1 time series analysis of fmri ii: noise, inference, and model error douglas n. greve
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Time Series Analysis of fMRI II: Noise, Inference, and Model Error
Douglas N. Grevegreve@nmr.mgh.harvard.edu
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fMRI Analysis Overview
Higher Level GLM
First Level GLM Analysis
First Level GLM Analysis
Subject 3
First Level GLM Analysis
Subject 4
First Level GLM Analysis
Subject 1
Subject 2CX
CX
CX
CX
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
Raw Data
Raw Data
Raw Data
Raw Data
CX
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• Hemodynamic Response Function (HRF) Model• Modeling the Entire BOLD Signal• Contrasts
• Noise Propagation• Inference• Design Efficiency• HRF Model Errors
Overview
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Review: Neuro-vascular Coupling
• Delay• Dispersion – spreading out• Amplitude
Stimulus
Delay
Dispersion
Amplitude
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Review: Fundamental ConceptAmplitude of neural firing is monotonically related to the HRF amplitude.
Interpretation: if the HRF amplitude for one stimulus is greater than that of another, then the neural firing to that stimulus was greater.
Emphasis is on estimating the HRF amplitude by constructing and fitting models of the HRF and the entire BOLD signal.
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Parametric Models of the HRF
tth
tettht
,0)(
,)(2
Dale and Buckner, 1997, HBM 5:329-340. =2.25s, =1.25s
Gamma Model:
Parameters: • Delay• Dispersion • AmplitudeModel converts a neural
“impulse” into a BOLD signal (convolution)
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Review: Estimating the Amplitude
m=Slope=Amp
Forward Model:y=xh*m
16 Equations 1 Unknown
yxxx Thh
Thm
1
xh
yy = xh =
16x1 16x1
h(t1,,)h(t2,,)h(t3,,)…
t
etth2
1)(
m
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Review: Contrast Matrix• Two Conditions (E1 and E2)• X = [xhE1 xhE2], E1 E2
• Hypothesis: Response to E1 and E2 are different• Null Hypothesis (Ho): E1 –E2= 0
Statistic:C*c1 c2E1 E2
c1* E1 + c2E2=0
c1= +1, c2 = -1C = [+1 -1] 1x2
Means E1 > E2
Means E2 > E1
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Noise
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Observed Never Matches Ideal
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Sources of Temporal Noise• Thermal/Background – Gaussian; reduce by temporal averaging and/or spatial smoothing• Scanner Instability – drift, instability in electronics• Physiological Noise
• Motion – motion correction, motion regressors• Heart Beat – aliasing, external monitor, nuisance regressors (RETROICOR)• Respiration, CO2– aliasing, external monitor, nuisance regressors (RETROICOR)• Endogenous (non-task related) Neural Activation
• Model Errors• Behavioral/Cognitive Variability• Wrong assumed shape
• ?????
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Noise Composition
• Physiological Noise contributes the most in cortex/gray matter.• First level (time series) noise generally less than intersubject
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fMRI Noise Spectrum Voxel-wise, no smoothing
fMRI Noise gets much worse at low frequencies.
Thermal Noise Floor
Block Period=60s
Longer Shorter
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Full Model with Noise
),0(~ , , 2nnNnnsynXy
y – observables – signal = Xn – noise, Model: Gaussian, 0-mean, stddev n, =I for white noise
n
s=Signal=X
n=Noise
y=Observable
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Noise Propagation
• Assumed shape is the same• Actual shape is the same• Noise is different• Amplitude estimates (slopes) are
different ()• Measurement noise creates
uncertainty in estimates
Experiment 1 Experiment 2
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Full Model with Noise
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2
1
2
2
)ˆ( ,)ˆ( :Unbiased
Variance Residual ˆˆˆ
Estimate) (Noise Residual ˆˆEstimate Signal ˆˆ
EstimatesParameter )(ˆ and :Unknowns
),0(~ , ,
nnTrue
T
n
TT
n
n
EEDOF
nn
synXs
yXXX
NnnsynXy
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Contrasts and the Full Model
ate)(multivariTest -F ˆˆˆF
e)(univariatTest - tˆ)(
ˆ
ˆˆ
t
Cin Rows J
Estimate VarianceContrast ˆ)(1ˆˆ
Contrast ˆˆ
Variance Residual ˆˆˆ
EstimatesParameter )(ˆ),0(~ , ,
1JDOF,
21DOF
212
2
1
2
T
nTT
nTT
T
n
TT
n
CXXC
C
CXXCJ
CDOF
nn
yXXX
NnnsynXy
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Propagation of Noise
ˆ)(1ˆˆ
),0(~ , ,
212
2
nTT
n
CXXCJ
NnnsynXy
Noise in the observable gets transferred to the contrast through (XTX)-1.
