Ranking and Averaging Independent Component Analysis by Reproducibility (RAICAR)

Transcription

1 Human Bain Mapping 29: (2008) Ranking and Aveaging Independent Component Analysis by Repoducibility (RAICAR) Zhi Yang, 1,2,3 Stephen LaConte, 2 Xuchu Weng, 1 and Xiaoping Hu 2 * 1 Laboatoy fo Highe Bain Function, The Institute of Psychology, Chinese Academy of Sciences, Beijing , People s Republic of China 2 Biomedical Imaging Technology Cente, The Wallace H. Coulte Depatment of Biomedical Engineeing, Geogia Institute of Technology and Emoy Univesity, Atlanta, Geogia College of the Humanities and Social Sciences, Gaduate Univesity of Chinese Academy of Sciences, Beijing , People s Republic of China Abstact: Independent component analysis (ICA) is a data-diven appoach that has exhibited geat utility fo functional magnetic esonance imaging (fmri). Standad ICA implementations, howeve, do not povide the numbe and elative impotance of the esulting components. In addition, ICA algoithms utilizing gadient-based optimization give decompositions that ae dependent on initialization values, which can lead to damatically diffeent esults. In this wok, a new method, RAICAR (Ranking and Aveaging Independent Component Analysis by Repoducibility), is intoduced to addess these issues fo spatial ICA applied to fmri. RAICAR utilizes epeated ICA ealizations and elies on the epoducibility between them to ank and select components. Diffeent ealizations ae aligned based on coelations, leading to aligned components. Each component is anked and thesholded based on between-ealization coelations. Futhemoe, diffeent ealizations of each aligned component ae selectively aveaged to geneate the final estimate of the given component. Reliability and accuacy of this method ae demonstated with both simulated and expeimental fmri data. Hum Bain Mapp 29: , VC 2007 Wiley-Liss, Inc. Key wods: fmri; independent component analysis; data analysis INTRODUCTION Contact gant sponso: National Institutes of Health; Contact gant numbe: RO1EB00200; Contact gant sponso: National Science Foundation of China; Contact gant numbes: , ; Contact gant sponso: Chinese Ministy of Science and Technology; Contact gant numbe: 2003CB *Coespondence to: Xiaoping Hu; Wallace H. Coulte Depatment of Biomedical Engineeing, Emoy Univesity, Wooduff Memoial Reseach Building, 101 Wooduff Cicle, Suite 2001, Atlanta, GA xhu@bme.gatech.edu Received fo publication 11 Octobe 2006; Revised 27 Apil 2007; Accepted 1 May 2007 DOI: /hbm Published online 27 June 2007 in Wiley InteScience (www. intescience.wiley.com). Independent Component Analysis (ICA) is a data-diven appoach that is widely used in functional magnetic esonance imaging (fmri) [Calhoun et al., 2006; McKeown et al., 1998; Van de Ven et al., 2004]. The commonly used spatial ICA (sica) consides the fmri dataset as a linea mixtue of spatially independent components that ae mixed by thei espective time couses. ICA has the advantage of being able to detect spatially distibuted netwoks and tempoal dynamics in the bain without assuming a known esponse [De Luca et al., 2006; McKeown et al., 1998]. It is thus suitable fo exploatoy analysis of fmri data, whee the geneal linea model (GLM) might be hampeed by the lack of an appopiate a pioi esponse model. VC 2007 Wiley-Liss, Inc.

