Optimization of Designs for fMRI
IPAM Mathematics in Brain Imaging Summer School
July 18, 2008
Thomas Liu, Ph.D.
UCSD Center for Functional MRI
T.T. Liu July 18, 2008
T.T. Liu July 18, 2008
Why optimize?
•
•
•
•
Scans are expensive.
Subjects can be difficult to find.
fMRI data are noisy
A badly designed experiment is
unlikely to yield publishable results.
• Time = Money
If your result needs a statistician then you should design a
better experiment. --Baron Ernest Rutherford
T.T. Liu July 18, 2008
What to optimize?
• Statistical Efficiency: maximize
contrast of interest versus noise.
• Psychological factors: is the design
too boring? Minimize anticipation,
habituation, boredom, etc.
T.T. Liu July 18, 2008
Which is the
best design?
It depends on the
experimental
question.
E
T.T. Liu July 18, 2008
Possible Questions
• Where is the activation?
• What does the hemodynamic response
function (HRF) look like?
T.T. Liu July 18, 2008
Model Assumptions
1) Assume we know the shape of the HRF
but not its amplitude.
2) Assume we know nothing about the
HRF (neither shape nor amplitude).
3) Assume we know something about the
HRF (e.g. it’s smooth).
T.T. Liu July 18, 2008
General Linear Model
Design
Matrix
Data
y
Additive
Nuisance Gaussian
Matrix
Noise
=
Xh
+
Parameters of
Interest
T.T. Liu July 18, 2008
Sb + n
Nuisance
Parameters
Example 1: Assumed HRF shape
Stimulus
Convolve w/ HRF
Design matrix depends
on both stimulus and
HRF
T.T. Liu July 18, 2008
 0  1
  
 0  1
0.5 1
  
 1  1
y   1 h1  1
  
1
1
  
0.5 1
 0  1
  

 0 
 
1
1 
1

 
2
2 
 0 
3

 
4
3
b1   
5  1 
b   
6 2
1

 
7
 2 
.5 
8

 

9

.2


Parameter
= amplitude of response
Design Regressor
The process can be modeled by convolving the activity
curve with a "hemodynamic response function" or HRF
=

HRF
Predicted neural activity
Predicted response
T.T. Liu July 18, 2008
Courtesy of FSL Group and Russ Poldrack
Example 2: Unknown HRF shape
 h1
 h2
 h3
 h4
Note: Design matrix
only depends on
stimulus, not HRF 
T.T. Liu July 18, 2008
0

0
0

1
y  1

1

0
0


0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
1
0


0
1
1
0


 h1 
0  1
h2 

0  1
h3  
0   1
h4  
1
1
1
1



1

1
1 
1

 
2
2 
 0 
3

 
4
3
b1   
5  1 
b2   
6
1

 
7
 2 
.5 
8

 

9

.2

Unknown shape and
amplitude
FIR design matrix
FIR estimates
T.T. Liu July 18, 2008
Courtesy of Russ Poldrack
Example 3: Basis Functions
If we know something about the shape,
basis function expansion : h  Bc
we can use a
4 basis functions
5 random HDRs using
basis functions
5 random HDRs w/o
basis functions
T.T. Liu July 18, 2008
Here if we assume basis functions, we only need to
estimate 4 parameters as opposed to 20.
Test Statistic
Stimulus, neural activity, field strength, vascular state
parameter estimate
t
variance of parameter estimate
Thermal noise, physiological noise, low
frequency drifts, motion
Also depends on Experimental Design!!!
T.T. Liu July 18, 2008
Efficiency
1
Efficiency 
Variance of Parameter Estimate

T.T. Liu July 18, 2008
Covariance Matrix

 
cov hˆ1, hˆ 2
var hˆ

cov hˆ N , hˆ 2
 var hˆ
1

ˆ , hˆ

cov
h
2 1
cov hˆ  

cov hˆ N , hˆ1



 

2




 
Known

Efficiency 
T.T. Liu July 18, 2008


cov hˆ1, hˆ N 

cov hˆ 2 , hˆ N 


var hˆ N 

1
  
Trace cov hˆ
Known as an A-optimal design
Efficiency
Example 1 :
Efficiency 
1

Var hˆ 1
Example 2 :
Efficiency 
T.T. Liu July 18, 2008


1


Var hˆ 1  Var hˆ 2  Var hˆ 3  Var hˆ 4
What does a matrix do?
y  Xh
N-dimensional vector

