Caution on causality analysis of ERP data

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1 4h Inernionl Conference on Mechronics, Merils, Cheisry nd Copuer Engineering (ICMMCCE 05) Cuion on cusliy nlysis of ERP d Jikn ue,, Snqing Hu,b, Jinhi Zhng, Wnzeng Kong College of Copuer Science, Hngzhou Dinzi Universiy, Hngzhou, Chin eil:xueikn@63.co, beil:sqhu@hdu.edu.cn Keywords: Grnger cusliy, ew cusliy, Even-Reled Poenil. Absrc. The even-reled poenil (ERP) echnique is one of he os widely used ehods in cogniive neuroscience reserch o sudy he physiologicl correles of sensory, percepul nd cogniive civiy ssocied wih processing inforion [,]. An ERP eeg segen d usully involves hree subsegens: pre-siulus,siulus nd pos-siulus. Obviously, he heic odels for pre-siulus (or pos-siulus) nd siulus should be differen. However, he generl pproch for ERP nlysis in he lierure lwys views he whole segen d s one odel. In his pper we use Grnger cusliy (GC) nd recenly proposed ew cusliy (C) ehods o deonsre h C is ore sensiive hn GC o revel rue cusliy influence bsed on ERP d. Inroducion In he lierure Grnger cusliy (GC) hs been widely pplied o deec he direcionl influence of syse coponens in ny differen res, such s econoics, clie sudies, geneics, nd neuroscience.the bsic ide of GC ws originlly conceived by Wiener in 956 [3], nd ler forlized by Grnger in 969 [4] in he for of liner regression odel. The ide cn be briefly described s follows: If he hisoricl inforion of ie series Y significnly iproves he predicion ccurcy of he fuure of ie series in ulivrie uoregressive (MVAR) odel, hen GC fro ie series Y o cn be idenified. Alhough GC hs reendous pplicions in ny res, his success hs lso been ccopnied by criicis fro differen prospecives [5] nd [6]. The criicis of GC hs os been cenered round he philosophicl debe on he relionship beween GC nd rue cusliy. In 0, Hu e. l [7] proposed new cusliy (C) ehod which describes he proporion h Y occupies ong ll conribuions o.as deonsred by nuber of illusrive exples in [7] which include experienl EEG d, C is uch ore sensiive hn GC o revel rue cusliy (or rend of rue cusliy). In his pper we will provide ore evidence o show h C ehod is beer hn GC ehod o revel rue cusliy. GC nd C Mehods Consider wo sochsic ie series which re ssued o be oinly sionry. Individully, under firly generl condiions,ech ie series dis n uoregressive represenion,,,, ε,, (), ε, nd heir oin represenions re described,,,,, η,,,,,, η, () where 0,,,, he noise ers re uncorreled over ie,ε i nd η i hve zero ens nd vrinces of σ(ε i ),nd σ(η i ), i,. The covrince beween η nd η is defined by σ(η,η )cov(η,η )[8]. For prcicl syse, generl pproch for deerining he order of he MVAR odel is he AIC Akike Inforion Crierion [8], [9]. 05. The uhors - Published by Alnis Press 670

2 GC in Tie Doin ow consider he firs equliies in () nd (), if σ (η ) is σ (ε ) less hn in soe suible sense is sid o hve cusl influence on. In his cse, he firs equliy in () is ore ccure hn in () o esie.oherwise, if σ (η ) σ (ε ), is sid o hve no cusl influence on. In his cse,wo equliies re se. Such kind of cusl influence, clled GC [0], [], is defined by σ ε F ln (3) ση Obviously, F 0 when here is no cusl influence fro o nd F >0 when here is. Siilrly, he cusl influence fro o is defined by σ ε F ln (4) σ η C in Tie Doin Bsed on he firs equliy in (), we cn see