1. Hidden Process Models with applications to fMRI data
Rebecca Hutchinson
August 2, 2009
Joint Statistical Meetings, Washington DC
Oregon State University

2. Introduction Hidden Process Models (HPMs): Example domain:
A probabilistic model for time series data.
Designed for data generated by a collection of latent processes.
Example domain:
Modeling cognitive processes (e.g. making a decision) in functional Magnetic Resonance Imaging time series.
Characteristics of potential domains:
Processes with spatial-temporal signatures.
Uncertainty about temporal location of processes.
High-dimensional, sparse, noisy.

3. fMRI Data Features: 5k-15k voxels, imaged every second.
Hemodynamic Response
Features: 5k-15k voxels, imaged every second.
Training examples: trials (task repetitions).
Signal Amplitude …
Neural
activity
Time (seconds)

4. Study: Pictures and Sentences
Press Button
View Picture
Read Sentence
Read Sentence
Fixation
View Picture
Rest
t=0
4 sec.
8 sec.
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.

5. Goals for fMRI To track cognitive processes over time.
Estimate hemodynamic response signatures.
Estimate process timings.
Modeling processes that do not directly correspond to the stimuli timing is a key contribution of HPMs!
To compare hypotheses of cognitive behavior.

6. v1 v1 v2 v2 1 2 1 2 v1 + N(0,s1) v2 + N(0,s2)
Process 1: ReadSentence
Response signature W:
Duration d: 11 sec.
Offsets W: {0,1}
P(): {q0,q1}
Process 2: ViewPicture
Response signature W:
Duration d: 11 sec.
Offsets W: {0,1}
P(): {q0,q1}
Processes of the HPM: v1 v2 v1 v2
Input stimulus :
sentence
picture
Timing
landmarks :
Process instance: 2
Process h: 2
Timing landmark: 2
Offset O: 1
(Start time: 2+ O) 1 2
One configuration c of process instances 1, 2, … k: 1 2 
Predicted mean: v1 v2
+ N(0,s1)
+ N(0,s2)

7. HPM Formalism HPM = <H,C,F,S>
H = <h1,…,hH>, a set of processes (e.g. ReadSentence)
h = <W,d,W,Q>, a process
W = response signature
d = process duration
W = allowable offsets
Q = multinomial parameters over values in W
C = <c1,…, cC>, a set of possible configurations
c = <p1,…,pL>, a set of process instances
= <h,l,O>, a process instance (e.g. ReadSentence(S1))
h = process ID
= timing landmark (e.g. stimulus presentation of S1)
O = offset (takes values in Wh)
C= a latent variable indicating the correct configuration
S = <s1,…,sV>, standard deviation for each voxel

8. HPMs: the graphical model
Configuration c
Timing Landmark l
The set C of
configurations
constrains the joint
distribution on
{h(k),o(k)} " k.
Process Type h
Offset o
Start Time s S
p1,…,pk
observed
unobserved
Yt,v
t=[1,T], v=[1,V]

9. Encoding Experiment Design
Processes:
Input stimulus :
Constraints Encoded:
h(p1) = {1,2}
h(p2) = {1,2}
h(p1) != h(p2)
o(p1) = 0
o(p2) = 0
h(p3) = 3
o(p3) = {1,2}
ReadSentence = 1
ViewPicture = 2
Timing
landmarks : 1 2
Decide = 3
Configuration 1:
Configuration 2:
Configuration 3:
Configuration 4:

10. Inference Over C, the latent indicator of the correct configuration
Choose the most likely configuration, where:
Y=observed data, D=input stimuli, HPM=model

11. Learning Parameters to learn:
Response signature W for each process
Timing distribution Q for each process
Standard deviation s for each voxel
Expectation-Maximization (EM) algorithm to estimate W and Q.
E step: estimate a probability distribution over configurations.
M step: update estimates of W (using reweighted least squares), Q, and s (using standard MLEs) based on the E step.

12. Process Response Signatures
Standard: Each process has a matrix of parameters, one for each point in space and time for the duration of the response (e.g. 24).
Regularized: Same as standard, but learned with penalties for deviations from temporal and/or spatial smoothness.
Basis functions: Each process has a small number (e.g. 3) weights for each voxel that are combined with a basis to get the response.

13. Evaluation Select 1000 most active voxels.
Compute improvement in test data log-likelihood as compared with predicting the mean training trial for all test trials (a baseline).
5 folds of cross-validation.
Average over 13 subjects.
Standard
Regularized
Basis functions
HPM-GNB
-293
2590
2010
HPM-2
-1150
3910
3740
HPM-3
-2000
4960
4710
HPM-4
-4490
4810
4770

14. Interpretation and Visualization
Timing for the third (Decide) process in HPM-3:
(Values have been rounded.)
For each subject, average response signatures for each voxel over time, plot result in each spatial location.
Compare time courses for the same voxel.
Offset: 1 2 3 4 5 6 7
Stand.
0.3
0.08
0.1
0.05
0.2
0.15
Reg.
Basis
0.5
0.03

15. Standard

16. Regularized

17. Basis functions

18. Time courses
Basis functions
Standard
The basis set
Regularized

19. Related Work fMRI Machine Learning 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 can be encoded by extending factorial HMMs with links between the Markov chains.

20. Conclusions Take-away messages: Future work:
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.
Future work:
Automatically discover the number of latent processes.
Learn process durations.
Apply to open cognitive science problems.

21. References
John 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,
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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, modeling4CAPS.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 CMU-CS