1. Optimization of Designs for fMRI
IPAM Mathematics in Brain Imaging Summer School
July 18, 2008
Thomas Liu, Ph.D.
UCSD Center for Functional MRI

3. Subjects can be difficult to find. fMRI data are noisy
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
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
If your result needs a statistician then you should design a better experiment. --Baron Ernest Rutherford

4. Statistical Efficiency: maximize contrast of interest versus noise.
What to optimize?
Statistical Efficiency: maximize contrast of interest versus noise.
Psychological factors: is the design too boring? Minimize anticipation, habituation, boredom, etc.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

5. Which is the best design?
It depends on the experimental question.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise. E

6. Where is the activation?
Possible Questions
Where is the activation?
What does the hemodynamic response function (HRF) look like?
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

7. Assume we know the shape of the HRF but not its amplitude.
Model Assumptions
Assume we know the shape of the HRF but not its amplitude.
Assume we know nothing about the HRF (neither shape nor amplitude).
Assume we know something about the HRF (e.g. it’s smooth).
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

8. Parameters of Interest
General Linear Model
Additive
Gaussian
Noise
Design
Matrix
Nuisance
Matrix
Data
y = Xh Sb n
Parameters of Interest
Nuisance
Parameters
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

9. Example 1: Assumed HRF shape
Stimulus
Convolve w/ HRF
Parameter = amplitude of response
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Design matrix depends on both stimulus and HRF

10. 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
Courtesy of FSL Group and Russ Poldrack

11. Example 2: Unknown HRF shape
Unknown shape and amplitude
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Note: Design matrix only depends on stimulus, not HRF

12. FIR design matrix
FIR estimates
Courtesy of Russ Poldrack

13. Example 3: Basis Functions
5 random HDRs using basis functions
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
5 random HDRs w/o basis functions
Here if we assume basis functions, we only need to estimate 4 parameters as opposed to 20.

14. Test Statistic
Stimulus, neural activity, field strength, vascular state
Thermal noise, physiological noise, low frequency drifts, motion
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Also depends on Experimental Design!!!

15. Efficiency
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

16. Covariance Matrix Known Known as an A-optimal design
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Known as an A-optimal design

17. Efficiency
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

18. What does a matrix do? k-dimensional vector N-dimensional vector
N x k matrix
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
The matrix maps from a k-dimensional space to a N-dimensional space.

19. Matrix Geometry
Geometric fact: The image of the a k-dimensional unit sphere under any N x k matrix is an N-dimensional hyperellipse.
Xv =  u v 1
 u
1 1
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
k-dimensional unit sphere
N-dimensional hyperellipse

20. Singular Value Decomposition
Xv =  u
 u
1 1
2 2 v 1 2
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

21. Assumed HDR shape Parameter Space Data Space Parameter Noise Space1 v
Parameter
Space
 u
1 1
Data
Space
Parameter
Noise Space
Xv =  u Xh
Good for
Detection h h
The probability of detection refers to the probability of detecting an activation when there is an activation. This is what we mean by detection power. It depends on what is called the non-centrality parameter which is defined here.
By making this parameter larger, we increase our probability of detecting a response.
The non-centrality parameter is essentially the expected power of our measured signal divided by the noise power.
Note that in contrast to estimation efficiency, we do need to assume a shape for the hemodynamic response in order to calculate estimation efficiency.
It turns out that we can maximize detection power when the columns of Xperp are as parallel as possible -- which is just another way of saying that blocked designs will generally maximize detection power.
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.

22. No assumed HDR shape Parameter Space Data Space Parameter Noise Space
Xv =  u
 u
1 1 v 1
The probability of detection refers to the probability of detecting an activation when there is an activation. This is what we mean by detection power. It depends on what is called the non-centrality parameter which is defined here.
By making this parameter larger, we increase our probability of detecting a response.
The non-centrality parameter is essentially the expected power of our measured signal divided by the noise power.
Note that in contrast to estimation efficiency, we do need to assume a shape for the hemodynamic response in order to calculate estimation efficiency.
It turns out that we can maximize detection power when the columns of Xperp are as parallel as possible -- which is just another way of saying that blocked designs will generally maximize detection power.
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.

23. Some assumptions about shape
If no assumptions, then use equal singular values.
The probability of detection refers to the probability of detecting an activation when there is an activation. This is what we mean by detection power. It depends on what is called the non-centrality parameter which is defined here.
By making this parameter larger, we increase our probability of detecting a response.
The non-centrality parameter is essentially the expected power of our measured signal divided by the noise power.
Note that in contrast to estimation efficiency, we do need to assume a shape for the hemodynamic response in order to calculate estimation efficiency.
It turns out that we can maximize detection power when the columns of Xperp are as parallel as possible -- which is just another way of saying that blocked designs will generally maximize detection power.
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.

24. Knowledge (Assumptions) about HRF
None
Some
Total
Make singular values as equal as possible
Maximize singular values associated with subspace that contains HDR
Maximize singular value associated with right singular vector closest to HDR
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

25. Singular Values and Power Spectrum
The probability of detection refers to the probability of detecting an activation when there is an activation. This is what we mean by detection power. It depends on what is called the non-centrality parameter which is defined here.
By making this parameter larger, we increase our probability of detecting a response.
The non-centrality parameter is essentially the expected power of our measured signal divided by the noise power.
Note that in contrast to estimation efficiency, we do need to assume a shape for the hemodynamic response in order to calculate estimation efficiency.
It turns out that we can maximize detection power when the columns of Xperp are as parallel as possible -- which is just another way of saying that blocked designs will generally maximize detection power.

