures. Thus, Thus, in order to extract extract the rhythmi rhythmicc information we map the EEG to the time-frequency domain by means of Morlet wavelets [19], whereas the slow cortical MRP are extracted by the application of a low pass filter, in form of a simple moving average filter. 2.2 2.2

Ex Extr trac acti tion on

Let  X  = [x[1], . . . , x[T ]] denote the EEG signal of one single trial (ST) of length  T , recorded from the two bipolar channels C3 and C4, i.e.   x[t] = [C3[t], C4[t]] T . The label information about the corresponding motor intention of a ST is denoted by  Y   ∈ {L, R}. For information obtain from observations until time  s  ≤  T , we will make use of subscript   |s  throughout this paper, e.g.   X |s  refers to [ x[1], . . . , x[s]]. This observational horizon becomes important with respect to the causality of the feature extraction process, especially in order to ensure the causality of filter operations we have to restrict the algorithm to a certain observational horizon. Note that  X |T  denotes a completely observed ST. However for notational convenience we will omit the index   |T  in case of complete observations. Considering ERD as a feature for ST classifications we model the hand-specific time course of absolute   µ-rhythm amplitudes amplitudes over both sensorimotor cortices. Therefore we utilize the time-frequency representations of the ST at two different frequency bands ( α, β ), ), obtained by convolut convolution ion of the EEG signal signal with comp complex lex Morlet wavelet waveletss [19]. [19]. Using the notation Ψα , and Ψβ  for a wavelet centered at the individual spectral peak in the alpha (8-12Hz) and the beta (16-24Hz) frequency domain, the ERD feature of a ST, observed until time  s is calculated as: ERD|s   = with

erd [1] |s



, . . . , erd|s [s] ,

 

 |(C3  ∗ Ψ )[ ]|   ∗ Ψ )[ ]|   ||(C4 (C3  ∗ Ψ )[ ]|  t α t |s β t |s |(C4|s ∗ Ψβ )[t]| |s

erd|s [t] =

α

 

.

(1)

In a similar manner we define the ST feature for the MRP by convolution with a moving average filter of length 11, abbreviated as MA(11).

mrp [1] mrp [ ]    (C3  ∗ MA(11))[ ] 

MRP|s   =

|s

with mrp|s [t] =

,...,

|s

s ,

t (C4|s  ∗ MA(11))[t] |s

 

.

(2)

According to (1) and (2) the  k -th labeled, observed STs for training, i.e. X (k), Y  (k)  maps to a STs in feature space, namely (MRP (k) , ERD(k) ).



3



Probab Probabili ilisti stic c Classi Classifica ficatio tion n Model Model

Before we start with the model description, we briefly introduce two concepts from Bayesian decision theory. theory. Therefore let  p (x|µy , Σy ), y  ∈ {L, R}  denote the PDFs of two multivariate Gaussian distributions with different means and covariance matrices ( µy , Σy ) for two classes, denoted by   L   and   R. Giv Given en the two two class class-cond -conditio itional nal distributio distribution n models, models, and under the assumption of a class prior of   P (y) =   21 , y   ∈ {L, R}, and given an observation   x, the posterior class distribution according to Bayes formula is given by  p(y |x, µL , ΣL , µR , ΣR ) =

  p(x|µy , Σy ) .  p(x|µL , ΣL ) + p(x|µR , ΣR )

 

(3)

Furthermore the discriminative power between these two distributions can be estimated using the Bayes Bayes error of misc misclassi lassificati fication on [5]. In case of distinct class covarianc covariancee matrices, matrices, the Bayes error cannot be calculated directly. However, by using the Chernoff bound [5] we can derive an upper bound and finally approximate the discriminative power  w  between the two distributions by 2w  ∼ =  1 −   min

0≤γ ≤1

 

  p(x|µL , ΣL )γ  p(x|µR , ΣR )1−γ dx.

