Akaike Causality in State Space Part I - Instantaneous Causality Between Visual Cortex in fmri Time Series

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1 Akaike Causaliy in Sae Space Par I - Insananeous Causaliy Beween Visual Corex in fmri Time Series K.F. Kevin Wong, Tohru Ozaki December 21, 26 Absrac We presen a new approach of explaining parial causaliy in mulivariae fmri ime series by a sae space model. A given single ime series can be divided ino wo noise-driven processes, which comprising a homogeneous process shared among mulivariae ime series and a paricular process refining he homogeneous process. Causaliy map is drawn using Akaike noise conribuion raio heory, by assuming ha noises are independen. The mehod is illusraed by an applicaion o fmri daa recorded under visual simulus. Keywords: Akaike causaliy, noise conribuion raio, sae space model, common source, parial causaliy, funcional MRI, primary visual corex, middle emporal corex, poserior parieal corex. 1 Inroducion For he purpose of causaliy analysis in mulivariae ime series daa Akaike (1968) decomposes power specral densiy ino componens, each coming from an independen noise of mulivariae auoregressive model (VAR). Conroversy on Akaike noise conribuion raio (NCR) causaliy mainly focuses on he validiy of causaliy when residuals are spaially highly correlaed, ha phenomenon can be refleced from large covariance enry in noise covariance marix. When driving noises have a high correlaion insananeously, independence assumpion of noise is no adequae, a non-zero noise covariance is essenial o improve he ime series model. This indispensable covariance suggess ha wo corresponding ime series are driven by similar noises, which apparenly showing causal relaionship insananeously from one o he oher, wihou a clue who is causing whom. The causaliy being discussed is known o be insananeous causaliy. Geweke (1982) esed he likelihood raio in order o decide he significance of insananeous causaliy. One deficiency is an unclear cu of causaliy when 1

2 he model order is geing high, so ha feedback of insananeous causaliy hrough auoregressive process occurs and insananeous causaliy plays a role more han jus insananeously. We propose an alernaive way o look a he insananeous causaliy by sae space model (Wong, 25). In paricular, we assume, insead of indireced causaliy beween wo variables, a direced causaliy from a laen variable o he wo variables. We will model he fmri by a linear auoregressive model plus a homogeneous variable in a sae space framework. 2 fmri daa under visual simulus The daa seleced as an example o illusrae our new mehod is obained from a recen research of Yamashia e al. (25). The ime series of BOLD signal of a healhy subjec under a visual simulaion was obained in an fmri scanning machine. A black screen is presened o he subjec for 3 seconds, hen whie dos appeared on he black screen and flew ouwards from cener of he screen for 3 seconds. The wo screens swiched in every 3 seconds. A deailed experimenal procedure and pre-processing procedure can be found in Yamashia e al. (25). Yamashia e al. (25) seleced hree regions of ineres, primary visual corex (V1), visual corex area 5 (V5) and poserior parieal corex (PP). They are repored o respond o human aenion o visual moion. (Büchel & Frison, 1997) The primary visual corex (V1) is an enrance of visual simuli. Through V1 informaion is furher ransmied o oher visual areas, such as visual areas V2, V3, V4 and V5. The visual area V5, also known as visual area MT (middle emporal), is a region of exrasriae visual corex ha is hough o play a major role in he percepion of moion. Poserior parieal corex (PP) is anoher disincive corical area appearing o be imporan for spaial processing and he conrol of eye movemens, may also have a cenral role in visual aenion. We are ineresed in how is conneciviy among hese areas in responding o visual simulus. In figure 1 we show he ime series daa on a ime axis in second. The daa se conains four disconinuous segmens. Each segmen has 27 ime poins covering 27 seconds. Yamashia e al. (25) analyzed he ime series by VAR and adding he informaion of onse of simulus as an exogenous variable o he model. They repored ha srong conneciviy exiss from V1 o V5 and from V5 o PP a a period of 6 seconds, which is he ime beween saring ime of wo consecuive simuli. 3 Mehod and Resul We inend o fi he ime series o a sae space model and plo a causaliy map based on he model. A laen variable is included in sae vecor in 2

