Hidden Process Models

1
Hidden Process Models

• Rebecca Hutchinson
• May 26, 2006
• Thesis Proposal
• Carnegie Mellon University
• Computer Science Department

2
Talk Outline

• Motivation fMRI (functional Magnetic Resonance
  Imaging) data.
• Problem new kind of probabilistic time series
  modeling.
• Solution Hidden Process Models (HPMs).
• Results preliminary experiments with HPMs.
• Extensions proposed improvements to HPMs.

3
fMRI Data High-Dimensional and Sparse

• Imaged once per second for 15-20 minutes
• Only a few dozen trials (i.e. training examples)
• 10,000-15,000 voxels per image

4
The Hemodynamic Response

• fMRI measures an indirect, temporally blurred
  correlate of neural activity.
• Also called BOLD response Blood Oxygen Level
  Dependent.

Signal Amplitude
Subject reads a word and indicates whether it is
a noun or verb in less than a second.
Time (seconds)
5
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.

6
Motivation

• 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.

7
The Thesis

• It is possible to
• simultaneously
• estimate the parameters and timing of
• temporally and spatially overlapped,
• partially observed processes
• (using many features and a small number of noisy
  training examples).
• We are developing a class of probabilistic models
  called Hidden Process Models (HPMs) for this task.

8
Related Work in fMRI

• General Linear Model (GLM)
• Must assume timing of process onset to estimate
  hemodynamic response
• Dale99
• 4-CAPS and ACT-R
• Predict fMRI data rather than learning parameters
  of processes from the data
• Anderson04, Just99

9
Related Work in Machine Learning

• Classification of windows of fMRI data
• Does not typically estimate hemodynamic response
• Cox03, Haxby01, Mitchell04
• Dynamic Bayes Networks
• HPM assumptions/constraints are difficult to
  encode in DBNs
• Murphy02, Ghahramani97

10
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.

11
HPM Formalism (Hutchinson06)

• HPM ltH,C,F,Sgt
• H lth1,,hHgt, a set of processes
• 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 configurations
• c ltp1,,pLgt, a set of process instances
• lth,l,Ogt, a process instance
• h process ID
• associated stimulus landmark
• O offset (takes values in Wh)
• ltf1,,fCgt, priors over C
• S lts1,,sVgt, standard deviation for each voxel

Notation parameter(entity) e.g. W(h) is the
response signature of process h, and O(p) is
the offset of process instance p.
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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 time O 1 sec (Start
time ?2 O)
?1
?2
One configuration c of process instances
?1, ?2, ?k (with prior fc)
?1
?2
?
Predicted mean
N(0,s1)
v1 v2
N(0,s2)
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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
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Inference

• Over configurations
• Choose the most likely configuration, where

• Cconfiguration, Yobserved data, Dinput
  stimuli, HPMmodel

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Learning

• Parameters to learn
• Response signature W for each process
• Timing distribution Q for each process
• Standard deviation s for each voxel
• Case 1 Known Configuration.
• Least squares problem to estimate W.
• Standard MLEs for Q and s.
• Case 2 Unknown Configuration.
• 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) and Q (using standard MLEs) based
  on the E step.
• Standard MLEs for s.

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Case 1 Known Configuration

• Following Dale99, use GLM.
• The (known) configuration generates a TxD
  convolution matrix X

DShd(h)
d(1)
d(3)
d(2)
Configuration p1 h1, start1 p2 h2,
start2 p3 h3, start2
1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1
0 1 0 0 1 0 0 0 0 0 0 1 0 0 1
t1 t2 t3 t4
T
For this example, d(1)d(2)d(3)3.
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Case 1 Known Configuration
V
d(1)
d(3)
d(2)
V
1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1
0 1 0 0 1 0 0 0 0 0 0 1 0 0 1
W(1)
d(1)

d(2)
W(2)
Y
T
W(3)
d(3)
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Case 2 Unknown Configuration

• E step Use the inference equation to estimate a
  probability distribution over the set of
  configurations.
• M step Use the probabilities computed in the
  E-step to form weights for the least-squares
  procedure for estimating W.

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Case 2 Unknown Configuration

• Convolution matrix models several choices for
  each time point.

Configurations for each row
d(1)
d(3)
d(2)
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0
1 ...
t1 t1 t2 t2 t18 t18 t18 t18
3,4 1,2 3,4 1,2 3 4 1 2
TgtT
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Case 2 Unknown Configuration

• Weight each row with probabilities from E-step.

d(1)
d(3)
d(2)
Configurations
Weights
e1 e2 e3 e4
3,4 1,2 3,4 1,2
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

Y

W
e1 P(C3Y,Wold,Qold,sold) P(C4Y,Wold,Qold,s
old)
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Learned HPM with 3 processes (S,P,D), and d13sec.
S
S
P
P
D?
D?
observed
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Results Model Selection

• Use cross-validation to choose a model.
• GNB Gaussian Naïve Bayes
• HPM-2 HPM with ViewPicture, ReadSentence
• HPM-3 HPM-2 Decide

Subject A B C
GNB 0.725 0.750 0.750
HPM-2 0.750 0.875 0.787
HPM-3 0.775 0.875 0.812
GNB -896 -786 -476
HPM-2 -876 -751 -466
HPM-3 -864 -713 -447
Accuracy predicting picture vs. sentence (random
0.5)
Data log likelihood
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Synthetic Data Results

• Timing of synthetic data mimics the real data,
  but we have ground truth.
• Can use to investigate effects of
• signal to noise ratio
• number of voxels
• number of training examples
• on
• training time
• cross-validated classification accuracy
• cross-validated data log-likelihood

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Recall Motivation

• 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.

