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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

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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

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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

)(

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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