Standard P300 scalp. Fz, Cz, Pz Posterior Electrodes: PO 7, PO 8, O Z

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1 Brain Computer Interface Step Wise Linear Discriminant Analysis and Linear Regression Instructor: Dr. Ravi Sankar Student: Kun Li & David Seebran

2 Brain-Computer Interface Utilize the electrical l activity i of the brain as the carrier of the communicated signal can all be viewed as methods for providing the user with control over the variance of the EEG. The methods by which such control over the variance has been achieved often focused on controlling the spectral composition of the EEG.

3 Krusienski, et al swork Standard P300 scalp locations: Fz, Cz, Pz Posterior Electrodes: PO 7, PO 8, O Z

4 Data Collection The participant i sat upright in front of a video monitor, focused attention on a specified letter of the matrix on the display and silently counted the number of times the target character intensified, until a new character was specified for selection. The rows and columns were intensified for 00 ms with 75 ms interval. S Sequence = a complete cycle of six row and six column intensifications. ifi ti Epoch = 5 sequences Session = 36 epochs Therefore, each session contains (6+6)*5*36 = 6480 stimuli For each channel used in the analysis, 800-ms segments of data (92 samples) following each intensification were extracted for the offline analysis. The data segments were concatenated by channel for each intensification, creating a single feature vector corresponding to each stimulus.

5 Linear Discriminant Analysis The purpose of Discriminant i i Analysis is to classify objects (people, customers, things, etc.) into one of two or more groups based on a set of features that describe the objects. Statistically, The classification rule is to assign an object to the group with highest conditional probability (minimize the total error of classification).. This is called Bayes Rule.

6 If there are g groups, the Bayes' rule is to assign the object to group i where P( i x) > P( j x), for j i Practically, P(i x) is hard to obtain, but it can be computed by using P(x i) Pi ( x ) = jj P( x i) P( i) P( x j) P( j)

7 If we assume that each group has multivariate Normal distribution and all group has the same covariance matrix, we get what is called Linear discriminant Analysis formula: μ T μ μ T i i k i i i f = μc x μc μ + ln( P ) 2 Assign object tkk to group i that t has maximum f i

8 Example Curvature Diameter Factory ABC ABC produces very expensive and high h quality chip rings that their qualities are measured di in term of curvature and diameter. Result of quality control by experts is given in the table below Quality control results Passed Passed Passed Passed Not Passed Not Passed Not Passed

9 Here comes a new chip ring that has curvature 2.8 and diameter Will this one pass the quality control? We make x = features of all data. Each row represents one object; each column stands for one feature. y = group of the object of all data. Each row represents one object and it has only one column.

10 so x = y =

11 Let x i = features data for group i, then x x = = Let u i = mean of features in group i, which is average of x i, then μ [ ] = μ [ ] 2 =

12 u = global mean vector, that is mean of the whole data set. μ = [ ] Let x i o = mean corrected data x o o = 2 x =

13 covariance matrix of group i c i = ( x ) x o T o i i n i pooled within group covariance matrix g Crs (,) = n i c i (,) rs n = i

14 P = prior probability vector P i = n i N then P 4 7 = 3 7 Discriminant Function = μ T μ μ T + i i k i i i f C x C l( ln( P) 2

15 We should assign object tkk to group i that t has maximum f i. For the chip ring we want to classify ( curvature 2.8 and diameter 5.46 ) f = f 2 = Therefore, it belongs to group 2 and can t pass the quality control.

16 Mathematically, the idea of LDA is to minimize the within group scatter and maximize the between group scatter. three scatter matrices, called within-class, between-class and total scatter matrices are defined as follows: k ( ) ( j) T Sω = ( x c j n )( x c ) X j= x j k S b = nj c c c c n j = n ( j ) ( j ) T ( )( ) S t n = ( xi c )( xi c ) i= Where c (j) is the centroid of the j-th class, and c is the global centroid. T

17 It follows from the definition that : S = S + S t b ω In the lower-dimensional space resulting from the linear transformation G, the scatter matrices become: S G S G S L T ω = L T ω b = G S G b S L t = T G SG t

18 The optimal transformation,, of LDA is computed by solving the following optimization problem (Duda et al., 2000; Fukunaga, 990) G trace S S LDA L L = arg max{ ( b ( t ) )} When deals with binary-class problems, The optimal transformation G F is given by F G = S c c + () (2) t ( )

19 Linear Regression linear regression model has the following form: T f ( x ) = x ω + b A popular approach for estimating and is the least squares, in which h the following objective function is minimized: i i T L ( ω, b ) = X ω + be y = f ( x i ) y 2 2 n 2 2 Where X = [x,x 2, x n ] is the data matrix, e is the vectors of all ones, and y is the vector of class labels. i= i

20 Assume that tb both th{ {x i } and d{ {y i } are centered. The bias term b becomes zero and and we look for a linear model by minimizing: The optimal is given by: T L( ω) = X ω y 2 2nn 2nn ω = S ( c c ) = G n 2 + () (2) 2 t 2 n2 2 F

21 Choosing w A combination of forward and backward stepwise regression is implemented. Starting with no initial model terms, the most statistically significant predictor variable having a p-value <0., is added to the model. After each new entry to the model, a backward stepwise regression is performed to remove the least significant variables, having p-values >0.5.