3 PP V1 V ime/seconds Figure 1: fmri BOLD signals under visual simuli 3

4 order o ge rid of a common dynamic which is driving he hree corex areas simulaneously. Neverheless, hree individual driving noises represening corresponding corex areas perain muual causaliy, hrough a feedback sysem provided by a ransiion marix. Le y denoe he observed daa and x he unobserved sae. We assume ha x depends on is pas values hrough a linear sochasic model, conaining a dynamical noise erm, and ha y follows from x hrough a linear observaion model, conaining an observaion noise erm; hen he following sae space model applies: x = F x 1 + Gw (1) y = Hx + ɛ (2) Equaions (1) and (2) are commonly known as sysem equaion and observaion equaion, respecively. w denoes he dynamical noise erm of he sysem equaion, assumed o follow a mulivariae Gaussian disribuion w N (, Q ), while ɛ denoes he observaion noise erm of he observaion equaion, assumed o follow a univariae Gaussian disribuion ɛ N (, R). Kalman (196) inroduced a filering echnique for sae space models which can efficienly calculae he condiional predicion and condiional filered esimaion of unobserved saes. A comprehensive inroducion o sae space models and Kalman filering has been provided by Kalman (196), Harrison & Sevens (1976), Harvey (1989), Grewal & Andrews (21). Since we aim a decomposing he ime series ino a common source componen and a paricular source componen, we choose a special srucure for he sae space model, such ha he las elemen of he sae vecor x represens he common source componen, and he former elemens of he vecor form a 3-variae AR model. By his we should have a canonical form (Aoki, 199) for he 3-variae AR and a coefficien for he common source along he diagonal of F. The 3-variae AR should capure main characerisics of he ime series bu he common source should only capure insananeous and simulaneous dynamic, herefore he coefficien for he common source should be small, for insance.5. The model parameers in Equaions (1) and (2) are esimaed from given daa by he maximum-likelihood mehod. Given a se of parameers, compuaion of he likelihood from he errors of he daa predicion hrough applicaion of he Kalman filer is sraighforward; see Mehra (1971), Åsröm & Kallsrom (1973), Sorenson (1985) and Valdés-Sosa e al. (1999) for a deailed reamen. A maximum likelihood esimae for he sae space model 4

5 is as follows F = , G = Q = R = , H = ,, Akaike informaion crierion(aic), a value for comparing saisical models by weighing likelihood funcion and number of model parameers, is (= ) for he sae space model, comparing o (= ) for a VAR(4) wih full marix of noise variance. I suggess ha he sae space model should be a more suiable model o he ime series. In figure 2(a) we show he specra of fmri of, from lef o righ, PP, V1 and V5, based on he esimaed sae space model. Each specrum is consiued of 4 colors, which corresponding o 4 sysem noises of he sae space model. By he sae space srucure we have already assume green, red, yellow and black are respecively PP, V1, V5 and common source. Through F, G and H, he 4 noises conribue o he ime series disincively, showed by he model specra. Among he 3 specra, he one of V1 has he 5

6 (a) (b) % 1% 1% 8% 8% 8% 6% 6% 6% 4% 4% 4% 2% 2% 2% (c) % % % % 1% 1% 8% 8% 8% 6% 6% 6% 4% 4% 4% 2% 2% 2% % % % Figure 2: Model specra, NCR causaliy map and parial NCR causaliy map of sae space 6

