Kernel PCA Feature Extraction of Event-Related Potentials for Human Signal Detection Performance

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1 Kerel PCA Feature Extractio of Evet-Related Potetials for Huma Sigal Detectio Performace Roma Rosipal, Mark Girolami Departmet of Computig ad Iformatio Systems, Uiversity of Paisley Paisley, PA1 2BE, Scotlad eoard J. Trejo Huma Iformatio Processig Research Brach, NASA Ames Research Ceter Moffett Field, CA Abstract I this paper, we propose the applicatio of the Kerel PCA techique for feature selectio i high-dimesioal feature space where iput variables are mapped by a Gaussia kerel. The extracted features are employed i the regressio problem of estimatig huma sigal detectio performace from brai evet-related potetials elicited by task relevat sigals. We report the superiority of Kerel PCA for feature extractio over liear PCA. 1 Itroductio I may safety-critical applicatios (e.g., air traffic cotrol, power plat operatio, military applicatios) cotrol is based o the ability of huma operators to detect ad evaluate task-relevat sigals i the preseted visual data. Performace quality of operators varies over time, ofte fallig below acceptable limits, ad may result i errors with potetially serious cosequeces. The likelihood of such errors could be reduced if physiological methods for assessmet of huma performace were available. A fudametal part i the developmet of such a method is to costruct a model reflectig the depedece betwee selected physiological metrics of metal workload (e.g., Evet-Related Potetials (ERPs)) ad the performace characteristics of a huma operator (reactio time, accuracy, ad cofidece). Prior research has demostrated that liear regressio ad oliear eural etworks ca model the relatioships betwee ERPs ad performace (see [1, 2, 3] ad ref. therei). However, whe we attempt to develop such a model we are cofroted with the curse of dimesioality, which arises from the complexity of physiological data. For example, 225 data samples (dimesios) are required to describe a sigle 1.5s segmet of ERP data from three electrodes. To address this problem, we ca accept two geeral assumptios about the real world data sets. First, there exist some correlatios amog iput variables; thus dimesioality reductio or so-called feature extractio allows us to restrict the etire iput space to a sub-space of lower dimesioality. Secod, i may practical problems we ca assume a smooth mappig from iput to output space; thus we

2 ca ifer the values of the output for poits where o iput data are available. This ca be doe by a appropriate regularizatio techique. I this study, we have used the recetly proposed Kerel PCA [4] method for feature selectio i kerel space. This allows us to obtai features (oliear pricipal compoets) with higher-order correlatios betwee iput variables, ad secod, we ca extract more compoets if the umber of data poits is higher tha their dimesioality [4]. The idea behid Kerel PCA [4] is based o computatio of the stadard liear PCA i a high dimesioal feature space F (with dimesio ), ito which the iput data x R N are mapped via some oliear fuctio Φ x. To this ed, we compute a dot product i spacef usig a kerel fuctio, i.e. K x y Φ x Φ y. This kerel trick allows us to carry out ay algorithm, e.g Support Vector Regressio (SVR) [5], that ca be expressed i the terms of dot products i spacef. Next, we used selected features to trai SVR ad Kerel Pricipal Compoet Regressio to estimate the depedecy betwee ERPs ad the performace of the idividual subjects. The results suggest the superiority of (oliear) Kerel PCA for feature extractio over liear PCA i some cases. 2 Methods 2.1 Kerel PCA, Multi-ayer SVR ad Kerel PCR The PCA problem i high-dimesioal feature spacef ca be formulated as the diagoalizatio of a -sample estimate of the covariace matrix Ĉ 1 Φ x i Φ x i T where Φ x i are cetered oliear mappigs of the iput variables x i R N, i 1. The diagoalizatio represets a trasformatio of the origial data to ew coordiates defied by orthogoal eigevectors V. We have to fid eigevalues λ 0 ad o-zero eigevectors V F satisfyig the eigevalue equatio λv ĈV Realizig, that all solutios V with λ 0 lie i the spa of mappigs Φ x 1 Φ x Schölkopf et. al. [4] derived the equivalet eigevalue problem λv Kv where v deotes the colum vector with coefficiets v 1 v such that V v iφ x i ad K is a symmetric matrix with K i j Φ x i Φ x j : K x i x j. Normalizig the solutios V k correspodig to the o-zero eigevalues λ k of the matrix K, traslates ito the coditio λ k v k v k 1 [4]. Fially, we ca compute the projectio of Φ x oto the k-th oliear pricipal compoet by q x k : V k Φ x v k i K x i x (1) We the select the first r oliear pricipal compoets, e.g. the directios which describe a desired percetage of data variace, ad thus work i a r-dimesioal sub-space of feature space F. This allows us to costruct multi-layer support vector machies [4], where a preprocessig layer extracts features for the ext regressio or classificatio task. I our study we focus o the regressio problem. Geerally, the SVR problem (see e.g.[5]) ca be defied as the determiatio of fuctio f x w which approximates a ukow desired fuctio ad has the form f x w w Φ x b where b is a ukow bias term, w F is a vector of