k-dimensional vector
N x k matrix
The matrix maps from a k-dimensional
space to a N-dimensional space.
T.T. Liu July 18, 2008
Matrix Geometry
Geometric fact: The image of the a kdimensional unit sphere under any N x k matrix
is an N-dimensional hyperellipse.
Xv1= 1u1
1u1
v1
k-dimensional
unit sphere
T.T. Liu July 18, 2008
N-dimensional
hyperellipse
Singular Value Decomposition
Xv1= 1u1
v2
v1
1u1
2u2
The right singular vectors v1 and v 2 are transformed
into scaled vectors 1 u1 and  2 u2 , where u1 and u2 are
the left singular vectors and 1 and  2 are the singular values.
The singular values are the
k square roots of the
eigenvalues of the kxk matrix XT X.
T.T. Liu July 18, 2008
Assumed HDR shape
Parameter
Space
Data
Space
Xv1= 1u1
v1
h0
Parameter
Noise Space
Xh0
1u1
h0
Good for
Detection
Efficiency here is optimized by amplifying the singular
vector closest to the assumed HDR. This corresponds to
maximizing one singular value while minimizing the
others.
T.T. Liu July 18, 2008
No assumed HDR shape
Parameter
Space
Data
Space
Xv1= 1u1
Parameter
Noise Space
1u1
v1
Here the HDR can point in any direction, so we don’t
want to preferentially amplify any one singular value.
This corresponds to an equal distribution of singular
values.
T.T. Liu July 18, 2008
Some assumptions about shape
If no assumptions, then use equal singular values.
If we know the the HDR lies within a subspace spanned by a
set of basis functions, we should maximize the singular values
in this subspace and minimize outside of this subspace.
T.T. Liu July 18, 2008
Knowledge (Assumptions) about HRF
None
Some
Make singular
Maximize singular
values as equal as values associated
possible
with subspace that
contains HDR
T.T. Liu July 18, 2008
Total
Maximize singular
value associated
with right singular
vector closest to
HDR
Singular Values and Power Spectrum
The singular values  i are the k square roots of
the eigenvalues of the
The eigenvalues of the
T
kxk matrix X X.
kxk matrix XT X asymptotically
approach the the power spectral coefficients of the first
T
row of X X - - i.e. the power spectrum of the design.
*(assume mean has been removed from design)
T.T. Liu July 18, 2008
Knowledge (Assumptions) about HRF
Some
None
Equally
distributed
singular
values = flat
power
spectrum
Total
Set of dominant
singular values =
spread of power
spectral
components
f
One dominant
singular value =
one dominant
power spectral
component
f
Power Spectra
T.T. Liu July 18, 2008
f
Frequency Domain Interpretation
Boxcar
Randomized ISI
single-event
Regressor
Low-pass
FFT
T.T. Liu July 18, 2008
Adapted
From S. Smith and R. Poldrack
Low-pass
Which is the
best design?
E
T.T. Liu July 18, 2008
Given an assumed
shape h0,
which design
maximizes Xh 0 ?
Xv1 = 1 u1

v1
E
T.T. Liu July 18, 2008
h0
1 u1
Xh0
Which design will have
the most equal spread
of singular values?

E
T.T. Liu July 18, 2008

Which design will have
the flattest power spectrum?
f
Which design will have
a tailored spread
of singular values?

Which design will have
a shaped power spectrum?

E
T.T. Liu July 18, 2008
f
Detection Power
When detection is the goal, we want to answer the
question: is an activation present or not?
When trying to detect something, one needs to specify
some knowledge about the “target”.
In fMRI, the target is approximated by the convolution
of the stimulus with the HRF.
Once we have specified our target (e.g. stimuli and
assumed HRF shape), the efficiency for estimating the
amplitude of that target can be considered our detection
power.
T.T. Liu July 18, 2008
Theoretical Trade-off
HDR
Estimation
Efficiency
Equal Singular values
One Dominant
Singular value
Detection Power
T.T. Liu July 18, 2008
T.T. Liu July 18, 2008
1
0.5
0.5
90
0
45
0
0
0.5
1
Detection power R
(c) k = 10
1
0
90 63 45
0
0
0.5
1
Detection power R
(d) k = 15
1
0.5

0.5
90 72
0
(b) k = 5
1
Estimation Efficiency
Estimation Efficiency
Estimation Efficiency
(a) k = 2
Estimation Efficiency
Theoretical Curves
45
0
0
0.5
1
Detection power R
0
90 75

