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- p. 1/12
Statistics 203: Introduction to Regressionand Analysis of Variance
Functional Data
Jonathan Taylor
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 2/12
Today’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 response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 3/12
Functional data
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 4/12
Functional 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 response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 5/12
A 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·1 are incountry # 1 and Y·2 are in country # 2 with
Yij,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 (µi would 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 response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 6/12
Functional t-test
■ For each t, and each pair compute
δt =1
n
n∑
i=1
Yi1,t − Yi2,t,
■ Also, compute
σ2(δt) =1
n − 1
n∑
i=1
(Yi1,t − Yi2,t − δt)2
■ Natural to use
Tt =δt
σ(δt)
to test H0,t : δt = 0, i.e. at age t the “country effect” is 0.■ Test for no effect H0 = ∩tH0,t: use
Tmax = maxt∈[0,18]
|Tt|.
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 7/12
Smooth noise
■ Problem: what is the distribution of Tmax? It is not χ2!Depends on properties of ε.
■ Maybe we model each noise εij as 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 εt is completely determined by
µt = E(µt), Rt,s = Cov(εt, εs)
■ If n is large (so Tt is almost Gaussian itself) then thedistribution of
εmax = maxt∈[0,18]
εt
depends on Var(εt) (known as Rice’s formula).
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 8/12
Another approach
■ Perhaps a more typical FDA approach would be “dimension”reduction. That is, take each curve Yij,t and express it as alinear combination of basis functions bk(t) – the projection ofYij,t onto the basis
Yij,t =∑
k
cij,kbk(t) + rij,t
Sometimes 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,k which 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 response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 9/12
Penalty example: second derivative
■ Suppose we choose to express each curve Yij,t as a linearcombination of cos waves of the form
bk(t) = cos (2πkt/18) , 1 ≤ k ≤ m.
■ Then
b′′k(t) = ckbk(t),
∫ 18
0
bk(t)bj(t)dt = δjkc′k.
■ This means that
∫ T
0
(∑
k
akb′′k(t)
)2
dt =∑
k
a2kc2
kc′k.
■ Smoothing spline problem to estimate δ (penalty on integralof δ′′(t)2) becomes a ridge problem.
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 10/12
Smoothing spline and ridge
■ Suppose we look at the two-sample problem in the Fourierbasis but we impose a second derivative penalty. Let
Yij,t =
m∑
k=1
cij,kbk(t)
be the “dimension reduced” Yij,t’s and
δt =∑
k
δkbk(t)
be a linear combination of sine waves.■ Problem findδt that minimizes
Lλ(δ) =
n∑
i=1
(Yi1,t − Yi2,t − δt
)2
dt + λ
∫ 18
0
δ′′(t)2 dt.
■ Both integrals reduce to sums involving δk’s and cij,k’s:equivalent to a ridge problem.
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 11/12
fMRI
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 12/12
What is fMRI?
■ fMRI (Functional Magnetic Resonance Imaging) is aspace-time recording of “metabolic activity” in the humanbrain
■ “paradigm” (motor task, i.e. finger tapping or cognitive task,i.e. face recognition) increases nerve cell activity in areasassociated with the “paradigm”
■ increased nerve cell activity increases metabolic demand foroxygen, increases metabolic activity results in a laggedincrease in oxygenated Hg (hemoglobin)
■ relationship between input “paradigm” and BOLD ismodelled through a transfer function, the HemodynamicResponse Function (HRF)
BOLDx,t = βInputx · (HRF ∗ Input)t +
k∑
i=1
βi,xXi + Errx,t.
- p. 13/12
Block design – finger tapping
0 50 100 150 200 250
0.0
0.2
0.4
0.6
0.8
1.0
Time
Pro
toco
l
- p. 14/12
Hemodynamic response function
0 5 10 15 20
−0.
20.
20.
61.
0
Time
HR
F
- p. 15/12
Convolved design – finger tapping
0 50 100 150 200 250
050
100
150
200
Time
Pro
toco
l
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 16/12
Components of Errx,t
■ physiological noise◆ cardiac noise◆ repiratory noise◆ basal metabolism
■ motion artifacts■ saturation of signal■ other sources of error: lumped into “noise term” εx,t
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 17/12
Full model for finger tapping data
■
BOLDx,t = βInputx · (HRF ∗ Input)t +
6∑
j=1
Tj,tβMotionj,x +
2∑
j=0
βTimej,x tj + εx,t.
■ Usually, model is fit voxel-by-voxel, usually spatial ignoringcorrelation between εx1,· and εx2,·.
■ Most common model is a two-stage procedure: first find a(smoothed) AR(1) coefficient image at each voxel in thebrain: estimate standard errors and coefficients with thisAR(1) value at each voxel.Sometimes, even correlation within time series at a singlepoint is ignored and model is fit by OLS.
■ Many people work on “improving” this basic model.
- p. 18/12
Voxel in the motor cortex
0 50 100 150 200 250
560
570
580
590
Time
BO
LD s
igna
l
- p. 19/12
Marginally significant voxel
0 50 100 150 200 250
640
650
660
Time
BO
LD s
igna
l
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 20/12
Real experiment: reward anticipation
■ B. Knutson, my psychologist collaborator is interested inreward anticipation.
■ Subjects play simple video game, and are told whether theycan win, lose or draw on each round.
■ If we contrast “win” (reward) vs. “draw” (neutral), we will findareas that are sensitive to anticipating reward.
● Today’s class
● What is fMRI?
● Block design – finger tapping
● Hemodynamic response
function● Convolved design – finger
tapping
● Components of Errx,t
● Full model for finger tapping
data● Voxel in the motor cortex
● Marginally significant voxel
● Real experiment: reward
anticipation
● Combining subjects: fixed
effect analysis
- p. 21/12
Combining subjects: fixed effect analysis
■ For each subject i, compute
TRwAnt,ix =
βRwAnt,ix
SE(βRwAnt,i
x ).
■ Transform subject-specific comparisons to Tailarach space(common coordinates).
■ Form group map
Tx =1√n
n∑
i=1
TRwAnt,ix .
■ Analogous to the role of Tt in “growth curve” example.■ Where to set the threshold? The uncorrected 0.05 threshold
is 1.96. The generalizations of Rice’s formula set thethreshold around 4.5 or so. Let’s look at the difference.