2 Yang et al. Despite its advantages, ICA has seveal theoetical and pactical limitations, including its inability to detemine the numbe of components and to ode these components [Hyvainen et al., 2001]. In standad ICA, the numbe of components is assumed to be equal to the ank of the voxel 3 time data matix (this is usually the numbe of time points collected since it is the smalle of the two dimensions). This assumption is geneally not appopiate in fmri, whee the numbe of signal souces is usually much less than the length of the fmri time seies [Beckmann and Smith et al., 2004]. Even though some ICA decomposition algoithms pemit fewe components, the numbe of components geneally needs to be specified befoe the decomposition, making the esults use-dependent (e.g. [Kiviniemi et al., 2003; Van de Ven et al., 2004]). Anothe limitation is that thee is no standad appoach fo odeing components, which may necessitate manual inspection of hundeds of components and make compaison of diffeent ICA esults poblematic. Algoithmically, ICA decompositions utilizing gadient-based optimization ae stochastic and ae based on iteatively updating the unmixing matix whose initial values ae usually geneated andomly [Himbeg et al., 2004]. The andomness of the initialization intoduces andomness into the ICA decomposition; consequently, a single decomposition is not eliable. Seveal goups have attempted to addess the above issues. McKeown and Sejnowski [1998] used an obsevation maximum likelihood method to estimate the numbe of components. Thei method does not extact the actual numbe of components but can be used as a means to make compaisons acoss methods [Esposito et al., 2002; Fomisano et al., 2004]. Beckmann and Smith [2004] have developed a pobabilistic ICA method implemented in the MELODIC package that estimates the numbe of components using pobabilistic PCA. Othes [Esposito et al., 2001; Gu et al., 2001] intoduced methods fo odeing the ICA components fo example by spatial chaacteistics such as the numbe of voxels and connection popeties. Moitz et al. [2003] poposed anking the independent components by the elationship between thei powe spectum and the stimulus fequency. Lu and Rajapakse [2003] developed a constained ICA algoithm and odeed components by the kutosis of thei pobability density distibutions. LaConte et al. [2001] odeed components based upon thei epoducibility acoss diffeent epochs in eventelated expeiments. Himbeg et al. [2004] poposed an appoach fo assessing both the algoithmic and statistical eliability of estimated independent components, via clusteing of epeated ICA ealizations and visualization of the clustes. In the pesent wok, a simila philosophy is utilized to develop an appoach, RAICAR (Ranking and Aveaging Independent Component Analysis by Repoducibility), fo anking the components, detemining the numbe of eliable components, and impoving thei estimates though aveaging. RAICAR is a famewok that makes use of spatial epoducibility to evaluate ICA components; it can be used to deal with algoithmic vaiability aising fom the optimization and/o data vaiability aising fom measuement noise. Although not the focus of the pesent wok, data vaiability can be examined by studying the epoducibility of ICA components with esampled data. In this pape, we focus on algoithmic vaiability by investigating the epoducibility of the component maps with epeated applications of ICA using diffeent initialization values. To deal with algoithmic vaiability, RAICAR aligns the components of individual ICAs and anks and selects the numbe of components based on thei epoducibility. The undelying assumption is that spuious components exhibit geate fluctuation acoss ealizations than stable ones. Theefoe, the epoducibility of each component (i.e. the esilience against the influence of andomness) eflects its elative eliability, and also allows estimation of how many components ae of sufficient stability to be etained. The final estimation of the etained spatial souces is obtained by selectively aveaging them acoss the ealizations, impoving thei quality and deceasing thei stochastic natue. Application to simulated and expeimental data demonstates that this appoach leads to intepetable and eliable esults. METHODS RAICAR Algoithm The basic idea of ICA is to decompose a data matix into seveal independent souces that ae linealy combined. Based on this idea, the spatial ICA employed in fmri data analysis teats spatial patten maps as the independent components, which ae mixed togethe accoding to thei coesponding time couses [McKeown et al., 1998]. This notion can be expessed as X ¼ MS, whee X is the obseved dataset, M is the mixing matix of time couses, and S epesents the independent spatial maps. Suppose an obseved dataset X (with T time points and V voxels) consists of C spatial souces. X is a T 3 V matix; M is T 3 C while S is C 3 V. The aim of ICA is to estimate both M and S simultaneously. The independence assumption is utilized in ode to pefom the simultaneous estimation. That is, M and S ae detemined by maximizing the independence of the components. Nongaussianity can be used to quantitatively epesent this independence, and seveal measues of nongaussianity have been poposed [Hyvainen and Oja, 2000]. Algoithmically, the ICA model is often ewitten as S ¼ WX, whee W is a squae full-ank unmixing matix and M ¼ W 1 [McKeown et al., 1998]. In most ICA algoithms (e.g. FastICA [Hyvainen, 1999]), W is iteatively updated until maximum nongaussianity is achieved. With the iteative pocedue being a gadient-based optimization, W is usually initialized with andom numbes at the beginning of the iteation, intoducing andomness into the decomposition [Himbeg et al., 2004]. The implication of having a andom initialization is that diffeent initial 712