7 highes power inensiy a around.2, abou 5-second period oscillaion, which can also be seen clearly from he daa. In figure 2(b) we show he NCR causaliy map, which obained by normalizing he specra in (a). A each frequency he specral power inensiy is squeezed ino % o 1%, so ha he raio of conribuion from each noise variance can be seen clearly a each frequency. Since mos power inensiy is dense beween o.6 inerval, we shall explain causaliy based on his inerval. The black color is he noise in driving he ime series simulaneously by assumpion. We can see his common source explains over 5% of power inensiy a Hz in all he specra. Also, i has shared over 5% of power inensiy of he lower frequency region of V1. Noe ha his common source has been inroduced o he sae space model hrough an AR process of coefficien.5, which meaning his noise is no providing an addiional degree of characerisic roo o he ransiion marix, bu sparing more room for he correlaed residuals from AR. In figure 2(c) we show he parial NCR causaliy map, when he conribuion of common source, ie black, is eliminaed. These remaining colors can ell he causaliy from hese independen noises o he ime series. V1 is showing up around low frequency range, saying ha causaliy from V1 o PP and V5 is significan. PP is causing V1 and V5 a lile, mosly a he neighborhood of.5 (2-25s period oscillaion), and a he same ime, V5 is causing PP a lile and V1 negligibly nohing. We compare he above resul o he causaliy resul from AR, of which noise variance marix is diagonal, esimaed by leas square mehod. AIC of an AR(4) wih diagonal noise variance is (= ), a value much greaer han he AIC of sae space model, meaning his AR(4) is less suiable o he ime series. ( y (1) y (2) ) = y (1) 1 y (2) 1 y (3) 1 y (1) 2 y (2) 2 y (3) 2 y (1) 3 y (2) 3 y (3) 3 y (1) 4 y (2) 4 y (3) 4 + η (1) η (2) η (3) 7

8 η (1) η (2) η (3) N, By his AR(4) we plo model specra in figure 3(a) and NCR causaliy map in figure 3(b). To our surprise his NCR causaliy map is so similar o ha in Figure 2(b). On one hand i has proven ha he common source componen was added o lessen he squares of residual bu no o ake away any model characerisics by our assumpion, and on he oher hand we can assure our resul in sae space is consisen. (a) (b) % % % % 8% 8% 6% 6% 6% 4% 4% 4% 2% 2% 2% % % % Figure 3: Model specra and NCR causaliy map 4 Discussion We proposed a new mehod o apply Akaike causaliy in sae space framework so ha he only limiaion of Akaike causaliy is solved when residuals of VAR is highly correlaed. Correlaion beween noises in VAR is sored ou as an addiional independen noise homogeneously driving mulivariae ime series in a sae space framework. By comparing he AIC we found ha he sae space model fis beer han he VAR. The idea in his paper can be furher exended. Besides a common source componen for all ime series, some pairwise or uple-wise common source 8

9 componens can be added ino he model. For insance in his paper, in addiional o he common source of black color, we can inroduce one more color for a common source of V1 and V5 bu no PP, so ha he common source wih he visual corex can be furher eliminaed. More generally we should add also he oher wo combinaions of pairwise common source componens. However, ime series in real applicaion ofen share common characerisics. A common characerisics grealy shared by wo ime series may also appear in oher ime series, ha i should no be negligibly zero even hough is srengh is small. Therefore pairwise common variables could be easily absorbed by an overall homogeneous variable. See Tanokura & Kiagawa (23) for a similar reamen. Like any oher causaliy heory, Akaike causaliy has o be based on a model. The goodness of an esimaed model affecs very much he causaliy conclusion. Therefore, before drawing any causaliy conclusion, much effor on finding a suiable model is necessary. 5 Appendix 5.1 Sae space model and is ARMA represenaion Here we will give he ARMA represenaion of sae space model. Referring o Equaion (1) and (2), le F, G and H are of size m m, m k and l m respecively. Then F has m eigenvalues, hus here is a characerisic polynomial of order m, so ha we can ransform F linearly o zero by Cayley Hamilon Theorem. F m φ 1 F m 1 φ 2 F m 2 φ m 1 F φ m I =. By his a linear sae space model can be ransformed o a VARMA in erms of observed daa y and noises η. y φ 1 y 1 φ 2 y 2 φ m y m Θ η + Θ 1 η 1 + Θ 2 η Θ m 1 η m+1 + Θ m η m (3) Le I be ideniy marix. Θ = ( HG I ) Θ 1 = ( H (F φ 1 I) G φ 1 I ) Θ 2 = ( H ( F 2 φ 1 F φ 2 I ) G φ 2 I ). Θ m 1 = ( H ( F m 1 φ 1 F m 2 φ m 2 F φ m 1 I ) G φ m 1 I ) Θ m = ( φ m I ) ( ) ( ) w j Q η j = N (, Σ), Σ = R ɛ j 9

