genovese_bayesian

genovese_bayesian - A Bayesian Time-Course Model for...

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Unformatted text preview: A Bayesian Time-Course Model for Functicnal Magnetic Reacnance Imaging Data Ghristopher Ft. Genny ese Functional Inagnctic resonance imaging if'leli is a new technique for studying the worltings oftlle actirt human brain. During an MRI experiment. a sequence of magnetic resonance images is acneiletl while the subject perform specific behavioral tasits. I:Ifitariges in the marl-sored signal can be used to identify and cltalactel'ire the brain activity resulting from tasit [:fldl'ffll'lllilm‘t' The data obtained from an fl'stRi cspcrinlenl are a realisation of a complelt spatiolelnporat process with many soutees of variation. both biological and technological. This article tlescl ibes a nonlinear Bayesian hierarchical medal for flu-till data and ]:Ittesenls inferential math-ads that enable investigators to directly target their scientific Questions of nnerest. many of which are Itlaeeessibte tu current methuds The article describes uplimisaliell and plateriur sanipiing l'f'L'Il'lJIlqulda tu- lit the model. bulb uf which must be applied triatly thousands {If limes fer a single datasbt. The ill-allel is used tu analyze data frem a psyehclegteul esperiment and la test a specific prettietien at it eugnititre theory. Eli‘t' WEEDS: Bayesian inference; Functional acute-imaging; Hierarchical mtsdels; Magnetic resnnanee. t. IMTFICJ DUCTIDN Functional magnetic resonance imaging {fhiitll enables cognitive psychologists and neuroseientists to study the till- man brain in scuba. During an MRI experiment. the sub- ject performs a carefully choreographed sequence of he- has-iora] tasks while magnetic resonance {MR} images cf his nr her brain are acquired at regular intervals- The tasks are designed to esercise specific motor. sensory. or cogni— tis-e processes. and the measured Mill signal contains ini‘or- tnatiUn abetlt the nature and Iueatiun til" the neural activity that results when these prneesses are engaged- F‘sychnln- gists‘hope to use fMFti data to build and test theoretical models of human engnilien. but let dn this diey must snlye a quintessentially statistical problem- in this article I de- 1telnp a new statistical model and new inferential methods i'nr i‘l‘s-lRI data- An fldRI experimental design specifies the taslts tltat each Subject is it: perfurtn. as well as their timing and dura- tinn- Each distinct taslc defines nne esperimental cnnditinn, which is usually replicated seyeral times during the esper— iment. Each eentigucus bluclt. cf lime in which the sub- jeet perfnnns a specified task is called an eprn‘is within the associated condition. in “block trial” designs. the subject repeatedly perfunns the taslt dunng each ep-tich; in “sin- gle trial“ designs. the subject performs the tasit once per epoch. The choice ul' tasks is critical tn the interpretatien nl' fMRI experimental results, because inferences frnm the data are commonly based on “subtractiye” logic. The tasks are designed in differ in the specific enmpnnents tJf prneess- ing that they |[are helieyed tol require. An observed dilter- ence inactivation between [w'tt conditions is then attributed to the eempnncnts ni' prneessing that diiter. C'hristupher I-t. [lanes-cab is Associate I-‘rulessnr. Department at" Staris— tics. Carnegie Mellun Llniytersity. Pittstalrgh. PA l52|3 IiE-mail: genes-ere [insides-alteedai. 'The anther would like tn thanlt Patricia Carpenter. Marcel last. and "I'llmtsthyI Heller I'nr pan-riding the data used in this article and Fitch Kass. Hill Eddy. and Marsha [.nyetl far helpful discussinus and cert-intents. The aulhnr would alsct like In thank the assaeiale eidilnr and an anenyrnnus referee fer their valuable enmlnents. This research was sqtppnned h].- .‘nin- tienal Science Foundation grants HMS 'idtldflfl? and SEE little Isl? and the t'ienter far the Neural Basis n’f Cngnitinn. 5131 Han i'llel experiment yields a sequence cf three- dimensional images of tlte subject‘s brain. Each image com- prises measurements of the tyIi-t signal over a grid of small. regular ynlume elements called parcels. The MR signal fer a spiral in a single image is related to the transyerse magneti- zatien ot' a particular nuclear spin. usually hydrugen. ascr- aged user the ease] and a small time intersal- But in func- tional MRI. we are not interested in the images per se. but rather in small. task-related changes in the measured Mil signal cyer time cattsed by idealised neural aetiyity The se- quence ni' MR images yields a time series nf measurements for each trot-tel. from which we can identify and character- iae the brain‘s respnnse tc basic-related aetiyity. 1ililo-.ttels typ- ically range frnn'l 3—50 mar", with cnrrespnnding images containing ltitl.titltl—4tfll.tltiil s'osets. The time to acquire an entire image typically ranges from less than I seennd in mere than it secends, depending en the aequisitjnn scheme. There is. however. a trade-off between spatial and temporal restilutienj. images with large ynsels can be acquired mere quickly than images with small ynsels. The statistical problem in fMl-‘tl is to identify and char- acterize the task-related signal changes in a way tltat helps scientists test their medals and predictions about how the brain wurlts. ICurrent metiteds l'ur analyzing EMF-l] data at— taelc what I call the farsnfr'earr'nn plnaiein: Hnw can data front an fMitI eaperiment he used to identify the parts of the brain that activate during performance of a given brain funetiun'? Lueali'aatiun in MRI is a prublcm ui' classifica- tion, where each meal is classified as active or iterative with respect to a comparison between two conditions. The assumptinn is that in an active 1mite] there is greater neu- ral actiyity in response to one condition than to the other. and hence neurons within that volume are il't‘nrtll‘i'titl in the prneessing distinet tn the nne ennditinn. Thus in a well-a designed fMRi caper-intent with well—chosen conditions, it is possible to localize yery specific brain functions- The unit ul' analysis fur must current i'l's-IRI methuds is the vessel time series; that is. far cnrnputatinnal and ether ._ _L ._ ._ _ ____ ___ u”;- atlliltl Ansottcan statistical Aaaaclatlan Jelltl'tal at the Antarlcan Statistical Aaacclatlcn September 2cm, trot. as. He. #51. Applications and (taco Etutlies E92 reasons, dilferent vortels are treated independently. Most methods uses statistical hypothesis tests to identify “signif- icant” task-related signal changes- For eaample, the com- monly used two sample .t test classifies as active those vort- els for which the mean l'vtit signal during the epochs of one condition is significantly different than the mean leR signal during the epochs of another condition. |Either fre- quently used tests include the Kolmogorov—Smimov test lAguirre, Earahn, and D'Esposito HHS}. tlte split-sample t. test (Friston, Frith. and Frackowiak 1994}, tests of nonzero correlation between the time series and a hand reference curve lEandettini, Jesmanowica, 1Ii'il'ong, anti Hyde [993}, 1'. and F tests in AHD‘JA [Cohen et al. 1994} and more general linear models {Worslcy and Friston I995}, and, for periodic designs, tests for large power in a frequency band lilr'riston. Jeaaard, and Turner [994; Weissltoff ct al- 1993}- Hontest- ing classification methods {e.g-, variattts on cluster analy— sis} have also been considered {Polinc and Masoyer I994: Weaver, fiaykin, Burr, Riordan, and Maerlender $94}- It is tempting to begin an assault on the statistical prob- lems of fMRI by trying to improve the testing procedures used for classification, but two limitations of the localiza- tion paradignt must be addressed. l-‘irst. tlte data obtained from an il'vIRI experiment are a realisation of a very com- ples spatiotemporal process rwith many sources of variation, both biological and technological. The noise in the data is complicated {see Sec. 2}, and the task—related sign at changes are generated by a response that varies over the brain and has both nonlinear and nonadditive features- The problem is further complicated by the highly irregular tissue bound- aries in the brain and the presence of confounding factors such as large blood vessels. Although the various hypothesis tests and classification procedures often give reasonable results and can reliably detect large signal changes the sintplistic assumptions un— derlying these methods introduce nontrivial inel'lieicncy into the inferences- This has evoked two responses in the fit-tit] literature: {I} the development of better preprocessin g algo- rithms to correct tlte data for "artifacts" {c.g., trendsl prior to statistical analysis (Eddy, Fitzgerald, flenovcsc, Mockus, and Holt “1196] and ill the search for new classification procedures that address some of the problems of simpler methods- The latter include adjusting the voselwise tests to improve the shape of identified active regions {Formsn et a]. 