 

(4)

In case of Gaussian distributions the above integral can be expressed in a closed form [5], such that the minimum solution can be easily obtained (see also [16]). Based on these two necessary concepts, we will now introduce our probabilistic classification method method. . Therefore There we first the distribution. class class-cond -condition itional al distribution distribut of instance each each feature featwe ureestimate at each each time instance asfore multivar multivariate iatemodel Gaussian Hence at eachion time the class means and the class covariance matrices in the feature space, based on the mapped training train ing STs, i.e. i.e. ERD(k) , MRP(k) . Thus Thus fro from m erd(k) [t] we obtain the following two classconditional sets of parameters: µy [t]

=

Σy [t] =

  E erd [ ]   Cov erd [ ] (k)

t

(k)

(5)

Y  ( k) =y

t

Y  ( k) =y

, y  ∈ {L, R}.

 

(6)

For convenience we summarize the estimated model parameters for the ERD feature as Θ[t] : = (µL [t], ΣL [t], µR [t], ΣR [t]), whereas Ξ[t] : = (ηL [t], ΓL [t]), ηR [t], ΓR [t]) denote the class means and the covariance matrices obtained in the similar manner from mrp (k) [t]. Given an arbitrary observation observation  x  from the appropriate domain, applying Bayes formula as introduced in (3), yields a posterior distribution for each feature: 4 R

  mrp erd Θ[ ]  Ξ[ ]

 p y  p y

, ,

t ,   erd ∈ t ,   mrp ∈

2 R .

 

(7) (8)

 

Additionally, according to (4) we get approximations of the discriminative power   w[t] and v [t] of the ERP resp. MRP feature at every time instance. In order to finally derive the classification of an unlabeled single trial at a certain time  s  ≤  T , we incorporate knowledge from all preceding samples   t  ≤  s , i.e. we make the classification Therefore we firs firstt apply (7) based on the causally causally extract extracted ed features features:: ERD|s  and MRP|s . Therefore and (8) given the observations erd|s [t] resp. mrp|s [t] in order to obtain the class posteriors based on observations until   s   ≤   T . Secon Secondly dly we combine combine these class posteriors posteriors with one another across time by taking the expectation under the distributions  w   and  v , i.e. c(y, s

 )= t≤s









w [t] ·  p y erd|s [t], Θ[t]  + v [t] ·  p y mrp|s [t], Ξ[t]   . t≤s w[t] + v [t]



 

(9)

As des descri cribed bed in [16 [16]] thi thiss yie yields lds an evi eviden dence ce accum accumula ulatio tion n ov over er time time about about the decisi decision on process. Strictly process. Strictly speaking Eq. (9) gives the expectation expectation value that the ST, observed observed until until time   s, is generated by either one of the class models (L or R), until time   s. Due Due to the the submission requirements of the competition the final decision at time  s  is C [s] = 1 − 2 · c(L, s),

 

(10)

where a positive or negative sign refers to right or left movement, while the magnitude indicates the confidence in the decision on a scale between 0 and 1.

4 4.1

Appl Applic icat atio ion n Compet Competiti ition on data data

The EEG from two bipolar channels (C3, C4) was provided with bandfilter settings of 0.5 to 30 Hz and sampled at 128 Hz. The data consist consist of recording recordingss from three different different healthy healthy subjects subje cts.. Except Except for the first data set set,, each each con contai tains ns 540 labeled labeled (for traini training) ng) and 540 unlabeled trials (for competition) of imaginary hand movements, with an equal number of  lef leftt and right right hand hand trials trials (firs (firstt dat dataa set pro provid vides es just 320 tri trials als each). each). Each Each tri trial al has a durati dur ation on of 7 s: aft after er a 3 s prepar preparati ation on period a vis visual ual cue is presen presented ted for one second second,, indicating indic ating the demanded demanded motor intention intention.. This is followed followed by another another 3 s for performing performing the imagination task (for details see [2]). The particular competition data was provided by the Dept. of Med. Informati Informatics, cs, Inst. for Biomed. Biomed. Eng., Univ. of Tech echn. n. Graz. The specific competition task is to provide an on-line discrimination between left and right movements for the unlabeled STs for each subject based on the information obtained from the labeled trials. trial s. More precisely precisely,, at every time instanc instancee in the inter interva vall from 3 to 7 seconds seconds a strictly strictly causal decision about the intended motor action and its confidence must be supplied. After competition deadline, based on the disclosure of the labels  Y  (k) for the previously unlabeled STs the output   C (k) [t] of the methods were evaluated using the time course of the mutual information (MI) [17], i.e.   1 MI[t] =  log 2 (SNR[t] + 1) (11) 2 2 E C (k) [t] Y  ( ) =L − E C (k) [t] Y  ( ) =R SNR[t] =   (12) 2 Var C (k) [t] Y  ( ) =L + Var C (k) [t] Y  ( ) =R