The properties of (XTX) are important!SingularityInvertibilityEfficiencyCondition
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• M*A = I, then A=M-1
• Complicated in general• Simple for a 2x2
Review: Matrix Inverse
m11
2x2
M = m12
m21 m22
= m11* m22 - m12* m21
m22
2x2
M-1 =-m12
-m21 m11
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Review: Invertibility
1.0
2x2
M = 2.0
0.5 1.0
= 1.0*1.0 - 2.0*0.5 = 1-1 = 0
1.0
2x2
M-1 =-2.0
-0.5 1.0
IMPORTANT!!!• Not all matrices are invertible• =0• “Singular”
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Review: Singularity and “Ill-Conditioned”
1.0
2x2
M = 2.0
0.5 1.0
= 1.0*1.0 - 2.0*0.5 = 1-1 = 0
• Column 2 = twice Column 1• Linear Dependence
Ill-Conditioned: is “close” to 0Relates to efficiency of a GLM.
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Review: GLM SolutionIntercept: b
Slope: m
Age
x1 x2
y2
y1
y = X*=X-1*y
y1y2
1 x11 x2
bm= *
1X =
x11 x2
x2X-1 =
-x1-1 1
1
= x2-x1Non-invertible if x1=x2Ill-conditioned if x1 near x2Sensitive to noise
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Noise Propagation through XTX
cant)(Insignifi 1 ,0
,ˆ
,)(
singular) becomes X As( 0 As
ˆˆ
t
ˆ)(ˆ ˆˆ
2
1
DOF
212
pt
XX
CXXC
C
T
nTT
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XTX for Orthogonal Design
1000*0*
0 :ityOrthogonal
)(arbitrary 10
00
shifted , ], [
VV
xxxx
Vxxxx
VV
xxxxxxxx
XX
xxxxX
hAhBhBhA
hBhBhAhA
hBhBhAhB
hBhAhAhA
TT
TT
TT
TTT
hBhAhBhA
A B BA
xhA
xhB
25
Orthogonal Design (Twice as long)
4000*0*
00
0 :ityOrthogonal
(doubled) 20
(shifted) ], [
VV
VV
xxxxxxxx
XX
xxxx
Vxxxx
xxxxX
hBhBhAhB
hBhAhAhA
hAhBhBhA
hBhBhAhA
TT
TTT
TT
TT
hBhAhBhA
A B BA
xhA
xhB
A B BA
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XTX for Fully Co-linear Design
0**,
0) ifty (Colineari
)(arbitrary 10
] [
VVVVVVVV
XX
Vxxxx
Vxxxx
xxxxX
T
TT
TT
hBhAhBhA
hAhBhBhA
hBhBhAhA
A+V A+V A+V A+V
xhA
xhB
Auditory and VisualPresented Simultaneously
Auditory Regressor
Visual Regressor
Singular!Does not work!
Note: DOF is OK
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XTX for Partially Co-linear Design
19**,
0)but,(arbitrary 9
)(arbitrary 10
similar)(but very ] [
QQVVVQQV
XX
Qxxxx
Vxxxx
xxxxX
T
TT
TT
hBhAhBhA
hAhBhBhA
hBhBhAhA
xhA
xhB
Working Memory: Encode and Probe
Encode Regressor
Probe Regressor
Note: textbook (HSM) suggests orthogonalzing. This is not a good idea.
This design requires a lot of overlap which reduces the determinant, but this does not mean that it is a bad design.
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Co-linearity and InvertibilityWhat causes co-linearity?• Synchronized presentations/responses• Many regressors
– Derivatives– Basis sets (eg, FIR)– Lots of nuisance regressors
• Noise tends to be low-frequency• Task tends to be low-frequency
• Only depends on X
Note: more stimulus presentations generally improves things
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Number of Regressors
y = Ntpx1
How many regressors can you have?How many regressors should you have?