conribuions o,, which include,,,,, nd he noise er η k, where he influence fro,, is cusliy fro s own ps vlues. Ech conribuion plys n iporn role in deerining,.if,, occupies lrger porion ong ll hose conribuions, hen hs sronger cusliy on, or vice vers. Thus, good definiion for cusliy fro o in ie doin should be ble o describe wh proporion occupies ong ll hese conribuions. So bsed on his generl guideline C fro o is defined s [8] n,, h, h, h Siilrly, C in ie doin fro o is defined by n,, h, h, h h, h, An ERP eeg segen d usully involves hree subsegens: pre-siulus,siulus nd pos-siulus. Obviously, he heic odels for pre-siulus (or pos-siulus) nd siulus should be differen. However, he generl pproch for ERP nlysis in he lierure lwys views he whole segen d s one odel. ex, we will deonsre h one should be cuion when one views he whole d s one odel o ke conclusion on cusliy. Min Resuls In his secion we will provide wo exples o discuss why we should be cuion when we nlyze cusliy of ERP d which y involve uliple segens of differen odels. Exple We consider he following oin regression odel: η,,,,,, ,,, η, where 0,,,,0000, he noise ers η i,, i, re uncorreled over ie, hve zero ens nd vrinces of σ (n )0.55 nd σ (n ).When 0,,,,5000,,, 0,when 500,500,,000,,, 0.5.Thus,odel (7) cn be considered o be regression odel of ie-vrin coefficiens for he whole ie period, lhough i is regression odel of (7) (5) (6) 67

3 ie-invrin coefficiens for ech of wo ie periods: 0,,,,5000 nd 500,,0000. Th is, odel (7) becoes η,, ,, η,, when 0,,,,5000, nd odel (7) becoes, 0.5, 0.8, η, (9), 0.5, 0.8, η, when 500,,0000. To clcule cusliy fro, o, in odels (7)~(9),for ech specific odel we genere d se of 00 relizions. For ech relizion, we esie odels (uoregressive represenions odel () nd oin represenions odel ()) wih he order of 8 by using he les-squres ehod nd clcule GC nd C. Then we obin he verge vlue cross ll relizions nd ge GC nd C fro o where he order 8 fis well (see Figs. () nd (b) fro which one cn see GC nd C keep sedy when he order of he esied odel is greer hn 8). Fro Figure one cn clerly see h i) GC fro o in odel (8) is he se s h in odel (9), h is, GC fro o hs nohing o do wih preers, nd, in wo odels. This is clerly poined ou in (ii) of Propery [8].Due o he fc h odel (7) is cobinion of odel (8) nd odel (9) in ers of wo differen ie periods, if he rue cusliy fro o in odel (8) is he se s h in odel (9), hen he rue cusliy fro o in odel (7) should be equl o h in odel (8) or odel (9). However, GC ( 0.39) fro o for odel (7) is uch sller hn h ( 0.94) in odel (8) or odel (9).This fc cully once gin srongly deonsres h GC vlue does no revel rue cusliy ll. ii) oe h, in () includes hree prs:,,,,, (8) nd he noise er η,.if,, occupies lrger porion ong ll hese hree prs, hen hs sronger rue cusliy on,or vice vers. Since, in odel (8) includes wo prs:-0.8,- nd he noise er η,,nd, in odel (9) includes hree prs:0.5,-, -0.8,- nd he noise er η,, he porion which -0.8,- occupies in wo prs:-0.8,- nd he noise er η, obviously is lrger hn h porion which -0.8,- occupies ong hree prs:0.5,-,0.8,- nd he noise er η,. Then one cn see h he rue cusliy fro o in odel (8) is surely lrger hn h in odel (9). Since odel (7) is cobinion of odel (8) nd odel (9) in ers of wo differen ie periods, hen he rue cusliy fro o in odel (7) should be in beween he rue cusliy fro o in odel (8) nd