3 ukow coefficiets ad w Φ x is a dot product i space F. I [7] the followig regularized risk fuctioal is show to compute the ukow coefficiets b ad w: R reg w 1 Err ε η 2 w 2 (2) where Err y i f x i w, η 0 is a regularizatio costat to cotrol the tradeoff betwee complexity ad accuracy of the regressio model ad Err ε is Vapik s ε-isesitive loss-fuctio [7]. I [7] it is show that the regressio estimate that miimizes the fuctioal (2) has the form: f x a a a i a i K 1 x i x b where a i a i are agrage multipliers [5]. Combiig the Kerel PCA preprocessig step with SVR yields a multi-layer SVR i the followig form [4]: f x a a a i a i K 1 q x i q x b, where compoets of vectors q are defied by (1). However, i practice the choice of appropriate kerel fuctio K 1 ca be difficult. I this study, a polyomial kerel of first order K 1 x y x y is employed. We are thus performig a liear SVR o the r-dimesioal sub-space of F. The advatage of liear SVR over ordiary liear regressio is the possibility of usig a large variety of loss fuctios to suit differet oise models [5], e.g. the proposed Vapik s ε-isesitive fuctio is more robust for oise distributios close to uiform. However, i the case of Gaussia oise the best approximatio to the regressio provides a loss fuctio of the form y i f x i c y i f x i c 2. Therefore, we used a Kerel Pricipal Compoet Regressio 1 techique which miimizes the followig risk fuctioal R pcr c 1 y i f x i c 2 The solutio f x c has the form f x c r k 1 b k q x k b 0 r k v k 1b k i K x i x b 0 c i K x i x b 0 where c i r k 1 b kv k i ad q x r k k 1 are agai defied by (1). The coefficiets ca be foud by solvig the ormal equatios for least squares estimatio. b k r k Data Sample Costructio We have used ERPs ad performace data from a earlier study [2]. Eight male Navy techicias experieced i the operatio of display systems performed a sigal detectio task. Each techicia was traied to a stable level of performace ad tested i multiple blocks of trials each o two separate days. Blocks were separated by 1-miute rest itervals. A set of 1000 trials were performed by each subject. Itertrial itervals were of radom duratio with a mea of 3s ad a rage of s. The etire experimet was computer-cotrolled ad performed with a 19-ich color CRT display. Triagular symbols subtedig 42 miutes of arc ad of three differet lumiace cotrasts (0.17, 0.43, or 0.53) were preseted parafoveally at a costat 1 A more detailed descriptio of a Pricipal Compoet Regressio is give i [6].

4 eccetricity of 2 degrees visual agle. Oe symbol was desigated as the target, the other as the o-target. O some blocks, targets cotaied a cetral dot whereas the o-targets did ot. However, the associatio of symbols to targets was alterated betwee blocks to prevet the developmet of automatic processig. A sigle symbol was preseted per trial, at a radomly selected positio o a 2-degree aulus. Fixatio was moitored with a ifrared eye trackig device. Subjects were required to classify the symbols as targets or o-targets usig butto presses ad the to idicate their subjective cofidece o a 3-poit scale usig a 3-butto mouse. Performace was measured as a liear composite of speed, accuracy, ad cofidece. A sigle measure, PF1, was derived usig factor aalysis of the performace data for all subjects, ad validated withi subjects. The computatioal formula for PF1 was PF1 = 0.33 Accuracy Cofidece Reactio Time usig stadard scores for accuracy, cofidece, ad reactio time based o the mea ad variace of their distributios across all subjects. PF1 varied cotiuously, beig high for fast, accurate, ad cofidet resposes ad low for slow, iaccurate, ad ucofidet resposes. ERPs were recorded from midlie frotal, cetral, ad parietal electrodes (Fz, Cz, ad Pz), referred to average mastoids, filtered digitally to a badpass of 0.1 to 25 Hz, ad decimated to a fial samplig rate of 50 Hz. The prestimulus baselie (200 ms) was adjusted to zero to remove ay DC offset. Vertical ad horizotal electrooculograms (EOG) were also recorded. Epochs cotaiig artifacts were rejected ad EOG-cotamiated epochs were corrected. Furthermore, ay trial i which o detectio respose or cofidece ratig was made by a subject was excluded alog with the correspodig ERP. Withi each block of trials, a ruig-mea ERP was computed for each trial. Each ruig-mea ERP was the average of the ERPs over a widow that icluded the curret trial plus the 9 precedig trials for a maximum of 10 trials per average. Withi this 10-trial widow, a miimum of 7 artifact-free ERPs were required to compute the ruig-mea ERP. If fewer tha 7 were available, the ruig mea for that trial was excluded. Thus each ruig mea was based o at least 7 but o more tha 10 artifactfree ERPs. This 10-trial widow correspods to about 30s of task time. The PF1 scores for each trial were also averaged usig the same ruig-mea widow applied to the ERPs, excludig PF1 scores for trials i which ERPs were rejected. Prior to aalysis, the ruig-mea ERPs were clipped to exted from time zero (stimulus oset time) to 1500 ms post-stimulus, for a total of 75 time poits. 3 Results The preset work was carried out with Gaussia kerels; K x y e x y 2, where is the width of the Gaussia fuctio. The desired output PF1 was liearly ormalized to have a rage of 0 to 1. We traied the models o 50% of the ERPs ad tested o the remaiig data. The described results, for each settig of the parameters, are a average of 10 rus each o a differet partitio of traiig ad testig data. The validity of the models was measured i terms of the proportio of data for which