X = Ntpx10
xh1 xh2
111…
123…
Tx1
Tx2
Tx3
…
Ty1
Ty2
Ty3
…
Tz1
Tz2
Tz3
…
Rx1
Rx2
Rx3
…
Ry1
Ry2
Ry3
…
Rz1
Rz2
Rz3
…
DOF = Rows(X)-Cols(X) > 0 #Equations > #UnknownsMore regressors = more colinearity (less efficiency)
XTX is 10x10
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Hemodynamic Response FunctionModel Error
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HRF Model Error• Systematic deviation between the assumed shape and the
actual shape• Sources
– Delay, Dispersion– Form (undershoot)– Duration (stimulus vs neural)– Non-linearity
tth
tettht
,0)(
,)(2
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HRF Delay Error
• Delay error of 1 sec• Loss of amplitude/slope (Bias), smaller t-values• Larger “noise” – ie, residual error.
– Cannot be fixed by more acquisitions (bias)• Scaling still preserved
xh
y
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HRF Bias vs Duration and Delay Error
• As stimulus gets longer, bias gets less
2s
5s
10s
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HRF Bias vs Duration and Shape Error
• HSM Fig 9.10• As stimulus gets longer, overall shape gets similar even if
individual shapes are very different
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Does Neural Activation Match Stimulus?• May be shorter – eg, due to habituation, not needing as much time to process
information• May be longer – eg, emotional stimuli• May not be constant
2
5
10
36
Does Model Error Matter?• More False Negatives (Type II Errors)
– Loss of amplitude– More noise (usually trivial compared to other sources)
• Scaling still preserved• Real Problem: Systematic Errors across …
– Subjects, Brain Areas, Time …
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When Does Model Error Matter?
Actual
EstimationDifference
Group 1 Group 2
• Groups have same true amplitude but different delays• Estimated amplitudes are systematically different• False Positives (or False Negatives)• Groups could be from different: populations (Normals vs
Schizophrenics), times (longitudinal), brain regions
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Adding Derivatives
Tdmh
h ddxxX
,
t
etth2
)(
39
Still have a Problem
Tdmh
h ddxxX
,
• Fit to observed waveform is better (residual variance less)• But now have two Betas• xh and derivative are orthgonal, so m does not change (bias is
not removed).• No good solution to the problem (maybe Calhoun 2004)
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End of Presentation
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Noise Propagation
• Centered at 2.0 (true amplitude)• Variance depends on
– noise variance, number of samples, a few other things
• Uncertainty• Type II Errors (false negatives)• Type I Errors (false positives)
true (Unknown)
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“Student’s” t-Test
t-value: tDOF
DOF = Rows(X)-Cols(X) = TimePoints-ParametersSignificance (p-value): area
under the curve to the right
DOFt
true
NullHypothesis
Probability “under the null” (True=0)
False Positive Rate (Type I Error)
NullHypothesis
How to get
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Optimal Rapid Event-related design
• Lots of overlap in jittered design• Should push apart so no overlap?• No overlap means fewer stimuli (fixed scanning time)• How to balance?• How to schedule?
Periodic Design (N=3)
Jittered Design (N=21)
xhA xhB
xhA xhB
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)(1 Efficiency
/ˆ
ˆ
ˆ)(
ˆt
1
221DOF
TT
nnTT
CXXC
CCXXC
C
Optimal Experimental Design
• Efficiency only depends on X and C• X depends on stimulus onset times and number of presentations• Choose stimulus onset times to max • Can be done before collecting data!• Can interpret as a variance reduction factor
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Finite Impulse Response (FIR) Model
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Finite Impulse Response (FIR) Model
1 0 0 0 0 0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 01 0 0 0 0 0 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 0
y = X*First Stimulus OnsetFirst Stimulus Onset + 1TR
Second Stimulus Onset
Average at Stimulus OnsetAverage Delayed by 1TRAverage Delayed by 2TRAverage Delayed by 3TRAverage Delayed by 4TR
First Stimulus Onset + 2TR
Second Stimulus + 1TR
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