1995}, using the distribution of the estremes of ran- dom fields to select a global threshold for voselwise tests [Worsley ltilviil, and applying randomisation tests to protect against violations of the assumptions while maintaining a nearly specified type 1 error for testing the otnnibus hypoth- esis over the brain (Holmes, Blair. Watson, and Ford 199m. In addition, Lange and Eeger (I???) developed a spectral- domain model for tlte vosel tinte series that allows variation in the response and thus adapts better than the sitnple tests. lnferences under their model are based on tests using pa— rameter estimates- The second and more critical limitation of localization methods, even with the aforementioned improvements, is that they address primarily a single question: Where did accrual of the Amerlcan Etatlsllcal .sssoelation. September soon the activity occur? Although this is an important step in the analysis, many recent fMRI esperiments are motivated by scientific questions titat relate to more comples rela- tionships in the pattern of responses than localisation can reveal, including the impact on the responses of changing task demands, dissociations among regions in the tempo- ral pattern of responses, and functional connectivity among distributed components in a processing network- In particu- lar, such questions often arise when cognitive psychologists test and refine their cognitive theories with fit-tit] data. lCon- sider two specific esamples. First, in the esamplc experi- ment discussed later, the relevant cognitive theory predicts that neural activity will increase monotonically across four selected conditions- A. hypod‘tcsis test for classifying mono- tonlcity could be constructed by combining the results of one-sided pairwise tests between successive conditions Isee the pairwise a map in Fig. il-IibilI, but because equality does not preclude monotonieity, the null hypothesis is compos- ite. The error rates of the combined test are also difficult to compute, providing an unsatisfying assessment of uncer- tainty- Second, a common practice in the llt-‘IRI literature is to compare conditions by couttts of active vosels, as classi- fied by a t map like the one just discussed. Although these counts estimate the cstent of activation. they provide no useful measure of associated uncertainty with which to pro- vide a baseline against chance fluctuations. Nor is it feasible to construct simple testing procedures for the difference in the counts because the null hypotheses are highly composite and the natural test statistics have intractable null distribu— tions in general. ' An Esample Esperimem The brain hart a set of general mechanisms, known as working memory [Haddeley l‘i'Shl, for maintaining infor- mation during processing. ll‘llorlting memory resources are limited, and performance can suffer when the demands of a task escced the supply. For esal'ople, some sentences are more difficult to understand than others because they re— quire morc working memory resources; in parsing compleat sentences, any nested clauses, modifiers, or unresolved am— biguities must be held in memory until they can be assigned a role in the meaning of the sentence. Research on language is thus intimately tied to theories of working memory- Cognitive theories provide an abstract representation of the processes that underlie task performance and make specific, quantitative predictions; for esample, what pro- cesses will be used and when, what distribution of re— sponses will result, and how long subjects will take to complete processing. a current cognitive theory of work— ing memory [lust and Carpenter 1992'. Just, Carpenter. Keller. L-dey. and Thutborn revel posited a hierarchy of resource pools specialised to particular types of infon'na- tion. in the contest of senteuce processing, this theory pre- dicts that sentence. with certain syntactic constrttcts will require monotonically increasing amounts of working mem- ory and that these working—memory demand-s will matti- fest themselves in the brain as monotonically increasing activation. Gemvese: Functional Hegneiic Heeonenoe Imaging Data Here Ieunsider art fiinti esperiment designed to test this theory and. in particular. to stud;-r how working memory utilisation changes 1with sentence ditheultv. During the es— perirnent. the subject reads a sequence of visually presented sentences and responds to a question about each sentence by pushing an appropriate hutton. This esperintent. part of a study described thust et al. {19%}. manipulates the dif— fieultp of the sentences presented to the subject. The es- periinental design specifies ti task erinditions arranged in 33 task epochs. as illustrated in Figure 1: {Fill simple rest. which serves as a tatiter between tasks and as a baseline control: {P} maintaining visual fixation on a marketl point in the center of the visual field: {Tr}I reading strings of consonants—trivial sentence processing with no semantic content; and three conditions [T1.T3.T3} titat involve read- ing and comprehendin g increasingly diflieult sentences- The sentences at task levels l—3 are distinguished by difiercnt syntactic and semantic structures that increase the cognitive load required to understand them. Note that by the suhtrae- live logic of the esperiment. Tr and 1" conditions serve as controls for the T-L’s. During each epoch. the subject pro- cesses several sentences of the given type in succession; the order of the epoclts is randomised. The data are described in Section at. E. MODELING THE DATA Let 1i’[t} be the observed MR signal at time t from a specific vosel. where t : il.fl,..-.['T -- on for it :2- ll- My voselwise tnodel decotnposes this time series into Font distinct components. [ll lets) — In + dtr) + rate; pt. 1. E} + oasis]. 333' EBEI IlvlFi Signet [filtilffll‘y units] EEIZI sen 220 _-.I. _.. . . I] 2th] I393 where timed. and the function tit-l are model parameters and e is a parameterisod noise process with mean ti and vari- ance I- Once the distribution of E is specified. this equation deten'rtines the likelihood for the model. The specification of deeper levels in the hierarchy is given in Section Fl- The four additive components in iii are called the haseiine sig- nal. drit't profile. activation prolile. and noise- In the hallow- ing sections l clarify the role and parameteriaation of each component. Throughout. I consider two dificrcnt strategies for fitting the model: masimization. in which 1 compute es- timates and approximate standard errors by posterior op- timisation. and sampling. in which i ohtain draws from the posterior distribution via Markov chain Monte lE‘arlo [MCMC} techniques- Baseline Sig net The real-valued parameter it in ill represents tlte mag- nitude of the baseline signal at the given vosel. deiined as the mean signal over time in d'te ahsenee of activation and ttoisc. The baseline signal can vary by an ordd' of mag- nitude across the imaged volume. Although some of this variation reflects difierenees in nuclear density across the brain tissue, much of it arises Frotrt other sources. including diiterential position and orientation with respect to the re- ceiver electronics. inhomogeneities in the receiver electron- ics. and local magnetic anomalies in the tissue. Nonetheless. the baseline p is usuallyr well determined from the data. Drift Profile The measured MR signal at a vosel tettds to drift over the course at an iMRl esperitnent. and the magnitude of these F 'T1 FlTEFl Tr FtTfl-Fl F Tt FlT'IFlTEiFiTEFl F "TEIFl Tr HTEHTlFl F TE'FITE FlT‘rFl Tr Pi lJl and SW Tin'te Iseeot‘tds} Hip-ere 'i. The Espoiaaeotai Desi-pa tor the Workin Memory Esperia'toot. The sis conditions are labe'Ed H. F. 1?. Ti. T2. and T3 es in the test. The corresponding epochs ere dismayed in sits-roaring oane's atone the horizontai era's nail:- the time scale. Also plotted is the time course For one vasei' with the posterior mean corp-s superimposed Note the sniper dol'i‘ at the ins-pinning of the ospen'meot that scene to stabilize after ahead 3 minutes and the apparent uncorrected movement artifact in the test epoch BEH- changes eften far exceeds bath the ambient neise level and the amplitude cf the task—related signal change. “the drift prefile slit] in {I} represents signal drift as a funetien ef time. Clur empirical study ef many flviltl datasets suggests that the drift prefilc eshibits diversc shapes and is nftcn highly nenlineat and heteregeneeus ever time. it tettds be be smeeth but has uceasienal sharp changes that may be related tn subject mevement- The shape and magnitude cf the drift can vary drastically acress vesels. Te capture these diverse shapes while avnidin g ennfnund- ing drift with task-related signal changes. I niedel d[ti as a spline nf same degree D [deEnc-r IQTEJ. Fer masin‘tizatien, rtirj betengs te a fitted spline space with a large number leg, 'l'i ef tracts and is regulariretl similarly re a smeeth— itlg spline [Hastie and Tibshirani lililfll: scc Sccliun 3. This strategy is cemputatienally efficient and is better able tn capture temperally heteregeneeus changes than a pelyne— mial basis- Fur sampling, the number and pesitiens cf ltnets are meclel parameters with the near-her ef knets kept small [typically enc In five]. Altheugh mcrc eernpulatiunally in- teusive titan the fised—lrnet appreacli, this strategy adapts bctter tn the diversity uf prelile shapes, particularly in its ability te fit hath simple and eempler farms threugh shrink— age aeruss spline spaces. In beth cases, I censtrain rtirj te be erthegnnal tn the baseline with respect tn the empirical inner pie-duet. ta| ti = EL“: ape] = a. 2. t Activatien F'retile lvltist current fh'IRl studies rely en the blend esygena— tien level dependent tHflLlJi effect fflgawa el al. [5391} te identify the signs nf neural activity. Neural activity initiates a lecalisect inflew ef esygenatetl bleed re the active area. a hemea'yrrrtrrtic respense. Because the magnetic prepcrtics nf esygenated and densygenated blend dilfer {Thulbnern1 Waterten. Matthews. and Radda 1932}. this respense is de- tectable in the MR signal. Figure 2 shews the basic shape ef the hcmudynamie signal change as a l'unctieut ef time fer a Ian Paflflrr'lta'lca Journal at the ems-lean Statistical Asseeiatiun. EBp’aarnbe-r EDUIL'I respense te a single taslc epnch- It also describes my current parameterisatien fer the respense curve, which attempts te capture these basic features while allewing fer variatien. l medel the respense in an iselatcd epech el' taslt. pcr— t'tirmanec beginning at time tn as prnpertienal tn the “bell” funetirrn lift imfi]- The parameter veeter ii determines the shape that net the amplitudel ef the respense te an lse- tater] epech ef task perfermance. ’l'hese sltape parameters can vary I't'urn vesel te vusel, alluwing the rnudcl 11:: fit changes in the underlying respense acress the brain- The veeter H centains frnn'l twe tn eight cempnnents, as illus- trated in Figure 2. Let i} and ll be the times at which taslt. perfurrnancc be- gins and ends. These times are given by the design and are censiderert te be lrnewn- The bell functien takes the farm lift; 3. ii] = estisaviftflsncl 'illrlsirsyiitfleresl — illlstaff: Hemli: {El {Ill {4} i-‘attacltlfifleE-Gl — Halli: — Hallie]- ll-l'-:lI.-'_'ay"iiii Ell-DEF] = Hell“: _ if1 _ EDL’IHE}! and lleapliiafltn—i} = fits ' I'll! — it — HUI. ill, fig-IE — IE'H.il,flE;,r'E — Emil, . .. ,UlpE'Ey'lE — Ell-ll: {5} where Ha (its) is a cubic spline, with an internal Isnet at if! t tit; (If? I- tip], that increases {decreases} smeethly frem fl te l [1 In D] ever it]. ll and has sere derivative eulsidc [[l', l]. The hi”, tcrm is cseludcd when l'tL1- dip is fit- This parameterizatinn is sufficiently Hesiblc te handle a wide variety ef designs, and it enables the fit te capture subtle changes in the respense shape. The aetivatien preiite. air) in lll. cenibines the respenses I'rem every taslt cpech in the cspet'imcnt, either additively er rtenadditively. in the additive regime, riftgir.H,-'y_i — ir Ek Pallet — ti-tfi]. where the sum is ever epechs. t]: is the eenditien asseciaterl with epuch a. and ti, is the start Lag-En .Il-t'acit Fitsel Lug-Ell" Decay , FEIII Elle She-w Im'flmU-Dm} Figure 2. .4. Typical Shape tier the Hem-Myriam Hespenee in I'l'L-I'Hl Date re a Single Epeeh c-l‘ Taalr Farther-manna. The time 4111er which the task is pain'errnerf is marlreci, and tha- eer't-e shelve the pattern at signal change that era-suits. This came is a grantee-miss ball function, as described in the teas. The labels indicate the rate at the vari'eee shape parameters in the medal. The rise. tail, and filt'E-W parameters here alfeer the shape at the eerrespert-a'ivg part at the curve Genny-ace: Functienal Magnetic Flesenanca Imaging Date time ef epecii ill. The y.."s. called the raspenrr'veness param- eters, specify the amplitude ef the signal change asseciated with each experimental cenditien as a prepertien ef the baseline signal. By default. I eenstrain is 2 ll fer every ceaditien c. E .2 Neise Di stributien The neisc in ll‘vlfll data is net sittlple. Irttpertat'tt fea- tures ef the neise distrihutien include subject mevement. signal drifts. spatial cerrelatiens. etttliers. and physielegi- ea] effects. The neise distributien is alse sensitive te the scheme used te acquire intages. Fer the results in this ar— ticle. I aceeunt fer drift httt use enty a simple white-neise precess te medel suit}. tvly seftvvare is capable ef substitut- ing ether neise families Ie.g.. heavy-tailed distributiens er specific autn-rcgressivc meving average [AI-tidal} precessesl in the medal. hut se far i have net explercd vvays te select the neise family as part ef the fit. lde. hevvever. preprecess the data with FIASCD Ifunetienal image analysis sefttvare. cemputatienai elie] [Eddy et al- 1996} te adjust fer seme cf the mere cemptex neise seurces. such as subject mevement (Eddy, Fitzgerald. and Hell Itifidll- Altheugh these prece- dttres are net as geed as including these seurccs ef variatinn directly inte the tnetlet. they are quite elfective in practice. El. PHIDH IN FDFIMATIDH The use ef prier inferntatien is a critical aspect ef in— ference. particularly in cemplcx er high-dimensienal preh- lems. because it enferces substantive censtraints and re- stricts the parameter space la a reasenabte Ierttl. I'vl'y' Ittedel fer fisle data is evetparameterized. and I need elfcctivc prier infermatien in help distinguish the cetnpenents. Fer— tunately. each eempn-nent ef rtty rttedel has itself lacett the ehject ef research in the MR literature. and my experience werlting with these data ltas yielded further insights. The stmcturc ef the fi't-‘IRI daLa depends critically en the scan- ner. experimental paradigm. and image acquisitieu scheme: hence Iittia g the medel requires seine careful tverit fer each site and even each dataset. In this sectien l describe the available prier infermatien fer each tnedet cempenent and illustrate hevr 1 use it it] arte such case. 3-1 Baseline Signal Uncertainty aheut the baseline signal p. arises primar— ily frem five seurces that relate te varying tissue cempe- sitien acress the brain and spatial etl'ects durittg scantting and recenstructien- Hevvcvcr. because Iti. is usually vvctl de— termined frem the data. inferences aheut yt are net very sen— sitive tn the cheice efprier. tier simplicity.l use a scaled t. distributien centered en a fixed value tag. which prevides a censervative assessment ef my prier uncertainty. The value at as can he set separately fer each vexel with sceut im- ages ebtained prier tn the experiment. This is ideal. but by default 1 set en te a typical large signal intensityr in the brain. which. because ef the arbitrary scaling ef the signal in predueirtg the data. must he chescn fer each scanner. 695 3 .2 Unit F' refi la The prier far at. must give vveight te the ebserved prnpcrtics—gencral smeethness 1.-vith several petcntial changepeints. seme ef which may be sharp- svhile disceer- aging spurietts structure ie.g.. escillatery behavieri that may he eenl'eunded with activatien. Let If he the number eI tin- tcrnal} ltnets fer the spline dirt] in the interval [i]. [T -- l]r’_"t:. and let s. with it e s. r: -- - c; are e: [T- 112-. be the hunt pesitiens. l'vly prier fer sift] depends an my fitting strategy. t-‘er maximisatien. l talte heth If and re te be fixed a prieri and fit the drift 1tvith an analeg ef a stneething spline. I typically talte K Le he prepertienal te T. and unless there is reasett te expect unusual effects leg, large subject mevemcnts] at the begituting er end ef the study. talte the sums te be equally spaced. Given h“ and re. the drift prefilc d lies in a specific vecter space. The prier registrilftle. s) is prepertienal te s—tireiete where eta] — a... it} asp] ar + It}: |a”|2tri at and p,” 3?:- i] is a fixed censtant that determines the relative penalty ascribed te nerm attd curvature ef d. Here it a [t mediates the nvcrall lcvcl ef smee-thncss in the preille. with smeethness increasing as .it gets smaller. I cheese the .3. that yields a targeted effective degrees ef freedem. defined as the trace ef the snteetlting matrix tllastie and 't‘ibshirani little}. btscausc this ferm is ntest eiiicient te cempute. This requires an initial nnise-levcl estimate and extra iterative step in the eptimiratien. I can include .it explicitly in the pesterier maximizatien. but this has net preven te be werth the enmputatienal cest- Fer sampling. Hus. and it are made] parameters- Because the ltl'lfll pesitiens can be chesen adaptively. the number ef l-tnets is usually smaller in this case. The prier adrigfd} is [if the ferm :rrdfig.{d|ft'.tte. It] - stir-sift”. fit} - rr[.3t|.ft':| - rtift']. where the first term is as given earlier. The cenditienal prier fer the hunt pesitiens is derived frem a Dirichlet fen. .. .ee. ii distrihutien fer the scparatiens a. — a._1 {with rte and HHI I being fixed endpeints} where, by default, e.