3 ICA Ranking and Aveaging by Repoducibility conditions can lead to diffeent esults in the gadientbased optimization. RAICAR pefoms the ICA decomposition K times (K ealizations). In ou implementation we use the FastICA algoithm with diffeent initial conditions, 1 yielding K mixing matices, M 1, M 2 M K, and coespondingly K sets of souces, S 1, S 2 S K. The numbe of components, C, is set to the estimated ank of the data matix duing the decomposition (the default fo FastICA is the numbe of eigenvalues of X lage than 10 7 fo fmri C often equals the smalle dimension of the data matix, X) and is the same fo all ealizations. Theefoe, each ealization leads to C components, heeafte efeed to as a ealizationcomponent (RC) and indexed by a component numbe (anging fom 1 to C) and a ealization numbe (anging fom 1 to K). To examine the epoducibility of the RCs, the fist step is to constuct a coss-ealization coelation matix (CRCM). This matix is C K 3 C K with the following stuctue: 2 3 R 11 R 12 R R 1K. R R R K R KK Figue 1. Each submatix in the coss-ealization coelation matix (CRCM), R ij, is the spatial coss-coelation matix between ealizations i and j. The cicled dot epesents the global maximum in the CRCM. Afte finding this maximum, the mth ow in each submatix R ai and the nth column in each submatix R ib ae seached fo a coesponding submatix-specific maximum. These maxima ae indicated by dots with thei positions given in paentheses. The R 0 ij sði ¼ 1; 2;...; K; j ¼ 1; 2;...; KÞ ae C 3 C submatices whose elements ae the absolute value of the spatial coelation coefficients (SCC) fo all pais of components fom ealizations i and j. Thus, the SCC is defined as the absolute value of the Peason s coelation coefficient between component maps. Note that the CRCM is symmetic, and submatices R ii ae identity matices and ae ignoed in the subsequent algoithm. Afte constuction of the CRCM, the next step is to align the ealizations. Since thee is no pedetemined ode fo the ICA components, a given component s position may appea in any of the C positions fom ealization to ealization. To align acoss ealizations, the following pocedue is epeatedly applied (see Fig. 1). Fist, the global maximum in the CRCM is identified. Let us assume that the maximum is located at the mth ow and the nth column (denoted as [m, n] heeafte) in submatix R ab (thee is an identical value in R ba since the CRCM is symmetic). This maximum allows us to establish coespondence between component m in ealization a and component n in ealization b, defining the stating point of an aligned component, with ealization components RC ma and RC nb. In the emaining ealizations, the RCs having the maximum 1 Fo the esults epoted hee, the andom initial conditions wee geneated by using the compute s system time to geneate a andom seed. Fom this seed, a andom matix was geneated, which detemined the initial conditions fo each ealization. coelation with these RCs ae identified. Specifically, the mth ow of the submatices R ai (i efes to all the ealizations othe than a o b) and the nth column of the submatices R ib ae seached, fo espective maxima. The positions of the maxima ae denoted [m, p i ]inr ai and [q i, n] in R ib, as shown in Figue 1. In many cases, the two maxima fo each ealization coespond to the same RC (i.e. p i ¼ q i fo ealization i); in this case, component p i of ealization i is assigned to the aligned component defined by [m, n] in R ab. In the case that p i = q i, the SCCs of components p i and q i of ealization i ae compaed, and the one with the lage SCC is assigned to the aligned component defined above. The ows and columns that contain the enties and thei diagonal eflections ae subsequently eliminated fom the CRCM befoe the next epetition stats. The pocedue is epeated C times until C aligned components ae identified. Each aligned component entails K RCs, one fo each ealization, and coss-coelation between these aligned RCs poduces K(K 1)/2 SCCs. Afte the alignment, a histogam of the SCCs in the CRCM (uppe tiangle) is geneated. 2 This histogam is bimodal, with modes nea 0 and 1, epesenting a lage numbe of components that ae not coelated and a small numbe of components that ae highly coelated, espec- 2 Ou implementation used 100 bins with SCC values anging fom 0 to 1 using default histogam function in Matlab TM (Mathwoks, Natick, USA). 713

4 Yang et al. tively. To eliminate insignificant SCCs, an SCC theshold is set at a point between the two modes. In this epot, the SCC theshold is found by smoothing the histogam and seaching fo the minimum. A epoducibility index fo each aligned component is then geneated by summing the SCCs, among the aligned component s RCs, that ae above the SCC theshold. Impotantly, as ou esults demonstate, the anking esult is not sensitive to the exact choice of SCC theshold, povided it lies in the valley of the histogam between the two modes. The epoducibility index is used to ank the aligned components in descending ode. If an aligned component is consistent acoss ICA ealizations, its epoducibility index will be high and thus its anking ode will be high. The epoducibility index is a measue of component eliability and can be used to detemine the numbe of epoducible components. In fact, given that the SCCs exhibit a bimodal distibution, most components will have a vey low epoducibility index, and a few components will have a elatively high epoducibility index. As we show in ou esults, the odeed epoducibility index geneally dops off shaply, allowing us to estimate the numbe of components, by keeping those components whose epoducibility index is above a cut-off point. Visual inspection of the odeed epoducibility plot allows the selection of the cut-off point by eye and is ecommended in pactice. The choice of a cut-off is often equied in many data eduction techniques (e.g. in dimensionality eduction using a plot of singula values) and is the esponsibility of the expeimente. In this wok, to be consistent acoss expeiments, the cut-off point is set to 50% of the index s maximum possible value ( KðK 1Þ 2 30:5). Anothe possibility includes finding the maximum slope in the epoducibility plot. And, even without selecting a cut-off, the numbe of components is geneally geatly educed compaed to C, making it possible to examine all of these components. To geneate the final components, the spatial maps and the coesponding mixing time couses of the RCs of each aligned component ae selectively aveaged. That is, only RCs that have at least one SCC highe than the theshold fo all of the othe the RCs of the given aligned component ae included when geneating the aveaged components. Simulation 1 A simulation study was conducted to evaluate the pefomance of RAICAR. Six nonovelapping spatial souces with equal aea wee geneated and mixed togethe accoding to the mixing time couses indicated in Figue 2. The mixing time couses had zeo mean and consisted of 162 time points. A slowly vaying global baseline was added to all the pixels, and the esultant time couse fo each pixel was futhe degaded by adding Gaussian white noise. The SNR fo each component was detemined by the vaiance of the mixing time couses, elative to that of the Gaussian white noise. Specifically, the SNR of the six mixing time couses wee 0.35, 0.29, 0.24, 0.20, 0.16, and Figue 2. Simulated spatial souces and thei mixing time couses. A: Spatial map of the numbeed souces. All the souces ae equal in aea. B: The coesponding mixing time couses of the souces. The bottom panel shows the added global baseline. 0.14, espectively. The SNR of the slowly vaying global baseline was set to With these values, the contast-tonoise atio (CNR : DS= noise ) of the mixing time couses anged fom 0.92 to 3.87, consistent with CNR values epoted in the fmri liteatue [Esposito et al., 2002; Huettel et al., 2004]. No pe-pocessing was applied to the simulated data. FastICA ( fo Matlab TM (Mathwoks, Natick, USA) was used to cay out the ICA decompositions. Thity diffeent ICA decompositions wee obtained with andom initial conditions (K ¼ 30) to pefom RAICAR. To examine the elevance of the esultant components, they wee matched to the tue souces accoding to thei tempoal coelations, and a eceive opeating chaacteistics (ROC) analysis was conducted based on the spatial maps. To test eliability, we epeated the above pocedue 10 times. Simulation 2 In Simulation 2, we supeimposed the souces fom Simulation 1 onto esting state fmri time seies. The esting state data wee acquied using a 3T Siemens Tio scanne (Siemens Medical Solutions, Malven, PA), with TR ¼ 2 s, TE ¼ 34 ms, and flip angle ¼ 908. The esting state data 714