    

k

k







k



k





More precisely precisely,, since the gene general ral ob objecti jective ve of the competiti competition on was to obtain obtain the single trial classification as fast and as accurate as possible, the maximum steepness of the MI was considered as final evaluation criterion, i.e. MI[t] . t≥3.5 t − 3s max

 

(13)

Note, that the feature extraction relies on a few hyperparameters, i.e. the center frequency and the width of the wave wavelet lets, s, as we well ll as the leng length th of the MA filt filter. er. All those those param parameeters were obtained by model selection using a leave-one-out cross-validation scheme of the classification performance on the training data.

 

4.2

Result Resultss and and Disc Discuss ussion ion

As proposed in section 3 we estimated the class-conditional Gaussian distributions cf. (5) – (8). The resulting resulting poster posterior ior distrib distribution utionss were then combi combined ned according according to (9) in order to obtain the final classification of the unlabeled STs. After disclosure of the label information our method turned out to succeed with a MI steepness (cf. (13)) of 0.17, 0.44 and 0.35 for the individual subjects. Table 4.2 summarizes the results in terms of the achieved minimum binary classification error, the maximum MI, and the maximum steepness of MI for each subject and each each compet competitor itor in the competiti competition. on. min. error rate[%] max. MI [bit] max. MI/t [bit/s] O3 S4 X11 O3 S4 X11 O3 S4 X11 10.69 9 11 11.4 .48 8   16.67   0.60 0.6027 27 0. 0.60 6079 79 0. 0.48 4861 61   0.1698   0.43 0.4382 82 0.34 0.3489 89 1.   10.6 2. 14.47 22.96 22.22 0.4470 0.2316 0.3074 0.1626 0.4174 0.1719 3. 13.21 17.59   16.48   0. 0.5509 0.3752 0.4675   0.2030   0. 0.09 0936 36 0. 0.11 1173 73 4. 23.90 24.44 24.07 0.2177 0.2387 0.2173 0.1153 0.1218 0.1181 5. 11.95 21.48 18.70 0.4319 0.3497 0.3854 0.1039 0.1490 0.0948 6.   10.69   13.52 25.19 0.5975 0.5668 0.2437 0.1184 0.1516 0.0612 7. 34.28 38.52 28.70 0.0431 0.0464 0.1571 0.0704 0.0229 0.0489 Table 1: Ov Overall erall ranked ranked results of the competin competingg algo algorithm rithmss (first row corresponds corresponds to the proposed propose d method) on the competition competition test data. For three differen differentt subject sub jectss (O3, S4 and X11) the table states different performance measures of classification accuracy (min. Error rate, max MI, steepness of MI), where the steepness of the MI was used as the objective in the competition. For a description of the 2.–7. algorithm please refer to [2]. The resulting time courses for the MI and the steepness of the MI are presented in the left panel of Fig. 1. For subject two and three, three, during the first 3.5 seconds (0.5 seconds after cue presentation) the classification is rather by chance, after 3 .5 seconds a steep ascent in the classification accuracy can be observed, reflected by the raising MI. The maximum steepness for these two subjects is obtained quite early, between 3 .6 −  3.8s. In opposite, opposite, for subject one the maximum is achieved at 4.9 seconds, yielding yielding a low steepness steepness value. value. Howeve However, r, a low value value is also found for the submi submission ssion of all other compet competitor itors. s. Neverthe Nevertheless, less, the MI constantly increases up to 0.64 Bit per trial at 7 seconds, which might indicate a delayed performance perform ance of subject one. The right panel in Fig. 1 provides the weights   w[t] and   v [t], reflecting the Bayes error of  misclassificati misclassifi cation on cf. (4), that were used for the temporal inte integrati gration on process. For subject two one can clearly observe a switch in the regime between the ERP and the MRP feature at 5 second seconds, s, as indica indicated ted by a cro crossi ssing ng of the tw twoo we weigh ightin tingg functi functions ons.. From this we conclude that the steep increase in MI for this subject between 3 and 5 seconds is mainly due to the MRP feature, whereas the further improvement in the MI relies primarily on the ERD feature. Subject one pro provides vides nearly nearly no discrimi discriminativ nativee MRP and the classificatio classification n is almost exclusively based on the ERD feature. For subject three the constant low weights at all time instances, reveal the weak discriminative power of the estimated class-conditional distribut distr ibutions. ions. Howeve Howeverr in Fig. 1 the advantage advantage of the integration integration process across time can clearly be observed, as the MI is continuously increasing and the steepness of the MI is surprising surpr isingly ly high eve even n for this subject. A comprehensive comparison of all submitted techniques to solve the specific task for data set IIIb of the BCI-competition is provided in [2] or available on the web   1 . Basically Basically this this evaluation reveals that the proposed algorithm outperforms all competing approaches.

