Hidden Process Models with applications to fMRI data

1
Hidden Process Modelswith applications to fMRI
data

• Rebecca Hutchinson
• Oregon State University
• Joint work with Tom M. Mitchell
• Carnegie Mellon University
• August 2, 2009
• Joint Statistical Meetings, Washington DC

2
Introduction

• Hidden Process Models (HPMs)
• 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
Hemodynamic Response
Features 5k-15k voxels, imaged every
second. Training examples 10-40 trials (task
repetitions).
Signal Amplitude
Neural activity
Time (seconds)
4
Study Pictures and Sentences
Press Button
View Picture
Read Sentence
Read Sentence
View Picture
Fixation
Rest
4 sec.
8 sec.
t0

• 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
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
N(0,s1)
v1 v2
N(0,s2)
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HPM Formalism

• HPM ltH,C,F,Sgt
• H lth1,,hHgt, a set of processes (e.g.
  ReadSentence)
• h ltW,d,W,Qgt, a process
• W response signature
• d process duration
• W allowable offsets
• Q multinomial parameters over values in W
• C ltc1,, cCgt, a set of possible configurations
• c ltp1,,pLgt, a set of process instances
• lth,l,Ogt, 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 lts1,,sVgt, 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
t1,T, v1,V
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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 ?
?2
?1
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

• Yobserved data, Dinput stimuli, HPMmodel

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 the probability distribution
  over C.
• 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
Models

• HPM-GNB ReadSentence and ViewPicture,
  duration8sec. (no overlap)
• an approximation of Gaussian Naïve Bayes
  classifier, with HPM assumptions and noise model
• HPM-2 ReadSentence and ViewPicture,
  duration12sec. (temporal overlap)
• HPM-3 HPM-2 Decide (offsets0,7 images
  following second stimulus)
• HPM-4 HPM-3 PressButton (offsets -1,0
  following button press)

14
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-fold cross-validation per subject mean 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
15
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 0 1 2 3 4 5 6 7
Stand. 0.3 0.08 0.1 0.05 0.05 0.2 0.08 0.15
Reg. 0.3 0.08 0.1 0.05 0.05 0.2 0.08 0.15
Basis 0.5 0.1 0.1 0.08 0.05 0.03 0.05 0.08
16
Standard
17
Regularized
18
Basis functions
19
Time courses
Basis functions
Standard
The basis set (Hossein-Zadeh03)
Regularized
20
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 (overview
  in Haynes06)
• 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.

21
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.
• Future work
• Automatically discover the number of latent
  processes.
• Learn process durations.
• Apply to open cognitive science problems.

22
References
John R. Anderson, Daniel Bothell, Michael D.
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Optimal experimental design for event-related
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1999. Zoubin Ghahramani and Michael I. Jordan.
Factorial hidden Markov models. Machine
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and Geraint Rees. Decoding mental states from
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http//www.ccbi.cmu.edu/project
10modeling4CAPS.htm. Kevin P. Murphy. Dynamic
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