5 ICA Ranking and Aveaging by Repoducibility Repoducibility anking of the simulated data. A: Bimodal distibution of the coelation coefficients. The majoity of the coelation coefficients lie in the lowe ange of , while the emaining fall in the uppe ange of These two anges ae sepaated by a boad valley (oughly ). The solid ed cuve is the smoothed histogam used to detemine the Figue 3. theshold. The aows indicate the thee SCC thesholds used. B: The epoducibility index plots geneated using the thee SCC thesholds. The half-maximum cut-offs ae shown with hoizontal geen lines, indicating six components in each case. The odes of the components deived with all thee thesholds ae also the same. wee coected fo motion and fo physiological noise using the AFNI RETROICOR plugin [Cox, 1996; Glove et al., 2000]. We applied RAICAR to thee combinations of data: esting state data only, esting state data with lowcontast simulated souces, and esting state data with high-contast simulated souces. Contast was contolled by scaling the magnitude of the simulated souces on a pixel-by-pixel basis. Fo the low contast case, the simulated mixing time couses had the same CNRs as in Simulation 1, and fo the high contast case, the CNRs of the simulated mixing time couses wee 2.7 times those in Simulation 1. Delayed Moto Task The data wee acquied on thee healthy ight-handed paticipants using single shot T2*-weighted EPI on a GE Signa 1.5T scanne (GE Medical Systems, Milwaukee, WI) TABLE I. The ode of the components coesponds to the descending ode of thei SNR Ode Repoducibility index SNR Figue 4. Detectability of RAICAR and individual ICA ealizations using the sixth simulated souce (with lowest SNR). ROC cuves of the individual ICA ealizations (black) and RAICAR esults (ed) ae shown. The light ed egion shows the spead of 10 RAI- CAR epetitions and the ed cuve shows thei mean. The individual ICA ealizations exhibit vaiable esults, while the epeated RAICAR esults ae all vitually identical and outpefom the majoity of individual ICA ealizations. 715