10 Auoregressive coefficiens of he VARMA are scalars which are coefficiens of he characerisic equaion of F. Moving average coefficiens Θ are formed by wo block marices, of sizes l k and l l, which depend on F, G and H only. This noise vecor η is sacked by w and ɛ verically. Noe ha he size of η is no necessary as same as ha of y. Alhough he auoregressive par is molded idenically for all variables in y, he moving average par refines each variable uniquely. 5.2 Akaike causaliy for VARMA and Sae Space Here we will give derivaion of Akaike causaliy of VARMA only. Akaike causaliy for sae space is sraighforward by combining his resul wih he formula in he previous subsecion. By Equaion 3 we will obain a power specral densiy marix P f for a VARMA. p F f (Φ) = I + Φ j e 2jiπf, F f (Θ) = j=1 q Θ j e 2jiπf, j= P f = F f (Φ) 1 F f (Θ) Σ F f (Θ) H{ F f (Φ) 1} H. A each frequency f, he diagonal elemens of P f are specral densiy of ime series and he off-diagonal elemens are cross specral densiy. If Σ is a diagonal marix, each diagonal elemens of P f is weighed sum of he diagonals of Σ. By his Akaike NCR causaliy is defined by he proporion of power from one noise variance o he power from all noise variance. NCR ( σ 2, y ) = specral densiy going o y from σ 2 oal specral densiy going o y from all variances Acknowledgemens The auhors would like o hank Dr Okio Yamashia and Prof Norihiro Sadao for providing he fmri daa, and special hanks o Prof Rolando Biscay for his commens and guidance. This work was suppored by he Asumi Inernaional Scholarship Foundaion, he Iwaani Naoji Foundaion, Research Insiue of Science and Technology for Sociey of he Japan Science and Technology Agency and he Japanese Sociey for he Promoion of Science hrough Kiban B no

11 References Akaike, H. (1968). On he use of a linear model for he idenificaion of feedback sysems. Annals of he Insiue of Saisical Mahemaics Aoki, M. (199). Sae Space Modeling of Time Series. New York: Springer- Verlag. Åsröm, K. J. & Kallsrom, C. G. (1973). Applicaion of sysem idenificaion echniques o he deerminaion of ship dynamics. In P. Eykhoff, ed., Idenificaion and sysem parameer esimaion. Amserdam: Norh- Holland. Büchel, C. & Frison, K. J. (1997). Modulaion of conneciviy in visual pahways by aenion: Corical ineracions evaluaed wih srucural equaion modelling and fmri. Cereb. Corex Geweke, J. F. (1982). Measuremen of linear dependence and feedback beween muliple ime series. Journal of he American Saisical Associaion Grewal, M. S. & Andrews, A. P. (21). Kalman filering: Theory and Pracice Using MATLAB 2nd ediion. New York: Wiley. Harrison, J. & Sevens, C. F. (1976). Bayesian forecasing (wih discussion). Journal of he Royal Saisical Sociey, Series B Harvey, A. C. (1989). Forecasing, srucural ime series models and he Kalman filer. Cambridge: Cambridge Universiy Press. Kalman, R. E. (196). A new approach o linear filering and predicion problems. Journal of Basic Engineering Mehra, R. K. (1971). Idenificaion of sochasic linear dynamic sysems. American Insiue of Aeronauics and Asronauics Journal Sorenson, H. W. (1985). Kalman Filering: Theory and Applicaion. IEEE Press. Tanokura, Y. & Kiagawa, G. (23). Exended power conribuion ha can be applied wihou independence assumpion. Tech. Rep. 886, The Insiue of Saisical Mahemaics. Valdés-Sosa, P., Jimenez, J. C., Riera, J., Biscay, R. & Ozaki, T. (1999). Nonlinear EEG analysis based on a neural mass model. Biological Cyberneics

12 Wong, K. F. K. (25). Mulivariae Time Series Analysis of Heeroscedasic Daa wih Applicaion o Neuroscience. Ph.D. hesis, Graduae Universiy for Advanced Sudies. Yamashia, O., Sadao, N., Okada, N. & Ozaki, T. (25). Evaluaing frequency-wise direced conneciviy of bold signals applying relaive power conribuion wih he linear mulivariae ime series models. Neuroimage