— = E. The cenditienal prier fer a is an espenential [tfitn]. where at. is chescn a pi'ie-ri as earlier. The prier fer If is a Feissen. truncated at an upper tt-eund Hum. This fermelatien causes mm. id put mere ntass en prefiies with less structure dis- ceuraging spurieus features. 3.3 Fiaspen sivanasa l cheese the prier fer y.r.',....r,, te match the available inferrnatien. l-‘er example. eutside ef large bleed vessels. respertsivcness rarely exceeds se fer current imaging cen- ftguratiens and is mere typically en the erder et' 2% fer a streng signal. I take each "r: 3 ti. because the BUILD mechanisnt leads te a nennegative apparent signal change taltheugh this eenstraint can be lifted}. The prier shy]. cen- trets the density ef each “yg. cenditienal an is. 3:- III and the determinatien ef vvhieh 'yg's are strictly greater than ILi. First. cenditienal en 1:. .':=- 0. I give fit.- a suitable gamma density by default {typical parameters l and fifii. Secend. I specify a discrete distributien ert sehsets ef eunditiens. with the full set excluded. Fer a given suhset .ri. C {1. .. ..f.'-'}. then enty *‘g. with r: E at are altevved te he nenrrere; the prier fer the GEE] uthers [given this subset] is a peint mass at t]. The inclusien nf such submtidels is critical. because the pesterier needs tn accnunt fer the substantial uncertaintyr as tn whether nr net a given vesei shnws any respnnse tn the specified tasks- 3.4 Shape The term ef the respense sltape described in Sectien I is based an an empirical characterisatien ef the heniudynaltlic respnnse functinn- In practice. the manifestatitm ef the re- spense in the data is influenced by the scanner strength. the vesel sise. the speed and structure cf acquisitien. and the nature but the taslt being perl'urttted. Fer crtarnple. in “single trial” studies. where the espcriment is designed tn detect the respense ef each individual tastt perfermance. resp-enses are weaker and faster with sherter attacks and decays. In “bitsslt trials“ nn the ether hand. where the esperiment is designed In detect the accumulated tespnnse nf repeated similar tri- aIs. the nespense functien is snteether with lenger attacks and decays. Lly default. I use independent garnma distribu— tinns as prinrs en the attaclt. decay. hfl'll'l lag parameters. and the undershnnt height- The fitted byperparameters are ebn- sen frent ene ef several templates depending en features cf the design and acquisitien. There is little prier infertnatien tn etinstrain the rise. fall. and shew parameters. but because they are naturally bnunded. I use unifurm distributiuns uver their range. 3-5 Neiee Parameters Fer the results in this article. I use a white—neise ntedel fnr the nnise [eseepting drift and metiun]. I put a gamma prinr cm the nnise preeisinn :ly'lr'll'! with fised hyperparam- eters i1.b and 21211.] by default] te appresintale the typical measured signal-te—neise Ievel fer the scanner that I used {a GE LET with the echn planar imaging sequence]. 4. MAKING INFEFIENGES FFiDl's'l FMFll DATA 'I'e illustrate the usefulness nf my made] in address- ing seientilie questiens beyend Iecalisatien. I censider the wnrlting memnry esperiment frem Sectinn I-l. Renal-1 that the primary geal at that es periment is re test whether activ. - tien increases menetenicaliy with sentence difficulty fretn Tr te Tl threugb T3- 1 analyse the data frent a single sub- ject [silt-113]. previded by Drs. Carpenter and Just. re lit the mudel with truth [be masimisatien and the sampling strate- gies- flur fneus here is net en the results per se.' rather. I wish te illustrate hew ettr medel can be used te directly ad- dress seicntitic qucstiuns I'ur which current methttds nfl'er nnly limited insights. a three-dimensieual image censistiug ef seven slices 'icach 5 nun thicIt separated by s l-rntn gap] was acquired every [-5 sccends during the espet‘imcnt. The sequence ul' images was recenstrueted and registered [see Flee- 5]. and each slice was clipped re a t'vl s fir] vesel regien centain- ing the brain. {Each vese] is 3.125 mm sstss mm in the slice plane and 5 mm thiclt.]I Figure 3 gives several linI-ted views at the data fer a target slice cheseu by the investiga- ters te intersect the twe principal language areas: Lireca‘s [predestien] and 1ili'erniche‘s licemprehensien]. .‘s'ete in par- JDtJFI'Ial ef the American Statistical Aseeclalien. September seue ticular hew the time ecurses at [he lac-undrst cf the brain {where the mean signal sharply rises] read It] eshibit mere preneunced fluctuatinns and nutliers. prnbahly the result nf residual mntinn [much nf it during the “rest” cnnditinn}- These edge effects are aise apparent in the variance map. which aise shews a brain-litre pattern bmause the intrinsic signal variance increases with signal level- The brealtdnwn uf signal by cnnditinn reveals snme systematic effects nf sentence cemplesity. altbeugh the difierences are semewhat ebscured by ether seurces ef variatien. Figure 1 sbews a single vesel tithe ceurse in detail. In fitting the mndel. ] use cubic splines with ltnnts at every fnur‘th grid paint and relative berm—curvature penalty p... : .0]. The stneething parameter s is... during sampiing} is set te target 5 effectiye degrees bf freednm- llCnnditinnal rm the firs being pnsitive. I give them an espnnential {5ft} prinr {a as change an average. a reasenabie elfect fer a lan- guage taslt en the scanner setup that was used]. and cempane the null and saturated sttbntedeis with weights .titl'i'fi and Lilli. I use a I'eur—parametcr shape cuntiguratien [lag-en. attaelt. lag-elf. decay. all in seennels] and En the gamma hyperparameters at [2. 4.3]. [4. I]. IE. 4.3]. and I4. .43] based en the estimated respenses fer sitnitar bleclt designs with earlier data. The garnnta prier en the neise precisien is given hyperparamcters {I-Et. Etltll tn match the distributien cf nuise levels predicted by preyinus studies en the same scanner. An esamiuatien ef the mudel tit reveals the feiIewing basic relatienships. The estimated {pesterier mean] base- lines are stable {median ptsstetitir standard deyiatitm ap- presimately .35 er sheet .1ss ef the baseline value inside the brain] and reveal the brain‘s anatetuical structure as es— pected; Figure 5 shuws ene slice. The drifts in these data are nnntrivial with an efl'ective signal change ttf apprusimately 5% baseline rm average fnr vnsels in the brain with nnly 23% cf this variatien esplained by a linear functien. The amplitudes ef the task-related signs] changes at are :rnestty very small. as wnuld be espected with a few regiuns cf fe- cal activatien- Ameng these vesels with a prehability big- ger than .titlI ef a- nnntrivial respense in at least ene cen- ditien. the 99th percentiles cf the pesterier means cf “y's fur T1 threugh T3 are apprnsimately .fl’lT. .ess, and use. It few ynsels eshihit large as {m-Ifi} in all fnur T cnndi- tinns that are Iii-rely artifacts frnm large bin-ed vessels. The ntedian pesterier means ef the shape parameters fer vesels with a nnntrivial respunsc are .23. Bill. .251. and 142 sec- nnds. These pnsterinrs are quite similar aernss 1u.-'i.'rr.els-—--the interauartile ranges ef these pesterier means are .03. slab. .U1. and 1.1b ascends—which is censistent with a cennnen physiulegical prucess underlying the respunse. Murcevcr. there is an apparent relatiunsbip between the estimated re- spnnsiyeness and shape parameters. either yisually er lin- ear] y intasitnutn abseiute cetrelatieu .sa between any pair]. The estimated nuise levels acress vesels fall mestiy in [we cnneentrated clusters cnrrespending tn ytisels in air and in the brain- The largest estimated nnise levels enrrespnnd tn veseis just eutside the brain er en its beundary. where ie~ cal tnevetnent and ether artifacts are inest preneunced. The brain veseIs en the edge esperieneed the greatest variance Genesaaa: Funeiienal Magnetic Flsstrnanea Imaelng Data Eiliee {flattens .- .\_ .-"" - H's... / . “IKE I Ehru' H‘- I Am I ~ harms: .~..a.i'-.I ' I '59? l - —'- — —— -----i - Tue-rm“ Time Churses fur inset Fusels Signal by Cenelitiun fur Inset ‘y'eseis a /J -- E f_--— F- II'I-Il'i -l-_- «earl—J; mm M1 W Hem“ —_—-* '-"' —.-— —-‘-- Image h'ltrp'ptu'nl - - . i J _ Signalhtiean QM We M r'mhl _____ -_--- if“ h I|I_-.'"I_il I We ., i -—v ':,_. WH'H'HH", _l' _d!__ __--- " ___H' . _i i m M W. q ___ en" “'_ “ _---‘ ' F...“ has“ am...“ - I} ate sue emu"! i "r- ___-_ - _ I ' __i Sigual ‘v'arianee —_____' _— . - -' '4'. a . 1 " a m .. J; " II: II. mt mmfllfifii .1 CH I I. I _ - J: 35:. W' I‘ll- -'-.'I|:| II — — _ I“ all. t....-- - um] I. '- a 4i" ill - " ILI':"|.I : I: H I." I: Tin-eats: L's-rallts-n 2h Elli] ants Figure .1 Linked watt-.5 er ma- flam the film- Tatyet Sites. Ail signal refutes are measured in inner MR unit's" determined by the seanner [lead— t‘ulate the inset has the the mean image. rerluetien under the fit {see Fig. 5 fur a mall] Figure I shews a made] lit flirt ene vase] titue—enurae. 