6 Yang et al. with the following imaging paametes: TR/TE ¼ 2,000/50 ms, flip angle ¼ 908, FOV ¼ 220 mm, matix ¼ , five oblique axial slices, stating fom the top of the head, slice thickness/gap ¼ 5/0 mm, and 126 volumes. The task consisted of ight-handed finge tapping in a delayed movement paadigm. The paticipants wee pesented with a visual cue indicating the finge tapping sequence, but did not move thei finges until a Go command was given [Catalan et al., 1998; Ogawa et al., 1998]. Fo each tial, the visual Cue and Go stimuli lasted fo 2 s each and the inteval between them was 12 s. The inte-tial inteval (ISI) was also 12 s. Each un consisted of nine such tials, each lasting 28 s. These data wee motion coected, baseline detended and masked to exclude the voxels outside the bain using AFNI [Cox, 1996]. RAICAR was applied, with 30 epetitions of andomly initialized ICA decompositions (K ¼ 30). Identification of the task-elated components was achieved by coelating the mixing time couse of the esulting components to the stimulus sequence, convolved with an ideal hemodynamic esponse function (HRF) geneated by AFNI. To examine the obustness of the epoducibility method, it was applied thee times, each time using thee diffeent SCC thesholds. Constant Foce Gip Task Subjects epeatedly gipped a wate-filled bottle with thei ight hand. The foce of the gip was gauged by the wate pessue in eal time and pesented to the subject so that he/she can adjust the foce to meet a taget level of 50% of his/he maximal voluntay contaction (MVC) level [Liu et al., 2000, 2002], calculated based on the maximal gip foce measued at the beginning of the expeiment. Subjects pefomed the gipping by following visual cues (geneated by a wavefom geneato [Wavetek Daton, San Diego, CA]) pojected onto the sceen above the subjects eyes in the magnet. Each visual cue was a ectangula pulse that indicated (taget amplitude fo 50% MVC and desied duation of 3.5 s) the desied contaction. The duation of each contaction was 3.5 s, followed by a 6.5-s esting inteval [Peltie et al., 2005]. fmri data wee collected on a 3T Siemens Tio scanne (Siemens Medical Solutions, Malven, PA), with 30 axial EPI slices (TR/TE ¼ 2,000/30 ms, voxel ¼ mm 3, flip angle ¼ 908). Motion coection and bain masking wee applied as pepocessing pocedues, and RAICAR was pefomed (K ¼ 30). To examine the sensitivity to the SCC theshold fo these data, we epeated RAICAR with two additional SCC theshold values. Investigation of the Impact of K The stability of the ank positions was studied as a function of K fo Simulation 1. We also investigated the vaiability of the RAICAR esults as a function of K (the numbe of individual ICA ealizations used by RAICAR). Fo Simulation 1 and both expeimental datasets, RAICAR was applied 20 times at diffeent K values anging fom 5 to 55 in steps of 5. All component maps wee matched to the esults epoted fo the above studies (which used K ¼ 30). Fo each K value, the vaiance fo each component in the 20 RAICAR epetitions was calculated and aveaged Figue 5. Repoducibility ankings obtained fom simulation 2. A: esting-state data only. B: esting-state data with low CNR souces. C: esting-state data with high CNR souces. The inceased CNR level shifts the components towads the left (inceasing thei ank). 716

7 Repoducibility anking of the event-elated, delayed moto data in one subject. Columns coespond to diffeent sets of ICA ealizations. The top ow shows the histogams of the SCCs fo the thee sets of ICA ealizations. The coelation coefficients ae distibuted in two modes, one nea zeo and the othe nea Figue 6. one. The bottom thee ows ae odeed epoducibility index plots fo thee diffeent thesholds. It can be seen that the numbe of components passing the cut-off (shown as hoizontal gay lines) do not vay significantly with the SCC thesholds and diffeent sets of ICA ealizations.