5 PF1 was correctly predicted with 10% tolerace, i.e 0 1 i our case. The performace of a Regularized Gaussia RBF (rgrbf) etwork [8] ad SVR traied o data pre-processed by liear PCA (PCA) i iput space was compared with the results achieved by multi-layer SVR (MSVR) ad the proposed Kerel Pricipal Compoet Regressio (KPCR) o features extracted by Kerel PCA. I both cases we used features (pricipal compoets) describig 99% of data variace. We used ε 0 01, η 0 01 parameter values i the case of SVR. The results achieved o subject A (592 ERPs), B(614 ERPs) ad C (417 ERPs) are depicted i Figure 1. O subjects A ad B we ca see cosistetly better results o features extracted by Kerel PCA (top ad middle left graphs). These superior results achieved usig Kerel PCA represetatio were also observed o the remaiig five subjects. I additio, we ca see that i all cases the SVR was superior to rgrbf o iputs extracted by liear PCA (right graphs). We have to ote that o all subjects we achieved similar results with features describig 98% of data variace. Without the PCA preprocessig step i feature space F we did ot icrease the overall performace. O the cotrary, o four subjects the performace was o average decreased by 0.5% o test proportio error Subject A KPCR MSVR Subject B KPCR MSVR Subject C KPCR MSVR x Subject A PCA+rGRBF Subject B PCA+rGRBF Subject C Figure 1: Compariso of the results achieved o subjects A, B ad C PCA+rGRBF x 10 4

6 4 Coclusios The selectio of appropriate features for regressio has bee ivestigated. O subjects A ad B we demostrated that (oliear) Kerel PCA provides a superior represetatio of the data set over that of liear PCA. However, o subject C the performace with features selected by liear PCA was slightly better. We have to ote, that i this case the dimesio of matrix K i feature spacef is lower (209) tha the iput dimesioality (225), thus we ca ot exploit the advatage of Kerel PCA to improve overall performace by usig more compoets i feature space tha the umber available i the iput space. We used features describig 99% of the data variace that for differet parameter represets 70 90% of all oliear pricipal compoets ad we showed that such a reductio of high-dimesioal feature space F does ot decrease the overall performace. Moreover, this ca be see as the deoisig techique assumig that the oise is spread i directios with small variace. O all subjects, we demostrated that the performace of SVR o features extracted by liear PCA was superior to Regularized Gaussia RBF. Ackowledgmets The first author is fuded by a research grat for the project Objective Measures of Depth of Aaesthesia ; Uiversity of Paisley ad Glasgow Wester Ifirmary NHS trust, ad is partially supported by Slovak Grat Agecy for Sciece (grats No. 2/5088/98 ad No. 98/5305/468). Data were obtaied uder a grat from the US Navy Office of Naval Research (PE60115N), moitored by Joel Davis ad Harold Hawkis. Dr. Trejo is supported by the Psychological ad Physiological Stressors ad Factors Project of the NASA Aerospace Operatios Systems Program (RTOP ), maaged by J. V. ebacqz. Refereces [1] Trejo J, Shesa MJ. Feature Extractio of ERPs Usig Waveletes: A Applicatio to Huma Performace Moitorig. Brai ad aguage 1999; 66: [2] Trejo J, Kramer AF, Arold JA. Evet-related Potetials as Idices of Display-moitorig Performace. Biological Psychology 1995; 40: [3] Koska M, Rosipal R, Köig A, Trejo J. Estimatio of huma sigal detectio performace from ERPs usig feed-forward etwork model. I: Computer Itesive Methods i Cotrol ad Sigal Processig, The Curse of Dimesioality. Birkhauser, Bosto, [4] Schölkopf B, Smola AJ, Müller KR. Noliear Compoet Aalysis as a Kerel Eigevalue Problem. Neural Computatio 1998; 10: [5] Smola AJ, Schölkopf B. A Tutorial o Support Vector Regressio. Techical Report NC2- TR , NeuroColt2 Techical Report Series, [6] Jollife IT. Pricipal Compoet Aalysis. Spriger-Verlag, New York, [7] Vapik V. The Nature of Statistical earig Theory. Spriger, New York, [8] Che S, Chg ES, Alkadhimi K. Regularised orthogoal least squares algorithm for costructig RBF etworks. It Joural of Cotrol 1996; 64.