5

Conc Conclu lusi sion on

We proposed proposed a general general Bay Bayesian esian framew framework ork for temporal combinatio combination n of sets of simple simple classifiers class ifiers based on different different feature features, s, whic which h is appl applicabl icablee to any kind of sequenti sequential al data 1

ida.first.fhg.de/projects/bci/competition_iii/

 

Figure 1: Left panel: time courses of the mutual information (light, dashed) and the competition criterion - steepness of mutual information (thin solid) cf. (13)- for the classification of the unlabeled STs is presented. Right panel: the time course of the weights reflecting the discriminative power (cf. (4)) at every time instance for the two different features (ERD dark, solid; MRP - light dashed). In each panel the subjects O3, S4, X11 are arranged top down. providing a binary providing binary classifica classification tion problems. problems. More Moreov over, er, any arbitrary arbitrary number of features features can be combined in the proposed way of temporal weighting, by utilizing the estimated discriminative power over time. Furthermore the estimation of the Bayes error of misclassification is not strictly linked to the chosen parametric form of the class-conditional distributions. For arbitrary distributions the Bayes error can be obtained for instance by statistical resampling approaches, such as Monte Carlo methods. Howeverr for the successful Howeve successful appli applicati cation on in the BCI-competitio BCI-competition n 2005 we chose chose Gaussian Gaussian distribut distr ibution ion for the sake of simp simplici licity ty concerning concerning two two issues: issues: estimatin estimatingg their parameters parameters and obtainin obtainingg their their Ba Baye yess error. error. Not Notee tha thatt althou although gh the combina combinatio tion n of the classifie classifiers rs across time is linear, the final classification model is non-linear, as the individual classifiers at each each time instance instance are non-line non-linear.F ar.For or a discu discussion ssion about linear linear vs. non-linear non-linear methods in the context context of BCI see [10]. More precisely precisely due to the distinct distinct covarian covariance ce matrices matrices of  the Gaussian Gaussian distribut distributions ions the individual individual decisio decision n boundar boundaries ies are of quadratic quadratic form. In particula parti cularr to solve solve the competiti competition on task we com combine bined d classifier classifierss based on the temporal evolution of different neuro-physiological features, i.e. ERD and MRP. The resulting on-line classification model finally turned out to succeed for the single trial on-line classification of  imagined hand movement in the BCI competition 2005. Acknowledgement:   This work work was supported supported in part by the Bundesminis Bundesministeriu terium m f¨u urr Bildung und Forschung (BMBF) under grant FKZ 01GQ0415 and by the DFG under grant SFB 618-B4. S. Lemm thanks Stefan Harmeling for valuable discussions.

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