he rue cusliy fro o in odel (9). Fro Figure () C ( 0.76) in odel (8) is indeed lrger hn h ( 0.6) in odel (9). Moreover, C ( 0.68) in odel (7) is indeed in he inervl [0.6,0.76]. Therefore, in his exple, C beer revels he underlying rue cusliy hn GC. Exple We consider he siulion exple of wo univrie ie series described in he firs prgrph of siulion secion []. According o he descripion he wo chnnels hve even-reled poenils (ERPs) produced by one cycle of Hz sinusoidl wves which re cobined wih ongoing civiies.the single-ril pliudes (A i ) of he sinusoidl wve for Chnnel ( ) re chosen independenly in he inervl. The single-ril pliude (Bi) for Chnnel ( ) is he pliude (A i ) of Chnnel plus sndrd Gussin noise (α i ). The single-ril lency shifs (τ i,τ i ) for he ERP coponens of wo chnnels re lso considered nd uniforly disribued beween 0s nd 0s. For nlysis convenience, we furher ssue τ i τ i τ i in his pper. 500 rils (relizions), ech wih 0 d poins spling re of 00Hz, re genered. Ech ril is 600s long, 00s of which occurred prior o siulus onse (0s). The ERP for Chnnel srs bou 50s fer he siulus onse, while he ERP fro Chnnel is delyed by bou 0s. The ongoing civiy for boh chnnels is Gussin whie noise processes wih zero en nd 0.05 sndrd deviion (η,η ). These wo noise processes re uncorreled wih ech oher. Figure (c) shows he 500 siuled relizions (rils) for boh chnnels. 67

4 For wo ERP coponens we cn obin C vlue fro o.then by verging ll hese C vlues for 500 rils we ge one verge C vlue n Th ens ERP coponen is inly deerined by ERP coponen. The rue cusliy fro ERP coponen o is close o (i.e.,he sronges cusliy). This resul is rel. By esiing he uo-regression odel of whose order is chosen s 3 bsed on AIC crieri we cn obin GC vlues fro o for 500 rils nd hen ge he verge GC F which is fr wy fro (i.e., he sronges cusliy by GC definiion). So, GC vlue of 3.964cnno relly revel he sronges cusliy. For oher ie poins which do no belong o ERP ( ),, nd, re ow noise ers fro which one cn esily ge n F 0.oe h he relion beween nd hs wo differen odels in he whole ie poins. However, he widely used pproch in ERP nlysis in he lierure is o esie one odel in he whole ie poins.the wo odels definiely cnno be represened by one esied odel. Thus he rue cusliy fro o in he whole ie poins y be copleely differen fro he cusliy clculed bsed on he esied odel.since wo differen odels re involved in he whole ie poins, we need o develop new ehod o discuss he rue cusliy fro o which is lef for furher sudy in our fuure work. For C ehod, he cusliy fro o in he ie poins se which only include ERP ( ) is nd he cusliy fro o in he oher ie poins se which do no belong o ERP ( ) is 0, so C vlue fro o in he whole ie poins se should be in (0,0.9995). In fc fer esiing he oin regression odel for wih he order of 4 bsed on AIC crieri we cn ge n which indeed belongs o (0,0.9995). For GC ehod, he cusliy fro o in he ie poins se which only include ERP ( ) is nd he cusliy fro o in he oher ie poins se which do no belong o ERP ( ) is 0, so GC vlue fro o in he whole ie poins se should be in (0,3.964). In fc fer esiing he oin regression odel wih he order of 4 nd he uo-regression odel for wih he order of 4 bsed on AIC crieri we cn ge F.76 which indeed belongs o (0,3.964). Therefore, no er which ehod (GC or C) is used, he obined cusliy fro o in he whole ie poins se is in he rnge