4.1 Leealizing Genditiun Cuntrasls In a typieal IMRI analysis. interest eentei's an enntrasts atneng the task-related signal ehanges I'er Lhe experimen— tal eentlitiens. l-‘er example. with the sentence prrreessiug data. Figure 4th shnws a map at" tare-sample t statistics eetnparing the T3 and Tr euntlitiens. after linear detrend- ing ef the data and ether preprueessing [Etltiy et a]. 1995}. The map is tltresheldeti at i-t far elassifieatinu {an arhitrary value used by the investigators at the time an all at their tiatasets} ta identify active vessels fer later etmsitieratiun. With my meriel. I ean matte a similar eeinpar'lseu by exam- iltit’lg the distt'ihutien ef the enntrasl flap" — “fr-[5- Censentien- ally. 1retitelwise slalistiea are viewed as hunges te reveal any spatial strueture. I can use yarieus statisties tu summarise the pesterinr distrihutien. t-‘nr esample. the pesterinr mean at" em em. gives paint estimates withuut a measure ef uncertainty. Alternatively. the pl'ehahilities PHI-H — 11-, :2 ill‘r'} anti Pisa“ an. :7 MY} measure at the strength uf eiritienee in ftl't’fll' ef eaeh steering 1arithnut a measure at ei- feet size. These are useful fur identifying unsels at interest; the means deserihe what is happening at eaeli vase]. Large aetiyatinn is sparse. but far unsels at whieh the prehahil- ity at a nenaere y in same eenrlitinn is greater than -I'Jtll. the ptisteriti-I means range train —.11 to .135. with appres- imately half It-nsltit-‘e anel half negative- Meet at the paste- rier means are very small. because the pesterinr prubahility of the null suh-tntide] is large. [Fig filial] shaws a map at” Phi": — em 9 HIV}. It is straightfnrwarti tu- luult at ether eunlraata Erma-c ie.g.. the interaerien yrs, aw: + yr.) in the same way. 4.2 Assessing Menatanieity A natural apprtiaeh [L1- lhe rtlelteteltieity questien is tn aslt whether the amplitude at the response inerettses with the tJiiiit-ulty at” the taslt. [it other wards, is an e in; 3 step: 5 em fer a given yesel'? Figure slid] sltnws a map ef these “nieimtenieity prnhahiiities“ Pfih. yTl s; 7T: 5 WEN} as cemented from the data. 1arith a pertiun mag- EQEI tel 11.11 ts} deurnal e1 1he American Statinfieel Aeeneiaiiert. September EGG-CI ‘IEI {bl Fwd—#3 I .- "fife-e2 I */1 i I t . tdl Figure 4. Farleus Flashes Derived Frem the Writing Heater}! Data Her the Target Stine. the sue-tr eurrtns septa-e an each map errata-sens the brain as facilitate carnpartsen acress panels: thts setters is arse Included en the ease-tine map in Figure 5. tn seen panel. the gray-scare refers re a ether-arr: quantity: 3. Iarenararities F' {a T3 rm :- unrt- This large names-.- at heart}; errata vans-rs resutts treat a pesteer mass far the smashes-rang rasnnnshransss parameters eerreenrratae' at D. This gray-scale as: true parter shew-s the presenting values a a tran'rrtenat t map inrestretnest at the arbitrary but eff-err asset mine It! :l:4. l'i're t‘l-Eh'lflrl'l'a'll' significance levers suggested by lrleerjr e'e net git-e the expected arm: refers. mast hit-rely because at elem-plenty in the rleree dialrlbulicli'l that as unacceurrred liar by the reel. Tha- grayeeala re.- lhr'a pane! sheets the! values. e. meanest-city probabilities l-"'1'*'.--,u-~1 L3 1.1-? L3 "nu-f e frTr '1'}. Tile grey-scale ifinr this panel she-we the prebahrliry values. cl. .Lel‘t panel: hit-enormile probabilities tier target site extracted frerrr the right side of the image {left slide at the basin} in (all. Clearer i'cisnri'fil'estiens are dismissed in the text. Right panel: Cams-spenub'rrg regime in the slice anjr'eeenl tel era-3' eels-w the target slice. See the brain sehernerie in Figure 5‘. nifiecl in Figure did left panel- tr'tn interesting variant. net shut-tn, is Fla-1;, :3 “st-.3 3 “ft'r'l :2 il,"r"]'r — tl|'ii"}. which in principle can distinguish areas recruited specifically I'er se— mantic precessingl The picture reveals stteng suppert fer mnnetnnieity' in cluster 1 {in-wer right} but much less in cluster 2 [middle right}. There are several msslhle rease-ns fer this distinctinn, ene nf which is discussed later. The brain’s respense is net restricted tn the 1rein:elnriza: cacatlinate system ef the images at: it is pessihle that re- stricting my analysis in vettelvrise ceinparisens may eb- sctlre in'tpertent effect-t. Fnr this reasen, it is enmmen in IMRI te censider the spatial extent nf aetit-‘atien as 1well as the 1mL'utel'iIrit-te amplitudes. The typical apnItJ-aeh is tn cem- pei‘e centiitielts 1ria the diflist'ertee in can nts el' vertels classi- fied active in each centlitien [relative in a enntrnll. witheut an accenteanjrlng measure ef uncertainty at the individual snhject level. With my ntedel. I can make urener ceinpar- isnns {if the extent [if aetitratinn subject by subject. Fer eaeh reitel 't-‘, let N”. be the indicateref the event {Tl-H, :- win-Id, fnr i = 1121.1, and let N:- : Sufi-1,. fn-r each i- I use the pesterier diatrihulien til {; r“N- ,Nl-g} te address the rnene- tenicity quesiien. In general. the trim. learnt-'31.} all hate different distributiens. but the ennvelutien can be eeinputerl efficiently using the fast Feurier trans-then. The jeirtt preh— ahilit1-r niass t'iinetien el‘eaeh vecter [him i '3“ i '31,.) is sup- perted en the lattice {l}. 1}“ and can be estentlcil by Li‘s In a larger lattice cantaining {ll-t 13 . .. ,t’}fi. where if is the te— tal nantber ef resels being considered. Because ] assume the 1rrtirrels he be independent, I run separate simulatian fer different t-‘esels, but I ean still ehtain the distributien ef {f‘t’1._-"t"g. N3} by multiplying the individual Feurier trans- ferrns ever the largtiT lattice and inverting the transferm. 'l'e reduce the cemputatienal burden. I take ‘v’ te be much smaller than the tetal nuinher ef vesels. because there is euls negligible mass arrayr frcnn fl l'c-r any ef the film's fer the 1rast majerity' nf 1Irr.-r.e|s. With these data. nnlgr 14'! tres- Geno-tress: Functional l'u'l'agne-tir: Fleaooarios Imaging Data els have posterior probability bigger than -lldl away front it]. d. [J]. and I take if = 23:55. An alternative strategy is to simulate draws from the distribution of Ii'v'1,f'v'g..-"t"3] by generating and adding {fa-fl... .-'v':._., Nfi..]'s_ This is computa- tionally efficient for both small and large v. but does add some uncertainty to the estimated probability. In my exam- ple. PIP-.53 “is this “3 WHY} m .ti’f, which appears weakly consistent with monotonicity in extent. sits mentioned in Section ]. there is no good way to address this question by combining voxelwise classifications Ie.g.. based on t tests}. These monotonicity results are consistent with the pre- dictions of a resource-usage model. As demands increase. new units are recruited to the taslc. both within a voch {thus explaining the observed monotonicity .in amplitude}I and across voxels [explaining monotonicity in estentl. Be- cause the experimental taslts focus on language compre- heitsion. one would expect the strongest eifects to arise in 1liliernictte‘s area. Indeed. the investigators attribute ctttsler 1 in Figure d-{dl to a part of this area. .f'ts the brain schematic in Figure Fl illustrates, 1|ilifernielte‘s area cuts across several slices as it follows the folds of cortex. It appears that Wer- nieI-:e's area folds baclc up into the target slice at a point somewhat anterior to cluster I, which may account for clus- ter 2. This interpretation is reinforced by the monotonic activation at the same site in the slice directly below the target slice [see right panel of Fig. sl-{dl}. Returning to the target slice, monotonic activation is also shown in cluster 3. This is probably the bottom part of Broca‘s area. a brain area associated with the production of language [sec brain schematic in Fig. 3i. Finding monotonicity here could be the result of sobvocalixation on the part of the subject {in- creasing with sentence difficulty} or. perhaps more lilcely, a reflection of shared function across these two intimately tied language areas- Such redundancy is a fairly common feature of brain organisation. 