Hidden Process Modelswith applications to fMRI data

Rebecca HutchinsonOregon State University

Joint work with Tom M. Mitchell

Carnegie Mellon University

August 2, 2009

Joint Statistical Meetings, Washington DC

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.

2

fMRI Data

…

Sign

al

Am

plitu

de

Time (seconds)

Hemodynamic Response

Neural activity

Features: 5k-15k voxels, imaged every second.Training examples: 10-40 trials (task repetitions).

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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

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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.

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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:

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)6

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 possible 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)

C= a latent variable indicating the correct configuration

= <1,…,V>, standard deviation for each voxel7

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

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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:

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Inference• Over C, the latent indicator of the correct

configuration

• Choose the most likely configuration, where:

• Y=observed data, =input stimuli, HPM=model

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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 the probability distribution over C.– M step: update estimates of W (using reweighted

least squares), , and (using standard MLEs) based on the E step.

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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.

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Models

• HPM-GNB: ReadSentence and ViewPicture, duration=8sec. (no overlap)– an approximation of Gaussian Naïve Bayes classifier,

with HPM assumptions and noise model

• HPM-2: ReadSentence and ViewPicture, duration=12sec. (temporal overlap)

• HPM-3: HPM-2 + Decide (offsets=[0,7] images following second stimulus)

• HPM-4: HPM-3 + PressButton (offsets = {-1,0} following button press)

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 477014

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

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Standard

16

Regularized

17

Basis functions

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Time courses

Standard

Regularized

Basis functions

The basis set

(Hossein-Zadeh03)19

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.

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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.

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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/.

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.

John-Dylan Haynes and Geraint Rees. Decoding mental states from brain activity in humans. Nature Reviews Neuroscience, 7:523–534, July 2006.

Gholam-Ali Hossein-Zadeh, Babak A. Ardekani, and Hamid Soltanian-Zadeh. A signal subspace approach for modeling the hemodynamic response function in fmri. Magnetic Resonance Imaging, 21:835–843, 2003.

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.

Kevin P. Murphy. Dynamic bayesian networks. To appear in Probabilistic Graphical Models, M. Jordan, November 2002. 22