beween he cusliy (0) obined in he ie poins se which does no belong o ERP ( ) where ongoing civiy re involved nd he cusliy obined in he ie poins se which belongs o ERP ( ) where ERP coponens re involved. This y be why reserchers in lierure lwys conduc cusliy nlysis on ERP d by using he whole ie segen d which includes no only ongoing civiy bu lso ERP coponen d. GC y no correcly revel he rue cusliy underlying in he whole ie segen d C fro o Model (7) Model (8) Model (9) GC fro o Model (7) Model (8) Model (9) 0.4 () (b) (c) (d) Figure. () C fro o s funcion of he order of he esied odels for odels (7) (9). (b) GC fro o s funcion of he order of he esied odels for odels (7) (9).(c) 500 relizions (rils) of siuled d for wo chnnels. Boh lency vribiliy nd pliude vribiliy re considered. (d) Residuls fer subrcing heir own verge vlues (of ech single ril) fro ech single ril for boh chnnels. Conclusions Order of he Esied Models Order of he Esied Models In his pper, on one hnd, we poined ou h C beer revels he underlying rue cusliy hn GC when one nlyzed segen d involves wo or ore subsegens of differen odels. On he oher hnd, we explined why os reserchers in he lierure y conduc cusliy nlysis for he whole nlyzed ERP d s one esied odel lhough i y include wo segens: ongoing civiies segen nd ERP coponens segen which hve wo differen odels. The bove wo exples y deonsre h C esure y be beer o revel rue cusliy hn GC ehod. 673

5 Acknowledgeen This work ws suppored in pr by he nionl url Science Foundion of Chin under Grn o.64730, url Science Foundion of Zheing Province, Chin, under Grn o.z3f03000, Inernionl Science & Technology Cooperion Progr of Chin under Grn o.04dfg570,key Lborory of Coplex Syses Modeling nd Siulion,Minisry of Ed ucion, Chin. References [] P. Cheolsoo, L. Dvid, R. veed, A. Alirez, nd P. M. Dnlio, Clssificion of Moor Igery BCI Using Mulivrie Epiricl Mode Decoposiion, eurl Syses nd Rehbiliin Engineering,vol., no., pp. 0, Jn. 03. [] S.J. Luck, An Inroducion o he Even-Reled Poenil Technique,Second ediion. Cbridge, Mss.: The MIT Press, 04. [3]. Wiener, The Theory of Predicion. In E.F. Beckenbch, edior,modern Mheics for Engineers, Chp. 8. McGrw-Hill, ew York,956. [4] C. W. J. Grnger, Invesiging Cusl Relions by Econoeric Models nd Cross-specrl Mehods, Econoeric, vol. 37, no. 4, pp , Aug [5] J. Perl, Cusliy: Models, Resning nd Inference, Cbridge Universiy Press, 009. [6] P. Spires, C. Glyour, nd R. Scheines. Cusion, Predicion, nd Serch, Second Ediion, The MIT Press, 00. [7] S. Hu, G. Di, G. Worrell, Q. Di, nd H. Ling, Cusliy nlysis of neurl conneciviy: Criicl exinion of exising ehods nd dvnces of new ehods, IEEE Trns eurl eworks, vol., no.6, pp , 0. [8] H. Akike, A new look he sisiclodel idenificion, IEEE Trnscions on Auoic Conrol, vol. 9, no. 6, pp , Dec.974. [9] A. Seghoune, Model selecion crieri for ige resorion, IEEE Trns. on eurl eworks, vol. 0, no. 8, pp , Aug [0] M. Ding, Y. Chen, nd S. L. Bressler, Grnger cusliy: Bsic heory nd pplicions o neuroscience, in Hndbook of Tie Series Anlysis,B. Scheler, M. Winerhlder, nd J. Tier Eds. Weinhei, Gerny:Wiley-VCH, pp , 006. [] J. Geweke, Mesureen of liner dependence nd feedbck beween uliple ie series, J. Aer. S. Assoc., vol. 77, no. 378, pp , Jun. 98. []. Wng, Yong. Chen, nd M. Ding, Esiing Grnger cusliy fer siulus onse: A cuionry noe, euroige, vol. 4, pp ,