22 Problem According to the choosing rules, the more tests we make, the higher the likelihood of falsely rejecting the null hypothesis. Suppose we set a cutoff of p=0.05. If H 0 (H 0 is the null hypothesis) is always true, then we would reject it 5% of the time. But if we had two independent tests, we would falsely reject at least one H 0 : -(-.05) 2 = , or almost 0% of the time. If we had 20 independent tests, we would falsely reject at least one H 0 : -(-.05) 20 = 0.645, almost 2/3 of the time.

23 Thank You!!!

24 References [Farwell and Donchin (988)] Farwell LA, Donchin E. Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials. Electroenceph Clin Neurophysiol 988;70: [Fabiani et al., 987] Fabiani M, Gratton G, Karis D, Donchin E. Definition, identification, and reliability of measurement of the P300 component of the event-related brain potential. Adv Psychophysiol 987;2: [Krusienski DJ, et al] Toward enhanced P300 speller performance, Krusienski DJ, et al., J Neurosci Methods (2007) [BCI, 2003] BCI Competition II Data Set IIb. Results (Blankertz and Curio); (2003) ii/results/. [BCI, 2005] BCI Competition III. Data Set II Results; (2005) de/projects/bci/competition iii/results/. [Blankertz et al., 2004] Blankertz B, M uller KR, Curio G, Vaughan TM, Schalk G, Wolpaw JR, et al. The BCI competition 2003: progress and perspectives in detection and discrimination of EEG single trials. IEEE Trans Biomed Eng 2004;5(6): [Blankertz et al., 2006] Blankertz B, M uller KR, Krusienski DJ, Schalk G, Wolpaw JR, Schl ogl A, et al. The BCI competition III: validating alternative approaches to actual BCI problems. IEEE Trans Neural Syst Rehabil Eng 2006;4(2): [Kaper et al., 2004] Kaper M, Meinicke i P, Grossekathoefer U, Lingner T, Ritter H. BCI competition 2003-data set IIb: support vector machines for the P300 speller paradigm. IEEE Trans Biomed Eng 2004;5: [Spencer et al., 200] Spencer KM, Dien J, Donchin E. Spatiotemporal analysis of the late ERP responses to deviant stimuli. Psychophysiology 200;38:

25 [Vaughan et al., 2003] Vaughan TM,McFarland DJ, Schalk G, Sellers E,WolpawJR. Multichannel data from a brain computer interface (BCI) speller using a P300 (i.e., oddball) protocol. Soc Neurosci Abs [Donchin et al., 2000] Donchin E, Spencer KM, Wijesinghe R. The mental prosthesis: assessing the speed of a P300-based brain computer interface. IEEE Trans Rehabil Eng 2000;8:74 9. [Sellers and Donchin, 2006] Sellers EW, Donchin E. A P300-based brain computer interface: initial tests by ALS patients. Clin Neurophysiol 2006;7: [Bostanov, 2004] Bostanov V. BCI competition 2003-data sets Ib and IIb: feature extraction from event-related brain potentials with the continuous wavelet transform and the t-value scalogram. IEEE Trans Biomed Eng 2004;5: [Meinicke et al., 2002] Meinicke P, Kaper M, Hoppe F, Huemann M, Ritter H. Improving transfer rates in brain computer interface: a case study. NIPS :07 : [Thulasidas et al., 2006] Thulasidas M, Cuntai G, Wu J. Robust classification of EEG signal for brain computer interface. IEEE Trans Neural Syst Rehabil Eng 2006;4():24 9. [Serby et al., 2005] Serby H, Yom-Tov E, Inbar GF. An improved P300-based brain computer interface. IEEE Trans Neural Syst Rehabil Eng 2005;3: [Fukunaga, 990] Fukunaga, K. (990). Introduction to statistical pattern classification. USA: Academic Press. [Duda et al., 2000] Duda, R., Hart, P., & Stork, D. (2000). Pattern classification. Wiley. [Jieping Ye] Jieping Ye, Least Squares Linear Discriminant Analysis, Department of Computer Science and Engineering, g Arizona State University ity, Tempe, AZ USA [Kardi Teknomo] Kardi Teknomo, Linear Discriminant Analysis Numerical Example,