4.3 Model 'v'alidatiori “the results of the model tit—the similarity in response shape across voxels. the small spread in noise levels within the brain, the anatomical location of the activation clusters—are consistent with what one would expect to see if the model were true. I conducted exploratory resid- ual analyses to seelt out potential sources of model mis- fit- For example. examining the residuals as a function of time showed few discernable patterns for most voxels- Cle- casional large outliers are scattered throughout the brain, as are wild signal tlnctuations where the smooth drift could not quite keep up. 1'tfoxcls near the edge of the brain. par- ticularly those showing strong activation. do exhibit some nontrivial residual atttocorrelation. I attribute this to a com- bination of systematic signal Iluetuations not fully capture-d by the smooth drift term, and variation in the true response magnitudes across epochs of the satne condition tthe fits here ttsed a single magnitude pararnetcr par condition}. It may also suggest incorporating a low-order autoregressive component into the noise process. I also examined residual distributions. If the model Iits well. then the residuals should be approximately iid normals; even if the normal model is E399 not entirely believable. this serves as a useful diagnostic. Quantite—quanlile plots suggest that the normal approxima— tion is reasonable for voxels inside the brain. when: the signal is high and the skewness negligible. For the voxels outside the brain. where the MR. signal is very low. tltc plots reveal a ltnovvn laelt of fit; the noise distributiott is somewhat skewed. because the reconstructed data are the modulus of a complex normal. as a quantitative summary of this. I performed voxelwise chi-squared goodness-of—t'it tests of the residual distribution against the corresponding nortnal. For the voxels inside the brain, approximately Iii-F: of the null hypotheses were rejected at the .dl level. and the voxels with rejected hypotheses showed no obvious pattern. Figure 5 displays the results of some of these residual anal- yses for the target slice. Comparison with Figure 3 may be informative. I also used the Bayes information criterion tHIC‘l {Schwara lsii'ti} to compare my model to the model underly- ing a widely used fl'nthI analysis. BIC provides a composite index that balances fidelity to the data and complexity of the model in the lit: lower scores are preferred. {For certain priors. ElC approximates the log Bayes factor between two models} So my model will succeed in this assessment only if the additional parameters that it uses are balanced by a sulliciently large gain in likelihood. The comparison model is based on the following common procedure te.g.. Jttst et a]. 199$: {Ill linearly detrend the data; [2,1 fit to die detrcnd residuals a piecewise-constant function that is constant over taslt_cpochs that comprise a single condition; and f3} test the resulting regressith parameters via i. or i-' tests. l'vly model has a lower score in all but lll voxels scattered through— out the brain- Among those voxels for which the compari— son model seored lower. the median difference is 1.55. with only 3 having a ditierence greater titan 1d. Among the re- maining voxels. the median difiercncc is I'LLSo in favor of my model. In sum. these analyses suggest that my model fits the data reasonably well. Flut several sources of lack of fit need to be addressed. First. the response amplitudes can vary among laslt epochs of the same condition. whereas as [it here. my model uses the same amplitude across such epochs- Second. when the durations of last-t performance differ drastically among con- ditions. the response shape can also vary across conditions within voxel. This problem did not arise with this design. but I have seen_other cases in which it does. Third. the noise distribution appears to be somewhat heavy tailed. and the occasional large outliers can have some efi'ect on the fit- I deal with this by simple windsoriaation in FIHSCU. but it would be desirable to account for this in the noise model itself. Finally, given the complexity of the data-collection process. it is not surprising that the time courses can oc- casionally exhibit odd changes—jumps. clips. spiltes—for which I have no explanation. All but the last of these is— sues can be handled by straightforward generalisations of my model. which I am cu1Tenlly pursuing. 5. GGMPUTATIU'NHL TEEHHIDUEE To analyze data with my methods. I must fit the model to FBI]- Residual QQ—Nermal Plats Jaurnal at the American Statistical Asseeiafinn. September enact II] It] {I 'H] TT...|____.. Li a irr' r1 ta ta Cflt'll‘lliie'ul 21]. figure 5. Linked Wart-'5 r.Itr the Residuals and Fti' Fer the Target Slice. Aft sl'gnal' train-as are area's—tired in teeat MR traits. many thnusands nf yesei time-series with yaryin g structure. There is little opportunity fer manual adjustment during the iit1 an I require Her-Little and eiiieient nuItIerieal methads- The raw data frnm the MR scanner are enlieeted in the Fnurier domain and thus require preprneessing tn reconstruct the images and eerreet fer several seurees at bias and miseali- hratinn- {Cnmputatinnal cnst currently precludes integrating these seurces at nariatien directly late the made].} I prepre- eess the data using FIr‘tSCfl {Eddy et ai- 1996} and take the resulting time-series at images as input tn the made] fitting and sampling seftware BRAIN {Bayesian resptmse analysis and inference Fri-r neurcuimaging}, a ptihlie-dnmain package written by the anther. {The BRAIN heme page can he at:— eesseti I'rem httpfg‘rwwwstat.emu.edufa-genns'esefihrainfii tier ntasimiaatien. 1 use direct numerical eptirniaatien at the leg unnru'rnaliaed pnsterinr far eaeh respn-nsitreness suhmedel. The result is a set of estimates and an appreai— male ees'arianee Irratriit derived frern the inverse ehseryed Fisher infnrn‘tatinn at the n'tnde n-htained them the cam- puted Hessian. I estimate the pesterier prnhahililies {if the snhmedels by eembining the suhmertel prier prehahilities and apprnttitnate Hayes faetttrs enmputecl with a farm tif the Laplace appreitinntt'teu iDiCieeie. Kass. Eaftery. and Wasaennan I995}- I than average ever the suhmndels using these prehahilities as weights. Fer MCMC sampling. 1 use a mix ef Metrepcrtis alt-cl Gibbs steps with a fitted scan er— der aertiss enmpenents {Smith and Rnherts 1993; Tierney 1994]. Bet‘ere reeerding thtiiptit. l pert-arm an eptienal pres— can In adjust the initial h-‘letrnpnlis jumping distrihntinns and a peried ei' burn-in {by default. iiiflfl sarttpies per pa- rameter] tn- equiiihrate the chain- The maximum [Masterinr estimates are used as the starting paint. The eptiena] presean attempts he find eil‘eetire widths fer the Mctrepnlis jtnn ping distrihntinns. Initial jumping widths are derived frem the sueeessire eunditienai 1.-'ar'tant‘;e£-. el' the parameters. which are cent puted 1:ia Chutes it:-r facteriaatitI-n nf the apprnitimate enyarianee matrix- The chain is then run fer a fined number at” scans during which heuristic per- fnrtnanee criteria are reenrded [e_g_. rejectinn rate and path fiennveae: Functienal Magnefie Fieaenane-e Imaging Data lengthl. This run is divided inte blectts. and the widths fer a bleach are adjusted using previeus hie-cits Le imprevc pcrfer- mance measured either by deviatinn ef rejectien rate frem a target [e.g., fill-“ad er path length (Rubens. flelman. and IL'riltts 1997'}. TII‘r'ith sn many chains In he run, efiieiency is critical, and I run tltent as leng as is telerahle. Iva default is lflllflfl sam- ples per cetnpenent after burn-in- but very—high—reselutieu images necessitate shnrter runs. 1IrIr'hen feasible, [ run the chains Innger and subsarnple te reduce cerrelatiens in the recereed sequence. i have several ways te speed up the cem- pu Latiens, including paralleliaing the cemputatien, eliminat- ing uninteresting vesels eutside the head. and erdering the eemputatien based en the preliminary estimates. l[Cine area that needs develepment here is cenvergence di— agnesis. because multiple chains and graphical inertiteting are incenvcnicnt in practice. I currently uSc enly rudimen- tary measures nf chain pet'fnrmance during analysis, am wet-king te itnpreve this- fits part ef a “quality centrel“ ef- fert. I studied tlte peri'ertnauce et' my sampling scheme en a cellectien nf vesel time series frem several experiments. IGraphical diagnestlcs. cerrelatiens arneng parameters. and varieth standard cenvcrgence diagnestics tGewlcs and Gar- Iin 1995) based are parallel chains with different starting peints suggest that the chains are misting quite well, and equilibrating sufiieiently and alse that the nerma] appresi- matien is reasenable in must cases. hienethetess. mere sys- tematie study is needed- 5.1 Default Sampling Scheme The baseline parameter p. ntedlates a number ef the etlter parameters and se has a cempIicated cemplctc cnnditicrnal. I use a symmetric Tandem-wall: Mett'epelis chain fer yr, but as part ef the tneve. I multiply all ef the y’s by rt'ftt [candidate te current} te- maintain thc activatien preiitc. The eemplete eenditienals 'fer the drift and respensiveness pa- rameters cart be sampled directly. 