27
Proposed Work

• Goals
• Increase efficiency.
• fewer parameters
• better accuracy from fewer examples
• faster inference and learning
• Handle larger, more complex problems.
• more voxels
• more processes
• fewer assumptions
• Research areas
• Model Parameterization
• Timing Constraints
• Learning Under Uncertainty

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Model Parameterization

• Goals
• Improve biological plausibility of learned
  responses.
• Decrease the number of parameters to be estimated
  (improving sample complexity).
• Tasks
• Parameter sharing across voxels
• Parametric form for response signatures
• Temporally smoothed response signatures

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Timing Constraints

• Goals
• Specify experiment design domain knowledge more
  efficiently.
• Improve the computational and sample complexities
  of the HPM algorithms.
• Tasks
• Formalize limitations in terms of fMRI experiment
  design.
• Improve the specification of timing constraints.
• Develop more efficient exact and/or approximate
  algorithms.

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Learning Under Uncertainty

• Goals
• Relax the current modeling assumptions.
• Allow more types of uncertainty about the data.
• Tasks
• Learn process durations.
• Learn the number of processes in the model.

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HPM Parameter Sharing (Niculescu05)
Special case HPMs with known configuration.
Parameter reduction from d(h) V to d(h)
V.
Scaling parameter per voxel per process.
New mean for voxel v at time t
No more voxel index on weights.
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Extension to Unknown Timing

• Simplifying assumptions
• No clustering. All voxels share a response.
• Voxels that share a response for one process
  share a response for all processes.
• Algorithm notes
• Residual is linear in shared response parameters
  and in scaling parameters, so minimize
  iteratively.
• Empirically, convergence occurs within 3-5
  iterations.

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Iterative M-step Step 1

• Using current estimates of S, re-estimate W.

d(1)
d(3)
d(2)
s110 0 0 0 0 0 0 0 0 0 0 s21 0 0 0 0 0 0
s11 0 0 0 0 0 0 0 0 0 0 0 s21 0 0 0 0

W(1) W(2) W(3)
New shape TV x 1
No more voxel index here. Single column of
parameters describing the shared responses.
TxD for v1

Y
s120 0 0 0 0 0 0 0 0 0 0 s22 0 0 0 0 0 0
s12 0 0 0 0 0 0 0 0 0 0 0 s22 0 0 0 0

TxD for v2

Replace ones of convolution matrix with shv.
Repeat for all v.
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Iterative M-step Step 2

• Using current estimates of W, re-estimate S.

d(1)
d(3)
d(2)
s11 s1V s11 s1V s11 .... s1V s21
s2V s21 ... s2V sH1 .... sHV
w110 0 0 0 0 0 0 0 0 0 0 w210 0 0 0 0 0
w12 0 0 0 0 0 0 0 0 0 0 0 w22 0 0 0 0

Y
d(1)

Each column has the scaling parameters for a
voxel. The parameter for each process is
replicated over its duration.
Original size data matrix.
Original size convolution matrix. Ones replaced
with W estimates.

Need to constrain these parameter sets to be
equal.
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Next Step?

• Implement this approach.
• Anticipated memory issues
• Replicating the convolution matrix for each voxel
  in step 1.
• Working on exploiting sparsity/structure of these
  matrices.
• Add clustering back in
• Adapt for other parameterizations of response
  signatures

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Response Signature Parameters

• Temporal smoothing
• Gamma functions
• Hemodynamic basis functions

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Temporally Smooth Responses

• Idea Add a regularizer to the loss function to
  penalize large jumps between time points.
• e.g. minimize (Y-XW)2 lSt(Wt-Wt-1)2
• choose l by cross-validation
• should be a straightforward extension to the
  optimization code
• Concerns
• this adds l instead of reducing the number of
  parameters!

38
Gamma-shaped Responses

• Idea Use a gamma function with 3 parameters for
  each process response signature (Boynton96).
• a controls amplitude
• t controls width of peak
• n controls delay of peak
• Questions
• Are gamma functions a reasonable modeling
  assumption?
• Details of how to fit parameters in M-step?

n
Signal Amplitude
t
a
Seconds
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Hemodynamic Basis Functions

• Idea Process response signatures are weighted
  sum of basis functions.
• parameters are weights on n basis functions
• e.g. gammas with different sets of parameters
• learn process durations for free with variable
  length basis functions
• share basis functions across voxels and processes
• Questions
• How to choose/learn basis? (Dale99)

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Schedule

• August 2006
• Parameter sharing.
• Progress on model parameterization.
• December 2006
• Improved expression of timing constraints.
• Corresponding updates to HPM algorithms.
• June 2007
• Application of HPMs to an open cognitive science
  problem.
• December 2007
• Projected completion.

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References
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