45
0
0
0.5
1
Detection power R
Efficiency vs. Power
Estimation Efficiency
1.2
1
A
random
B
0.8
0.6
C
0.4
0.2
0
4 2 1
8
16
32
Periodic Single Trial
D
-0.2
0
T.T. Liu July 18, 2008
0.2
0.4
0.6
0.8
Detection Power
1
1.2
Efficiency with Basis Functions
T.T. Liu July 18, 2008
Knowledge (Assumptions) about HRF
Some
None
Experiments where you want
to characterize in detail the
shape of the HDR.
Total
Experiments where you have
a good guess as to the shape
(either a canonical form or
measured HDR) and want
to detect activation.
A reasonable compromise between 1 and 2.
Detect activation when you sort of know the
shape. Characterize the shape when you sort of
know its properties
T.T. Liu July 18, 2008
Question
If block designs are so good for detecting
activation, why bother using other types
of designs?
Problems with habituation and anticipation
Less Predictable
T.T. Liu July 18, 2008
Entropy
Perceived randomness of an experimental design is an
important factor and can be critical for circumventing
experimental confounds such as habituation and anticipation.
Conditional entropy is a measure of randomness in units of bits.
Rth order conditional entropy (Hr) is the average number of
binary (yes/no) questions required to determine the next trial
type given knowledge of the r previous trial types.
2
Hr
T.T. Liu July 18, 2008
is a measure of the average number of possible outcomes.
Entropy Example
AA N AA N AAA N
Maximum entropy is 1 bit, since at most one needs to only ask
one question to determine what the next trial is (e.g. is the next
trial A?). With maximum entropy, 21 = 2 is the number of
equally likely outcomes.
A C B N C B AA B C N A
Maximum entropy is 2 bits, since at most one would need to
ask 2 questions to determine the next trial type. With maximum
entropy, the number of equally likely outcomes to choose from
is 4 (22).
T.T. Liu July 18, 2008
T.T. Liu July 18, 2008
Efficiency  2^(Entropy)
T.T. Liu July 18, 2008
Multiple Trial Types
1 trial type + control (null)
AA N AA N AAA N
Extend to experiments with multiple trial types
ABABNNANBBANANA
BADBANDBCNDNBCN
T.T. Liu July 18, 2008
Multiple Trial Types GLM
y
=
Xh
+
Sb + n
X = [X1 X2 … XQ]
h = [h1T h2T … hQT]T
T.T. Liu July 18, 2008
Multiple Trial Types Overview
Efficiency includes individual trials and also contrasts
between trials.
Rtot 

K
average variance of HRF amplitude estimates


for
all
trial
types
and
pairwise
contrasts


tot 
T.T. Liu July 18, 2008
1
average variance of HRF estimates 


for
all
trial
types
and
pairwise
contrasts


Multiple Trial Types Trade-off
T.T. Liu July 18, 2008
Optimal Frequency
Can also weight how much you care about individual
trials or contrasts. Or all trials versus events.
Optimal frequency of occurrence depends on weighting.
Example: With Q = 2 trial types, if only contrasts are of
interest p = 0.5. If only trials are of interest, p = 0.2929.
If both trials and contrasts are of interest p = 1/3.
Q(2k1 1)  Q (1  k1 )  k
2
p
T.T. Liu July 18, 2008
1/2
1
Q(2k 1)  Q
1
Q(Q 1)(k1Q Q  k1 )
2

(1  k1 )
1/2
Design
As the number of trial types increases, it becomes more
difficult to achieve the theoretical trade-offs. Random
search becomes impractical.
For unknown HDR, should use an m-sequence based
design when possible.
Designs based on block or m-sequences are useful for
obtaining intermediate trade-offs or for optimizing with
basis functions or correlated noise.
T.T. Liu July 18, 2008
Optimality of m-sequences
T.T. Liu July 18, 2008
Clustered m-sequences
T.T. Liu July 18, 2008
Genetic Algorithms
QuickTime™ and a
decompressor
are needed to see this picture.
T.T. Liu July 18, 2008
Wager and Nichols 2003
Genetic Algorithms
QuickTime™ and a
decompressor
are needed to see this picture.
T.T. Liu July 18, 2008
Wager and Nichols 2003
Genetic Algorithms
QuickTime™ and a
decompressor
are needed to see this picture.
T.T. Liu July 18, 2008
Wager and Nichols 2003
Genetic Algorithms
QuickTime™ and a
decompressor
are needed to see this picture.
T.T. Liu July 18, 2008
Kao et al. available at
http://aaron.stat.uga.edu/~amandal
Topics we haven’t covered.
The impact of correlated noise -- this will change the
optimal design.
Impact of nonlinearities in the BOLD response.
Designs where the timing is constrained by psychology.
T.T. Liu July 18, 2008
Summary
• Efficiency as a metric of design
performance.
• Efficiency depends on both
experimental design and assumptions
about HRF.
• Inherent tradeoff between power
(detection of known HRF) and
efficiency (estimation of HRF)
T.T. Liu July 18, 2008