1 first sample 'y and tlten the drift prefile cenditienal en 1-, because the nennegativ- ity censtraint en the respensiveness cemplicates its distri- butien. The eenditienal distribtttien I'er y given everything but the drift is a multivariate nerrnaI truncated tn the pes- itive nrthant. Te satnple frnm this distributien, I draw the eempencnts nf er ene at a time frem successive univari- ate ceneitienal distributiens. The Ghelessy facterieatiens ef the cevariancc matris and its inverse allnw deriving the mean attd variance ef these cenditienal distrihtttiens itera- tively. I then draw i'rem a univariate truncated nermal using the inverse distributien functien methed when the mean is sufficiently large te ensure precisien in cemputing the ner— maI distribttlien functien and a rejectien rnetherl {based en an cspencntial appresimatien tn the nerrnal tail,'I etherwise- The drift prefilc can then be drawtt as a whele frem its cemplete ceuditienal. The shape parameters capture ntest cf the nenlinearityI in the Inedcl. I ehensc frem ameng tvve different types ef Metrepelis meves fer these parameters: a leg netntat tandent walk. in the parameters individually. and ceupled jumps in related pairs [lag-en and attack, Iag-efi' and decay, etc-J. As an esample ef the latter1 I use ene meve 7TH type that keeps lag-en + attach cen stant while varying their relative size and artethcr U'llil changes the sum while Iteep— ing the relative size censtant. These diverse meves previde an autematic reparametet'ieatien vnsel tn vesel that reduces the cerrelatien ameng the parameters and itnpreves mlsing. The smeething hyperpatamctcr leg .3. f ellews a ncrrtftal ran— dern walk, and the neise preeisien is drawn frem its cum— plete cenditienal- 5.2 lvleclel Jumping Fer varying tlte structure ef the ntedel in discrete ways. I use the reversible jump framcwerlc {Green I‘Jil'fil. In par- ticular, this altews meves acrnss submedets in the ry's and acress spline spaces fer the drift. because these centpenents maintain their interpretatien in every submedcl. I average ever the medals tn aeceunt fnr uncertainty in the structure. Fer ertample. at each sampling iteratien. I update y by a Gibbs step as described earlier. attd thee with same preh- ability attempt a submedel jumping nreve: inclusien ef a stern enmpenent and remeval ef a nnnIern cempenent. beth equally Ilitely. If a jump is te he made. then I select an ap- prepriate cuntpenent c at tandem and generate a candidate fer bnth err, and yt- The latter is mevcd because switching syn tn nr frnm fl affects which measurements previde infer- matien ahent the baseline. Let I. and I; dennte the number ef acquisitlens in the screed and candidate cenditiens. The simplest rneve Laltcs [tr-my} in [It] —-ll'g[l. + “,-'}],fii1+t-,:_r],rt.tli fer remeval and [petij tn [fl] - igiy‘fll t Igfl + slime}, where .3 is a tandem respenslveness candidate independent ef lrs- lvlising can be impreved by randemly perturbing the cetnpenents here. Beth nf these fellntv the template given by Green [1995] and satisfies detailed balance. Similarly. t'er drill. I take a Gibbs step every iteratlen and then Tandeme decide amnng meves that change a tract pu- sit'ten. add :1 Intel. and I'emeve a hunt. When ene is adding, retrieving. er ttteving a Itnet. the affected Itnet is selected at randetn, and the basis is then refermattcd tcr maite it eas— ier te update that eempenem. Having a knet invefvcs ran- dumly perturbing the selected ltnet within the heunds ef its neighher. The simplest way te add er rcmeve a Itnet is te change a single ceeffieient, setting it te fl when remeving er drawing 'It frent a distributien independent ef the pre- file when adding. This attains detailed balance but dnes net mis very wcll1 because enly a small petlttrbatien be a single cempenent leads tn an acceptable change in the prefile. I lift this prebtetn by alse updating the ether centpenents ef the prefile as part ef the were. I parameteri'ac the drift with er- thenet’malieed spline bases that makes such changes mere efficient. The dimenslen-matchlng requirement is satisfied, mining is itttpreved. and detailed balance is maintained. Er. DISCUSSION tviy rhech has several netablc advantages ever traditienal metheds fer analyzing fIvIP-l data. It can capture eerltpli- cated drifts and changes in the shape ef the hetnedyearmc respense. In centrast te tnest cttrtent tnetheds. the medel eempencnts are estimated simultaneeusly rather than seri— ally- The fit te the data can thus be mere precise than the F'UE implicit lits underlying mest classificatien tests. By preduc— ing estimates et' meaningful parameters rather than just test statistics, my methttd facilitates quantitative interpretatien ef the results. Mereever. the medel prevides useful mea- sttres ef uncertainty in centrast tn the diiificult—te—use errer prupcrties cf classiticatien inetheds. In additien. my rnedel handles eemples designs, including beth bleelt and single trial, and can include impurtant features ef the respense such as the undersheet dip and nenlinear cembinatien ef clesely spaced respenses. Finally. the medel acceunts fer what scientists ltnew abeut the underlying precesses gener- ating the data and can easily incetperate new infermatien as it becemes available. The inferences that can be derived under ettr medet can address questiens ef leealizatien and thus suhsutne the tra- ditienal classifieatien-hased metheds- In my epinien, hew- ever. the primary centrtbutien ef nty appteach is that it maltes accessible te direct analysis a wide range ef ntere general clucstiens as well. In this article I have tackled ene such questien. menutenicity. r‘tltheugh ether metheds te-g., I'andemisatien tests]| might he fasltiened te handle such spe- cific cases. my medel can address a mttch wider range ef questiens under a single framewerlt. including qttestiens aheut changes in the respense aeress cenditicns and aeress vesels. abeut temperal patterns in the respense that distin— guish different Types ef precessing. and sheet functienai cennectivity ef brain regiens- This desihiiity ailews scien— tists te directly target the anestiens that they want te answer with accurate and relevant measures ef uncertainty that help them evaluate the analysis. Several weaknesses in my appreach remain te be ad- dressed- The iirst is that fitting the mttdel, particularly via MCMC simulatien, is very cemputatienalty intensive. re- quiring en the erder ef a day te anater a single subject‘s data. With careful parallelisatien arid impreveincnts in al- gerithms. hewever. I espeet this preblcm te became less severe ever time. Secttnd. the neise medel used fer the re- sttlts lll this article is stilt rather simple. and lwill estend it te deal with the neise cemplcsity. especially physielegical variatiens and spatial dependence. Third. structural inde- pendettce ef the parameters aeress vesels is Iii:er a sim- plistic assumptien. and 1 can gain precisien by cembining aeress vesels with similar f u nctienal preperties. ecceunting fer this spatial structure dynamically is a challenging preb- lem. as I discuss later. Finally, inferences under the medet depend semewhat en the priers. and it may be infermattve te systematically evaluate the nature uf d-ns dependence. In my experiments I have fcund that specific shape ef the prints has enly a small impact en the results prevtded that the basic range ef the parameters is suitably censtrained. lvly geai has been te include generally accemed inferrns- tien abqu the basic precesses. se the priers that I use reflect reas-enahty uneentreversial censtraints- Henetheless. further medel validatien remains a prierity. The basic implementatiett efettr medei presented in this article can be extended in several directiens. First, vari- atiens in the respense amplitude can he allewed aeress epechs and within each cenditien te capture this variatien in the data tfienevese. Nell. and Eddy 1539?}. Secend. varia- Jeurnal at the emerlcen Statistical aeeeciat'en. September seee liens in the respense shape can be allewed aeress cenditicns within a vesel: this is impertartt fer designs in which the cenditjens nceur en vastly dilferent time scales. Third, the medel can he parameterised by an i‘tth‘Wt decempesitten cf the respense amplitudes te accennnedate facterial de- signs. Finally. rather than treat vesels as independent units. we need te aceeunt fer spatial structure in the hemedynamic respense- The shape ef the respense funetien, the amplitude ef the respense. the impact ef phystelegical variatiens. and ether such features eshibit ceinpiicated dependence aeress vesels. Medciing these relatienships increases the preeisien ef inferenecs because multiple vesels eentrihute inferma- tien abeut features that they have in cemmen. It alse fa- cilitates addressing ccntplicated questiens abcut the spatial pattern ef respenses. The taslt is net tn segment the image per se, but rather tn identify regiens with censistent physie- legic and functienat preperties. a particular cltallenge here is that tissue beundaries in the brain are cenveluted and piecemeal. se metheds based en Marhev raridern tields {Lie- mart arid Cieman I934: ll3eman and McClure I937: Jehn- sen1 Weng, l-lu1 and Chen ISIS!” will he must efi'eetive if given eneuglt anatetnicat prier inferinatien. which is difli- cult. 1 am currently werlting te adapt the methed described by .iehnsen. Eevvsher, Jase-seals. and Turttingten {1995} that ineerperatcs disjeint regien dcs-cripters as parameters at a deeper level in the hierarchy. Airether issue is tltat great variability in the physical