8 Yang et al. ove all voxels and all components. The aveage vaiance as a function of K was used to measue convegence. RESULTS AND DISCUSSION Simulation 1 The simulation data wee decomposed into 162 components by FastICA (C ¼ 162). Figue 3 shows the histogam of the SCCs and the plot of the odeed epoducibility index. The SCC histogam (Fig. 3A) follows a bimodal distibution. The smoothed histogam is shown as the ed cuve in Figue 3A, and has a minimum at 0.73, which was used to theshold the SCCs. In addition to 0.73, two othe SCC thesholds, 0.60 and 0.80 (maked with aows in Fig. 3A), wee also used to test the sensitivity of the analysis to the SCC theshold. Out of 162, the numbe of eliable components detemined using the half-maximum cut-off (217.5) was six fo all SCC thesholds. Each of these six components uniquely matches one of the oiginal souces in tems of both spatial patten and mixing time couse. As shown in Table I, the ode in which they appea in Figue 3 coincides with the ode of deceasing SNR of the souces. Theefoe, the pesent method successfully extacted both the numbe and the ode of the components in the simulated data. While the ank-ode may not coincide with the SNR ode in geneal, it does eflect the stength of the component. The odeed epoducibility indices geneated fom the thee SCC thesholds ae plotted in Figue 3B. In these plots, the fist six components (those above the cut-off) exhibit the same anking and have epoducibility indices that ae essentially independent of the SCC theshold. In othe wods, the SCC theshold has a negligible effect on the anking of the epoducible components and thei numbe. The ROC cuves geneated fo the sixth souce in the simulation ae shown in Figue 4. RAICAR outpefoms nealy all the individual ICA ealizations. The wide spead in the detectability (aea unde the cuve) among individual ICAs clealy indicates the stochastic natue of FastICA, which can not be neglected. In contast, the spead of the 10 epetitions of RAICAR is elatively small. The tempoal coelation coefficients between the oiginal souce and the mixing time couse estimated by diffeent methods ae 0.89 fo the best individual ICA, 0.31 fo the wost individual ICA, and 0.91 fo RAICAR. Identical esults wee obtained with two othe epetitions of the anking pocess. It is impotant to note that the souces in ou simulation studies ae not exactly independent since they ae mutually exclusive (e.g. Souce 1 does not spatially ovelap with any othe souce). This lack of complete independence may lead to difficulties in ICA decomposition but is pobably moe eflective of eal fmri data whee independence is not guaanteed. Fotunately, ou esults show that some dependence in the data did not pevent us fom obtaining good esults. Simulation 2 Ten epoducible components wee obtained in the esting state data (see Fig. 5A), eflecting functional connectivity [Biswal et al., 1995] and stuctued noise. When the simulated souces wee added to the esting state data, they wee ecoveed by RAICAR and anked in the ode of descending SNR. As shown in Fig. 5B,C, the incease of contast of the simulated souces shifted thei position in the ank. At low contast, the simulated souces wee located away fom the top anks and dispesed among the esting-state components, while at high contast, they became the top components. This may indicate that at the high CNR levels, the simulated components have CNR highe than o compaable to those of the esting components. These obsevations indicate that RAICAR woks well in the pesence of fmri backgound noise and the anking povides a useful measue of the stength of the components. Delayed Moto Task Fo the delayed moto data, the numbe of components in each individual ICA ealization was 123 fo all thee subjects (C ¼ 123). The thesholds detemined by the SCC histogams wee 0.73, 0.75, and 0.71, espectively, fo the thee subjects. Figue 6 shows RAICAR esult of one of the subjects. Simila to the simulated data, the SCCs (top ow) ae distibuted in two modes, one nea zeo and the othe close to one. The bottom thee ows show the epoducibility index plots, obtained using thee diffeent SCC thesholds, with thee diffeent sets of ICA ealizations. As Table II shows, the numbe of components detemined by the cut-off index does not vay substantially with eithe the SCC theshold o the set of ICA ealizations. Howeve, thee is a slight vaiation in the numbe of components detemined since the dop-off of the epoducibility index is moe gadual in these data than in the simulated data. Table II also shows that Go and Cue components of the delayed movement task have anks that ae not highly dependent on the SCC theshold o the set of ICA ealizations. Simila conclusions hold fo the esults fo the othe two subjects. The insensitivity to the SCC theshold indicates that it is a paamete that does not have to be caefully selected, and the insensitivity to the specific set of Th. TABLE II. The estimated numbe of components (Num. IC) and the positions of the task-elated components (go, cue) ICA set 1 ICA set 2 ICA set 3 Go Cue Num. IC Go Cue Num. IC Go Cue Num. IC

9 Seveal components extacted by RAICAR in the delayed moto dataset. Component maps ae displayed on the tanspaent glass bain with coesponding time couses shown in blue; the ed and black cuves illustate the ideal esponse of Cue and Go task, espectively. Components 1 and 2 ae due to cadiac noise; Component 10 is task-elated activation in the sensoy and eye Figue 7. field egions; Component 12 shows the activations coesponding to the visual motion contol; Component 13 is elated to head motion; Component 14 coesponds to the Cue task; Component 18 shows the activations in pefontal cotex; Component 19 is activated by the Go task.

