Hidden Process Modelsfor Analyzing fMRI Data
Rebecca Hutchinson
Joint work with Tom Mitchell
May 11, 2007Student Seminar Series
In partial fulfillment of the Speaking Requirement
Carnegie Mellon University Computer Science Department
2
Introduction
• Hidden Process Models (HPMs): – A new probabilistic model for time series data.– Designed for data generated by a collection of latent
processes.
• Potential domains:– Biological processes (e.g. synthesizing a protein) in
gene expression time series.– Human processes (e.g. walking through a room) in
distributed sensor network time series.– Cognitive processes (e.g. making a decision) in
functional Magnetic Resonance Imaging time series.
3
t
td1
…
dN
…
Process 1:
t
td1
…
dN
…
Process P:
…
d1
…
dN
Prior knowledge:
An instance of Process 1 begins in this window.
An instance of Process P begins in this window.
An instance of either Process 1 OR Process P begins in this window.
There are a total of 6 processes in this window of data.
4
t
td1
…
dN
…
Process 1:
t
td1
…
dN
…
Process P:
…
d1
…
dN
Process 1 timings:
Process P timings:
…
More questions:-Can we learn the parameters of these processes from the data (even when we don’t know when they occur)?-Would a different set of processes model the data better?
5
Simple Case: Known Timing
• If we know which processes occur when, we can estimate their shapes with the general linear model.
• The timings generate a convolution matrix X:
1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 1… … …
p1 p3
t=1t=2t=3t=4…
P
T
p2
6
Simple Case: Known Timing
T
D
=
1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 1… … …
p1 p3p2
p1
p3
p2
D
W(1)
W(2)
W(3)
Y
7
Challenge: Unknown Timing
T
D
=
1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 00 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 1… … …
p1 p3p2
p1
p3
p2
D
W(1)
W(2)
W(3)
Y
Uncertainty about the processes essentially makes the convolution matrix a random variable.
8
Our Approach
• Model of processes contains a probability distribution over when it occurs relative to a known event (called a timing landmark).
• When predicting the underlying processes, use prior knowledge about timing to limit the hypothesis space.
9
fMRI Data
…
Sign
al
Am
plitu
de
Time (seconds)
Hemodynamic Response
Neural activity
Features: 10,000 voxels, imaged every second.Training examples: 10-40 trials (task repetitions).
10
11
Study: Pictures and Sentences
• Task: Decide whether sentence describes picture correctly, indicate with button press.
• 13 normal subjects, 40 trials per subject.• Sentences and pictures describe 3 symbols: *,
+, and $, using ‘above’, ‘below’, ‘not above’, ‘not below’.
• Images are acquired every 0.5 seconds.
Read Sentence
View Picture Read Sentence
View PictureFixation
Press Button
4 sec. 8 sec.t=0
Rest
12
Goals for fMRI
• To track cognitive processes over time. – Estimate process hemodynamic responses.– Estimate process timings.
• Allowing processes that do not directly correspond to the stimuli timing is a key contribution of HPMs!
• To compare hypotheses of cognitive behavior.
13
HPM Modeling Assumptions
• Model latent time series at process-level. • Process instances share parameters
based on their process types. • Use prior knowledge from experiment
design. • Sum process responses linearly.
14
Process 1: ReadSentence Response signature W:
Duration d: 11 sec. Offsets : {0,1} P(): {0,1}
One configuration c of process instances 1, 2, … k: (with prior c)
Predicted mean:
Input stimulus :
1
Timing landmarks : 21
2
Process instance: 2 Process h: 2 Timing landmark: 2
Offset O: 1 (Start time: 2+ O)
sentencepicture
v1v2
Process 2: ViewPicture Response signature W:
Duration d: 11 sec. Offsets : {0,1} P(): {0,1}
v1v2
Processes of the HPM:
v1
v2
+ N(0,1)
+ N(0,2)
15
HPM FormalismHPM = <H,C,,>
H = <h1,…,hH>, a set of processes (e.g. ReadSentence)
h = <W,d,,>, a processW = response signature
d = process duration
= allowable offsets
= multinomial parameters over values in
C = <c1,…, cC>, a set of configurations
c = <1,…,L>, a set of process instances = <h,,O>, a process instance (e.g. ReadSentence(S1))
h = process ID = timing landmark (e.g. stimulus presentation of S1)
O = offset (takes values in h)
= <1,…,C>, priors over C
= <1,…,V>, standard deviation for each voxel
16
HPMs: the graphical model
Offset o
Process Type h
Start Time s
observed
unobserved
Timing Landmark
Yt,v
1,…,k
t=[1,T], v=[1,V]
The set C of configurations constrains the joint distribution on {h(k),o(k)} k.
Configuration c
17
Encoding Experiment Design
Configuration 1:
Input stimulus :
Timing landmarks :
21
ViewPicture = 2
ReadSentence = 1
Decide = 3
Configuration 2:
Configuration 3:
Configuration 4:
Constraints Encoded:
h(1) = {1,2}h(2) = {1,2}h(1) != h(2)o(1) = 0o(2) = 0h(3) = 3o(3) = {1,2}
Processes:
18
Inference• Over configurations
• Choose the most likely configuration, where:
• C=configuration, Y=observed data, =input stimuli, HPM=model
19
Learning
• Parameters to learn:– Response signature W for each process– Timing distribution for each process – Standard deviation for each voxel
• Expectation-Maximization (EM) algorithm to estimate W and .– E step: estimate a probability distribution over
configurations.– M step: update estimates of W (using reweighted
least squares) and (using standard MLEs) based on the E step.
– After convergence, use standard MLEs for
20
Uncertain Timings• Convolution matrix models several choices for
each time point.