ge— etnetry ef the cerebral certes. aeress individuals makes it diflicult te eempare and cembine analyses aeress subjects. The mest cemmen methed fer cembining flv'lRI data aeress subjects is te ntap the subjects’ brains ente a cemtnen ce— erdinate system called the Talairaeh atlas fTalairach and Tnuneus. [933] and then average in this eenrdinate system. The Talairaeh atlas was derived frem a detailed study ef sis human brains. and the mapping fer a given subject is cemputed using enly a few gress measurements ef that sub— ject's brain- This averaging precedure is far frtrm satislae- tery, hewever. because large intersuhject variatinns remain. Une way in which my mede] can be used In cembine results applies when the questien el' intere5t invelves enly a func- tien t3" ever the {tetale parameter space that dues net depend en the explicit eenrdinate system cf the image fer a given subject; fer esample. G might be the integrated respense ever a prespecified anti anatemicatly defined regien ef in— terest er the indicater that twe taslts yield distinct tempersl patterns ef respense- Suppese fer simplicity that t”? depends enly en y. and that the J subjects in the esperttnent cen- tributc data TL. .. ,YJ. [f ene is willing te assume that these data are drawn iid freni snnte pepulatien distributien. then ene can estimate the pepulatien distributien ef t? by cembining the distribtttiens ef r3 given ‘t’t'. fer esamplc. Etfltvll e Hart 23;. stetvi Ya. ii’t'rcciw'tl Jase HE'S. Revised December JWPJ FlEFEFiENGES Aguirre. 5.. Earahn. E.. and Il'Espcsile. lvl. H.995]. “fit Critique cf Ilte Ust‘. cf the KeItnLt-gurtw Stnit'trev Ill-CS] Statistic fer the Analysis {If BOLD flt'lRl Data." .‘l-rlrtgttcn'r demurrer-tr t'r-I Medicine. 39. Still-SHE. Geneva-nee- Funefielnel Magnetie Flesunarreu Imaging Data Buddelej. It. [19363. Whining Henlerjr'. New "t'iJrit Diti'utli Unis-emit].- Free—e. Hitnderrini. P. A... Jeanienuwiea. A... Wang. F.. t"... enel Hyde. .T. “993]. “ ‘Ieeessing Strategies for fime-E‘etnse Elam. Sets in Ftlneliennl MR] eF lhe Fllll'lililll Bruin." .irfrignelie' He.trirltint'e in Medicine. 3ft. Iii-I - I73. Cullen. 1.. Farm-an. 5.. B1fi'I-T-l. T.. Casey. ii. Serve-1 Schreihe r. F1. and Null. I} “994]. "neti'r'inien ui I‘Teirentsl fairies in u. Nuns-pine“ kaing Meme-T}- Tesla With Ftllleii-flnnl MRI." Hears": 3min Meaning. 1. 193- Sills-i. Eel-wins. M. 162.. and lEarlin. B. 1’. il'il'StEI]. "Marlins Chain Meme Cele Cen- eergenee Fiiagnnsries: A rump-native. Review." .iriurnni urine American inanime .‘I iitn'itrlitm. '5' |. ES 3-90-11. eie Hem. I73. [19?3i. .41 Fun-nerd (Junie in .E'niinex. New Turk. Springer- ".I'el'lng. DiCieeiu. T. J.. Kalle-t. R. F... Reflery. fie. :tnei Wessermnn. T.. NWT]. "Cem- pLIIiug Bayes Fen-ms by Genie-mine Simuiatimi and .nsymlitmie n]:- prusiuullinus." .I'rJni'n-rn' rii'riae nnrerit'nn Smrr'erieni .I‘l.'-.'-'n-t'nrJ'|'em. 9.3. W3- EH5. Eddy. W. F. Filzgel'eld. M.. GE'IHJ'I'DHE. C. F'... Meekue. ii... and Hell. D. “996i. “Fins-Iimml [rnsge Analysis Sflit'ni-nre— Cempmmienal Dlie.“ in i'n'n'eeriiner r'n Cmnnnrniirerni Sitrfl'.'-'Ii-'_'.'-'-. "'-"n|. IE. ed. i'l. Frill. Heidel- berg: i‘iljsieiI-‘e'efleg. pp. 39' 45' Eddy. W. F.I Fillgeralri. M- and Mn". D. C. “995:. "[m [arm-ed Image Reg- istrfllifln Using Flt-IIriEI‘ Tnterlmlati IZIII.“ Ji-‘i'trgnefr'e Rel-unrult'e in .Herl'ie'r'ne. 36. 923 9'31 Fermen. 5.. when. I. {1. Fingeialil. H. Eddy. W. Minttln. M. and Hell. [‘1'.C.i|1i|'}5i.“1mpnwetl Msesmnem fli Signil'lum'lt Clnlnge in F'Llnutit‘utnl Megitetie Hesutlenee mil-ill [lee ei at lCluster Size Tichfihu-iti." .‘ri‘tignerr'e Reennnnt‘e in Medicine. 33. Iii-E I541 Prime-n. K. J.. Frith. E'. El. and Fleekmeinit. R. F}. J. “994). “'Time- Dependent Changes in Ffieelive Enumeliviw Measured with PET." Hu- nmrl Hrrrin Mel-rifting. I. Fig—75'. Fristnn. B1- 1.. Jezrard. 11. and Turner. R. IIl'JEi-i]. "Analysis n’r' anetinnui MFll Time-Series." Hunnin 3min Hun-inlay. I. ISJ—I T-‘I. German. 5.. and Herman. D. [193.4]. "."itnehastie ReJaIetinIL. Eiihhs Distri- hutiens. :Ind Huyeeinn Restelutien et' Jinuges." FEE}? Triniimrrirmi. nun i'nrrenr Ann-rum rme' Ii-‘i'en'irr'ne inreiiinenn‘. I5. "I'll 4'41. fiemen. 5.. and hie-Cline. D. F.. [193% “'Stetistienl Methntls Fer Tame-- graphic “tinge: Heeenslruelinn." in Fmt'e'eri'i-Iml- Hf ri'ie 'ilfiI-I'r Eel-ser rear lire i3}. Ruiierin m" the iii. 52. Genet-'ese. If. P... H1111. D. II.'.'.. .-n1-:i Etldy. W. F. ll'J'SIT-‘J. "Estimating 'T'esl- Fleiesl Reliilhiliu‘ :II'I Fit-1TH T.“i'|r!'-:-'3'ne-'it' Resin-Irma- in Francine. 33. £19?- 507. firee n. P. l. III‘J'ZJFIII. "Revelsihle Jump Mil—{MC Cnmputetinn and Bayesian Mn-del mlel'li'lll'lfilil'ifl." BiereIrI'icrr. HE. 7| '4'}:- Hnslie. 'T.. and 'Tihsilirani. Ft. il'il'ilfli. Ii'r'enerniizerl' Hamlin-e Mrnien'e. New Twit: Chap-unlit and fill“. Helmes. it. |-'*.. Blair. H.132. Weiss". J. [L and Fartl. l. lIEIEl'fi]. ""ien- Pal-mnenie Analysis ef Ennis-tie. image.- Frem Funetienill Mapping Es- perin1ente." Jeln'nni' ej'Cerelmn' .ti'ieeti New nnti .‘I-ft'rnhel'r'enl. Ifi. T 22 Juhnsun. V. E.. Blemish-Jr. .T.. I anaemia. It . run] Tarkingmn. T. i | 995]. “AI-ill- gin-is and Retransnueiien elf Hedi-eel images Using Frinr Tninrmarien.“ in C'er Erwin-I. in Hint-Linn Errii'is'i'ir't. "e'litli.1.EEIE.C.GH.1:=-.I‘Ilihi..i.5. Hntlges. F0 3 R. E. Hess. and N. D Singpururells. New ‘i’erk: Swinger-firing. pp 1-19 213. JLIIInsfl-n. "-". E.. i-‘r'ung. W. H.. flu. PL. and Chen. C. 'T. [IFIE‘I i. “emit-Ute elf Instge He'stumtietl Using Gibbs Friers: Bt'lllll-litil'}' Medeiing. Tlentmeln ui Blurring. mm Seieetien et ilyperpnreunflers." iEEE i'ixinrrnt'rr'enn nn Pnrre'rn flflfl'flfl'ifi' nnn' .Hnenine inrel'iieene'e. ]."-. 412 2.15. Jusl. M. A... and Carpenter. P. A- [11392]. “A L'itpaeity' Theury uf Lem- prehenslun: individual litilferenees in Wurklng Mennuj-I' Fanningieni Eerie-n». U9. lEE—lslli'. Just. M. Earl-renter. 1-1.. Keller. T. .5... Eddy. W. F.. and Tl'lllii'lfl-I'l'l. FL Ft. lil'ii'iini. "Hrein Aelivetinn Mnelnlmeel by fienlenee l'ftnnprehensinn." .E'r'i- ilnee. 274. IN. 1.:tnge. H" and. Regen S (IQWJ. "Hun-Linear FnunrierTime Series All. hail }'fii.~'- fun Hem-en Brain Mapping 1'ij Funetiemul Magnene Resenenee imaging." {iii-ill: diseuseien]. Jti-nrntri efrire Rrrgr'tii Sterrr'err't'ni Siren-fr. Sen C. I16. ] Ell. flgewe. ."i-. 'l'enk. L1. Men-en. L'II.. Ellerrnenn. J.. Kim. H- Merltle. H_. and [FgurhiL EC. [JEWEL "Intrinsie Signal [fhnnges Aeenmpenying Sensnly E1imu]etinn: E'Llnetienel Biein Map-pine Using MRI." Pmeeunir'nes ei'riln Nerinnm' Anmfmn}. nlr'fi'eienees. ill]. 5951—5955. Feline. .i. [-i.. :tnei Mung-er. H. [1W4ifi'lilus1er Analysis in intlividuel Fune- 1i-nnel Fireln [rung-es. Some New Teehniqttes te- Enhenee the Fiensitlvity nf Aetieetinn EI-eteeti-nn Methtris." Hinneni Ere-in Mapping. '3. [EH—l | |. Relierls. G. 13.. Gelmsn. 9*. inn-ti Gill's. W. E. HEW]. "Week Convergcnm and Dutiunli Sealing Elf Rnlldflrn Well: Menage-lie hlgerithms.” el-Irirriii' elf Appneti i'irJiJ'n-t'nn'r'ry. 7". | 1U IEU. Sel'lwent- Cr. HMS}. “Estimating the Dimensiun ui e hltnl-el." Iiireflnnnis e-Li'h'rnrisrr'es. e. 4|Ei I—4IS4. Smith. A. l-'. M. and Reherls. Li. fl. {I993}. "Bayesian L‘emp-utetien via the Eiihhs Sampler and. Related. MUJEU'R L‘hein Mnnte Curie Eu'lethudet." Jennie! nj'rile Hus-fin! Eterin'ieni Srieierj'. tier. H. 55. 3—33. Taleireeh. J'.. and Tnunnus. F'. Ill'JHli-i. tl'nninnnr Siereermie stn'ns nj'rile Hirninn Brain. Tiiree-flr'nlensinnni Frepnrrie-nni.5'_I.-.I:rern.'.-1.n Alppmueh n1 Cerehrni imaging: Thieme. New Turk: MEIZIJE'HJ Publishers. lue. Thnlhnrn. FL. Wnleilen. .].. Matthews. lit. and Reel-tin. Ci. i 1952}. "flu-gens- iien {lependenee er the Tmnsverse Relaxatien Time of Water i‘rmens in 1'-'II'hLI]I.'. Bleed ill High Field." Bfr-‘E'lll't'fllie'E-i-r fi'ith-rrfr'i'it'i- eit'itr. 7' I4. 265- 37-1). 'l'ierne'y. L. [195143. "I'i-lerkue 'L‘h'elins t'er linphning Ptistefitir Liis1r1'hutituls." i'Tre .llnneis qi'b'rnrr'sries'. .12- ilTiJiI—l ET. Weaver. J. E... Haj-kin. Pt. ‘I’.. Eturr. R. E- Elie-Eden- H.. u.n::| J'i-leerlender. A. Iil'ZJ‘i-i]. "Frineipul {itilnntment Anni}.- sis et' Funetiune] MFll ut' Melnur'y.“ in Fmeeen'ings .1! the .Enr'ierj- I'm' .Hugnerie Resmtunee. .‘i'eenncf Ann-nu! .I'l-‘iie'e'riiig. p. Him- WeissitelT. R. M. Fisher. .].. Fleili'eenit. 1.. Davis. '1'. I...I K'flrfil'lg. K- It"... Ifelien. M. 5.. and ltesen. B. R. “993i. "lire-er Spectrum Analysis ef Funetiunnlij.I WHEth MR Dulir What’s in the Ne-ise’i." in i’irit'eeti- ings rifrile b'eeieiyferil-i'ngrleiit' Hesrmnriee in Medicine. Tweijliir .rlrlnntii Meeting. 11-. T. ‘i'r'nrsley. ill. I. “Will. "Estimating the Number ef Peaks in a Elan Held lising the Hedwige: lt'.'h:tr::|eteri.~ilie hf Eseursinn Sets. 1i-‘r'ith Applieelinns 1e Hedi-en] images." The Minnie ri-Hi'rurisries. 23. fidli—fifr'il. War-ale}. H. I.. and Fristnn. BL tl'i'ifii. "Analysis nf ’r'h-‘IRI Time Series Revisited—Again." .‘t’ennii'muleu. 2. ITJ—I ii]. ...
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