10 Yang et al. Figue 8. The epoducibility ank obtained fom the constant foce gip dataset. 17 components wee above the epoducibility cut-off. ICA ealizations minimizes the stochastic natue of an individual ICA. Figue 7 shows eight components extacted by RAICAR. The two top-anked components ae elated to cadiac noise; Component 10 shows the task-elated activations on the sensoy and the eye field egions; Component 12 shows the infeio paietal egions (BA40/7) and the pecuneus, which ae elated to visual motion contol; component 13 clealy eflects atifacts due to head motion; component 14 contains activations in posteio paietal and pefontal egions, which ae elated to moto pepaation [Hanakawa et al., 2003]; component 18 pimaily shows activation in bilateal pefontal cotex, which is elated to moto pepaation and spatial imagey; component 19 is elated to the moto execution task. To compae the RAICAR Go map (component 19) and the individual ICA maps, we calculated thei spatial coelation coefficients with the GLM activation map coesponding to the Go task. As expected based on ou simulated ROC studies, the coelation of the RAICAR patten of activation was in the uppe ange of the individual ICA ealizations, even though in this case the ange was elatively naow. Specifically, the 30 individual ICA ealizations had a mean coelation of and standad deviation of while the RAICAR coelation was Constant Foce Gip Task Fo the constant foce gip data, the numbe of components in each individual ICA ealization was 71 (C ¼ 71). Using a theshold of 0.80, as detemined by the SCC histogam, the epoducibility ank was geneated and the numbe of components was detemined as 17. Figue 8 shows the epoducibility ank, and Figue 9 pesents 3D views of all components extacted by RAICAR. The fist component eflects egions that have been implicated in the default mode netwok [Raichle et al., 2001]. The second component is task-elated activation in the pimay moto aea. Components 4 and 5 ae also task-elated, likely coesponding to moto contol; inteestingly, thei mixing time couses follow the behavioal paadigm but ae modulated by a slow vaiation. Component 11 eveals some task-elated atifacts at the base of the bain. Othe components ae eithe atifacts o unexplained bain activation. To make a visual compaison between RAICAR and individual ICA maps, Figue 10 shows components fom both RAICAR and a andomly selected individual ICA. Fo the top components in the epoducibility ank, RAI- CAR maps and individual ICA maps do not exhibit geat diffeences. Howeve, as the ode appoaches the end of the epoducibility ank, individual ICA maps tend to be noisie than the RAICAR maps. Table III lists the epoducibility anks with diffeent SCC thesholds. Although the total numbe of components vaies with diffeent SCC thesholds, the numbe of epoducible components (above the epoducibility cut-off) dose not change substantially, again showing the insensitivity to the SCC theshold. Row and Column Equivalence of the CRCM Ou implementation handles the case that p i = q i, by choosing the RC with the lage SCC in the alignment pocess. To evaluate how fequent this case aises in pactice, we tacked its occuence duing the alignment fo the simulation 1 and the two expeimental data sets. Table IV shows the occuence ate of p i ¼ q i aveaged ove the epoducible components and 20 epeated applications of RAICAR. Fo all thee datasets, moe than 99% of the cases ae p i ¼ q i, and the standad deviations acoss 20 epetitions ae athe small. This means that the p i = q i cases ae ae in the epoducible components. Impact of K Figue 11 shows the convegence chaacteistics of RAI- CAR. Figue 11A gives the positions of the components identified by RAICAR fom data in Simulation 1 as a function of K. Fo K above 20, thee is no change in the positions. Figue 11B displays the aea unde the ROC cuve as a function of K, geneated fom the lowest SNR souce in Simulation 1. The aea unde the ROC cuve is constant when K is lage than 20. Panels C, D, and E of Figue 11 show the aveage vaiance among 20 RAICAR epetitions fo Simulation 1, the delayed moto dataset, and the con- 720

11 ICA Ranking and Aveaging by Repoducibility The 17 components fo the constant foce gip dataset. The component maps ae shown on the tanspaent glass bain with the coesponding time couses shown below. The ideal task esponse is shown (ed) fo components that ae highly coelated with it. The fist component may eflect the default mode Figue 9. netwok; Components 2, 4, and 5 aise fom task-elated activations, which include functional aeas fo moto contol and execution; Component 11 seems to be due to task-elated atifacts at the base of the bain. Othe components ae eithe atifacts o unexplained bain activation. 721

12 Yang et al. Figue 10. Compaison of the component maps extacted by RAICAR and a andomly selected individual ICA ealization. The top ow of each component shows the RAICAR map and the bottom ow shows the individual ICA map. Fo highe anked components, both esults tend to be simila. While fo lowe anked components the individual ICA maps tend to be noisie than the coesponding RAICAR maps. stant foce gip dataset as functions of K. When K ¼ 30, the aveage vaiance is no moe than in all thee datasets. This vaiance is negligible when compaed with the intensity of the component maps (mean ¼ 0, standad deviation ¼ 1), indicating that a K of 30 is sufficient fo RAICAR applied to the thee data sets examined. In pactice, the pope choice of K can be data dependent. The methods descibed hee allow uses to choose pope K value fo diffeent datasets. TABLE III. The numbe of epoducible components (using the half-max cutoff) is not sensitive to diffeent SCC thesholds SCC theshold Numbe total components Numbe epoducible components

13 ICA Ranking and Aveaging by Repoducibility Data set TABLE IV. Occuence ate of the p = q cases Geneal Discussion Pecentage of p ¼ q (mean 6 std) Simulation1 (6 comps) ( )% Delayed moto (21 comps) ( )% Constant foce gip (17 comps) ( )% The means and standad deviations ae geneated based on 20 RAICAR epetitions. RAICAR is an extension of AFRICA [LaConte et al., 2001] and is unique compaed to othe methods fo odeing independent components [Esposito et al., 2001; Gu et al., 2001; Lu and Rajapakse, 2003; Moitz et al., 2003], in that it elies upon epoducibility. Selective aveaging of RCs is used by RAICAR to geneate the final components. Othe consensus methods, such as clusteing o weighted aveaging, could be used to estimated the components fom epeated ICAs; similaly, it is possible to use measues othe than coelation coefficient (e.g. Fishe tansfom of the coelation coefficients o R 2 ) to geneate the epoducibility index. Finally, othe stochastic algoithms beyond FastICA can also be incopoated. These possibilities emain to be exploed. As mentioned befoe, RAICAR is demonstated hee only with algoithmic vaiability. It is also possible to conside data vaiability. Pocedually, the pimay diffeence would be to epeat ICAs with diffeent data subsets instead of o in addition to using diffeent initialization values. Ou esults suggest that the numbe and ode of RAI- CAR components is impotant. In paticula, the esults of Simulation 2 indicate that the numbe of components is consistent with the ICA model of mixed souces and thei Figue 11. Convegence of RAICAR. A: The positions of the components as a function of K, geneated fom simulation 1. The positions of the components do not change when K is lage than 20. B: The aea unde the ROC cuves as a function of K, geneated fom the lowest SNR souce in Simulation 1. The estimation eo of the component maps does not change substantially when K is lage than 20. C, D, and E: The aveage vaiance of the component maps as a function of the numbe of ICA ealizations (K), geneated fom the six souces in simulation 1, the 21 souces in the delayed moto dataset, and the 17 souces in the constant foce gip dataset. Fo all thee datasets, these cuves apidly appoach asymptotic levels. When K ¼ 30 (as used fo ou epoted esults), the aveage vaiance is no moe than