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0… … … 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 10 0 0 0 0 1 0 1 00 0 0 0 0 1 0 0 1... … …
P D
t=1t=1t=2t=2…t=18t=18t=18t=18…
T’>T
SConfigurations for each row:
3,41,23,41,2…3412…
21
Uncertain Timings
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0… … …
P D
e1e2e3e4…
S
Y=
W
3,41,23,41,2…
Configurations: Weights:
e1 = P(C=3|Y,Wold,old,old) + P(C=4|Y,Wold,old,old)
• Weight each row with probabilities from E-step.
22
Learned HPM with 3 processes (S,P,D), and d=13sec.
P PS S
D?
observed
Learned models:
S
P
D
D start time chosen by program as t+18
predicted
P PS S
D D
D?
23
ViewPicture in Visual Cortex
Offset = P(Offset)0 0.7251 0.275
24
ReadSentence in Visual Cortex
Offset = P(Offset)0 0.6251 0.375
25
Decide in Visual CortexOffset = P(Offset)0 0.0751 0.0252 0.0253 0.0254 0.2255 0.625
26
ViewPicture
27
ReadSentence
28
Decide
0 0.5 1 1.5 2 2.5 3 3.5
0 0 0 0 0.025 0.05 0.075 0.85
Seconds following the second stimulus
Multinomial probabilities on these time points
29
Comparing ModelsHPM Avg. Test Set LL
PS -1.0784 * 10^6
PSD -1.0759 * 10^6
PS+S-D -1.0742 * 10^6
PSD+D- -1.0742 * 10^6
PSDB -1.0741 * 10^6
PSDyDn -1.0737 * 10^6
PSDyDnDc** -1.0717 * 10^6
PSDyDnDcB -1.0711 * 10^6
5-fold cross-validation, 1 subject
P = ViewPicture
S = ReadSentence
S+ = ReadAffirmativeSentence
S- = ReadNegatedSentence
D = Decide
D+ = DecideAfterAffirmative
D- = DecideAfterNegated
Dy = DecideYes
Dn = DecideNo
Dc = DecideConfusion
B = Button
** - This HPM can also classify Dy vs. Dn with 92.0% accuracy. GNBC gets 53.9%. (using the window from the second stimulus to the end of the trial)
30
Are we learning the right number of processes?
• Use synthetic data where we know ground truth.– Generate training and test sets with 2/3/4 processes.– Train HPMs with 2/3/4 processes on each.– For each test set, select the HPM with the highest data log
likelihood.
Number of processes in the training and test data
Number of times the correct number of
processes was chosen for the test set
2 5/5
3 5/5
4 4/5
Total: 14/15 = 93.3%
31
Related Work
• fMRI– General Linear Model (Dale99)
• Must assume timing of process onset to estimate hemodynamic response.
– Computer models of human cognition (Just99, Anderson04)• Predict fMRI data rather than learning parameters of processes from
the data.
• Machine Learning – Classification of windows of fMRI data (Cox03, Haxby01,
Mitchell04)• Does not typically model overlapping hemodynamic responses.
– Dynamic Bayes Networks (Murphy02, Ghahramani97)• HPM assumptions/constraints are difficult to encode in DBNs.
32
Future Work
• Incorporate spatial prior knowledge. E.g. share parameters across voxels (extending Niculescu05).
• Smooth hemodynamic responses (e.g. Boynton96).
• Improve algorithm complexities.
• Apply to open cognitive science problems.
33
Conclusions
• Take-away messages:– HPMs are a probabilistic model for time series
data generated by a collection of latent processes.
– In the fMRI domain, HPMs can simultaneously estimate the hemodynamic response and localize the timing of cognitive processes.
34
ReferencesJohn R. Anderson, Daniel Bothell, Michael D. Byrne, Scott Douglass, Christian Lebiere, and Yulin Qin. An integrated theory of the mind. Psychological Review, 111(4):1036–1060, 2004. http://act-r.psy.cmu.edu/about/.
Geoffrey M. Boynton, Stephen A. Engel, Gary H. Glover, and David J. Heeger. Linear systems analysis of functional magnetic resonance imaging in human V1. The Journal of Neuroscience, 16(13):4207–4221, 1996.
David D. Cox and Robert L. Savoy. Functional magnetic resonance imaging (fMRI) ”brain reading”: detecting and classifying distributed patterns of fMRI activity in human visual cortex. NeuroImage, 19:261–270, 2003.
Anders M. Dale. Optimal experimental design for event-related fMRI. Human Brain Mapping, 8:109–114, 1999.
Zoubin Ghahramani and Michael I. Jordan. Factorial hidden Markov models. Machine Learning, 29:245–275, 1997.
James V. Haxby, M. Ida Gobbini, Maura L. Furey, Alumit Ishai, Jennifer L. Schouten, and Pietro Pietrini. Distributed and overlapping representations of faces and objects in ventral temporal cortex. Science, 293:2425–2430, September 2001.
Marcel Adam Just, Patricia A. Carpenter, and Sashank Varma. Computational modeling of high-level cognition and brain function. Human Brain Mapping, 8:128–136, 1999. http://www.ccbi.cmu.edu/project 10modeling4CAPS.htm.
Tom M. Mitchell et al. Learning to decode cognitive states from brain images. Machine Learning, 57:145–175, 2004.
Kevin P. Murphy. Dynamic bayesian networks. To appear in Probabilistic Graphical Models, M. Jordan, November 2002.
Radu Stefan Niculescu. Exploiting Parameter Domain Knowledge for Learning in Bayesian Networks. PhD thesis, Carnegie Mellon University, July 2005. CMU-CS-05-147.