Today's classWhat is fMRI?Block design -- finger tappingHemodynamic response functionConvolved design -- finger tappingComponents of Errx,tFull model for finger tapping dataVoxel in the motor cortexMarginally significant voxelReal experiment: reward anticipationCombining subjects: fixed effect analysis- p. 1/12Statistics 203: Introduction to Regressionand Analysis of VarianceFunctional DataJonathan Taylor●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 2/12Today’s class■Functional data.■A functional t-test.■Mean, variance, covariance functions.■Functional data I care about – fMRI.●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 3/12Functional data●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 4/12Functional data■Last class, we talked about smoothing one function at a time.◆B splines;◆smoothing splines;◆kernel smoothers.■In a functional data setting we observe many functions atonce, we might also observe many covariates for each curve.■Example I will talk about later: functions are space-timeimages, covariates are simple: “motion correction” and “slowdrift.”●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 5/12A two-sample functional t-test■Suppose we observe a group of n paired curves: maybe thegrowth curves of twins separated at birth over the first 18years of life but raised in different countries with a bigdifference in standard of living.■Nutrition is known to have an effect on population height:twins’ curves might be different.■We can describe this data as(Yi1,t, Yi2,t), 1 ≤ i ≤ n, 0 ≤ t ≤ 18 where twins Y·1are incountry # 1 and Y·2are in country # 2 withYij,t= µi,t+ δt· 1{j=1}+ εij,t.■The measurement noise ε can be assumed independentacross subjects and set of twins put probably is dependent intime (µiwould be a random effect curve for the i-th set oftwins), and δ is the “nutrition effect”■Although we never observe the whole curve, we should thinkof actually having an entire curve.●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 6/12Functional t-test■For each t, and each pair computebδt=1nnXi=1Yi1,t− Yi2,t,■Also, computebσ2(bδt) =1n − 1nXi=1(Yi1,t− Yi2,t−bδt)2■Natural to useTt=bδtbσ(bδt)to test H0,t: δt= 0, i.e. at age t the “country effect” is 0.■Test for no effect H0= ∩tH0,t: useTmax= maxt∈[0,18]|Tt|.●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 7/12Smooth noise■Problem: what is the distribution of Tmax? It is not χ2!Depends on properties of ε.■Maybe we model each noise εijas a smooth Gaussianprocess on [0, 18].■What is a Gaussian process? A random function such thatfor every collection {t1, . . . , tk} the random vector(εt1, . . . , εtk)is multivariate normal.■A Gaussian process εtis completely determined byµt= E(µt), Rt,s= Cov(εt, εs)■If n is large (so Ttis almost Gaussian itself) then thedistribution ofεmax= maxt∈[0,18]εtdepends on Var( ˙εt) (known as Rice’s formula).●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 8/12Another approach■Perhaps a more typical FDA approach would be “dimension”reduction. That is, take each curve Yij,tand express it as alinear combination of basis functions bk(t) – the projection ofYij,tonto the basisYij,t=Xkcij,kbk(t) + rij,tSometimes the remainder rij,t= 0 and no dimensionreduction is used.■In any case, once you have expressed each curve in a givenbasis, the two-sample t-test problem becomes a standardregression problem involving the cofficients cij,kwhich is the“new data.”■With this basis approach, it is possible to impose penalties(i.e. on the second derivatives, etc.) as long as you knowhow to compute the penalties in your specific basis.●Today’s class●What is fMRI?●Block design – finger tapping●Hemodynamic responsefunction●Convolved design – fingertapping●Components of Errx,t●Full model for finger tappingdata●Voxel in the motor cortex●Marginally significant voxel●Real experiment: rewardanticipation●Combining subjects: fixedeffect analysis- p. 9/12Penalty example: second derivative■Suppose we choose to express each curve Yij,tas a linearcombination of cos waves of the formbk(t) = cos (2πkt/18) , 1 ≤ k ≤ m.■Thenb00k(t) = ckbk(t),Z180bk(t)bj(t)dt = δjkc0k.■This means thatZT0 Xkakb00k(t)!2dt =Xka2kc2kc0k.■Smoothing spline problem to estimate δ (penalty on integralof δ00(t)2) becomes a ridge problem.●Today’s class●What is fMRI?●Block design –