26. Knowledge (Assumptions) about HRF
None
Some
Total
Equally distributed singular values = flat power spectrum f
Set of dominant singular values = spread of power spectral components f
One dominant singular value = one dominant power spectral component f
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Power Spectra

27. Frequency Domain Interpretation
Randomized ISI single-event
Boxcar
Regressor
Low-pass
Low-pass
FFT
Adapted From S. Smith and R. Poldrack

28. Which is the best design?
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise. E

29. Xv =  u 1 v
 u
1 1 h Xh
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise. E

30. To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise. f E

31. f
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise. E

32. 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.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
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.

33. Theoretical Trade-off
Equal Singular values
HDR
Estimation
Efficiency
One Dominant
Singular value
In event-related fMRI experiments we are confronted with two competing goals.
On the one hand, we are interested in detecting aactivations -- our ability to do this is referred to as statistical power or detection power
And on the other hand, we are interesting in estimating the shape of the hemodynamic response to the stimulus -- our ability to do this is measured by the estimation efficiency of the design.
Estimation efficiency and detection power are not the same thing, and in this talk we’ll be exploring the trade-offs between the two.
Detection Power

34. Theoretical Curves Estimation Efficiency Estimation Efficiency
In event-related fMRI experiments we are confronted with two competing goals.
On the one hand, we are interested in detecting aactivations -- our ability to do this is referred to as statistical power or detection power
And on the other hand, we are interesting in estimating the shape of the hemodynamic response to the stimulus -- our ability to do this is measured by the estimation efficiency of the design.
Estimation efficiency and detection power are not the same thing, and in this talk we’ll be exploring the trade-offs between the two.
Estimation Efficiency
Estimation Efficiency

35. Efficiency vs. Power Estimation Efficiency Detection Power
In event-related fMRI experiments we are confronted with two competing goals.
On the one hand, we are interested in detecting aactivations -- our ability to do this is referred to as statistical power or detection power
And on the other hand, we are interesting in estimating the shape of the hemodynamic response to the stimulus -- our ability to do this is measured by the estimation efficiency of the design.
Estimation efficiency and detection power are not the same thing, and in this talk we’ll be exploring the trade-offs between the two.
Detection Power

36. Efficiency with Basis Functions
The probability of detection refers to the probability of detecting an activation when there is an activation. This is what we mean by detection power. It depends on what is called the non-centrality parameter which is defined here.
By making this parameter larger, we increase our probability of detecting a response.
The non-centrality parameter is essentially the expected power of our measured signal divided by the noise power.
Note that in contrast to estimation efficiency, we do need to assume a shape for the hemodynamic response in order to calculate estimation efficiency.
It turns out that we can maximize detection power when the columns of Xperp are as parallel as possible -- which is just another way of saying that blocked designs will generally maximize detection power.

37. Knowledge (Assumptions) about HRF
None
Some
Total
Experiments where you want to characterize in detail the shape of the HDR.
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
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

38. Question
If block designs are so good for detecting activation, why bother using other types of designs?
Problems with habituation and anticipation
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Less Predictable

39. 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.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
is a measure of the average number of possible outcomes.

40. Entropy Example A A N A A N A A A N A C B N C B A A B C N A
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 A A 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).
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

41. To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

42. Efficiency  2^(Entropy)
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Efficiency  2^(Entropy)

43. Multiple Trial Types 1 trial type + control (null) A A N A A N A A A N
Extend to experiments with multiple trial types
A B A B N N A N B B A N A N A
B A D B A N D B C N D N B C N

44. Multiple Trial Types GLM
y = Xh Sb n
X = [X1 X2 … XQ]
h = [h1T h2T … hQT]T
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

45. Multiple Trial Types Overview
Efficiency includes individual trials and also contrasts between trials.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

46. Multiple Trial Types Trade-off
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

47. 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 = If both trials and contrasts are of interest p = 1/3.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

48. 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.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

49. Optimality of m-sequences
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

50. Clustered m-sequences
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

51. Genetic Algorithms Wager and Nichols 2003
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Wager and Nichols 2003

52. Genetic Algorithms Wager and Nichols 2003
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Wager and Nichols 2003

53. Genetic Algorithms Wager and Nichols 2003
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Wager and Nichols 2003

54. Genetic Algorithms
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.
Kao et al. available at

55. 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.
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.

56. 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)
To be more precise about what I mean by detection power and estimation efficiency, I’ll briefly review the framework of the general linear model.
We have here the observed data vector y being equal to a signal
Term a nuisance term and a noise term.
The design matrix X times the hemodynamic response vector h.
For the purposes of this talk, we will assume that the columns of X are simply shifted versions of the stimulus pattern, so that Xh is simply the convolution of the the stimulus with the hemodynamic response.
The term Sb represents nuisance effects, such as a constant term, linear trends, etc.
Finally, we’ll assume that the noise term is additive white Gaussian noise.