14 Yang et al. Figue 12. Compaison of the estimated numbe of components fo RAI- CAR and ICASSO. The compaison was conducted using the souces fom Simulation 1 with diffeent time seies lengths. RAICAR estimates the coect numbe of souces at all lengths, while ICASSO esults vay with data length. When the length exceeds 440, ICASSO estimates could not be obtained due to lage memoy equiements. positions can be manipulated by changing thei CNR. Simila obsevations can be made fom ou expeimental esults. In the delayed moto task, the task-elated components wee anked between 10 and 19. This is likely due to the low CNR event-elated signals. In contast, in the constant foce gip task, the task-elated components esided mainly at the top pat of the epoducibility ank, consistent with the high CNR in the continuous hand gip data. These obsevations futhe suppot the notation that the epoducibility ank is meaningful and the epoducibility can be used as a measue of stength. The ICASSO method [Himbeg et al., 2004] was developed to validate and visualize ICA esults by clusteing the independent components. Thei findings suppot the pemise taken hee that andomness exists in gadientbased ICA algoithms and can be educed to poduce bette esults. Ou appoach beas similaity to ICASSO in epeating the ICA decomposition and extacting components fom the multiple ICA ealizations. Impotantly, though, thee ae substantial diffeences between RAICAR and ICASSO. (1) In RAICAR, the matix-based alignment is much less demanding in tems of computational equiements than the hieachical clusteing used in ICASSO. In fact, ICASSO equies fa moe memoy and CPU opeations, and its pactical use often equies data eduction with PCA as a pepocessing step. As pointed out by McKeown et al. [1998], some ICA components of inteest may compise only a few pecent of the total vaiance and be inadvetently eliminated by PCA. Note that the significant eduction in computational buden makes RAICAR pactical fo lage datasets that may aise fom long time seies o inte-session/subject data. (2), ICASSO identifies the esultant component fom each cluste using the centotype while RAICAR uses selective aveaging of the aligned components. This diffeence is expected to lead to diffeent spatial and tempoal esults. (3) The metics fo evaluating the quality of components fo both methods ae diffeent. ICASSO povides two metics the quality index and R-index, whose intepetation equies expet knowledge. RAICAR uses a spatial epoducibility index which is intuitive to undestand and use. To povide a moe diect compaison of the two methods, they wee both applied to ou simulated souces. In Figue 12, the estimated numbes of components fom the two methods ae shown fo data geneated with diffeent time lengths. The RAICAR esults give a coect estimation fo all data lengths while ICASSO estimation is less accuate, vaying acoss the data lengths. Moeove, as shown in Table V, ICASSO equied much longe pocessing time, and, fo data with moe than 440 time points, ICASSO could not be successfully un (due to excessive memoy equiements) on a wokstation with 6 GB RAM. Fom a pactical point of view, RAICAR does not equie difficult use decisions in tems of paamete selection. The paametes needed ae (1) the numbe of ICA ealizations, (2) the SCC theshold, and (3) the epoducibility index cut-off. We have demonstated that RAICAR is not sensitive to the fist two paametes, allowing flexibility in thei selection, and the cut-off can be chosen by visual inspection of the epoducibility anking. Theefoe, we expect RAICAR to pefom well fo all uses, egadless of expeience level. CONCLUSION We have intoduced an ICA method, RAICAR, based on epoducibility to impove the decomposition and inte- TABLE V. Compaison of computation time between RAICAR and ICASSO Time length RAICAR 3 min 5 min 40 s 11 min 15 min 36 s 21 min 26 min 22 s ICASSO 52 min 2 h 43 min 9 h 14 h The esults ae geneated on a wokstation with RedHat TM Entepise Linux WS4, 3.6G hypetheading CPU, and 6 GB RAM. 724

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